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This is to certify that the
dissertation entitled

THERMALLY ACTIVATED ESCAPE FROM A
PERIODICALLY MODULATED OPTICAL TRAP

presented by

James Ryan Kruse

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Physics

 

 

’Eim Glut...

/Major Professor I; Signature

May 5, 2004

 

Date

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6/01 c:/CIRC/DateDue.p65-p.15

THERMALLY ACTIVATED ESCAPE FROM A PERIODICALLY MODULATED
OPTICAL TRAP
By

James Ryan Kruse

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

2004

ABSTRACT
THERMALLY ACTIVATED ESCAPE FROM A PERIODICALLY MODULATED
OPTICAL TRAP
By

James Ryan Kruse

Thermally activated escape is a ubiquitous process that underlies a wide variety of
physical, chemical, and biological phenomena, such as diffusion in crystals, chemical
reactions, and protein folding. More than 60 years ago, Kramers developed a quantitative
theory of thermal activation based on a model of a Brownian particle in a metastable
potential well coupled to a thermal reservoir. Fluctuations of the heat bath lead to a finite
probability for the particle to escape. Since the escape rate is exponentially sensitive to
an external force, modulation of the potential may be used as means to control the escape
rate. At large modulation amplitude the rate is enhanced as the barrier is reduced,
approaching a bifurcation point where it disappears. Recent theoretical work has

predicted a power-law scaling of the mean escape rate W with the amplitude of periodic
modulation A, ln(W)oc (AC — A)“ , where Ac is the bifurcation amplitude. The critical

exponent ,u depends on modulation frequency, and is expected to be ,u = 3/2 for a quasi-
stationary system. As frequency increases, the exponent changes to ,u = 2 when a
nonadiabatic regime is entered. This thesis reports the first observation of a nonadiabatic
critical exponent and explores the frequency dependence of the bifurcation amplitude.
The physical system consists of an overdamped Brownian particle in a three

dimensional bistable optical potential under periodic modulation. Two visible laser

beams are focused through a high-power microscope Objective into an aqueous solution
containing a 0.6 pm Silica Sphere. Interaction of the optical fields and sphere creates two
stable potential minima, a fraction of a pm apart, with an intervening energy barrier. The
surrounding medium causes positional fluctuations of the particle in the vicinity of one of
the stable points, until a particularly large fluctuation drives the Sphere over the barrier
where it is captured by the other stable point. The potential is tilted periodically by
modulating the laser power at rates from 0.2 Hz to 2 kHz. A camera records the image of
the sphere at 5 ms intervals. Real-time digital image processing is used to obtain the

sphere position 7(t) as a function of time and the time evolution of the particle is

analyzed to calculate the full three dimensional potential. Knowledge of the potential
enables the components of the relaxation time of the particle in the potential to be
calculated. Over-barrier transition rate statistics in the static and modulated potentials are
evaluated.

An adiabatic regime is identified at modulation frequencies below 0.5 Hz. The
loss of adiabaticity is seen at higher frequencies, consistent with the onset of internal

relaxation times that cause the particle to fall out of equilibrium at these driving rates. In
. . . . . S. 3:03 , ,
the weakly nonadiabatic regime it 18 found that ln(W)oc (AC — A)2 . The critical

amplitude A? is seen to vary linearly with frequency, as predicted by theoretical

considerations. These results are the first experimental observations of nonadiabatic

Brownian dynamics as a bifurcation point is approached.

T 0 my parents

iv

ACKNOWLEDGEMENTS

I would like to thank my advisor, Brage Golding, for the Opportunity tO work on
this project from its inception, and his boundless Optimism that I could complete it.

I would also like to thank Lowell McCann for his work on this project. The basis
for many of the techniques described in this thesis were developed during his initial
experiments, and his visits in the subsequent years gave me much needed boosts of
enthusiasm.

Ann Kirchmeier, Darla Conley, and Debbie Simmons made the bureaucratic
aspects of university research totally transparent. With every new building there are
problems, and the Biomedical and Physical Sciences building had it share. I would like
to thank Sandy Teague and Marc Conlin for finding solutions when they existed, and
simply letting me complain when they did not.

Tom Palazzolo, Jim Muns, Tom Hudson, and the rest of the Physics and
Astronomy machine shop staff patiently answered my questions and gave me the
confidence to make those small parts that every experiment needs. They also allowed me
tO develop a new hobby, a relaxing diversion from the lab. Special thanks go to Tom
Palazzolo for not banning me from the student shop after I filled it with sawdust.

I would like to thank my labmates: Connie Meinke, Zhongning Dai, An-Ping Li,
Mahdokht Behravan, and Murari Regrni for their outside perspectives and the lessons on
growth of heteroepitaxial diamond films. I also thank Reza Loloee and Baokang Bi for
their assistance and encouragement.

Finally I would like to thank my family, my parents Jim and Suzeahn Kruse, my

sister Michelle, and Eric, Jayme, and Libby. They were always there for me.

TABLE OF CONTENTS

 

 

 

 

 

LIST OF TABLES - -- - - IX
LIST OF FIGURES - - X
CHAPTER 1 INTRODUCTION 1
CHAPTER 2 DRIVEN SYSTEMS WITH A F LUCTUATING FORCE ................... 4
2.1. FLUCTUATIONS OF A CLASSICAL PARTICLE ............................................................. 5
2.1.1. Damped Harmonic Oscillator ......................................................................... 5
2.1.2. Randomly Fluctuating Particles in a General Potential .................................. 7

2.2. THERMALLY ACTIVATED TRANSITIONS ................................................................... 9
2.2.1. Mean Rate of Transitions .............................................................................. 10
2.2.2. Optimal Path ................................................................................................. 13

2.3. TRANSITIONS IN A PERIODICALLY DRIVEN SYSTEM ............................................... 16
2.3.1. Stochastic Resonance .................................................................................... 16
2.3.2. Logarithmic Susceptibility ............................................................................ 26
2.3.3. Strong Modulation ........................................................................................ 29

2.4. SUMMARY .............................................................................................................. 34
CHAPTER 3 OPTICAL TRAPPING 36
3.]. EXPERIMENTAL REALIZATIONS OF OPTICAL TRAPPING ......................................... 37
3.1.1. Radial Trap .................................................................................................... 37
3.1.2. Three Dimensional Trap ............................................................................... 38
3.1.3. Trapping Optics ............................................................................................ 41
3.1.4. Other Trapping Geometries .......................................................................... 44

3.2. TRAPPED PARTICLE POSITION TRACKING .............................................................. 45
3.2.1. Camera Particle Tracking ............................................................................. 45
3.2.2. Single Photodiode Particle Tracking ............................................................ 46
3.2.3. Quadrant Photodiode Particle Tracking ........................................................ 48
3.2.4. Comparison of Tracking Methods ................................................................ 50

3.3. PRIOR EXPERIMENTS USING OPTICAL TRAPS ......................................................... 52
3.3.1. Measurement of Spring Constants from the Fluctuation Spectrum .............. 52
3.3.2. Measurement of Potential Energy from the Position Distribution ............... 54
3.3.3. Measurement of Trapping Force from Stokes Damping .............................. 56
3.3.4. Hydrodynamic Interactions Between Colloidal Particles ............................. 59
3.3.5. Synchronization of Transitions in A Bistable Potential ................................ 60

3.4. SUMMARY .............................................................................................................. 64
CHAPTER 4 EXPERIMENTAL METHODS 67
4.1. TRAPPING OPTICS .................................................................................................. 71
4.1.1. Optical Bench ................................................................................................ 72
4.1.2. Lasers ............................................................................................................ 73
4.1.3. Stabilizers ...................................................................................................... 73

vi

4.1.4. Beam Power Monitoring ............................................................................... 76

 

 

4.1.5. Polarization Optics ........................................................................................ 77
4.1.6. Beam Expanding Optics ............................................................................... 79
4.1.7. Beam Steering ............................................................................................... 82
4.1.8. Microscope .................................................................................................... 83
4.1.9. Optical Alignment Procedure ....................................................................... 85
4.1.10. Sample Cell ................................................................................................... 87
4.2. PARTICLE TRACKING ............................................................................................. 90
4.2.1. Camera Interface ........................................................................................... 91
4.2.2. Data Acquisition and Analysis ...................................................................... 93
4.2.3. WinMi12 — Data Acquisition Program .......................................................... 94
4.3. OPTICAL STABILITY ............................................................................................. 120
4.3.1. Laser Power Stability .................................................................................. 120
4.3.2. Focal Plane Position .................................................................................... 123
4.3.3. Lens Mount Expansion ............................................................................... 132
4.3.4. Pointing Fluctuations Caused by Air Motion ............................................. 137
4.4. SUMMARY ............................................................................................................ 14]
CHAPTER 5 EXPERIMENTAL RESULTS 144
5.1. MEASUREMENT OF PARTICLE DYNAMICS IN SINGLE AND Two BEAM TRAPS ...... 145
5.1.1. Single Beam Trap ....................................................................................... 146
5.1.2. Two Beam Trap .......................................................................................... 150
5.1.3. Determination of Transition Rates .............................................................. 153
5.2. DOUBLE-WELL POTENTIAL CALCULATION .......................................................... 156
5.2.1. Extraction of the Potential and Potential Surfaces ...................................... 157
5.2.2. Parameterization of the Potential ................................................................ 159
5.3. DETERMINATION OP OPTIMAL PARTICLE TRAJECTORIES ..................................... 170
5.4. TRANSITION RATE DEPENDENCE ON MODULATION AMPLITUDE AND FREQUENCY
............................................................................................................................. 175
5.4.1. Transition Rate in a Periodically Modulated Potential ............................... 177
5.4.2. Locating the Adiabatic and Nonadiabatic Regimes .................................... 180
5.5. SUMMARY ............................................................................................................ 194
CHAPTER 6 DISCUSSION 196
6.1. THEORETICAL PREDICTIONS OF ADIABATIC CRITERIA ......................................... 196
6.2. COMPARISON TO EXPERIMENTALLY DETERMINED FREQUENCY REGIMES ........... 198
6.3. SUMMARY AND CONCLUSIONS ............................................................................. 204
6.4. SUGGESTIONS FOR FURTHER WORK ..................................................................... 206
APPENDIX A QUANTITATIVE THEORY OF OPTICAL TRAPPING ............. 208
A.1 RAYLEIGH REGIME ............................................................................................... 208
A2 RAY OPTICS REGIME ............................................................................................. 21 1
A3 INTERMEDIATE REGIME ....................................................................................... 214
APPENDIX B . DEPENDENCE OF TRAPPING FORCES ON OPTICAL SYSTEM
215

 

vii

APPENDIX C COHERENCE MEASUREMENTS OF THE MILLENNIA LASER

 

 

 

219
APPENDIX D TRAPPING IN AIR AND VACUUM 225
DJ APPARATUS ......................................................................................................... 226
D.1.1. Trapping Optics .......................................................................................... 226
D.1.2. Trapping Cell and Particles ......................................................................... 227
D.1.3. Trapping in a Vacuum ................................................................................ 229
[)2 RESULTS .............................................................................................................. 230
D2]. Spectrum vs. Power and Pressure ............................................................... 231
D.2.2. Curvature vs. Pressure ................................................................................ 234
D3 SUMMARY ............................................................................................................ 236
APPENDIX E CALCULATION OF THE POTENTIAL ENERGY AND
TRANSITION STATISTICS USING PROFILES 238
13.1 THREE DIMENSIONAL POTENTIAL ENERGY SURFACE ............................................ 239
132 ONE DIMENSIONAL POTENTIAL ENERGY PROFILES ............................................... 242
E3 DWELL TIME AND TRANSITION TIME .................................................................... 244
REFERENCES 246

 

viii

LIST OF TABLES
TABLE 4.1 GAUSSIAN PROPAGATION SPREADSHEET .......................................................... 81

TABLE 4.2 FUNCTION GENERATOR SETTINGS TO MINIMIZE BURST MODULATION ERRORS

.................................................................................................................................. 1 17
TABLE 4.3 TYPICAL BATCH ACQUISITION FILE ................................................................ 1 19
TABLE 4.4 BEAM POWER FLUCTUATION AMPLITUDE ...................................................... 123

TABLE 5.1 STATIC POTENTIAL PARAMETERS FOR DATA USED IN SECTIONS 5.1, 5.2, AND 5.3

.................................................................................................................................. 146
TABLE 5.2 SINGLE BEAM OSCILLATION FREQUENCIES AND DAMPING RATES .................. 149
TABLE 5.3 DOUBLE-WELL PARAMETERS FROM QUADRATIC FITS TO ONE DIMENSIONAL

PROFILES ................................................................................................................... 162
TABLE 5.4 BARRIER HEIGHT MEASUREMENT RESULTS WITH SPATIAL AVERAGING ......... 162

TABLE 5.5 RESULTS OF 3 POINT QUARTIC FIT TO STATIC TILT POTENTIALS IN FIGURE 5.12

.................................................................................................................................. 165
TABLE 5.6 STATIC POTENTIAL PARAMETERS FOR DATA USED IN SECTION 5.4 ................ 177
TABLE 5.7 RESULTS OF FITS OF EQUATION (5.7) T0 W VS. F DATA ................................. 184
TABLE 5.8 POWER LAW FIT RESULTS FOR THE NONADIABATIC REGIME ........................... 190

TABLE 5.9 POWER LAW FIT RESULTS FOR THE NONADIABATIC REGIME, WITH y = 2 AND r0 =
1n(2 f ) .................................................................. ERROR! BOOKMARK NOT DEFINED.

TABLE 8.] TRAPPING FORCES FOR VARIOUS TRAPPING PARAMETERS ............................. 218

ix

LIST OF FIGURES
FIGURE 2.1 PROBABILITY DISTRIBUTION OF AN ARBITRARY POTENTIAL ............................. 9
FIGURE 2.2 ESCAPE OVER A POTENTIAL BARRIER ............................................................. 10

FIGURE 2.3 QUADRATIC APPROXIMATIONS TO THE EXTREMA OF A BISTABLE POTENTIAL. 1 1

FIGURE 2.4 COMPARISON OF KRAMERS THEORY TO MEASURED TRANSITION RATES ......... 12
FIGURE 2.5 QUARTIC POTENTIAL WITH A TILT MODULATION ............................................ 17
FIGURE 2.6 SIGNAL TO NOISE RATIO FOR STOCHASTIC RESONANCE .................................. 21

FIGURE 2.7 NET PROBABILITY CURRENT IN A PERIODICALLY MODULATED SAW TOOTH
POTENTIAL .................................................................................................................. 24

FIGURE 2.8 DIRECTED DIFFUSION OF A BROWNIAN PARTICLE ........................................... 26
FIGURE 2.9 PERIODIC STABLE AND UNSTABLE STATES AT SMALL DRIVING AMPLITUDE 27
FIGURE 2.10 PERIODIC STABLE AND UNSTABLE STATES AT LARGE DRIVING AMPLITUDE .. 30

FIGURE 2.1 1 PERIODIC STABLE AND UNSTABLE STATES AT LARGE DRIVING AMPLITUDE

AND HIGH FREQUENCY ................................................................................................ 32
FIGURE 3.1 RADIAL OPTICAL TRAP ................................................................................... 38
FIGURE 3.2 THREE DIMENSIONAL OPTICAL TRAP .............................................................. 39
FIGURE 3.3 BASIC TRAPPING SYSTEM ............................................................................... 42
FIGURE 3.4 CAMERA TRACKING SYSTEM .......................................................................... 46
FIGURE 3.5 PHOTODIODE TRACKING SYSTEM .................................................................... 47
FIGURE 3.6 QUADRANT PHOTODIODE TRACKING SYSTEM ................................................. 49
FIGURE 3.7 TRAPPING FORCE VS. POSITION IN THE RAYLEIGH SIZE REGIME ...................... 53

FIGURE 3.8 THREE DIMENSIONAL POTENTIAL ENERGY HISTOGRAM OF A PARTICLE IN A

BISTABLE POTENTIAL ................................................................................................... 56
FIGURE 3.9 RESIDENCE TIME HISTOGRAM IN A STATIC BISTABLE POTENTIAL .................... 61
FIGURE 3.10 RESIDENCE TIME HISTOGRAMS UNDER MODULATION ................................... 63
FIGURE 4.1 OPTICAL TRAP FORMED BY TWO FOCUSED LASER BEAMS ............................... 67

FIGURE 4.2 OPTICAL SYSTEM SCHEMATIC ......................................................................... 72
FIGURE 4.3 LS-lOO AND LS-PRO SCHEMATIC ................................................................. 74
FIGURE 4.4 BEAM WAIST OF A CONVERGING GAUSSIAN BEAM ......................................... 79
FIGURE 4.5 GAUSSIAN BEAM PROPAGATION THROUGH A LENS ......................................... 80
FIGURE 4.6 MICROSCOPE AND PARTICLE TRACKING APPARATUS ...................................... 84
FIGURE 4.7 CELL CONSTRUCTION ..................................................................................... 90
FIGURE 4.8 DALSA CA-Dl TIMING DIAGRAM ................................................................... 92
FIGURE 4.9 WINMIL2 USER INTERFACE ............................................................................ 96
FIGURE 4.10 LOOKUP TABLE ............................................................................................ 97
FIGURE 4.1 1 BLOB ANALYSIS ......................................................................................... 102
FIGURE 4.12 ELECTRONIC COMPONENT DIAGRAM .......................................................... 108
FIGURE 4.13 ORIGIN OF SKIPPED FRAMES DURING DATA ACQUISITION ........................... 109
FIGURE 4.14 13 X 13 PIXEL ARRAY USED IN THE 2 POSITION CALCULATION .................... 1 1 1
FIGURE 4.15 Z CALIBRATION .......................................................................................... 1 13
FIGURE 4.16 THE OBJECTIVE CLAMP ............................................................................... 124
FIGURE 4.17 OBJECTIVE TO CELL DISTANCE CHANGES WITH OBJECTIVE CLAMP HEATING
.................................................................................................................................. 126
FIGURE 4.18 THE OBJECTIVE CLAMP WITH WIDE SHELF .................................................. 128
FIGURE 4.19 OBJECTIVE TO CELL DISTANCE CHANGES WITH REDESIGNED OBJECTIVE
CLAMP ....................................................................................................................... 129
FIGURE 4.20 CELL CLAMP .............................................................................................. 131
FIGURE 4.21 LENS MOUNT HEATER ................................................................................. 133
FIGURE 4.22 DEFLECTION OF A BEAM THROUGH A SINGLE LENS ..................................... 134
FIGURE 4.23 DIAGRAM OF THE BEAM DEFLECTION AS IT PROPAGATES THROUGH THE
OPTICAL SYSTEM ....................................................................................................... 134
FIGURE 4.24 X POSITION VS. LENS MOUNT TEMPERATURE .............................................. 135

xi

FIGURE 4.25 X BEAM POSITIONS WITH PLEXIGLAS BOX .................................................. 137

FIGURE 4.26 BEAM SEPARATION VS. TIME ...................................................................... 138
FIGURE 5.] TIME SERIES AND POSITION DISTRIBUTIONS FOR A SINGLE-BEAM TRAP ........ 147
FIGURE 5.2 SINGLE BEAM POTENTIAL ENERGY SLICES .................................................... 148

FIGURE 5.3 ONE DIMENSIONAL POTENTIAL ENERGY PROFILES OF A SINGLE BEAM TRAP. 149
FIGURE 5.4 X PROFILE WITH VARYING BEAM POWER FOR A TWO BEAM TRAP ................. 150
FIGURE 5.5 THREE DIMENSIONAL POTENTIAL ENERGY SLICES FOR A TWO-BEAM TRAP... 151
FIGURE 5.6 DYNAMICS OF OVER-BARRIER TRANSITIONS IN A BISTABLE POTENTIAL ....... 152
FIGURE 5.7 DETERMINATION OF TRANSITION EVENTS ..................................................... 153

FIGURE 5.8 SENSITIVITY OF TRANSITION RATE TO CHANGES IN HALF-WIDTH 8X AND
MIDPOINT X0 ............................................................................................................. 154

FIGURE 5.9 DWELL TIME DISTRIBUTION FOR A STATIC POTENTIAL .................................. 156

FIGURE 5.10 CALCULATION OF ONE DIMENSIONAL PROFILES FROM THE THREE

DIMENSIONAL HISTOGRAM ........................................................................................ 159
FIGURE 5.1 1 ONE DIMENSIONAL PROFILES WITH QUADRATIC FITS .................................. 161
FIGURE 5.12 QUARTIC FITS TO TILTED x PROFILES .......................................................... 164
FIGURE 5.13 MEASUREMENT OF TILTED POTENTIAL UP To A = A, .................................. 167
FIGURE 5.14 A0 vs. OFFSET VOLTAGE ............................................................................. 169
FIGURE 5.15 CONSTRAINTS FOR PREHISTORY CALCULATION .......................................... 171
FIGURE 5.16 PREHISTORY DISTRIBUTION AND OPTIMAL PATH ........................................ 173
FIGURE 5.17 X AND 2 COMPONENTS OF THE OPTIMAL PATH VS. TIME ............................. 174
FIGURE 5.18 INTERWELL TRANSITIONS UNDER PERIODIC MODULATION .......................... 178
FIGURE 5.19 TRANSITION RATE MEASUREMENT FROM FIT TO DECAYING PEAKS ............. 179

FIGURE 5.20 PHASE DIAGRAM OF EXPERIMENTALLY OBSERVED MODULATION REGIMES 181

FIGURE 5.21 MEAN TRANSITION RATE VS. FREQUENCY FOR SEVERAL MODULATION
AMPLITUDES .............................................................................................................. 182

xii

FIGURE 5.22 PHASE DIAGRAM DERIVED FROM W VS. F MEASUREMENTS ........................ 185

FIGURE 5.23 TRANSITION PROBABILITY VS. MODULATION AMPLITUDE FOR LOW

FREQUENCY MODULATION ......................................................................................... 186
FIGURE 5.24 1n(W) VS. A IN THE ADIABATIC REGIME .................................................... 188
FIGURE 5.25 1n(W) VS. A IN THE NONADIABATIC REGIME .............................................. 190
FIGURE 5.26 FREQUENCY DEPENDENCE OF THE CRITICAL AMPLITUDE ............................ 192

FIGURE 6.1 BOUNDARIES OF ADIABATIC AND NONADIABATIC REGIMES IN THE (F, A) PLANE

.................................................................................................................................. 196
FIGURE 6.2 COMPARISON OF EXPERIMENTAL DATA WITH THEORETICAL FREQUENCY

REGIMES .................................................................................................................... 198
FIGURE 6.3 POSITIONS OF STABLE AND SADDLE POINTS IN A MODULATED BISTABLE

POTENTIAL ................................................................................................................ 201

FIGURE 6.4 COMPARISON OF EXPERIMENTAL DATA WITH ADIABATIC CRITERIA ON THE X
AND 2 RELAXATION RATES ........................................................................................ 202

FIGURE A.1 GAUSSIAN BEAM INTENSITY ........................................................................ 209

FIGURE A2 A SINGLE RAY INCIDENT ON THE SPHERE AT ANGLE 013 PARTIALLY

TRANSMITTED WITH AN ANGLE OR REFRACTION a .................................................... 212
FIGURE A.3 CALCULATION OF FORCE FROM A SINGLE FOCUSED RAY ............................. 213
FIGURE C.1 SINGLE LASER OPTICAL TRAPPING APPARATUS ............................................ 219

FIGURE C2 MICHELSON INTERFEROMETER FOR COHERENCE LENGTH MEASUREMENTS . 221

FIGURE C.3 VISIBILITY VS. PATH LENGTH DIFFERENCE FOR THE MILLENNIA LASER ....... 223
FIGURE C.4 CHANGE IN THE POTENTIAL DUE TO INTERFERENCE ..................................... 224
FIGURE D.) VACUUM CELL SCHEMATIC .......................................................................... 228
FIGURE D.2 SCHEMATIC OF THE PUMPING SYSTEM ......................................................... 230
FIGURE D.3 FLUCTUATION SPECTRUM FOR CHANGING PRESSURE ................................... 232

FIGURE D.4 FLUCTUATION SPECTRUM OF THE UNDERDAMPED PARTICLE WITH CHANGING
BEAM POWER ............................................................................................................. 233

FIGURE D.5 POSITION OF THE SECOND AND THIRD SPECTRAL PEAKS VS. BEAM POWER .. 234

xiii

FIGURE D.6 X POTENTIAL ENERGY PROFILES WITH CHANGING PRESSURE ...................... 235

FIGURE D.7 X AND Y POTENTIAL CURVATURES VS. PRESSURE ........................................ 236
FIGURE E.1 PROFILES USER INTERFACE .......................................................................... 238
FIGURE E.2 TWO DIMENSIONAL POTENTIAL ENERGY PROJECTIONS ................................. 240

Images in this dissertation are presented in color.

xiv

Chapter 1 Introduction

Thermally activated escape over an energy barrier provides the basis for many
physical phenomena, including chemical reactions, nucleation in phase transitions, and
diffusion of atoms in crystals. For example, an atom on the surface of a crystal resides in
an extended periodic potential. Thermal fluctuations cause the atom to move over the
barrier separating two stable points; the atom diffuses along the surface. Eventually it
reaches a step (where the potential is much deeper) and stops diffusing, becoming part Of
the next crystal layer.

The probability of escape from a metastable state is exponentially sensitive to an
external force. A periodic field, weak compared to the barrier height, could thus be used
to control the rate and direction of diffusion in an extended system. Strong modulation
distorts the potential, eliminating the metastable point and ultimately leading to a
reduction of the stochastic aspects of the dynamics. Approaching this critical modulation
amplitude the escape rate dependence on the amplitude displays singular behavior.

The present experiments investigate the effect of periodic modulation on an
overdamped Brownian particle in a bistable potential. A sub-micron diameter Sphere is
trapped by two laser beams, creating two stable points in the trapping potential.
Suspended in water, Brownian motion causes the sphere to fluctuate around a stable point
until a large thermal fluctuation drives it over the intervening potential barrier.
Modulation of the laser power tilts the potential, changing the barrier height.

The particle position is obtained by analyzing particle images recorded by a CCD
camera. Each image is transferred to a computer and the three coordinates of the

particle’s center determined using pattern matching analysis. The image is analyzed in

1

real time at 200 frames per second, storing the position 17(1). Since there is no need to
save the image, this reduces memory usage, allowing the particle to be tracked for more
than 12 hours. The full three dimensional potential energy is calculated from 7(t)

assuming a Boltzmann distribution. The time series is analyzed to determine the
transition statistics and the time scales of the particle motions.

The beam power in a Single well is modulated electro—optically with varying
amplitude and frequency, tilting the potential. The amplitude is varied from zero (static
potential) to above the bifurcation amplitude, the amplitude at which the barrier
disappears.

The most probable trajectory for a transition is calculated for a static potential.
The particle moves in both x (the direction connecting stable points) and z (the direction
of beam propagation). The 2 restoring force is much weaker, with a relaxation time
larger than the other directions. This affects the crossover from adiabatic to nonadiabatic
modulation.

The effect of modulation is investigated as a function of modulation amplitude
and frequency by measuring the interwell transition rate. Multiple regions in phase space
are identified, including the adiabatic and weakly nonadiabatic regimes.

To ensure that the static potential does not change during modulation experiments
the stability of the optical system was improved using passive and active control
methods. The laser beam power fluctuations were reduced through stabilization.

This thesis is organized into six sections. Chapter 2 provides the theoretical
description of a Brownian particle in a driven system. Chapter 3 introduces the

experimental techniques of optical trapping and particle tracking, and discusses prior

optical experiments. Chapter 4 describes the experimental apparatus used to create and
drive the bistable potential, to determine particle position, and to establish long-term
system stability. Chapter 5 describes the data analysis and results, including
measurement of the most probable transition path and the power law dependence of the
activation energy in the adiabatic and nonadiabatic frequency regimes. Chapter 6
summarizes these results and makes suggestions for further work. Since there is
substantial reliance on computer control, acquisition, and analysis, detailed descriptions

Of the software algorithms and interfaces are provided in the body of the dissertation.

Chapter 2 Driven systems with a fluctuating force

The present experiment measures the dynamics of a Brownian particle undergoing
thermally activated transitions in a bistable potential. This chapter provides a theoretical
description Of the dynamics. In a static potential the particle behaves like a damped
harmonic oscillator. The addition of a random force (Brownian motion) causes the
particle position to fluctuate, exploring the potential. The position probability in the
potential is given by Boltzmann statistics.

In a metastable or bistable potential the particle makes thermally activated
transitions over the potential energy barrier AU. The rate of transitions is proportional to
the Arrhenius factor, exp(— AU / k 8T). In the Kramers theory, the constant of
proportionality is a function of the potential curvatures near the stable point and the
barrier top [1]. For large barriers (AU >> kBT), the fluctuation responsible for a transition
is rare and the distribution of paths the particle follows over the barrier is sharply peaked
around an optimal path [2].

Periodic modulation of the barrier height breaks the temporal symmetry; the
particle is more likely to escape when the barrier is low. In some systems, a small input
signal in a noisy system results in a large output, a phenomenon called stochastic
resonance. When the spatial symmetry of the potential or the temporal symmetry of the
modulation is broken the transition rates in opposite directions can be independently
adjusted, leading to a net current in extended systems.

Calculation of the activation energy under periodic modulation Shows three regions

of universal dependence of transition rate on modulation amplitude. For large amplitudes

the stable and unstable points approach as the potential tilts. In this regime there is a

power law dependence Of the change in activation energy (5R on the distance from this
critical amplitude A, 6Roc|Ac —A|/’. The power law exponent ,u depends on the

modulation frequency regime. In the adiabatic regime the modulation frequency Q is

much less than the relaxation rate 1,", the particle follows the potential, and ,u = 3/2.

When (2 2 t,"l the activation energy changes too quickly for the particle to respond and

,u = 2. For modulation amplitudes approaching the bifurcation amplitude (for which the

stable and unstable points merge), t7] —>0. Therefore every system becomes

I

nonadiabatic at some modulation amplitude.

2.1. Fluctuations of a Classical Particle

An overdamped particle in a stable potential U (7‘ ) is approximated as a damped
harmonic oscillator for small displacements from the stable point. In this experiment the
particle is confined in three dimensions. In each direction the damping rate F is much
larger than the eigenfrequency of the potential or, the motion is overdamped. A random
driving force is added and the amplitude of random fluctuations determined using the
power spectrum. For a general potential the equation of motion is written as a Langevin
equation, explicitly including the random force. The stationary position distribution is

found from the Fokker-Planck equation, resulting in the Boltzmann distribution.

2.1.1. Damped Harmonic Oscillator

To second order in displacement a stable potential (where U " > 0) is harmonic

for small displacements from the stable point. In one dimension the equation of motion is

mi + 7x + [or = 0 ,
where m is the mass, 7 is the damping constant and k is the Spring constant [3]. The

steady state solution is x(t) = A exp(— 1T+t)+ B exp(— Rt) , where

 

r1 = F/2 i J(r/2)2 — (03 and a) = ./k/m.

For an overdamped particle, F >> a), and the damping rates I} are approximated:

I“, = F/z i F/2,/I — (w/ZF)2 z r/z i F/2 i (oz/l".
The motion of a particle displaced from the stable point becomes

x(t)= A exp(— Ft)+ Bexp(— (02 /Ft), with a fast decaying rate F and a Slower relaxation
rate (oz/I“. For a 0.6 pm diameter silica sphere (m = 2.3 x107”) kg) in water (viscosity

7] = 0.001 kg/m 5) the damping rate F = 67r17a/m = 2.5 x 107 S". For the potential in the

present experiment the curvature in the y direction is a), = 2x105 5", so that

0.3/1“ = 1.6x 103 5'1 (Section 5.2). The F >> a) approximation is equivalent to removing

the inertial term from the equation of motion,
Fit + (02x = 0
with solution x(t)= A exp(— (02/1— I) for I >> l/F.
Driven by an external force, the equation of motion becomes
55 + Fir + (02x = F/m cos(Qt),

with steady state solution

(t): M" sin(Qt + ,8),
x \/(a)2 422)2 +I“ZO2

 

 

2_ 2
where ,6 = tan‘l L2— .
F!)

The amplitude due to the driving force depends on the driving frequency Q, the natural

frequency a), and the damping rate F. Calculating the power spectrum of the position

S_t(Q)= |JC(Q]2 , the mean amplitude for an arbitrary force fit) is calculated from the

5W [4]-

 

power spectrum 55 (Q) =

s,(..)=le_—S.(Q)

+ szz
in the overdamped system. For white noise the power Spectrum is a constant (for small

frequency), S; (Q) = 4kaBT . The position spectrum is a Lorentzian

with a comer frequency at Q = a) 2 /F , the relaxation rate.

2.1.2. Randomly Fluctuating Particles in a General Potential

The harmonic approximation is limited to a quadratic potential or small
fluctuations about the stable point of a more general potential. For larger displacements
the equation of motion is rewritten as the Langevin equation for an overdamped particle
in potential U(x) with random force at) [5-7], )dr(t) = -VU(x)+ fit).

There is a time scale rc such that the random force on the particle at time t is

independent of the force at time t + I}. Averaging over times larger than re the force is

delta correlated (5(t)§(t')) = 2D5 (t — t') with noise amplitude D.

The Spectral density of this noise, S: (Q), is independent of frequency (up tO

Q = 27r/rc ).

SQ()= 2 [ "”( g6);(0))dt =41)

Fluctuations that satisfy this condition produce white noise. (Noise with S: (Q) having

dependence on Q is called colored noise.) The fluctuating force has a Gaussian
amplitude distribution with zero mean [5].
The probability distribution for a particle in potential U(x) is found with the

Fokker-Planck equation. P(x,t | x0,t0) is the conditional probability that the particle will

be at position x at time t if it was initially at x0 at time to. The Fokker-Planck equation

relates this probability to the Langevin equation of motion [6]

OP _a(— VU(x)P) ”k T__ 6313
at ax ax

The stationary solution aP/at = 0 for the probability is

(2.1) P(=x) Poexp( U(x ——)),

kBT

the Boltzmann distribution.

Figure 2.1 shows a schematic drawing of the bistable potential used in this
experiment (solid line). The probability distribution (dashed) is peaked at the stable
points. The probability of the particle being found at the top of the barrier is proportional

to the Arrhenius factor exp(— AU / k 3T), where the barrier height AU is the difference

between the potential at the local minima and the barrier maximum.

 

 

 

 

 

X

Figure 2.1 Probability distribution of an arbitrary potential. The dashed line is the
probability distribution, the solution to equation (2.1) for the bistable potential
(solid line).
If the random fluctuations place the particle at the barrier top, it has an equal

probability to roll down in either direction. The transition rate is expected to also be

proportional to exp(— AU / k 3T ) The next section examines the transition rate and the

particle motions as it makes a transition.

2.2. Thermally Activated Transitions

The particle fluctuates in the regions around a in Figure 2.2. Eventually a large
fluctuation pushes the particle up the barrier to the point b and across to c. The rate
depends on the exponential of the activation energy, W = C exp(— R/ k B T). In a static
potential R = AU . The prefactor C depends on the shape of the potential at the stable
point and the banier.

For large barriers (AU / kBT >> 1) escapes are rare. The probability of a
fluctuation large enough to cause a transition is sharply peaked around an optimal (most
probable) fluctuation. The corresponding path, called the optimal path, depends on the

shape of the potential.

 

U(X)

 

 

 

 

 

Figure 2.2 Escape over a potential barrier. The particle moves randomly in the area
around the stable point a until a large fluctuation pushes it over the barrier of
height AU at b to the stable point e.

2.2.1. Mean Rate of Transitions

The Arrhenius factor demonstrates the temperature dependence of the transition
rate, but it does not give the exact rate for an arbitrary potential. In 1940 Kramers [I]
solved the Fokker-Planck equation to find the prefactor for an overdamped particle. The

transition rates are

 

W=Mliex -AU .
272T kBT

Given a known potential with curvatures at, at the stable point and 0);, at the barrier, this

gives the exact transition rate. For a bistable potential the curvatures in Figure 2.3 are

calculated using the Taylor series approximations

Ua(x)zU(a)+%wj(x—a)2
(2.2) 1
u.(x)~u(b)-gw:<x—b)2

at the stable point a and barrier b [6].

10

 

U(x)

 

 

 

 

Figure 2.3 Quadratic approximations to the extrema of a bistable potential. The dashed
lines represent the quadratic approximations in equation (2.2), centered at x = a
and x = b with curvatures cod and (ob, respectively.

In the present experiment the Brownian particle moves in three dimensions. For a
potential that is metastable only in the x dimension and stable in the y and 2 dimensions,

the rate equation for an overdamped particle becomes [8]

x x y z _
(2,3, W=meaw_a,XP( AU).

271T a); a); kBT

 

Here to; and a); are the curvatures of the potential in the I"h direction at a and b.

The Kramers theory has been investigated since its first formulation [9,10].
Corrections have been made for non-white noise [1 l] and finite barrier heights [12]. The
crossover from low damping to high damping adds an additional term to the prefactor,
peaked around 170) = 10 [12,13]. The transient solutions have also been studied [14-17].

There have been few tests of Kramer’s theory. A quantitative test requires
complete knowledge of the potential energy. Thus the tests only occur in metastable

potentials that can be calculated exactly [18] or that can be measured precisely.

ll

.’~
I
.‘.:A
LIOL

‘.‘, .
l

.m...

...I'.__

 

 

In the overdamped regime, the Kramers theory has been verified by McCann et al
[8]. A 0.6 pm diameter dielectric Sphere was optically trapped in a bistable potential and
driven by Brownian motion. The potential was formed by electric field gradient forces
on a dielectric sphere created by two focused laser beams. The static particle position
distribution is given by equation (2.1). The potential energy was calculated for three
dimensions and the potential curvatures were calculated by quadratic fits to the well and
barrier (Figure 2.3). The Stokes damping rate was calculated from the fluid viscosity and
particle radius.

The transition rate was modified in a controlled way by changing beam power and
spacing. The transition rate was measured (W‘ms) over four orders of magnitude and
compared to the value calculated from equation (2.3) (Figure 2.4). Prefactors calculated
for left-to-right (squares) and right-to-left (triangles) transitions agreed with the

measurements over the entire range (solid line in Figure 2.4)

 

Wk (S'II
k

0.1 r

 

 

 

001 . w .
0.01 or 1 to

wrneas (5‘1)

Figure 2.4 Comparison of Kramers theory to measured transition rates [8]. The points
each represent a data set where the prefactor and barrier height were calculated
from the potential and damping rate (Wk) and the transition rate measured
(W'ms). The boxes represent lefi-to-right transitions and the triangles right-to-left
transitions. The line with unity slope represents exact correspondence between
Wk and Wm”.

12

2.2.2. Optimal Path

The particle fluctuates around a stable point until a large, rare fluctuation drives it
over the barrier. The most probable path for the particle to follow is the result of the
most probable fluctuation that causes a transition [11]. These are called the optimal path
qopt(t) and Optimal fluctuation 9‘0p1(t), respectively.

For the overdamped particle in an arbitrary potential the Langevin equation of

motion for the coordinate q(t) is q(t) = K (q)+ :0), where K(q) contains the deterministic

forces (K (q) = —VU(q)) and at) is the random force. The spectrum of the noise ¢(Q) is
the Fourier transform of the correlation function ¢(t) = (47’ (015(0)) .

The particle makes a transition when there is a sufficiently large §(t). Following

the derivation in [19], the probability of this force is given by the functional

 

Pl:(t)]=exp - ‘ dezdt'FItJFI—z'ro')

2kBT
with noise amplitude kBT. F(t) is the reciprocal of ¢(t), satisfying
Idt'F(t-t')¢(t'—tl)=kBT§(t—-t,). For white noise F(t)=6(t)/2. For small noise
intensity the probability is exponentially small to have a force at) that causes a

transition. Also, the probabilities of different transition inducing forces are exponentially

different. There is an optimal fluctuation 50p, (t) around which the probability
distribution is peaked. The path the particle follows under this fluctuation is the optimal
path qopt (t)

The optimal path and optimal fluctuation maximize the probability functional and

minimize the activation energy,

13

qu(tl€(t)] = g I jdtdrgwo — t'k(t)+ Id: 2(0ch — K(q)— F(t)].

with the Lagrange multiplier xi(t). (The Lagrange multiplier relates 60p, (t) and qop,(t),

finding the probability maximum provided that the equation of motion is satisfied.) The
particle starts near the stable point qa for t ——> -oo and approaches the unstable point qb as
l —) 00 .

The transition rate depends exponentially on the minimum activation energy
functional that takes the particle from go to qb, W = C exp[— Rmin / k 3T ] Assuming white

noise (averaging over time much larger than the noise correlation time I.) the probability

functional is constant and Rm." is the minimum of the action,

RIq(t)I = % ojdtlé - K (0)]2 -

The path that minimizes the activation energy is the solution to

40p, (t) = K (qopt (t)). This is the time reversed path of a particle sliding from the unstable

point to the stable point in the absence of random noise. The process takes on the order
of one relaxation time t,.
Applying these results to this experiment, the bistable potential is approximated as

a one dimensional quartic,

B 2 C 4
2.4 = —— — o

The optimal path is

qopt(t) = 1"] B

C + exp[28(t + tc )]

 

 

l4

with constant tc. The transition is degenerate in time; it can occur for any I...

For a real system there is a distribution of paths that lead to a transition, grouped
around the optimal path. Stepping backwards in time, the paths that arrive at position q,-
at time tf are described by the prehistory probability density, described in [2]. The

distribution p,l (q,t;qf ,tf) is defined as the probability of a path q(t) that ends at point q;

at time tfhaving passed through point q at time t. Equivalently it is the probability of the

fluctuation q(t) that brings the system to (If from the stable point qa through q at time t.
ph (q,t;qf,tf )2 ph (q,t — If;qf,0)= Cexpl— p(q,t — tf;qf )/kBTJ, where p, analogous to
an activation energy, is defined as p(q,t;qf )= ,5(q,t;qf )— 5(q,0;qf ), where
filmq f )= min R[q(tlé(t)l-

,5(q,0;qf) is determined by the optimal path, which minimizes the activation
energy function. Therefore [5(q,t;qf )2 ,5(q,0;qf) and p(q,t;qf )2 0. When q
coincides with qopt(t), ,0(qOpt (t),t;qf )2 0. The peak of the prehistory probability density
is pk (qopt (t),t;qf ,0), following the optimal path.

In the next section a periodic driving term is added to the force K(q;t). The
activation energy gains a time dependent term, changing the transition rate. Transitions
are exponentially more likely to occur when the barrier is a minimum, synchronizing the
transitions with the field. This lifts the time degeneracy of the optimal path and leads to

control of the stochastic system [20].

15

2. 3. Transitions in a Periodically Driven System

The introduction Of a time-dependent external field causes a strong, nonlinear
response in the dynamics of the system. When the amplitude Of the external field is
much less than the barrier height (A << AU) the system is weakly modulated. The
amplitude A can still be larger than kBT, and therefore affect the transition rate
exponentially strongly. The transitions become partially synchronized with the field. At
higher amplitudes (strong modulation), the potential becomes deformed until eventually a
bifurcation point is reached. The stable point and barrier merge and the barrier
disappears.

At modulation frequencies Q much smaller than the relaxation rate (Qt, << I ),
the particle has time to respond to the changes in the potential. The particle’s dynamics
depend on the instantaneous value of the potential and the system is adiabatic. At higher
modulation frequencies (Qt, 21) the system is nonadiabatic. The particle can no longer

equilibrate to changes in the potential.

2.3.1. Stochastic Resonance

A system with a weak periodic input and a strong noisy input is expected to have
a noisy output with a small periodic component. In some nonlinear systems the noise
enhances the signal, producing a large periodic output with small fluctuations. Similarly,
the output signal increases with increasing noise, experiencing a peak in the signal to
noise ratio at an optimal noise amplitude. This phenomenon is called stochastic

resonance [21 -25].

16

Stochastic resonance appears for an overdamped Brownian particle in a bistable
potential with a periodic tilt modulation [26]. Figure 2.5 Shows how a tilt in the potential
lowers the barrier for the left stable point (dashed line). The particle, driven by thermal
fluctuations, has an exponentially larger probability to make a transition than in the static
case. Alter the modulation phase changes by It (dotted line), the barrier increases and the
transition probability is reduced. The barrier height modulation for the two wells is trout

of phase.

 

U00!

 

 

 

 

Figure 2.5 Quartic potential with a tilt modulation. The potential undergoes a linear tilt,
represented in equation (2.5). The curves represent zero modulation (solid line), 0
phase (dotted line), and aphase (dashed line).

A Brownian particle in a symmetric bistable potential U(x) with periodic driving

of amplitude A and frequency Q has an equation of motion

(2.5) q = -%j— + A cos(Qt)+ F(t).
q

The potential has stable points at q].2 and a barrier at q b = 0.

The dynamics of the system are described using linear response theory [24]. (The

derivation up to equation (2.13) follows [26].) The input Acos(Qt) causes the

17

coordinate q(t) to oscillate at the same frequency (and its harmonics). Averaging over

many modulation periods,

(2.6) (q(t)) = Za(n)cos[th + ¢(n)],

n20
where a(n) and ¢(n) are the amplitude and phase shift for each harmonic component.

For small modulation amplitude A, the amplitudes a(n)oc A". The amplitude
a(l) can be written a(1)= Al 1(Q] , where 1(Q) is the susceptibility, relating the driving

force A cos(Qt) to the response (q(t)). The phase Shifi is

¢(1)= — arctan[;£ei%] .

The spectral density of the position fluctuations is the square of the Fourier

transform
1 ’ 2
Q(a))=1im —l Jdt q(t)exp(iwt* .
HP 47rr _r

The response in equation (2.6) gives rise to 6-peaks at the frequencies nQ with area
l/Ia(n)2. The signal to noise ratio (SNR) is defined as the area under the peak at

frequency Q divided by the spectral density in the absence of modulation Q”) (Q). For
small modulation amplitude, SNR = (1/4)A2|1(Q)/Q(0’(Q).

The dependence of this signal to noise ratio on noise intensity D has a peak, the

defining trait of stochastic resonance. To determine if a system displays this

phenomenon, one need only measure the susceptibility 1(Q) and Spectral density

18

Q”) (Q) However, these quantities are related through the fluctuation dissipation

theorem [27,28]

‘-

(2.7) Rez(w)=—:5P[dw.Q“”(w.)—.“i—..
0

K. h

a), —a)

(2.8) Im 1(0)): 32—? Q”) ((1)),

where P implies the Cauchy principal part of the integral.

For the symmetric bistable system defined above, the susceptibility, Spectral
density, and SNR can be calculated for small noise intensities (D << AU, with barrier
height AU). The particle spends most of its time at the stable points 611 and q2 with

intrawell fluctuations with decay rate Q],2 = U I'(ql‘z) and infrequent interwell transitions.

The spectral density is approximated as the sum of contributions from these fluctuations

(2-9) Q(O)(w)= WIQIO)((”)+ WzQim(w)+ erm(w),

where w,, (n = 1,2) is the population of the nth stable state. The susceptibility is also
approximated as three components, calculated from the spectral density in equations (2.2)
and (2.8).

In the symmetric bistable system the populations w,2 and curvatures Q" for the
stable states are equal. Therefore those spectral density and susceptibility components
are also equal (Q,‘°’((o)= Q§°’(w), 11 ((0): 12 ((0)). Assuming the potential is quadratic
at the stable points, its Taylor series expansion is truncated afier the fourth order in q.

The spectral density is approximated in terms of the spectral density of a driven,

overdamped harmonic oscillator, Ln (co) = (1/ 7t)D/ (U ,',' 2 + (02 ).

l9

Allowing for perturbations by the higher order corrections, the spectral density of

intrawell fluctuations becomes

(2.10) Ql°’(w)zL.(w)—7rLi(‘”iU3M‘Ljnbun l

,
4U;2 +0)“

where the primed variables represent Spatial derivatives evaluated at q,,.

The contribution to equation (2.9) from noise induced transitions is

 

(2.11) 9),“)

3

(Ar... .. (<q>:°’ —<q>‘.°’lw‘°’
_ l 2
7T

,
~ ’)
Wm) +a)~

where W0) is the sum of the Kramers rates for transitions out of each well in the absence
. o .
of modulation, WW 2 WI‘ZO’ + WZ‘IO’ = 2W3” where W120) cc exp[— AU/D]. (q); ) IS the

average value of the coordinate q in each well (neglecting interwell transitions). For
small noise amplitudes q remains near the harmonic region of the potential and

1 Du:
2 0:2 '

((1)20) = q.

 

SO far the derivation has assumed D << AU and W0) << Q,, where Q, the
relaxation rate. Transitions occur infrequently due to low noise amplitude, with a rate
much smaller than the relaxation rate. Keeping only terms to first order in D/AU and
zeroth order in Q/Q, (low modulation frequency), but without constraint on W°’/Q, the

Si gnal to noise ratio and phase Shift become

m2 (23le2 +9202

(2.12) SNR=402 Q2W(0’+QZD and

 

QZW‘O’ +920
(2.13) ¢=—arctan 3 ’ 2 .
9. QrW‘O’ +920

 

20

At low noise amplitude, R ~ 1/ D; as D increases, R decreases. For larger D the

WW)2 term grows, causing R to increase with D up to D ~ AU/2, where the assumption
of small noise amplitude fails. At large D the barrier effectively disappears. The particle
diffuses across the entire potential, causing the 6—peak at Q(Q) to spread out and
decreasing R.

Figure 2.6 is a plot of R/A2 vs. D in equation (2.12) for the quartic potential with
B = l and C = l in equation (2.4). The relaxation rate Q, = 2 and the barrier height
AU = 1A. The modulation frequency is set to Q = 0.01 and amplitude A = 0.1. With
increasing D the Signal to noise ratio falls until it reaches a minimum. It then rises to a
peak at D ~ AU/2, where the approximations break down. Extrapolating up to D = AU
shows R falling as the barrier disappears. This has been observed in a bidirectional ring

laser [29] and other systems (see references in [22,24,25]).

R/A2

 

20

15

10

 

 

 

 

Figure 2.6 Signal to noise ratio for stochastic resonance. The normalized signal to noise
ratio (SNR), calculated from linear response theory in the low noise limit, is
plotted vs. noise intensity D up to D = AU. The peak near D = 0.125 indicates
that the SNR is enhanced with increasing D up to a maximum. Additional noise
then reduces the SNR as the barrier is reduced.

21

For a thermally activated transition, the nonzero Signal to noise is
counterintuitive. If there is no noise the rate is zero as there are no transitions. This is
the case in discrete two state systems (such as thermal switching of magnetic spins).

However, the stable points of a continuous system move under modulation, introducing
components to 1(Q) through QLO)(Q) in equation (2.9).

In this experimental system it is easier to change the modulation frequency Q than
to change the noise amplitude D. For a one-dimensional bistable system the particle has
varying degrees of synchronization, depending on the modulation frequency. At low
frequency Q << Wm) the modulation is adiabatic. The particle sees a static potential with
barrier height AU = AU‘O) + 6U(t), where dU(t)z Acos(Qt) with amplitude A at A for
small A. The transition probability is highest When the barrier is lowest, but the particle
still has enough time to escape when the barrier is large.

At high frequency Q >> Q, the particle does not have time to respond to the
instantaneous potential. Instead it sees a potential averaged over the relaxation time
t, = F/a)2 , with modulation amplitude smaller than the adiabatic amplitude. The rate
approaches the unmodulated rate W” for increasing modulation frequency.

At intermediate frequencies, the probability increases for making a transition
while the barrier is low. If the particle does not make a transition, it tends to wait for a
whole period until the barrier is low again. The distribution of dwell times (the amount
of time the particle spends in each well before making a transition) has a decaying series
of peaks at the odd half-integer multiples of the modulation period. When Q = W”) the

synchronization reaches a maximum. This effect was experimentally observed for a

22

Brownian particle in a bistable optical potential by Simon and Libchaber [30] (see

Section 3.3.6).

Directed Diffusion

The dynamics of a symmetric bistable system under symmetric modulation is the
same for both stable points. By breaking the spatial or temporal symmetry, the average
escape rate for each well can be modified independently. For an extended periodic
potential this leads to a net probability current, the direction of which is determined by
the potential and modulation parameters.

This effect is illustrated by the probability distribution in a saw tooth potential,
periodically turned on and off (Figure 2.7). The slope of the potential is greater on the
left, but the barrier heights are the equal. The transition rates are equal and there is no net
current. The particle resides in the minimum of the potential with a probability
distribution given by equation (2.1). The potential is turned off and the particle diffuses
freely. When the potential is turned on again, the probability that the particle has
diffused past the left barrier is greater than the probability that it diffused past the right
barrier, resulting in a net probability current. Spatially asymmetric potentials are called
Brownian ratchets [31,32].

This effect was first considered by Magnasco in his investigation of motor
proteins [33]. In a living cell, chemicals are transported in vesicles on a microtubule
highway by motor proteins. The protein’s attachment to the microtubule is highly
asymmetric. The protein turns single molecules of adenosine triphosphate (ATP) into

adenosine diphosphate (ADP), corresponding to an on-off modulation of the potential,

23

resulting in a net motion. The simple model ignored most of the complexities involved in

biological systems, but yielded a Similar qualitative effect.

”‘i

 

 

 

time I, ‘ I \ [I ‘ I
l A \J A ‘J ‘.

Figure 2.7 Net probability current in a periodically modulated saw tooth potential. The
particle begins localized in one stable point with probability distribution
represented by the solid curve (top). When the potential (dashed line) is turned
off, the position probability distribution spreads out. When the potential is turned
on again there is a greater probability that the particle moved to the left than the
right. Averaged over multiple modulation periods this produces a net current to
the left.

The biased diffusion of a 1.5 pm diameter sphere in an optical ratchet potential
has been measured [34]. A single optical trap was scanned in a circle at 100 Hz with
synchronized modulation of laser power. The potential was alternated between the
ratchet potential and a circular potential around which the particle diffused freely. At the
maximum asymmetry the probability of moving in the clockwise direction was 15%
higher than moving in the counterclockwise direction.

Directed diffusion has also been observed in symmetric potentials with a zero-
mean asymmetric tilt modulation [35-37]. The symmetry is broken, in the case of an
asymmetric sawtooth potential, when the rate of change of the instantaneous barrier

height is different for the two wells. Stochastic resonance theory says that the maximum

synchronization occurs when modulation frequency is half of the escape rate. The

24

maximum synchronization frequency is different for the two directions resulting in a non-
zero current.

The symmetry is also broken if the minimum barrier height is different for the two
wells, as with a biharrnonic modulation with non-zero phase. In this case the escape rate
increases for the well with the lowest barrier, as has been observed in a bistable optical
potential [19]. A 0.6 pm diameter Sphere driven by Brownian motion was trapped in a
potential created by two laser beams. The power in a single beam was modulated with

the biharrnonic function
(2.14) AP = §P[sin(Qt)+ (l/2)Sin(2Qt + ¢,,)],
where OP is the change in beam power corresponding to 1/3 the barrier height, Q is the
modulation frequency (Q/27r = 1 Hz), and (15. 2 is the phase shift (Figure 2.8 inset). For
maximum asymmetry (4512 = 702) the instantaneous escape probability varied with the
phase of the modulation, with maxima corresponding to the minima in barrier heights
(Figure 2.8). The populations differed by 20%. An additional aphase shift interchanged
the populations.

These phenomena are described quantitatively using the susceptibility 1(Q)

defined using linear response theory. In a symmetric system, the susceptibility for

transitions to the lefi (1‘(Q1 and to the right I 1+(Q) are equal. Breaking either the

spatial or temporal symmetry of the system, the susceptibilities are not equal for certain
modulation frequencies. Since the transition rate depends exponentially strongly on

1(Q) it is called the logarithmic susceptibility [19,3 8-41]. Calculation of the logarithmic

susceptibility for certain modulation regimes shows power-law dependence on amplitude.

25

 

 

)— ] —I

i H, ‘ul ‘
003 Ht from 11:10

 

 

' I

g

L .
0
If)
(3

0.02

0.01 '

Probability of switching

 

 

 

 

0.00"
0 It 2R

Phase of modulation (radians)

 

Figure 2.8 Directed diffusion of a Brownian particle in a bistable potential. The curves
represent the least squares fit to the instantaneous escape rate as a function of
phase for the right-to-left and left-to-right transitions under biharrnonic
modulation (equation (2.14)) at maximum asymmetry (¢12 = 71/2). The peak in the
ri t-to-lefi rate shows a bias towards motion to the left. Inset: the waveform of
the right beam power modulation. [8]

2.3.2. Logarithmic Susceptibility

At modulation frequencies greater than the intrawell relaxation rate t, the particle
no longer has time to respond to changes in the potential; the modulation is non-
adiabatic. For small modulation amplitudes (A z D) the rapid modulation becomes an

additional fluctuation, increasing the effective temperature of the system. The logarithm
of the time averaged escape rate 1n(W) at A2 [13,18,42]. At higher amplitudes (but small
compared to the barrier height, A << AU ) the solution of the escape problem shows
ln(W)oc A [43]. As the modulation amplitude grows large compared to the barrier, the

potential becomes tilted and the stable and unstable states approach each other. In this

26

regime the system Shows a power-law dependence Of W on A, with an exponent that

changes with modulation frequency regime.
Following the derivation in [19] to equation (2.16), the escape of an overdamped
particle in a potential U(x) with periodic external modulation F(t) and fluctuating force
q = K(q;t)+ q(t), with

at) is described by the Langevin equation

K(q;t)= -VU(q)+ F (t),

where q is the coordinate. The modulation is periodic F (t): F (t + r,). In the case of

white noise, at) is 6 correlated, (§(0);f(t)) = 2D6(t).

The potential with modulation has a periodic attractor qa (t) = qa (t + 1,) (solid line

in Figure 2.9). In the absence of driving, q, is the stable point of the potential. There is

an unstable state ‘II (t): qb (t+ 1,) representing the potential barrier. An escape occurs

(dashed line) when the particle crosses qb (t) . The particle begins in the stable state at

t —> —00 and reaches qb(t) at t —) oo.

 

qb

q° VWV

Figure 2.9 Periodic stable and unstable states at small driving amplitude. The stable
state q, and unstable state qb approach as the potential in equation (2.4) is tilted.

The dotted line is the optimal path for escape over the barrier, occurring when the

 

 

states are closest.
27

In the regime where F(t) is nonzero but does not distort the system, the
modulation can still affect the escape probability exponentially strongly. The change in
the activation energy due to the modulation is (SR, and the change in the time averaged
escape rate is W/ W = exp(— 5R/ D). W0 is the escape rate in the absence of driving,
given by the Kramers expression (equation (2.3)).

The optimal path q0p.(t) minimizes the time-dependent activation energy

functional
(2.15) R[q(t)l=% I14 + U'(q)— F(t)]z dz-

In the static system the escape can occur at any time t, (Section 2.2.2). With periodic
modulation the time degeneracy of the optimal path is broken. The most probable escape
path occurs once per modulation period and the transitions become synchronized with the
field (Figure 2.9).

To first order in F, the correction 6R is calculated from equation (2.15) along the

most probable escape path qOpt (t) The escape path is an instanton with width on the

order of the intrawell relaxation time (dotted line in Figure 2.9). The correction to the
activation energy is

5R = min 6R(tc)
’c

(2.16) 6R(tc)= Olga — tc)F(t)dt a 21629»; exp(ithc)

-CD

1(Q) is the Fourier Transform of 1(t) and F, is the nth Fourier component of F (t) [39].

For sinusoidal driving F (t) = A cos(Qt), 6R = —|1(Q]A .

28

The term 1(Q) is the linear response of (SR to changes in F(t), and thus the
response Of the logarithm Of W . 1(Q) is called the logarithmic susceptibility (LS). For
white noise driven systems, the LS takes the form 1(t) = 453,} (t).

Using the LS, it is possible to derive both the exponent and the prefactor in the

escape rate. For a White noise driven system [43]

$0 = 2—17I—_£exp(— 5R(¢/Q)/D)d¢

where 6R(tc) comes from equation (2.16). In the limit of large modulation amplitude

(A/D >> 1 but A/AU << 1) the activation energy 6R, and thus the logarithm of the escape

rate, is linear with amplitude.

2.3.3. Strong Modulation

The third region of universality of the escape rate dependence on modulation
amplitude occurs when the amplitude is large enough that the distance between stable and

unstable states becomes small [44-46]. At frequencies small compared to t,, there is a
critical amplitude Afd at which these states merge (Figure 2.10). The potential becomes
flat and the restoring force disappears; the particle is free to diffuse. The relaxation rate
slows down until eventually Q >> t;I for all modulation frequencies, leading to the

universal behavior. (The derivation in this Section follows [45].)

The motion of an overdamped Brownian particle is described by the Langevin
equation q'=K(q;A,t)+§(t), where K(q;A,t)=K(q;A,t+rF) contains the periodic

driving with amplitude A and period ‘l'F. (The modulation frequency (up = Q in previous

29

sections.) The barrier is assumed to be much larger than the noise intensity D (kgT in this

’- '

experiment), and the mean interwell transition rate W<< (up, I

 

 

 

t

Figure 2.10 Periodic stable and unstable states under large amplitude adiabatic driving.
The stable state q, and unstable state qb merge as the quartic potential (Figure 2.5)

is tilted at the critical amplitude Afd. Half of a period later qb merges with the
other stable state (not shown).

The motion of the particle is adiabatic when I}. >> I, , the intrawell relaxation
time. The adiabatic stable and unstable states qub (t) are the solutions to K (qu’b;A,t)= 0
(Figures 2.9 and 2.10). The states merge when A = A2” , the adiabatic critical amplitude
(Figure 2.10). Without loss of generality the coordinates are offset so qj" (0) = q :d (0) = 0
and A: Af" att=nrp(n =0,:1, ...).

AS qzd and qzd approach the potential curvature decreases, changing the
restoring force. Near the critical amplitude aK/aq —> 0, the instantaneous relaxation

time t, = —(6K/6q) diverges, and the adiabatic approximation is violated.

The transition rate is affected exponentially strongly by the change in barrier

height. Assuming AU >> D (or W << (op), a transition is most probable for the small

30

interval about t= nrp. Expanding K (q;A,t) around I = 0 and A = A3" and keeping the

lowest order terms,

2

(2.17) K : aq2 + MA“ —ay3(w,.i)

Where 6Aad:A_Aad, a—lfl fl=§£9 and 72: _l abK

— 2 aq2 , (3A Zara); Otz

 

 

, are evaluated at

A = Af",t= 0, and q = qjd(0).

The adiabatic relaxation time becomes

(2'18) ’3‘" = %I(a7w,t)2 —afl6A“d["2

around the stable and unstable points qub = $1/ (2614:“1 ). For I: 0 and 6A“: 0, tfd
diverges. Using equation (2.18) the condition for adiabaticity is (othd << I. A second,

stronger condition limits the maximum rate of change of the relaxation time,

latfd/atl<<l. This condition is also expressed as tfd <<t, for t, =(aya), )""2, or

to, << lfldAad

 

/7 . The time scale t, limits the adiabatic approximation.

AS the stable and unstable points approach they become distorted (Figure 2.11).,
Solving equation (2.17) for K = 0 in as 5Aad —> 0, the extrema merge along the line
4.11:0): yroFt. The critical amplitude becomes frequency dependent,
A5." = Afd + you, /,6 , where ACS] is the critical amplitude in the nonadiabatic slow driving

regime. The condition wthd <<1 remains valid.

31

 

 

qb

 

qo

 

 

t

Figure 2.11 Periodic stable and unstable states at large driving amplitude and high
frequency. The quartic potential is tilted with A > A2”. The stable state qa and

unstable state qb distort, finally merging at the bifurcation amplitude A:' along
the line q = wat.

Concerning units: The subsequent derivations use a set of reduced variables.
Before explaining the derivation of the power law behavior, the length, time, and energy
scales will be connected to those in the current experiment. The one dimensional
equation of motion is mFJ'r(t)= —VU (x; A,t)+ q(t), where the potential energy U has
units of kBT, position x has units of microns (um), mass m is in kg, and Stokes damping

rate F is 5". Therefore K = -VU/mF.

Assuming the quartic potential (with modulation term) in equations (2.4) and

 

 

Aad 2 Aad .
(2.5), a = 30C" , fl = L, and y2 = mF ‘ (0F = c , where the stable pornt
ml" mF 6Cx (02 ml” 6Cx
s F s

x, : ,/B/3C at the critical amplitude Afd : 2193/2 /3J3C .
Transforming into dimensionless coordinates for position Q =at,q, amplitude
77 = ,6(Acs' —A)/7a)F , and time variable,r = t/t, and transforming the derivative

Q = dQ/ d r , the equation of motion becomes

32

Q = G(Q,I),r)+ 17(1), where G(Q,I7,T)= Q2 — r2 +1 — 77.
Since the system slows when the particle is most likely to escape, any correlation

time in the noise is smaller than the relaxation time. The noise is approximated as

"200.)‘3’2 [dt<:(t):(0)>.

6—correlated, (f(0)f(r)> = 2D6(r) where D 2 la/ 4

 

The activation energy R =D/DR is given by the minimum of the energy

functional

(2.19) R :fifJQ—QZ +22 —l+77)2dr

where again the boundary conditions are Q—> Q0 (1) as r —-> —oo and Q —> Qb(r) as

r —) oo .
In the adiabatic regime (77 >> 1) corresponding to slow modulation, escape is

most likely when r = 0 and thus the 12 term is neglected. The Optimal path that
minimizes R is the instanton Qop,(r)= (77—1)}; tanh((q—1)ié (r—z'0 )j, where to is
arbitrary. The escapes are distributed in time around r= 0. The activation energy

E =§(n—1)3/2 at (A5“ — A)”.

In the nonadiabatic regime the 22 term must be included. This lifts the time

degeneracy and synchronizes the escape, leading to a nonadiabatic correction in R . The

minimum of the activation energy occurs for To = 0. To first order,
~ 4 ‘2 71' 2 — 1’]
R “ 3(77-1I3/ +‘g‘(77-1)'2~

Numerical simulations (also performed in [45]) Show 7% agreement for 77 > 3.

33

Close to the bifurcation point A? (77 << 1) the escape occurs when the stable and

unstable states Q, (r) and Q, (r) are close to each other. Q“ (r) = Q, (r) z r in this regime,

allowing the integrand Of equation (2.19) to be linearized. The solution is

S=J§nzm<<h

The activation energy is quadratic in the distance from the bifurcation point (ACSI — A)2 in
the slow driving regime.

The width of this interval is proportional to (0,.- for coFt, <<1. The adiabatic

exponent ,u = 3/2 crosses over to the nonadiabatic exponent ,u = 2 at large modulation
amplitudes (A:d < A < A?) or large driving frequencies.

As A —>A:' the relaxation time t, becomes long enough to invalidate the
expansion of the driving force in equation (2.17). For long relaxation times the duration
of the most probable escape path exceeds the modulation period; the particle requires
many cycles to escape. In this case the activation energy shows the R at (A:l — A)”,2

dependence on modulation amplitude.

2. 4. Summary

An overdamped particle confined to a potential and driven by a random force is
analogous to a damped, driven harmonic oscillator. There is a relaxation time, the time it
takes a displaced particle to return to the stable point, which is a function of the potential
curvature and the damping rate. The random force causes the particle to explore the

potential resulting in a position distribution given by the Boltzmann distribution.

34

In the bistable potential the thermal fluctuations drive the particle over the barrier

with a rate proportional to exp(— AU / khT ) with a prefactor that depends on the shape of

the potential. The trajectories the particle follows are distributed around the optimal path,
caused by the most probable energy fluctuation that results in a transition. This path can
be directly measured from the prehistory distribution, the distribution of all paths that
result in a transition.

The optimal path minimizes the activation energy. With the addition of periodic
modulation this minimum occurs once per period and the transitions become
synchronized. The modulation frequency can be adjusted to maximize the transition rate,
leading to directed diffusion in a spatially or temporally asymmetric system.

Calculations of the activation energy for large amplitude modulation reveal power
law dependence on the distance to the critical amplitude, 5R oc (A,SI — A)”. For
adiabatically slow modulation (compared to the relaxation rate) the critical exponent is
,u = 3/2 and Af.‘ -—> Afd . In the nonadiabatic regime, ,u = 2 and A? is linear in frequency.

As the modulation amplitude increases the stable and unstable points approach and the
relaxation rate, which depends on the curvature of the potential, approaches zero. The
system becomes nonadiabatic for all modulation frequencies and there is a crossover of
the power law from ,u = 3/2 to ,u= 2.

This theory has been developed recently, and only verified through numerical
simulations [44-46]. The experiments presented in this thesis are the first to identify the

adiabatic and weakly nonadiabatic frequency regimes and confirm the critical exponent

,u = 2 and linear frequency dependence of the critical amplitude A f' .

35

Chapter 3 Optical Trapping

Prior experiments, beginning with the first radial optical trap [47] and the single
beam three dimensional trap [48], have developed methods to create the traps, measure
the potential created by the trapped particle in the beam, and to measure the position of a
trapped particle. This chapter describes these methods and their relationship to the
present work. Chapter 4 contains a detailed description of the experimental apparatus.

Optically transparent particles refract an incident laser beam, causing a
momentum transfer that pushes the particle to the region of highest electromagnetic field
gradient. These forces were used to trap particles in two and ultimately three dimensions.

The position of a trapped particle has been measured with a camera [8], a
photodiode [49], and a position sensitive quadrant photodiode [50]. The camera recorded
images of the particle under an external illumination source. These images were
analyzed to determine the particle center. The photodiode devices measured the trapping
beam scattered from the trapped particle. A calibration related the photodiode signal to
absolute position.

The particle driven by Brownian motion explores the potential. The position
fluctuations contain information about the potential that is extracted from either the
fluctuation spectrum [49] or the position distribution [8]. The potential was also
measured from the particle position under an external force [51].

Multiple laser beams are used to create multiple, independent traps. Two trapped
particles have been used to measure the position dependence of hydrodynamic forces
between the particles. A single particle in two beams, spaced a small distance apart so

the traps overlap, makes thermally activated transitions between the stable points.

36

Additionally, the trapping forces of the independent traps have been modulated with

beam power, synchronizing the transitions with the modulation.

3.]. Experimental Realizations of Optical Trapping

3.1.1. Radial Trap

In 1970 Ashkin discovered that micron-sized transparent particles in water were
drawn into the axis of a laser beam and accelerated in the direction of propagation [47].
He focused a 6.2 pm diameter TEMOO Argon ion laser beam horizontally into a glass cell
filled with water and a mixture of transparent latex spheres (polydisperse solution, 1 pm
mean diameter). A single “trapped” sphere moved in the beam’s direction of propagation
until hitting the wall of the cell, where it remained in the beam center. When the beam
was blocked the particle diffused freely.

The explanation invoked a force generated by momentum transfer due to
reflection and refraction of the laser beam. A latex particle (with index of refraction n =
1.58) in water (index n... = 1.33) deflects incident rays (Figure 3.1). With each deflection
a ray transfers momentum to the sphere, represented by the gray arrows, creating a force
pointing towards the ray and along the direction of propagation. If there are multiple rays
with different intensities there is a net force towards the more intense rays. The TEMOO
laser beam has a Gaussian radial intensity profile, with greatest intensity at the center of
the beam. Therefore the spheres were drawn towards the center of the beam as they were
pushed through the solution.

Ashkin and Dziedzic expanded on this work, levitating transparent particles and

oil drops in air and vacuum [52-54]. A single lens focused the laser beam upward into an

37

air-filled cell. The particles, either 1 to 40 um diameter silicone oil drops or 20 um
diameter fused glass spheres, were drawn into the center of the beam and accelerated
upwards away from the beam waist. The particle came to rest when the axial radiation
pressure and gravity balanced. For the oil drops [53] the equilibrium position varied with

drop diameter; the smaller drops came to rest higher in the beam.

 

Figure 3.1 Radial optical trap. A ray incident on a transparent sphere is refracted,
exchanging momentum with the sphere. If two parallel rays of different powers
enter the sphere, this force pushes the sphere along the direction of propagation
and towards the higher power ray.

3.1.2. Three Dimensional Trap

To create a stable, three dimensional trap the radiation pressure in the direction of
beam propagation must be opposed by an additional axial force. Using the slightly
diverging beams from the radial trapping experiment, a three dimensional trap was
created with two counter-propagating laser beams [47]. The latex particles were radially
trapped at the beam centers and axially trapped by opposed radiation pressure forces. In
1986, Ashkin et a1. [48] three dimensionally trapped latex and silica spheres in a single
laser beam focused with a large convergence angle. The resulting axial gradient
produced a force towards the focal point large enough to overcome the scattering force.

They successfully trapped particles ranging from 25 nm to 10 pm with input laser powers

of100 mW to 1.4 W.

38

Shown schematically in Figure 3.2, the focused beam was deflected by the sphere
(due to its higher index of refraction than the surrounding water). The momentum
exchanged with the particle when it moved axially or radially produced a force towards
the focal point.

Expressions were derived for the trapping and scattering forces in the Rayleigh
regime [48], where the particle diameter 2a is much smaller than the trapping beam
wavelength xi. Under this approximation the intensity variation across the sphere due to
time varying electromagnetic fields is assumed to be small. The electric field was treated

as static.

 

 

Net Force

 

e
&

 

—V

Figure 3.2 Three dimensional optical trap. The lens focuses the beam (represented by
two rays) to the focal point (black dot). The beam is refracted, exchanging
momentum with the sphere and creating a net force in the direction of the bold
arrow. For axial and radial displacements of the particle the net force is towards

the focal point.

39

The force created by light scattering off the sphere is

 

,
m“+2

3]) F _Ion,,1287r5a° m3-1 ‘
( ' scat — 3,14 ’

where 10 is the beam intensity, m was the effective index np/nb, with particle index tip and
index of the surrounding medium nb [48]. The force is in the direction of propagation.

In general, for a neutral, dielectric particle in a static electric field E there is an
induced dipole moment (I =aE, where a is the polarizability of the particle. The
electrostatic energy of this dipole is U = —d - E = —ar]E|2 . In a uniform electric field there
is no force on the particle. If the field is nonuniform, the force is proportional to the
spatial gradient of the energy, F = —VU = aV]E|2.

Substituting for the polarizability of a spherical Rayleigh particle [55] the gradient

force in the axial direction is

2
9

 

 

3 3 2
na m —l
3.2 F =— b VE
( ) grad 2 ( 2+2]|

with a negative sign showing that it opposes the scattering force. The maximum value of
the gradient force occurs when the particle is at an axial displacement z = mtg/J31 ,
where wo is the beam waist radius at the focal point. If, at that point F grad is larger than

F scat, the particle is stably trapped. This sets a constraint on sphere size, wavelength, and

beam waist radius through the ratio of forces R [48],

Fm" = 3‘5 "g is 21.
Fm,t 647!5 [m2 -l J a3w5

m2+2

R:

 

 

40

L111

1"

wu

For the system in [48] (A = 514 nm, 21m = 1.51, latex spheres with m = 1.24), setting
R = 3 (the maximum gradient force was three times the scattering force) the maximum
sphere diameter was 95 nm, for which the Rayleigh approximation (it >> a) is no longer
valid. The conclusion was that this did not imply a lack of stability for larger particles,
where ray optics representations like Figures 3.1 and 3.2 became more valid. Trapping of
1 pm and 10 um diameter spheres showed that this was the case.

The condition R 21 is independent of beam power. To insure that the sphere did
not diffuse out of the trap, they required the power be sufficiently large so the Boltzmann

factor exp(— U / k 3T ) << 1, where

U ___ "3,.3 (m2 —1]IE\2

 

2 m2+2
Requiring U/ka z 10, the minimum silica sphere diameter trapped by a 1.5 W input

beam is predicted to be 19 nm. The smallest sphere actually trapped was 26 nm diameter

silica, trapped with 1.4 W.

3.1.3. Trapping Optics

Ashkin et al. [48] used a laser emitting at A = 514 nm focused by a 100x 1.25 NA
microscope objective lens to create a three dimensional trap. Typical trapping systems,
shown in Figure 3.3, contain a high numerical aperture (NA) lens (usually a microscope
objective), a laser with an output of at least 10 mW, and beam steering mirrors, plus
additional optics to modify beam size, power, stability, and polarization. A detailed

description of our experimental apparatus is in Section 4.1.

41

Laser Stabilizer fl
I J--|_'—_l U

Beam Expander

 

 

 

Beam Steering
Mirror

 

Figure 3.3 Basic trapping system. The beam emitted by the laser is stabilized and power
controlled with the stabilizer. The beam diameter is expanded by two lenses. A
beam steering mirror changes the angle at which the beam enters the objective
lens. The objective focuses the beam to a diffraction limited spot in the water
filled cell.

Objective Lens

To maximize the axial forces the trapping lenses are chosen to have a large

numerical aperture, where the numerical aperture is defined as NA = n b sin ¢ with index

of refraction of the surrounding medium nb and full beam convergence angle 2¢. In the
Rayleigh regime the axial force depends on the NA to the fourth power (see Appendix
B). Therefore a 60x microscope objective lens with NA 0.8 will have 50% of the
maximum axial force from a 100x lens with NA 0.95, or 33 % of the maximum axial
force from a 100x oil immersion lens with NA 1.4.

The largest available numerical apertures are found in microscope objectives. An
objective is a system of lenses designed to magnify nearby objects with high angular
magnification. Microscope objectives are corrected for spherical aberrations and
dispersion, and an objective with 100x magnification focuses the beam to a diffraction
limited diameter (under certain conditions, see Section 4.1). When the objective is part
of a microscope, the position of the focal point in the cell is controlled in three
dimensions with the sample stage and microscope focus controls. If the beam entering
the objective is collimated (i.e., parallel), the beam waist will be in the focal plane of the

42

objective. A particle trapped at the beam waist will be visible when viewed through the
microscope eyepieces and auxiliary viewports. In Section 3.2.1 the particle image is
projected onto a camera by the trapping objective.

When a microscope is used, the laser beam is directed to the objective lens
without disturbing the viewing path. The beam is brought in through the rear
fluorescence port or a side port. A dichroic mirror reflects the laser wavelength into the

objective, but transmits the returning microscope illumination for subsequent viewing

(Figure 3.4).

Trapping Laser

The trapping laser is chosen for the wavelength, power, and stability necessary for
the application. The minimum trapping power reported by Ashkin et a1. [48] was less
than 1 mW. Typical trapping systems use at least 10 mW, although some applications
require more than 1 W (e.g., trapping very small particles). Trapping laser wavelengths
vary from blue to near infrared. Biological trapping systems use near IR where the
absorption by biological materials is lower than at the visible wavelengths. Experiments
with inorganic materials tend to use visible wavelengths for simplicity.

Standing waves in a laser’s resonance cavity create transverse electromagnetic
modes (TEM), variations in the intensity and phase perpendicular to the direction of
propagation. Lasers typically emit in the lowest order mode in both transverse directions,
TEMOO, which has a cylindrically symmetric Gaussian intensity profile. Higher order
modes have regions of zero intensity, for example the TEMo. mode has two bright
regions separated by a node along the x axis. The intensity gradient (and the force for

m > 1 in equation (3.2)) points away from the beam center. Typical trapping systems use

43

TEMOO emitting lasers, but a highly focused beam with cylindrically symmetric TEM;l

(donut) mode has also stably trapped a particle in three dimensions [56].

Beam power is controlled to change the trapping force or to prevent damage to
trapped objects [48]. This occurs through either a mechanical device, such as a polarizer,
or an electro-optic device like a Pockels cell. Some experiments varied the beam power
periodically to modulate the shape of the potential [8,30]. If constant-power is required,
changes in beam power due to laser power fluctuations are filtered using electro-optic

stabilizers.

3.1.4. Other Trapping Geometries

The system described in Section 3.1.3 represents only one way to create a stable
three dimensional trap. Traps have been created using lasers delivered through fiber
optics [57-60], aligned to create the counter-propagating beam trap first developed by
Ashkin [47]. For low numerical aperture focusing lenses the axial trapping force has
been increased using standing wave traps [61,62]. The laser beam entering the cell was
reflected by a mirror, interfering and creating a steep axial intensity gradient between the
nodes and antinodes.

Other experiments have changed the trapping laser. Stable three dimensional traps
have been created using diode lasers [63], corrected for the characteristic divergence
angle difference parallel and perpendicular to the junction. Multiple stable traps have
been constructed with vertical cavity surface emitting lasers (VSCEL) arrays, creating an
8 x 8 array of individually controllable traps [64].

Using conventional gas or solid state lasers, multiple traps have been created by
scanning a single beam across the objective’s focal plane with simultaneous modulation

44

of the power [65-68]. This makes multiple regions of high field gradient at the cost of
overall trapping power and possible high frequency modulation effects. Arrays of traps
have been created through static diffractive elements [69] and dynamic computer

generated holograms [70-72].

3. 2. Trapped Particle Position Tracking

There are three devices other groups [8,49,50,73] have used to track the particle
position: a CCD camera, a single photodiode, and a quadrant photodiode. The CCD
camera records images of the particle, illuminated by a light source, which are computer
analyzed to find the particle position. The photodiode systems measure the intensity
pattern of the trapping laser scattering off of the particle. The forward scattered
component, combined with the unscattered portion of the trapping beam, changes
intensity as the particle changes position. The CCD camera tracking system is
advantageous for recording large particle displacements, as is the case for a particle in the

two well trap used in the present experiment.

3.2.1. Camera Particle Tracking

McCann et al. [8] used a CCD camera to record the image of the particle in the
focal plane. This allowed them to track the particle when it made large excursions from
the trap center, necessary for their dual beam trapping experiment.

The particle was trapped by the laser, focused through the trapping objective lens
(Figure 3.4). A lamp emitted white light, focused into the cell by a condenser lens,
aligned to create a uniformly bright background. The particle was imaged onto the

camera by the trapping objective and an eyepiece. A laser rejection filter prevented beam

45

an

.\...~

b

N.

reflections from reaching the camera. The image was then transferred via a framegrabber
to a computer and a pattern matching algorithm used to find the center of the particle with
i10 nm resolution in x and y. The 2 position was measured by the method described in

Section 4.2, with :t40 nm resolution.

I. ........... . Lamp

Condenser

Cell

Trapping
Objective

 

Mirror

CCD Camera

Figure 3.4 Camera tracking system. The optical trap is formed by the laser, reflected up
by the mirror and focused through the trapping objective lens into the cell. The
particle is trapped in the trapping objective’s focal plane. The lamp is focused
onto the cell by the condenser lens. The particle is imaged by the trapping
objective. The image is sent to a CCD camera, where it is recorded or analyzed
by a computer. [8]

3.2.2. Single Photodiode Particle Tracking

Ghislain et al. [49] used a single photodiode to measure the position fluctuation
spectrum of a trapped sphere (Section 3.3.1). They trapped a particle in the focal plane of
the trapping objective (Figure 3.5). The light forward scattered from the particle (along
with any unscattered trapping beam) was collected by the imaging objective lens and

directed to the photodiode. The imaging objective and photodiode were positioned so the

46

beam diameter at the photodiode was equal to the radius of the detector. The photodiode

was positioned a distance 20 from the conjugate plane.

Photodiode

Imaging
Objective

Cell

 

Trapping
Objective

 

 

 

——7 Mirror
Figure 3.5 Photodiode tracking system. A particle is trapped in the focal plane of the

trapping objective. The photodiode intercepts the forward scattered laser beam,
expanded by an imaging objective lens over a distance Z0. [49]

The particle was assumed to be a spherical lens, and the change in intensity due to
an axial or radial displacement was calculated in the ray-optics limit. The change in

signal power AP; for a small axial displacement A2 is

AP: = [£3ng z 2%vc(mobjmp)2 gfi
where Pave is the average power intercepted by the photodiode and mob]- and mp are the
magnifications of the imaging objective lens and the particle, respectively. Z0 is the axial
separation between the image plane of the second objective and the photodiode. The
maximum signal occurred when the photodiode intercepted '/2 of the total optical power.

The intensity profile at the photodiode was cylindrically symmetric. If the

intensity profile were centered on the photodiode, position changes in the transverse

direction would be measured in distance from the trap center r 2 (ix2 + y2 , with no

47

discrimination between x and y. To measure the position changes in the x direction the

photodiode was positioned off center. The change in signal power AP, is

(11’ ,
Apt = —"“£ Ax z 3 RM (mobimp )3 ,
' dx 7: ' Rdct

where Rde. is the radius of the detector. The maximum signal occurred when the detector
was offset by Rdd. A photodiode centered in the beam with half of its active area masked
could also be used record x fluctuations, at the cost of halving the intercepted power Pm.
Ghislain et al, [49] used this technique to measure the force constants of a single
particle in an optical trap (Section 3.3.1). The position resolution was 1 nm in the radial
direction and 10 nm in the axial direction. Their system was designed to measure the
deflection in a single dimension, a projection along x and z. The three dimensional

motion could not be measured.

3.2.3. Quadrant Photodiode Particle Tracking

Gittes and Schmidt [50] used a quadrant photodiode system (Figure 3.6) to track a
trapped particle in three dimensions simultaneously. The quadrant detector consisted of
four active elements in a 2 x 2 array, labeled A, B, C, and D. The imaging objective lens
collected the forward scattered light, which was collimated by a second lens. The lens
focused the image from the back focal plane onto the quadrant photodiode. This
arrangement allowed Gittes and Schmidt to position the beam anywhere in the objective’s
field of view without affecting the intensity pattern. Changes in the particle’s radial
position deflected the bright spot in the intensity, changing the differential voltage

between elements. Motions in the axial (z) direction changed the overall brightness.

48

 

l

 

 

 

’ fE. Quadrant
'I‘D Photodiode

K

 

- -~—--Back Focal Plane
Imaging
Objective

Cell

 

Trapping
Objective

 

——7 Mirror
Figure 3.6 Quadrant photodiode tracking system. The quadrant photodiode is placed in a
plane conjugate to the back focal plane of the imaging objective. The dotted line
represents the imaging of the back focal plane. The quadrant photodiode, split
into four regions labeled A, B, C, and D measures the position of the peak in the
forward scattered intensity pattern. [50]

When the particle was centered in the trap the differential voltages
Vx = (B + D)—- (A + C) and V y = (A + B)— (C + D) were zero. A change in the x position

caused a change in V, while Vy remained zero and vice versa; the x and y signals were
decoupled. A change in the 2 position resulted in an overall intensity change,
Vz = A + B + C + D. For small particle displacements from the trap center, these signals

were linear.
For a static displacement in x, changes in the intensity (due to z motions)
appeared as motions in the x direction as well. When the intensity in all four elements

doubled, Vx also doubled. To compensate, the displacements were calculated by

§:(B+D)—(A+C)
V A+B+C+D

Z

Axoc

 

i=(A+B)—(C+D)
K A+B+C+D

 

Ayoc

49

AzocVz =A+B+C+D

The differential signal increased the sensitivity (3 1 nm) but required an analog
electronic circuit and introduced extra noise. As with the single photodiode, the linearity
only occurred for small displacements from the trap center. The signals for large
excursions (where the intensity pattern shifts a large amount on the photodiode) were
multivalued. Particles far away from the center of the trap could not be tracked.

Two quadrant photodiodes have been used to track two particles in two traps [74]
(Section 3.3.5). The traps were created with two orthogonally polarized beams. The
image in the back focal plane, a combination of the forward scattered light from both
trapped particles, was split by a beam splitter in place of the upper mirror in Figure 3.6,

and sent to their respective photodiodes.

3.2.4. Comparison of Tracking Methods

The quadrant photodiode differential voltage is linear with particle position only
over a small range of displacements (100 nm). With greater displacements the particle
moves out of the beam waist, no longer deflecting the trapping beam. In practice, the
limit to the measurable particle displacement depends on the acceptance angle of the
imaging objective (Figure 3.6). For a single particle in a dual beam trap, the position
may not be recorded as the particle makes transitions between stable points, depending on
the particle size, beam separation, and imaging objective NA. The camera tracking
system uses the microscope lamp to illuminate the particle anywhere in the objective’s
field of view. The particle is tracked regardless of the trap position.

In the experiments described in this thesis the potential was tilted by changing the

power in one of the beams. A filter in the camera system rejects the reflected laser light

50

and the image is unaffected. Unless compensated, in a quadrant photodiode system this
appears as a periodic shift in z for one beam only.

A camera tracking system marginally achieves the time resolution required by the
present experiments. To track the intrawell motions of the particle the sampling rate
should be many times the intrawell relaxation rate, in this case ~200 Hz. Meiners and
Quake [75] used a quadrant photodiode system running at 50 kHz to measure
hydrodynamic interactions (Section 3.3.5). However, to measure the much slower
interwell transitions in our experiment the 200 frame per second CCD camera used by
McCann et a1. [8] is sufficient.

A disadvantage of the CCD camera system is its image storage requirements. A
quadrant photodiode at 50 kHz requires 8- to l6-bits per channel, for a data rate on the
order of 250 kB per second. A 256 x 256, 8-bit grayscale camera running at 200 frames
per second produces more than 10 MB per second (36 GB per hour), requiring extensive
computer memory if the images are stored. In this experiment the memory requirements
have been reduced by analyzing the image in real time, then discarding the image and
storing the position.

The cost to implement a quadrant photodiode tracking system is approximately
$1000 for a multichannel analog to digital converter board, $1000 for the computer, and
$20 for the photodiode. To determine the particle position the voltages are digitized and
converted to position through calibration or calculation. By comparison a high speed
camera, a framegrabber, and a computer fast enough to analyze the images in real time
cost around $15,000. Specialized software was developed to perform the pattern

matching analysis that determines the sphere position from the images.

51

The camera system requires external illumination in addition to the trapping
beam. For higher frame rates, greater illumination intensity is required. These systems
are expensive and can have adverse thermal effects on the fluid surrounding the trapped
particle. In the photodiode tracking systems, the intense and highly localized trapping

laser illuminates the particle.

3. 3. Prior Experiments Using Optical Traps

This section describes prior work relevant to this experiment. (For an overview of
optical trapping experiments see [76,77].) The shape of the potential was calculated from
the fluctuation spectrum, the position distribution, or direct application of an external
force. Two types of dual beam trapping experiments dealt with problems similar to this
experiment, described in Chapter 4. The problems of creating two beams, tracking a one
or two particles, maintaining a constant beam spacing, and introducing beam power

modulation have been addressed here.

3.3.1. Measurement of Spring Constants from the Fluctuation Spectrum

Ghislain, et al. [49] measured the spring constants of a particle in an optical trap
by measuring the position fluctuation spectrum with the single photodiode setup (Section
3.2.2). These quantities were derived knowing only the comer frequencies and the
Stokes damping rate, calculated from the particle size and solution viscosity.

For small displacements from the trap center (Figure 3.7), the potential was

assumed to be harmonic so that the overdamped one dimensional equation of motion is

(3.3) Mt): —kx(t)+:(t).

52

where k is the spring constant, 7is the Stokes damping coefficient, and 5 (t) is the random
driving force. The damping coefficient for a spherical particle in a fluid is given by the
Stokes expression y = 6mm , where 77 is the viscosity of the surrounding medium and a is
the sphere radius. A three dimensional trap with radial symmetry has a radial and an
axial spring constant. Divided into separate equations of motion 75c(t)= —kxx(t)+¢f(t)
and )2’(t)= —kzz(t)+ fit), Ghislain et al. measured these spring constants simultaneously

by measuring displacements projected onto a line in x and z.

 

F

 

 

 

max

 

 

 

 

X

Figure 3.7 Trapping force vs. position in the Rayleigh size regime. For small
displacements in x the force F is linear. A maximum trapping force Fmax is
reached when the sphere center is at xmax, approximately one sphere radius from
the trap center.

Driven by the random force, the particle position fluctuates in three dimensions.

The power spectrum of fluctuations in a single dimension is the square of the Fourier

transform of the positions, S_,.(f)=|x(f)2, where x(f)= Idtx(t)exp(2m'ft) and the

-d3

Fourier transform of x(t) is — 2722fx( f )

53

The power spectrum of a white noise source is S: (f )=47k ET. The Fourier
transform of equation (3.3) gives (k —27ziyf)x(f)=;‘(f). Squaring both sides gives
47:27:03 +le§_.(f)= S: (f): 47kBT, where fl. = k/2727. Solving for Si,(f),
S,.(f)= k 8T / #2 7(fC2 + f2 ). In one dimension the spectrum is a Lorentzian with comer

frequency fl.
The power spectrum of the position fluctuations (and also the voltage fluctuations
from the photodiode) produced a spectrum with two comer frequencies, found at 40 Hz

and 400 Hz for a 2 pm particle trapped in a 150 mW beam. These corresponded to the
axial (z) and radial (x) spring constants k: =2x10'5 N/m and k". = 2x 10‘4 N/m ,

respectively. The axial intensity gradient, created by the beam converging to the beam

waist, is an order of magnitude smaller than the radial gradient.

3.3.2. Measurement of Potential Energy from the Position Distribution
A trapped particle in thermal equilibrium has a probability distribution p(x, y,z)

related to the potential energy distribution U (x, y, 2) by the Boltzmann distribution

1 U , ,
(34) p(x,y,Z)=ECXp(——-(-E—::-Z—)],
B

where kBT is the temperature (in units of energy), and constant A normalizes the total
probability to unity. By measuring the probability distribution, the potential energy for
every region the particle visits is calculated [8,78-80]. This technique makes no
assumptions about the shape of the potential, only that the particle is in thermal
equilibrium and the position is measured over times much longer than the slowest time

scale. For a single trap the slowest time scale is the axial relaxation time, on the order of

54

1 ms, while for a single particle in a bistable potential they are the interwell transition
times which depend on barrier height.

McCann, et al. [8] obtained the three dimensional potential energy for a particle in
a bistable optical potential. They measured the particle position F (t) at 5 ms intervals for
thousands of interwell transitions using the CCD camera tracking system in described
Section 3.2.3. The position data were converted into a three dimensional histogram

P(x, y,z) by segmenting the volume into elements of size AV = AxAyAz and counting

the number of times the particle was in each volume element.

From this discrete distribution the potential energy U (x, y, 2) was calculated at

each volume element. From (3.4),

U(x,y,z)= —kBTln[£(—)£I’;¥i§-)] — kBTln(Z).

0

The probability distribution p(x, y,z) is represented by the ratio of the discrete position
distribution P(x, y,z) and the total number of counts Po. A depends on the shape of the

potential, and creates a constant offset in the energy. Figure 3.8 shows a typical result for
a particle in a bistable potential. The color scale represents energy in 1 kBT steps with an
arbitrary offset. The red regions contain an average of 1100 counts, while the blue
regions contain only one count.

A stable trap holds a particle for an indefinitely long time; by definition, a stably
trapped particle can never explore outside of the trap. Therefore the absolute energy is
not measured with this method. To measure the potential in the well at greater distances
from the minimum, the particle must be tracked over intervals of time that increase

exponentially with energy.

55

 

Z (um)

 

 

 

0.2 Y (Pm)

 

 

 

15.0 0.2 0.4 0.6
X (m)

Figure 3.8 Three dimensional potential energy histogram of a particle in a bistable
potential. The xz, yz, and xy planes are slices through the minimum potential
energy (x = -0.21 pm) The data set (labeled 101403-03) contains 2,000,000
points with more than 16000 transitions between the stable points. The contours
represent 7 to 13 kBT in l kBT steps (with arbitrary offset), corresponding to
averages of 500, 180, 67, 25, 9, and 1 counts in the corresponding volume
element AV .

56

3.3.3. Measurement of Trapping Force from Stokes Damping

The maximum trapping force and force vs. position relations have been measured
using the velocity-dependent Stokes drag to apply external forces to a trappedparticle.
Figure 3.7 shows a typical trapping force vs. position curve. The force is linear for small
displacements from the trap center, then reaches a maximum force F max. Felgner et al.
[81] measured this maximum force by flowing water around a trapped sphere with
increasing velocity until the particle was pushed out of the trap.

Theoretical trapping force calculations (Appendix A) show that the force varies
linearly with beam power. Ashkin [82] grouped the geometrical factors (particle size,
beam convergence angle, etc.) into a dimensionless trapping efficiency Q, the ratio of the

force on the particle to the total incident photon momentum. Fmax =(an/c)Q, the

maximum force F max is proportional to the beam power P and the index of refraction of
the surrounding medium nb. For particles trapped in water (11,, = 1.33),
nb /c = 4.4 pN/mW.

F elgner et a1. translated the cell on a motorized sample stage, flowing the solution

around the particle with velocity v. At velocities such that 67r77avmax 2 (an/c)Q , the

Stokes force exceeded the maximum trapping force and the particle escaped. A camera
displayed the image of the particle on a monitor. When the particle escaped it moved
away from the trap in the direction of the fluid flow. The procedure was repeated to
reduce the uncertainty in vmax due to random fluctuations. This method was used to test
the predictions of the trapping force calculations [83]. Felgner et al. [81] measured
trapping efficiencies of 0.01 in the axial direction and 0.1 in the radial direction for 1.2

pm diameter silica spheres (corresponding to F max = 0.44 pN/mW P).

57

The force for small displacements (the linear region of Figure 3.7) was measured
using a similar technique. The flow velocity v was held below vmax and the time-
averaged position relative to the trap center measured. From the equation of motion

(equation (3.3)) the spring constant in the direction of motion was calculated [49,81,84].

Spring constants were on the order of 2x10“1 N/m in the radial direction and
5x10‘5 N/ m in the axial direction for a 1 mm diameter sphere.

Removing the assumption of linearity from equation (3.3), the equation of motion

becomes 67r77av=F(x)+¢f(t). Averaging over long times (such that (50)) =0), the
trapping force F (x) was measured from the particle velocity by Simmons et al. [51].

They measured the velocity of a 1 pm diameter silica sphere falling into a trap by first
trapping a particle, then displacing the trap by around 500 nm. From the particle’s
velocity they calculated the force in one dimension for positions beyond xmax in Figure
3.7, the region of maximum trapping force.

Sections 3.3.1 through 3.3.3 describe three methods of inferring the shape of the
potential from the particle motions. The fluctuation spectrum measurement uses a model
potential and the damping rate to calculate the spring constants. The sampling rate must
be large enough to observe the changes, greater than the relaxation rate. The Stokes
damping force measurement gives the force vs. position curve in a single dimension,
from a free particle to the stable point. Calculating the potential energy from the position

distribution gives the shape of the potential without a predetermined model potential.

58

3.3.4. Hydrodynamic Interactions Between Colloidal Particles

In a viscous solution, colloidal particles interact through motions of the solvent
(amongst other forces). The motions of two particles in two optical traps are correlated
due to these hydrodynamic interactions [74,75]. The position correlation functions
parallel and transverse to the beam separation were measured as a function of r.

The optical traps were created using a X = 1064 nm beam from a single solid state
laser. The beams were split by a polarizing beam splitter, creating two orthogonally
polarized beams. After passing through beam steering optics, the beams were
recombined and focused into the sample cell by a high power microscope objective. A
1 pm diameter sphere was trapped in each beam. Trap separation was measured with a
camera.

To track each particle independently, the forward-scattered light was collected by
an imaging objective. The beams were split based on polarization orientation by a
polarizing beam splitter (replacing the upper mirror in Figure 3.5). Each beam was sent
to its own quadrant photodiode, where positions were measured with nm resolution. To
prevent crosstalk and interference effects, Meiners and Quake [75] alternately chopped
the beams at 100 kHz. (The chopping was too fast for the particle to react. Instead it saw
a time averaged potential as described in Section 2.3.1 .)

The particle trajectories were recorded for around 107 data points. Henderson, et
al. discovered that the laser pointing instability caused the traps to move randomly in the
focal plane, over distances on the order of 10 nm with time scales of seconds. While
these fluctuations were too slow to affect the particle dynamics (the fluctuations were

slower than the ~l kHz relaxation rate), the beam motion changed the voltages measured

59

at the photodiodes. To reduce this effect, they split the dataset into 20,000 data point
blocks that were analyzed then averaged together.

The particle position autocorrelation function showed the expected decay, with a
decay rate corresponding to the relaxation rate. The motions of two closely spaced (< 10
sphere radii), trapped particles were anticorrelated. As a particle moves, the surrounding
fluid will flow in the opposite direction. If another sphere is close enough, it will be
caught in this flow and pushed in the opposite direction [74]. The peak position in the

crosscorrelation function depends on the trap strength and the friction on the particles.

3.3.5. Synchronization of Transitions in a Bistable Potential

Simon and Libchaber [30] measured the transitions of a 1 pm diameter particle
over the barrier of a bistable potential under tilt modulation. The bistable potential was
formed by two overlapping optical traps. An Argon ion laser beam with it = 488 nm was
split by a nonpolarizing beam splitter. Each beam reflected off a steering mirror to a
second beam splitter, where they were recombined then focused into the water filled cell
by a 100x objective lens. Each beam passed through a neutral density filter wheel to
independently control the beam power.

The particle image was projected onto a camera running at 30 frames per second,
then transferred to S-VHS video tape. Analysis was performed directly off the video
monitor. The residence time 2' (the time between interwell transitions) was recorded,
ignoring the crossing time (usually around 70 ms).

Inside the cell the beams were separated by approximately 1 pm. The particle

made thermally activated transitions over the barrier. The transition probability was

60

assumed to be constant in time, therefore the residence time probability distribution p(r)

was Poissonian (plotted in Figure 3.9), with normalized distribution p(r) = 2. exp(— Ar).

 

 

 

 

 

 

 

 

 

100
i .
ml - ..;
4 § 3
V d
l A 1
a- 5; j
1 Q. I:
’3 1
5 . ‘
v 6" 1
Q . 0.1 TTnl'I'I'r'r'l'
‘5 0 r 2 3 4 5 o 7 8
4 ‘ residence trme 1: (sec)
2..
41.1
0‘ [in lL—{l [in n
'''''' 1"VVIVIIVT'VV" TVVVT‘ VVUVVV' fiV—VY" 'V'VU‘V Irv—VT'V

residence time I (see)

Figure 3.9 Residence time histogram in a static bistable potential [30]. The probability
p(r) of making a transition is plotted vs. the time since the previous transition.
The probability decays exponentially with decay time ? = 1.3 s, with a cutoff for

short times. Inset: Logarithmic plot with fit corresponding to transition rate

A = 0.76 Hz.

Fitting the dwell time histogram for the N = 219 transitions (sampled over 15
minutes) binned at 1/8 second intervals yielded A = 0.76 i 0.05 Hz. The mean escape
time 17, = 1/1 = 1.3 :t 0.1 s. The distribution also showed a cutoff at short residence times.
The authors concluded that the particle relaxed to the potential minimum before making

another transition, with time scales of the relaxation time t,. The relaxation time was

estimated to be tr = 0.1 i 0.1 s. From Kramers’ theory

1 1 [ U J
——=——exp —— ,
1,, t, kBT

the barrier height was calculated to be 2.6 i 1.0 kBT.

61

The potential was modulated by varying the beam power in each beam with the
motor controlled neutral density filter wheels. The filter wheels were rotated 7t out of
phase to provide tilt modulation. (The change in the potential is represented in Figure

2.6). Approximating the double well potential as a quartic with a time dependent linear
tilt, the particle followed the Langevin equation 5: : Axcos(Qt +9)+ Bx—Cx3 +§(t),
where A is the tilt amplitude, with frequency Q and phase shift (9. The static parameters
B and C define the double well potential. 5‘0) represents the random force due to

Brownian motion. The filter wheel rotation was set so the amplitude was 7% of the total
beam power. Comparisons of residence time histograms above to those in the present
experiment show that the 7% change in beam power corresponds to approximately
1.5 kgT change in the height of the 2.6 kBTbarrier.

At modulation rates below the mean static transition rate, the transitions became
partially synchronized with the modulation. The particle was more likely to escape when
the barrier is low, but because the modulation was so slow there was still a finite
probability to escape at any other time. This was seen in the residence time distribution

(Figure 3.10a) as a peak at time i” E a/Q (half of the modulation period) overlapping

with the static exponential distribution.

At modulation rates above the mean static transition rate but below the relaxation
rate the particle almost always made a transition when the barrier was low. The potential
changed so quickly that if the particle did not make a transition immediately it would
wait a full modulation period until the barrier was low again. The dwell time distribution
(Figure 3.10b) showed a decaying series of peaks at odd half-integers of the modulation

period.

62

(a)

 

 

 

0 l 2 3 4 5 6 7 8
dimensionless time ‘t’l‘l:k

Figure 3.10 Residence time histograms under modulation [30]. The probability p(r) for
(a) modulation slower than the mean transition rate, f/rk = 1.54, showing a

synchronization peak along with the exponential decay at short times. (b) shows
the odd harmonics for faster modulation, f/rk = 0.38. Plots contain (a) N = 282

events and (b) N = 225 events with bin width 1/8 5.

When the modulation rate was one half of the mean static transition rate,
maximum synchronization occurred. The probability to escape during the first
modulation period was a maximum, and higher order harmonics were reduced or
eliminated. Simon and Libchaber considered this peak as confirmation of stochastic
resonance in their system.

At modulation frequencies higher than the relaxation rate (not studied in their
experiment), the particle would no longer have time to respond to the modulation. For
'5 << 1;, the particle would see a time averaged potential with a barrier height equal to the
static barrier. The residence time distribution would lose the harmonic structure,
reverting back to an exponential decay with a mean transition rate lower than the static
rate.

The experiment demonstrated the synchronization of a particle with the tilt

modulation of a bistable potential for a small range of frequencies and a single amplitude.

63

Increasing the modulation amplitude the harmonics in Figure 3.10b should shift towards
the first peak. Eventually the barrier disappears for some fiaction of the modulation
period, during which the particle’s motion will be diffusive.

The relaxation rate in their system was 10 Hz, but the largest modulation
frequency was 1.2 Hz. The range of frequencies from the mean transition rate to the

relaxation rate and beyond was unexplored. Theory predicts a crossover from adiabatic
to nonadiabatic dynamics as the modulation frequency approaches t;1 [45]. This

crossover also depends on the amplitude, since the relaxation rate changes as the potential
distorts.

A Brownian particle moves in three dimensions, yet only one dimension was
considered. Force measurements have shown that the z axis restoring force is an order of
magnitude softer than in x or y [49]. The creation of the barrier breaks the symmetry
between x and y, creating three relaxation rates. These rates affect the residence time
distribution in Figure 3.9, creating the cutoff at short times. Since a single relaxation rate
is inferred from this cutoff, it is unknown how each contributes to the dynamics. A better
method would be to measure the potential through another means, as described in

Sections 3.3.1 to 3.3.3.

3.4. Summary

Three dimensional optical traps are formed by the interaction of transparent
particles with a focused laser beam. The particle is drawn to the region of highest field
gradient with a force depending on the beam convergence, beam power, and particle size.

The present experiment uses two of these traps, positioned roughly one sphere radius

64

apart, to create a bistable potential with a barrier height that depends on the beam power
and trap spacing.

The position of the particle was determined from either the forward scattered light
or by traditional video microscopy. In the dual beam system two quadrant photodiodes
were used to track two particles simultaneously. However, for a single particle in a
bistable potential, it is difficult to determine which beam contains the particle. When
making transitions the particle will always leave the linear sensitivity region, obscuring
some of the dynamics of the transition. Camera based particle tracking is used in the
present experiment to measure the position. Temporal resolution below 5 ms is not
advantageous due to the intrawell relaxation rates of approximately 200 Hz.

The three dimensional potential energy was calculated using the position data.
This has an advantage over the Stokes damping measurements for large displacements
from the trap center or for more complicated (e.g., bistable) potentials. The absolute
depth of the potential could not be measured from the position data alone.

Simon and Libchaber [30] performed experiments combining these techniques.
They positioned two beams within one sphere diameter to create a bistable potential.
They also modulated the beam powers to tilt the potential, synchronizing the transitions
between stable points. The residence time distribution showed peaks at the odd half-
integer harmonics of the modulation period. The harmonics decayed faster as the
frequency approached half of the mean transition rate.

The present experiment improves on this system, increasing stability and
flexibility. The potential remains stable over a longer period, increasing the statistics and

ensuring changes in the dynamics are due to applied modulation and not drift. The

65

electro-optic beam power modulation extends the maximum frequency range above the
maximum relaxation rate, and can vary the amplitude beyond the bifurcation point.
Measurement of the particle position in three dimensions allows calculation of the full
three dimensional potential and its time scales. These changes allow the exploration of

the frequency and amplitude dependence of transitions in a bistable potential.

66

Chapter 4 Experimental Methods

To study the dynamics of a Brownian particle in a metastable potential, it is
necessary to create a representative model system. The physical embodiment developed
here is formed by a single Brownian particle trapped by the partially overlapping laser
beams depicted in Figure 4.1.

The system consists of a sub-micron diameter silica sphere suspended in water,
trapped by the bistable potential formed by the particle’s interaction with the beams. The
beams create a potential maximum that separates two minima, or stable points, of the
potential. A cross-section is shown in Figure 4.1, although the potential created is
actually three dimensional. Thermal fluctuations cause the particle to undergo
fluctuations near the bottom of the potential until eventually a large fluctuation causes the
particle to overcome the barrier, leading to an interwell transition. The potential barrier

height is controlled by the relative beam spacing and the beam powers on the scale of

 

U(x)

 

 

 

 

Figure 4.1 Optical trap formed by two focused laser beams. Representation of the (a)
two beam optical trap with 0.6 pm sphere and (b) potential energy (U(x)) cross-
section. The beams propagate in the z direction and are separated in the x
direction. Interaction of the beams and particle create a potential with barrier b
and stable points a and c.

67

zero to 10 kBT. To measure the transition statistics (and three dimensional potential
energy) the particle is imaged on a CCD camera and computer analyzed to obtain the

three coordinates of the particle as a function of time F (t) The resulting time series is

then analyzed to obtain several statistical measures of the transition dynamics, such as
instantaneous and average transition rates. Analysis of the time series also leads to an
accurate determination of the three dimensional potential surface.

Particles ranging in size from 25 nm to 10 um diameter have been stably trapped
in single beam optical traps [48]. Outside this range the thermal fluctuations overcome
the trapping forces. Smaller particles intercept a reduced intensity gradient while larger
particles require larger beam powers until the scattering forces overcome the trapping
forces. A HeNe laser emitting less than 20 mW, focused by a 100x objective lens can
stably trap a silica sphere of around 1 pm diameter in water. Two focal points separated
by approximately 0.5 pm create a bistable potential with barrier heights less than 10 kBT.
The transition rate is exponentially sensitive to the distance between beams.

The optical traps are created by focusing two lasers to diffraction limited spots
with a high power microscope objective lens. Radial and axial optical trapping forces
increase as the convergence angle increases. The Olympus 100x PLAN-APO objective
has a numerical aperture of 1.4, corresponding to a full angle beam convergence of 144
degrees for the outermost rays. To take advantage of the larger forces from these outer
rays, the 4.8 mm diameter entrance aperture of the objective lens is overfilled by the
4.9 mm diameter beams. Two pairs of lenses expand the beams from the diameter at the

laser output (0.98 mm) before they enter the objective lens.

68

The HeNe lasers with 20 mW output power exceed the minimum trapping
threshold of 5 mW (for a 0.6 pm diameter sphere). Exceeding 100 mW risks damaging
the microscope optics. Higher beam powers result in a trap with larger curvature and
thus higher barriers. For comparable barriers, higher beam powers must have smaller
beam separations, amplifying the effect of beam steering instabilities (which are
independent of power and position). For this reason the beam power entering the
microscope was set around 8 mW.

There are two limitations on the sphere size. At diameters smaller than the
wavelength of the illumination (1 = 0.5 um) the sphere is difficult to resolve. It requires
more power to trap and the traps need to be closer together for the desired barrier height.
For spheres above 1 pm diameter the beam separation becomes comparable to the sphere
diameter. The beams travel through the edges of the sphere, creating a third stable point
between the beam waists. The spheres used are 0.6 pm diameter silica (Bangs
Laboratories), large enough to resolve with a light microscope and small enough so no
third stable point appears at any trap separation.

For 8 mW per beam entering the microscope a 0.6 pm diameter silica sphere will
experience a potential barrier of approximately 6 kg?" with 0.5 pm beam separation.
These conditions lead to a mean interwell transition rate of 0.2 Hz. The separation
between the traps is controlled with a mirror in each beam. The mirrors change the angle
at which the collimated beams enter the objective lens, displacing the beams in the focal
plane.

The power in each beam is controlled independently by a laser power stabilizer.

A single beam is sinusoidally modulated at rates from 0.2 Hz to 2 kHz at amplitudes up

69

to and beyond the amplitude at which the barrier disappears. This allows us to modulate
the barrier of both traps at rates above the largest relaxation rate (500 Hz).

An interwell transition takes between 10 and 25 ms. A camera recording to
videotape has been used [30] to measure particle position with 33 ms resolution. The
faster photodiode tracking techniques in Section 3.2 are not viable due to the large
displacements of the particle from the beam center. The CCD camera used here operates
at 200 frames per second, recording the image of the sphere in the objective’s focal plane.
The images are analyzed by a computer in real time, extracting and storing the x, y, and 2
positions then discarding the image. This reduces the memory used by a factor of 1000,
allowing the particle to be tracked for more than 12 hours. The 5 ms integration time is
fast enough to capture interwell transitions.

The CCD camera attached to the side viewport of the microscope images the
particle in the objective lens’s focal plane. The digitized signal is sent to the data
acquisition card in the computer. A pattern matching routine is used to find the x and y
positions. The particle’s 2 position is calculated from the integrated brightness of the
image, centered on the x and y coordinates. The transparent spherical particle acts like a
lens, focusing the background illumination. For a small range of displacements the
image brightness changes linearly with 2 position. A calibration of the sum of the
image’s pixel values vs. the particle’s position relative to the focal plane allows x, y, and
z to be measured for every image.

The escape rate changes with the beam spacing and barrier height. For a barrier
of 6 kBT the average rate is approximately 1 Hz. For a static potential, the particle has a

constant probability of escape; the escape process is Poissonian. The statistical

70

uncertainty of a Poissonian process with N events is x/N . To get an uncertainty below
10%, the potential needs to remain stable for over 100 transitions in each direction or an
average of 200 seconds for the 6 kBT barrier. To study the dynamics under modulation of
the banier, the static barrier height should remain within :t 0.1 kg?" for the entire set of
varying frequencies or amplitudes, typically 1 to 2 hours.

This chapter describes the optics used to create and control the dual beam optical
trap, including the optics used to expand, position, and modulate the beams, the
microscope optics, and the sample cell. Particle tracking with the CCD camera will also
be explained. Finally the modifications made to increase the beam power and pointing

stability will be discussed.

4. 1. Trapping Optics

The optical system (Figure 4.2) is designed to expand the 0.98 mm diameter
beams from the two HeNe lasers to slightly overfill the 4.8 mm diameter entrance
aperture of the objective lens. Additional optical elements stabilize and modulate the
beam powers and provide independent beam steering in the objective lens’s focal plane.

The two independently steerable optical traps are created by two 17 mW HeNe
lasers emitting at 633 nm. Laser power stabilizers (SI and 82) filter out intensity
fluctuations and control beam power. Each beam travels the same distance before being
combined at the polarizing beam splitter, BSl. The lenses L1 through L4 expand the
beam from 0.98 mm diameter at the lasers to a 4.9 mm diameter collimated beam at the
entrance aperture of the objective. The beams are independently positioned in the

objective lens’s focal plane with the gimbal mounted mirrors G1 and G2. The 100x oil

71

immersion objective lens focuses each beam to a diffraction limited spot in a water filled

cell mounted on the microscope stage.

 

 

 

 

 

 

Las 1 81 Pl
|_—___'er—l............:| fill-23M] -------- Beam 1
=2-u _J - -Beam 2 ”mp
PDl Condenser
Laser 2 82 p2 Ll L2?Léz C,
:i-{ju -.j___]_§-s, L3 4 _
diva“ ._... _ _._._. _._.-.._.,t-Dl
D2 BSl 7J4
Microscope

Figure 4.2 Optical system schematic. Lasers 1 and 2 emit beams 1 and 2, respectively.
The beams travel through stabilizers SI and S2, lenses L1 and L2, steering
mirrors G1 and G2, are combined at beam splitter BSl, then lenses L3 and L4
before entering the microscope where it reflects off of dichroic mirror D1 and
enters the objective lens. The cell is on the microscope stage and illuminated
from above by the lamp and condenser. Beam samplers P1 and P2 pick off part
of the beam and send it to photodiodes PD] and PD2. The half-wave retarder
(M2) rotates beam 1’s polarization before it reflects off mirror Ml. Both beams
travel through the quarter-wave retarder M4 after L3.

4.1.1. Optical Bench

To damp floor vibrations, the lasers, optics, and microscope are mounted on an
optical vibration isolation table. Two 2000 pound optical tables, bolted together in an L
shape, are suspended on six air piston supports, damping vibrations above 10 Hz.

The optical system exists entirely on a 5 foot x 10 foot x 12 inch thick Technical
Manufacturing Corporation Cleantop II optical table. One end butts against the side of a
4 foot x 14 foot x 12 inch thick table, connected by steel yoke. Two Gimbal Piston air
supports are at each end and at the comer of the L. The 3/16 inch thick stainless steel top
is epoxied to a steel honeycomb. This provides rigidity without the mass of a solid table.
Tapped screw holes in a 1 inch rectangular array allow easy mounting of the optical

elements.

72

4.1.2. Lasers

The two independent optical traps are created by two Research Electro-Optics
LHRP-1701 Helium-Neon lasers (labeled Laser 1 and Laser 2 in Figure 1). The laser’s
output is 17 mW (minimum) at 633 nm (actual powers are greater than 20 mW) with
vertical polarization. The beam diameter at the output coupler of the laser is 0.98 mm
with a divergence of 0.82 mrad.

The lasers are mounted in Newport ULM-TILT cylindrical laser mounts. The
1.75 inch diameter laser head is clamped to a V-groove with a nylon screw. The V—
groove’s angular orientation in two axes is controlled by a pair of screw actuators and a
ball pivot. The ULM-TILT is mounted to a 10” tall, 1V2” diameter hollow stainless steel
post. The post’s base is clamped to the optical bench with two custom aluminum table
clamps. To eliminate the need to change the beam height before entering the microscope,

the laser is positioned so the beam is 8.5 inches above the table.

4.1.3. Stabilizers

Each laser is followed by a laser power stabilizer (S1 and S2 in Figure 4.2).
Inadvertent laser intensity fluctuations modulate the potential, contributing noise in
addition to the spontaneous thermal Brownian motion. The stabilizers reduce these
fluctuations by factors of 80 to 200 from DC to 10 kHz, with decreasing attenuation up to
2 MHz. The stabilizer also changes the laser power in response to an external voltage
signal. This is used to control the DC power in both beams and to modulate beam 1 at
frequencies ranging from 1/5 Hz to 2 kHz and amplitudes from 0.06 to 6 mW.

Labeled S1 in Figure l, the Cambridge Research & Instrumentation LS-100

stabilizer follows Laser 1. The LS-lOO uses a Pockels cell to rotate the polarization of

73

the linearly polarized incident beam (Figure 4.3). The angle of rotation is directly
proportional to the applied voltage. A calcite polarizer removes the horizontally
polarized portion, emitting an attenuated beam polarized perpendicular to the optical
bench. A beam splitter sends 3% of the transmitted beam to a temperature compensated
photodiode. The photodiode voltage is used as a feedback control of the Pockels cell
voltage, reducing noise over a 0 to 2 MHz bandwidth.

The LS-lOO is mounted on two 2 inch long, '/2 inch diameter stainless steel posts.
Each post is secured to a 2 inch long, 1 inch OD aluminum post holder, which is screwed
into a 3 inch x 2 inch x 3/8 inch aluminum base. Each base is clamped to the table with

two custom aluminum table clamps.

 

    
 

Beam splitter

 

Pockels Cell

 
    
   

Polarizer

 

 

Feedback
' hotodiode

 

 

 

 

 

 

Figure 4.3 LS-lOO and LS-PRO schematic. The beam enters the Pockels cell where the
polarization is rotated. In the LS-PRO only this is followed by a liquid crystal
element which rotates the polarization at frequencies below 10 Hz. The polarizer
removes the horizontally polarized component of the beam. Before exiting, the
beam passes through the field aperture and the beam splitter, where a small
fraction of the beam is sent to the feedback photodiode.

Proper alignment eliminates distortions in the beam profile due to diffraction and

maximizes stabilization, especially since the LS-IOO entrance aperture diameter is 2 mm,

only slightly larger than the beam diameter. The first step in alignment is to remove the

cover. The manual recommends using less than 25 mW while aligning the stabilizer,

therefore the 20 mW from the HeNe laser does not need to be reduced. Positioning the

74

front end of the LS-100 aligns the input beam with the entrance aperture. The height and
orientation of the rear end are adjusted so the beam passes through the Pockels cell, field
aperture, and output aperture. The beam should exit the stabilizer without beam profile
distortions. To maximize stabilization, the beam should be centered on the photodiode.

The maximum input power is limited to 25 mW by the photodiode response. For
higher power lasers, a neutral density filter should be installed between the beam splitter
and photodiode. With the neutral density filter, the stabilizer can accept up to 10 W of
laser power with reduced power resolution.

The LS-100 output power is controlled and modulated by the external power
supply, which applies from 0 to 250 V across the Pockels cell, causing a greater than 90°
rotation to the beam polarization. The laser power knob on the power supply controls the
beam power offset over the range set by the feedback gain knob. The feedback gain is set
to the maximum of 1000 V/mA. The power supply also accepts a low voltage external
signal supplied by a Stanford Research Systems D8345 Synthesized Function Generator
to control the DC beam power set point. The amplified external signal directly drives the
Pockels cell, allowing modulation of the beam power at frequencies from 0.2 Hz to 2
kHz.

Laser 2 is stabilized and controlled by a CRI LS-PRO stabilizer, labeled S2.
Similar to the LS-100, the LS-PRO uses a Pockels cell and polarizer to reduce the beam
power fluctuations in the range of 10 Hz to 2 MHz. A liquid crystal variable retarder, in
series with the Pockels cell, rotates the incident beam polarization to eliminate
fluctuations below 10 Hz (Figure 4.3). With a liquid crystal to set the DC beam power,

the voltage across the Pockels cell is greatly reduced, extending its lifetime.

75

The LS-PRO output power is controlled by an external control circuit. An AC
converter supplies the LS-PRO with 18 V DC. The beam power is externally controlled
through a 9-pin serial connector on the stabilizer body. ‘A custom-built control circuit
allows the user to vary the setpoint with a 10 turn potentiometer or with a 0 to 2 V signal
from a second SRS D8345 function generator. Due to the slow response of the liquid
crystal element, the maximum modulation rate is 10 Hz.

Since there is no external power supply, the LS-PRO has an ON/OF F button and a
MODE button on its chassis. When the LS-PRO is turned on or the MODE button
pressed the stabilizer cycles through its dynamic range to optimize the gain of the
external voltage to the beam power. The stabilizer is kept on to prevent the gain from
changing, invalidating the beam power to voltage calibration. Alignment of the LS-PRO

follows the same procedure as the LS-100.

4.1.4. Beam Power Monitoring

The beam power is monitored by measuring a small fraction of the total power,
sampled by the beam sampling optics P1 and P2 following the stabilizers. The beam
sampler is a 1 inch diameter antireflection coated glass window with a slight wedge
shape. It reflects a small portion of the incident beam (up to 10% at 45 degrees) without
distorting the transmitted beam. The beam samplers are mounted in 1 inch mirror mounts
screwed into 1/2 inch diameter stainless steel posts, secured in an aluminum post holder
mounted on a 3 inch x 2 inch x 3/8 inch aluminum base. The base is clamped to the table
with two custom aluminum table clamps. The mirror mount can rotate the beam sampler
and reflected beam, but this translates the beam and changes the alignment further down

the beam line.

76

The power in each sampled beam is measured by a Centronic OSD155T silicon
photodiode in a shielded box (PDI and PD2). The photodiode acts in photovoltaic mode
(i.e., unbiased) with a 1100 Q resistor in parallel. Coaxial cables connect each circuit
output to the 1 MS) terminated inputs on the LeCroy 9310AM digital oscilloscope. The
beam power vs. photodiode voltage and power vs. firnction generator offset are calibrated
for each beam. Using the math functions on the oscilloscope, the beam powers are
displayed vs. time.

The power vs. modulation voltage curves are calibrated for the LS-100 and LS-
PRO. A calibration of beam 1 power entering lens L4 (Pl) to the offset voltage of the
LS-100 (V1) has the linear relationship Pl 2 —6.31(mW/V)Vl + 7.98 mW , and is valid for
frequencies from 0 to 2 kHz above which the photodiode PDl response rolls off. The
LS-PRO’s power (P2) to voltage (V2) calibration is P =20.7(mW/v)V2 —0.l9mW.
These calibrations change when the gain is modified (through the LS-100’s power supply

or the LS-PRO’s automatic gain optimization).

4.1.5. Polarization Optics

To prevent the loss of 50% of the beam power when the beams are combined, the
polarizations are adjusted with a half-wave retarder so a polarizing beam splitter can be
used. To remove any polarization dependent effects the linearly polarized beams are
transformed to circular polarization with a quarter-wave retarder.

In beam 1 the beam sampler is followed by an Edmund Scientific zero-order
quartz half-wave (M2) retarder mounted in a rotation mount, held by a 1/2 inch stainless
steel post, aluminum post holder, and aluminum base clamped to the table with aluminum

table clamps. The half-wave retarder rotates the polarization of beam 1 from vertical to

77

horizontal. The beams are orthogonally polarized when they are combined at the
polarizing beam splitter (BSl) (Newport lOBCl6PC.4, 633 nm polarizing beam splitter
cube)

The polarizing beam splitter/combiner transmits 90% of a horizontally polarized
beam while reflecting 99% of a vertically polarized beam. A non-polarizing beam
splitter transmits and reflects 50% of each beam, losing 50% of the power from the 17
mW lasers and severely limiting the dynamic range. The beam splitter is mounted on a
prism table held by a '/2 inch stainless steel post, aluminum post holder, and aluminum
base clamped to the table with two aluminum table clamps. Motions of the beam splitter
add extra noise to beam 2 only. To improve stability the prism table is also held from the
side opposite beam 2 by a ‘/2 inch stainless steel post in an aluminum post holder. The
post holder is screwed into a Newport 340-RC clamp on a 1‘/2 inch stainless steel post,
clamped to the table with two aluminum table clamps.

To remove any polarization dependent trapping effects, the beams are circularly
polarized with a quarter-wave retarder (Tower Optical zero-order quartz quarter-
waveplate) positioned afier lens L3. The quarter-wave retarder is mounted the same way
as the half-wave retarder. The fast axis is aligned by setting each beam to the same
power. A polarizer at any orientation is put in front of a power meter. The power in each
beam is measured and the quarter-wave retarder rotated until these powers are equal. At
this point the orthogonally polarized beams are rotated by the same amount and must be
circularly polarized. The polarizer is rotated to confirm that the power doesn’t change for

either beam.

78

4.1.6. Beam Expanding Optics

Theoretical trapping force estimates [82,85-87] show that the trapping force
increases with the convergence angle and thus the size of the beam entering the objective
lens. The 0.98 mm diameter beam emitted by the RED HeNe lasers is expanded by two
pairs of lenses to 4.9 mm, overfilling the 4.8 mm entrance aperture of the Olympus
PLAN-APO 100x oil immersion lens.

The propagation of the Gaussian beam produced by the TEMOO mode is different
from geometrical optics [88]. The Gaussian beam is always either diverging from or
converging to a beam waist, the region where the beam diameter is smallest. Due to

diffraction, the beam does not collapse to a point. It has a minimum diameter

(10 = 4A/7t6l , where 6’ is the full angle beam divergence. The quantity (106 is constant.

The beam waist diameter is inversely proportional to the convergence angle.

do
0

m '2

Figure 4.4 Beam waist of a converging Gaussian beam. The beam propagates in the z
direction, converging at an angle Bto a beam waist diameter do.

 

At a distance 2 away from the beam waist the beam diameter is d 2 = d; + 6222.

Far away from the beam waist the diameter is approximated by the geometrical optics
expression d = 92. The extent of the beam waist region is characterized with a parameter

called the Rayleigh range 2R, defined as the distance from the beam waist to the beam
diameter d =\/§d0. This distance is 2R =d0/6’ and the beam diameter becomes

== aim/1+(z/2R)2 .

79

 

22 e: z’1—-—N

 

1*— x———>’<— f—véa— f —+i<—-x’—~

 

 

 

Figure 4.5 Gaussian beam propagation through a lens. The beam, propagating in the z
direction, expands from the beam waist a distance 22 from the lens. The beam is
focused to the beam waist at 2" . f is the focal length, x = (22 — f) and,

x' = (z; — f). Primed coordinates indicate distances on the right (rear) side of the
lens.

Gaussian beam propagation through a lens is considered in terms of the
Newtonian form of the thin lens equation for geometrical optics is — xx' = f 2 , where f is

the focal length of the lens, x is the distance from the front focal point to the object, and

I

x is the distance from the rear focal point to the image. For Gaussian beams this

becomes (22 — f le' — f )= f 2 — f02 , where 22 is the distance from the lens to the front
beam waist and z,’ is the distance from the lens to the rear beam waist. (Primed

coordinates are on the rear side of the lens.) The diffraction term f02 is defined

I
f—dO-ig- 22'
o “ RR'

'b‘a
The beam waist diameters on the left and right side of the lens are related by

0,62 (22 — f )2 d5 (2" — f). The relationship between d0 and d; is calculated in terms of

 

a = [fl/«22 —f)2 + 2,2? , d6 = ado , and 6' = O/a. The position of the new beam waist

Ski20=aWa-fl

80

A Gaussian beam can never be truly collimated (6 = 0). The beam is considered

collimated when the distance to the next beam waist, z,’ , is maximized. This occurs

when a=f/ 22?a and 22 =f+:,,. In this case, 2; =f+f2/22R.

Each beam is expanded by two pairs of lenses. The diverging beam from the laser
is doubled in diameter by lenses L1 and L2. After the beams are combined, both are
expanded by an additional factor of 2 by lenses L3 and L4. A single pair of lenses could
have been used to expand the beam, but the Ll-L2 pair is necessary to make independent
adjustments to the position and angle of each beam during alignment. The L3-L4 pair is
required for beam steering, discussed below.

Table 4.] Gaussian propagation spreadsheet (from Lenses4.xls). The beam waist
diameter d0, Rayleigh range zR and position 21 for each lens are calculated from
equations (4.1 to 4.4) in units of mm (angles are in radians). dLl, d12, dZG, dG3,
d34, and d4O are the distances between the laser and lens L1, L1 and L2, L2 and
the gimbal mounted mirror, the gimbal mirror and L3, L3 and L4, and L4 and the
objective lens’s entrance aperture, respectively. The lens spacings and focal
lengths were adjusted so the beam slightly larger than 4.8 mm at the entrance
aperture. The beam expands from 0.98 to 1.28 mm by the time it gets to the first
lens. The final beam diameter is 4.91 mm with a divergence angle of 0.3
milliradians. The objective focuses this to a 0.2 pm diameter beam waist with a
Rayleigh range of 50 nm inside the cell.

 

Beam diameter at apeture 4.91 mm
Divergence angle at aperture 3e—4 rad

 

 

81

Lens pair L1-L2 is positioned so L1 is 1000 mm from the lasers (Table 4.1). At
that distance the beam has expanded from 0.98 mm to 1.27 mm diameter. The lenses
have focal lengths fl = 100 mm and f2 = 200 mm, and are spaced 310 mm apart. The
output is a nearly collimated beam with diameter 2.41 mm. After being combined at the
polarizing beam splitter, the beams are expanded by the L3-L4 pair with focal lengths
f3 = 200 mm and f4 = 400 mm spaced the sum of their focal lengths (600 mm) apart. The
result is a collimated beam with diameter 4.91 mm entering the objective lens 400 mm

after lens L4.

4.1.7. Beam Steering

The potential barrier height is controlled by varying the distance between the
optical traps. Using a gimbal mounted mirror in each beam (G1 and G2), the beams are
independently positioned in the objective’s focal plane. The mirrors and lenses L3 and
L4 are positioned so that, under rotations of the mirrors, the beam stays centered on the
entrance aperture of the objective. The mirrors and entrance aperture are in confocal
planes. This prevents clipping of the beam during translation in the focal plane, which
would change the trap power and shape based on the trap position.

The general expression [85] for the distance between the gimbal mirror and the
first lens do; in terms of the lens spacing d34, the distance from the fourth lens to the

objective entrance aperture d4o, and the focal lengths f3 and fl is

f4d34 Td4o(d34 ".f4)
f4(d34 "f3)"'d4o(d34 "'f3 —f4).

 

d6} =f3

If the distance between the lenses is equal to the sum of the focal lengths (d34 =f3 +fi)

the expression simplifies to

82

f2. fr
(163:.— .3 .4-“d4o -
Ll H I. l

The tilt angle at the entrance aperture is 60 = —2(f3/f4 PO, where 60 is the

gimbal mirror mechanical angle. In the focal plane of the objective, the distance traveled

Ax: fEFLQO, where fan is the effective focal length of the objective lens.

(fEFL = 1.19 mm for the Olympus PLAN-APO 100x objective.)

For focal lengths f3 = 200 mm and f4 = 400 mm and lens spacings d34 = 600 mm
and (MO = 400 mm, dG3 = f3 = 200 mm. Each beam needs its own gimbal mirror,
therefore the mirrors are positioned before the beams are combined at the polarizing
beam splitter. To make room for the beam splitter mount, lens L3 was chosen to have a
focal length larger than 100 mm. From the selection of lenses including 100 mm, 125
nun, 150 mm, 200 mm, and 250 mm (each with a corresponding lens of twice the focal
length for L4), the 200 mm focal length was chosen. This provided a large enough
distance between the gimbal mirrors and L3 while keeping the total distance from the
gimbal mirror to the microscope 100 mm smaller than the 250 mm — 500 mm

combination.

4.1.8. Microscope

The beams are focused to diffraction limited spots in the cell by a 100x
microscope objective mounted in the Olympus IX-70 inverted microscope (Figure 4.6).
The microscope provides a framework for objective mounting, illumination, and sample
translation. The beam enters the microscope below the stage, reflects off of the dichroic
mirror (D1 in Figure 1) then enters the objective. The beam is considered “inside” the
Olympus IX-70 microscope when it is directly over the first edge of the microscope base,

83

260 mm from lens L4. The beam travels 80 mm to the dichroic mirror (Chroma
Technologies 632DCRB) which reflects the beam up into the entrance aperture of the
objective 60 mm away. The dichroic mirror is designed to reflect all polarizations

equally, preventing distortions of the circularly polarized beams.

  
  
 
   
 
  

Lamp

Condenser

 

Cell

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Computer
Stage
Objective
_ _ L. Dichroic Mirror
7 7
I“! Laser Rejection
_ : Filters
Genesrs i 0 A
f
CCD Camera Microscope

Figure 4.6 Microscope and particle tracking apparatus. The beam enters the microscope
below the stage, reflecting off the dichroic mirror into the 100x oil immersion
objective lens. The beam is focused to a diffraction limited spot in the water
filled cell. The condenser focuses the illumination into the cell. The objective
and 5x eyepiece focus the image onto the CCD camera. The dichroic mirror and
laser rejection filters remove the backscattered trapping beam. The image is
transferred to the computer via the Genesis framegrabber for analysis.

The beams enter the Olympus PLAN-APO 100x oil immersion objective. The
objective focuses the beam 0.2 mm past the lens with numerical aperture 1.4. (Numerical
aperture is defined in Section 3.1.3.) A layer of index matching oil between the objective
and the cell reduces the reflections at the glass interface and increases the numerical
aperture. The objective lens is flat field corrected (indicated by the PLAN designation).
The focal point remains in the same plane throughout the field of view. An uncorrected

lens would focus closer to the lens at the edges of the field of view. The objective is also

apochromatic (APO). Due to dispersion, an uncorrected lens would have a focal distance

84

that changes with wavelength. The apochromatic correction adds triplet lenses to make
red, green, and blue wavelengths focus at the same point.

A 0.55 NA condenser focuses the illumination fiom a 100 W halogen lamp onto
the trapped particle in the sample cell. The condenser is aligned for Kohler illumination,
which creates a uniform intensity over the field of view. The light captured by the
objective lens travels through the dichroic mirror and into the microscope. The light
scattering off the particle and the cell’s glass/water interface are much brighter than the
illumination. A pair of 3 O. D. laser rejection filters (Omega Optical 633-SIB) below the
dichroic mirror eliminate the scattered laser light.

The particle image enters the microscope where it is distributed to the viewports.
It is magnified by 1.5x before a beam splitter sends 80% of the light to the side viewport
where a 5x eyepiece focuses the light on the Dalsa CA-Dl CCD camera. The remaining
20% goes to the trinoc head (not shown in Figure 4.6) where 80% is sent to the Sony
DXC-151A color CCD camera and displayed on a Sony PVM-14N2U monitor, while the

other 20% goes to the eyepieces.

4.1.9. Optical Alignment Procedure

The trapping efficiency and stability are maximized for a properly aligned system.
The alignment of the 18 optical elements pictured in Figure 4.3 is broken down in order
of necessity, starting with lasers and mirrors, then stabilizers and beam samplers, and
finally lenses. The alignment is checked during each step using the reflections of a piece
of glass on the microscope stage. The final alignment is tested by tracking a trapped

particle.

85

The optical system is initially aligned with as few elements as possible. The
lasers, mirrors, and beam splitter (equivalent to a mirror for beam 2) are mounted and
placed to the proper positions. To simplify alignment, the beams travel in directions
parallel to the table’s edges. The mirrors are rotated so the beams enter the microscope.
In practice, one beam is aligned at a time.

The shape of the beam in the focal plane is used to verify the alignment at every
step. A piece of glass is mounted on the stage to provide a surface for reflections. The
laser rejection filters are removed so the reflections can be viewed on the Sony monitor.
The mirrors are rotated to center the beams in the objective’s field of view. At the same
time, the mirror positions are adjusted so the reflections are radially symmetric. This
assures that the beams are both centered on the objective’s entrance aperture and enter
vertically.

Next the elements that cause shifts in beam position are added. The beam
splitters in the LS-100 and LS-PRO stabilizers translate the beam up 3 mm. The beam
sampling elements translate the beams a distance that depends on the angle of the
reflected beam. After installation, the beams are realigned with the same procedure.
This step usually requires the laser mounts to be lowered.

Finally the lenses are added, starting with L1 in beam 1. The beam is aligned
using the reflections from the front and back of the lens. When the lens is properly
aligned, both reflections should overlap and travel back into the stabilizer. Due to
imperfections in the previous alignment steps, the beam may not be centered in the field

of view of the objective. The lens is translated perpendicular to the beam direction until

86

the reflection is centered and uniform on the Sony monitor. The lens mount is clamped
to the table.

This procedure is repeated with lens L2 in beam 1 and lenses L3 and L4 after the
beam splitter. With L3 and L4 in place the positions of gimbal mounted mirror G2 and
the beam splitter usually need to be adjusted. Once beam 2 is centered and uniform in
brightness the remaining lenses in beam 2 are aligned, beginning with L1.

The final step in alignment is tracking a particle in each trap. The three
dimensional potential energy will show if the beams are vertical or if one beam is deeper
in the focal plane. If the beams are not vertical, lens L2 is moved until the soft axis of the
potential energy is aligned with the z axis. If necessary, a trap is translated in z by
moving L2 closer to or farther from L1. After each translation the potential is measured

and the process repeated until the traps are vertical and in the same focal plane.

4.1.10. Sample Cell

The 0.6 pm diameter silica spheres are dispersed in water and placed in a glass
cell sealed with epoxy resin. The cell design allows light from the condenser to pass
through to the objective lens without obstruction. To limit convection the cell thickness
is less than 200 pm. The buffer solution places a slight negative charge on the spheres
and glass surfaces to prevent bonding. The cell is designed to be used with the objective
clamp (discussed in section 4.3.1).

Due to the short working distance of the objective, the bottom surface of the cell
is #1 cover glass. The 100x oil immersion objective has a working distance (distance
from the lens housing to the beam waist) of 0.2 mm. Only #1 cover glass, with a

thickness of 0.13 to 0.16 mm, is thin enough to be practical. #le cover glass, with

87

thickness 0.16 to 0.19 mm, allows only 20 to 30 pm of working distance above the glass
while the focal plane would be inside a piece of #2 cover glass (thickness 0.19 to 0.25
rmn). The objective lens focuses upwards in the Olympus IX-70 inverted microscope,
therefore only the bottom surface of the cell needs to be #1 cover glass.

The condenser focuses the illumination from the lamp to a 1 mm diameter spot.
The local heating causes convection currents in the cell which increase with cell
thickness. To limit this anisotropic force the cell is less than 0.2 mm thick. For thinner
cells the proximity of the glass surfaces would change the effective damping of the
particle motion. At this thickness the convection current forces are observed to be orders
of magnitude smaller than the Brownian forces.

The 0.6 pm silica spheres are dispersed in a 0.0001 M NaOH solution. The
hydroxyl groups bond with the surface, producing a net negative charge (0.5 OH/nm2 at
pH = 10) [89]. This prevents the spheres from colliding with and adhering to the glass
cell. Normally silica particles in deionized water will not bond, but the epoxy used in cell
construction leaches ions into the solution, eventually adhering the spheres to the surface.
The buffer solution extends the life of the cell from less than one week to four weeks.

The solution is created by mixing dry 0.6 pm spheres and the NaOH solution. A
22 gauge needle is used to scoop a small amount (enough to coat 1 mm of the needle)
into 40 ml of the NaOH solution. The solution is placed in an ultrasonic cleaner for 30
minutes to break up the clumps into single spheres. This solution is diluted 1:150 in
NaOH solution and placed in the ultrasonic cleaner for another 30 minutes.

Approximately 10 pl is deposited in the cell with a syringe.

88

The cell is constructed with an 18 mm diameter #1 cover glass on the bottom,
attached to a 3 inch x 1 inch, 1.25 mm thick glass slide with epoxy. The gap between is
wedge shaped with 100 pm to 200 um thickness. The glass slide spans a gap in the
sample stage so the cell can be translated without the stage colliding with the objective
clamp (Section 4.3.2). The 18 mm diameter cover glass was chosen to fit the objective
clamp’s 28 mm inside diameter. Therefore the objective clamp supports the cell on the
flat glass slide rather than the cover glass and epoxy.

The glass pieces are cleaned, assembled, then baked before being filled. The
glass is washed in Alconox detergent for 15 minutes in an ultrasonic cleaner. The glass is
rinsed with deionized water, then cleaned in acetone for 15 minutes, then methanol for 15
minutes. Devcon 5-Minute Epoxy is applied to 3%: of the circumference of the 18 mm
diameter cover glass. The cover glass is placed on the slide with a #1 cover glass shard
as a spacer (Figure 4.7a). The capillary action of the epoxy causes the open end to have
an approximately 200 um gap and the closed end a 100 um gap. The cell is baked at
50 °C for at least 12 hours to fully cure the epoxy.

The cleaned and baked cell is filled with 0.6 pm diameter spheres in the buffered
solution. After curing, the spacer is removed and the cell is rinsed in deionized water and
blown dry at least four times to remove deposited solvents. A small volume of 0.6 pm
sphere solution is drawn into a 3 cc syringe with a 22 gauge needle. The needle is placed
in contact with the slide at the open end of the cell. The wedge shape of the gap draws
the solution to the closed end of the cell. The cell is filled to within 3 to 5 mm of the top.

The cell is sealed with 5 Minute Epoxy (Figure 4.7b). If the epoxy touches the

solution, the cell lifetime is reduced to a few days. To prevent this, a minimum amount

89

of epoxy is applied. As the epoxy flows into the gap the escaping air forms a 1 mm wide
channel to the edge. The channel is filled when the epoxy is so viscous that it no longer

flows into the gap (around 3 minutes). If the cell is not airtight more epoxy is mixed and

 

 

 

 

 

 

 

 

 

 

 

applied to the edge.
h

a
U

b
O

l C l

E

Figure 4.7 Cell construction. (a) The 18 mm diameter cover glass is epoxied on 3A of its
circumference and adhered to the 3 inch x 1 inch glass slide. A #1 cover glass
spacer (0.13 mm thick) is placed in the open end to prevent the gap from closing.
(b) After curing, the cell is filled with solution (blue) then capped with more
epoxy. A 3 to 5 mm air gap is left so the epoxy does not contact the solution. (c)
cell side view.

4. 2. Particle Tracking

The particle position is the only output of this experiment. The potential energy
in three dimensions is calculated from the three dimensional position distribution, and the
potential parameters from one dimensional potential profiles. The interwell transition
statistics are measured from the x position fluctuations with 5 ms time resolution.

The particle is trapped in the transparent, water filled cell and illuminated from

above by the lamp focused by the 0.55 NA condenser (Figure 4.7). The objective lens

90

sends the image to the CCD camera on the microscope’s side viewport. The camera
exposes the image for 5 ms then transfers it to the computer for pattern matching
analysis. The analysis calculates the center of the particle in x and y, and the integrated
image brightness centered on (x, y) is used to calculate z.

The analysis takes less than one exposure time (5 ms), so the double-buffered
algorithm calculates the position and discards the image before the next one has finished
being transferred. The x, y, and z coordinates are stored in the computer’s memory until

the run is complete, possibly more than 12 hours.

4.2.1. Camera Interface

The CA—Dl camera is a 256 x 256 pixel, 8-bit CCD image sensor with 16 um
square pixels and 100% fill factor. It receives a synchronization signal from the Genesis
framegrabber, triggering the transfer of a single image. The camera transfers data at
15 MB per second, corresponding to a maximum frame rate of 200 frames per second.
The particle image is magnified by the 100x objective, a 1.5x magnifier inside the
microscope, and the 5x eyepiece before the camera. Calibration with a fixed grating
shows that each 16 um wide pixel is equivalent to 0.071 pm in the focal plane. The
camera’s fixed pattern noise is 0.5 pixel values.

The camera uses a charge transfer process to read out the image, minimizing the
inactive time. The incident photons from the objective’s focal plane create a charge in
the CCD elements. After a set exposure time (signaled by an RS422 pulse), the charge is
shifted from the photosensitive region to an optically masked storage region of the CCD.

While this secondary array is transferred the primary array is free to expose the next

91

image. The parallel design lowers the minimum time between frames to the frame

transfer time (5 ms).

 

_l | EXSYNC

l L— FVAL
I L_| l_| |__ LVAL

STROBE

l l l l l ]£.1|1,2[I.3|1.4| J 1231521231231 13,113,2E3l341 l l PIXELS

Figure 4.8 Dalsa CA-Dl timing diagram. The EXSYNC pulse triggers frame readout.
The frame valid signal, FVAL, is set to high for the duration. The line valid
signal, LVAL, is set high during each line readout. A single pixel value is output
over 8 single bit channels synchronized with the 15 MHz pixel clock, STROBE.
(This timing diagram is for a 3 x 4 pixel CCD array.)

 

 

 

 

 

 

The camera exposure and frame transfer is controlled through the RS422 (2 V
differential) external synchronization signal, EXSYNC. The falling edge triggers a frame
transfer (Figure 4.8). The primary CCD continues exposure until the next EXSYNC
falling edge. If EXSYNC is held low, the camera operates in free-run mode and will
transfer at the maximum frame rate. The EXSYNC signal is also used to synchronize the
beam power modulation and thus the camera is always operated in triggered mode. A
secondary control signal, PRIN, drains the charge from the CCD and can be used as an
electronic shutter.

The camera outputs the image along with signals to synchronize reception by the
framegrabber. The camera timing is illustrated in Figure 4.8. EXSYNC is sent from the
computer and set low for at least 100 nanoseconds to trigger frame readout. The frame

valid (FV AL) signal tells the data acquisition system that a frame is being transferred.

92

The line valid (LVAL) is set high as each line is transferred. Each 8-bit pixel is output

over 8 parallel channels, at the same rate as the 15 MHz STROBE signal.

4.2.2. Data Acquisition and Analysis

The Dalsa CA-Dl camera transfers its image to the Matrox Genesis imaging
system’s framegrabber for analysis. The image is analyzed on the Genesis processor
board and the host computer’s hardware using a series of C functions from the Matrox
Imaging Library (MIL). WinMi12, the data acquisition program, uses MIL commands to
acquire images and find the x, y, and 2 position of the particle in real time. After
acquisition WinMi12 ’5 analysis routines are used to calculate three dimensional
histograms and the intrawell dwell times (Appendix E).

The Genesis framegrabber and processor each occupy one PCI slot in the dual
processor data acquisition computer. Over a SCSI-2 type connector the framegrabber
accepts 32 bits at 40 MHz from a camera. This data is transferred to the processor board
over a ribbon cable (not through the PCI bus), during which formatting and zooming
operations occur. The image is stored in 64 MB of onboard SDRAM. Image analysis
can be performed with the TMS320C80 (C80) digital signal processor or the image can
be transferred to the computer memory and analyzed with the host CPU. The Pentium
Xeon 550 MHz processor outperforms the 50 MHz C80 DSP in pattern matching analysis
speed, therefore all images are transferred to the host memory for analysis. The
processor board has its own VGA display adapter which controls a second (overlay)

monitor.

93

4.2.3. WinMi12 — Data Acquisition Program

The data acquisition and analysis are performed by the custom Visual C++
program WinMi12. The program uses the MIL commands to interface with the Dalsa CA-
Dl camera, acquire images at 201 frames per second, and perform pattern matching
analyses at the same rate. The program sends instructions to the framegrabber to change
the region of interest and contrast of the incoming image, the model image and model
parameters are chosen, and the x, y, and 2 position data is saved in ASCII or binary

format.

Matrox Imaging Library

The Matrox Imaging Library is a set of C routines included in the Matrox Genesis
imaging system, used to control the camera interface and image acquisition. There are
also more than 100 image processing algorithms ranging from image rotation and pixel
value modification to Fourier transform, edge detection, and pattern matching. The
image processing commands are run on either the host computer CPU or the optimized
image processing hardware on the Genesis processor board.

MIL separates the individual hardware components into systems, all under control
of the MIL application. The available systems are: (1) the Genesis board for image
acquisition, display, and processing; (2) the Host CPU for image processing; and (3) the
Host VGA adapter to display the image on the computer monitor. Buffers and displays
are allocated on each system and reside in that system’s hardware. The systems

communicate over the PCI bus.

94

Camera Interface

To acquire images at the maximum rate without image distortion, the Genesis
framegrabber is given the camera timings in Figure 4.8. These values are stored in the
digitizer configuration format (dcf) file, CAD10256MSU3.dcf. This file is downloaded
to the framegrabber when WinMi12 is initialized.

The def file for the CA-Dl included with the MIL software was modified to
achieve the maximum frame rate using Matrox’s camera interfacing application,
Intellicam. Readout of the horizontal lines is triggered by an EXSYNC pulse. The line
transfer begins two STROBE pulses after this, a period called the back porch. 256 pixels
are transferred at the 15 MHz STROBE rate, followed by a front porch of 24 STROBE
pulses until the next sync pulse. 282 total pulses at 15 MHz give a horizontal frequency
of 53.2 kHz. The entire frame contains 256 horizontal readouts for a final rate of
207.8 Hz.

The dcf sets the external synchronization (EXSYNC) pulses sent to the camera
from the framegrabber. The time the signal is active (low) or inactive (high) can each be
set between 20 us and 1 second. To achieve the maximum frame rate, the active time is
set to 164 horizontal sync pulses, or 3.1 ms, and the inactive time is to 100 sync pulses, or
1.9 ms. Since the frame transfer is triggered by the falling edge of the EXSYNC signal,
the actual active and inactive times are arbitrary as long as they sum to 264 pulses. The

active and inactive times are controlled by MIL commands in WinMi12.

WinMi12 User Interface

The WinMi12 user interface (Figure 4.9) divides the program functionality into 6

sections. The controls are buttons (push buttons, radio buttons, and check boxes), edit

95

boxes, and text boxes. The upper right section, marked Area 1, sets the lookup table

(LUT) and region of interest (ROI) parameters. Area 2 controls the RunningCOG
routine. Area 3 contains the data acquisition controls. Area 4 controls the acquisition of
the pattern matching model as well as the model parameters. Areas 5 and 6, discussed in

Sections 5.3 and 5.2, respectively, control the intrawell dwell time (switcher) calculation

and the 3D potential energy calculations.

 
      
 
   
    
     
 
 
  
         
   
    

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Figure 4.9 WinMi12 user interface. The program functions are roughly divided into six
areas. Areal: LUT and ROI parameters. Area 2: RunningCOG. Area 3: data
acquisition and saving routines. Area 4: pattern matching model image

parameters. Area 5: intrawell dwell time analysis. Area 6: 3D potential energy
analysis.

96

Lookup Table

To increase contrast in an image with pixel values that do not span the full 0 to
255 range the values over the entire image are changed using a lookup table (LUT). This
occurs as the frame is transferred from the framegrabber to the processor board. For each
input value in the table there is an output value. The LUT generation function can be a

polynomial, trigonometric or logarithmic function. but WinMi12 uses a linear function.

 

 

 

 

 

 

 

 

 

 

 

 

 

. . LUT LUT .

Original Image index values Resulting Image
30 60 90 30 30 0 0 84 170 0
30 90 120 60 60 84 0 170 255 84

—> —>
30 90 120 60 90 170 0 170255 84
30 60 90 90 120 255 0 84 170170

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.10 Lookup Table. The pixel values in the 4 by 4 pixel original image are
transformed by the LUT. For each pixel value in the LUT index there is a
corresponding output value. The resulting image has nearly 3 times the dynamic
range.

The LUT scales pixel values from the index to the LUT value (Figure 4.10). The
LUT is defined in area 1 of the user interface in Figure 4.9. The beginning and ending
LUT index values (in the “From” edit boxes) represent the minimum and maximum
expected pixel values. The LUT values (in the “To” edit boxes) represent the final pixel
value range. Figure 4.10 demonstrates a four value lookup table. The original image’s
pixel values range from 30 to 120. The LUT expands the range to 0 to 255 to create the
resulting image. This increases the contrast and dynamic range, but the fixed pattern

camera noise is amplified.

97

Region of Interest

To increase analysis speed the transferred image is cropped to a region of interest
(ROI) around the sphere. The image of the particle is much smaller than the field of view
of the CA-Dl camera. Each camera pixel corresponds to 0.071 um in the objective lens’s
field of view. A 0.6 pm diameter sphere spans 15 pixels (with diffraction), and the traps
are spaced less than 0.5 pm (7 pixels) apart, for a total of 22 pixels in the x direction.
Analysis of the image outside of a 40 x 40 pixel R01 is a waste of processing time.

Set in area 1 in Figure 4.9, the ROI parameters are downloaded into the
framegabber before acquisition begins. As the image is acquired, only the pixels within
this area are saved to a buffer; the others are discarded. This increases the pattern

matching speed by a factor of 4.

Running Center of Gravity

RunningCOG, a modified version of the real time particle tracking routine, is used
to display the motion of a trapped sphere. This provides visual feedback when
positioning the beams (using the gimbal-mounted mirrors) to achieve the desired
transition rate. The particle position is determined with the pattern matching algorithm
and the model center is displayed as a white dot on the image. To center the model and
eliminate mismatches, the model image and model parameters are changed in the model
parameter section (Figure 4.9, area 4). RunningCOG is also used to track two instances
of the model image with the beamtracker subroutine, used to track beam positions
(Section 4.3).

The RunningCOG routine is controlled in area 2 of Figure 4.9. The start button

initializes the buffers. A “grab” buffer on the Genesis system receives the image from

98

the camera. This is transferred to the overlay buffer for display on the overlay monitor.
Two analysis buffers are allocated on the host CPU system, used in the blob analysis or
pattern matching analysis below. Three display buffers and two displays are allocated on
the host VGA system. The def file, LUT and ROI are downloaded to the framegrabber.

RunningCOG grabs an image, displays it on the selected computer monitors, and
finds the particle center. A Windows system timer is set to post a message to WinMi12 ’s
message queue every 5 ms. This message triggers a Grab command, transferring the next
available image to a buffer on the Genesis board. This image is copied to the Genesis
VGA memory for display on the overlay monitor and to the host CPU system for
analysis. If the Show buffer box is checked (Figure 4.9, area 1) the image is displayed on
the computer monitor. The x and y positions of the sphere, calculated through either blob
analysis or pattern matching analysis, are displayed in a separate window along with a
histogram of x positions. The average x and average y positions are displayed in the X
COG and Y COG edit boxes. The averages and position window are reset after the
number of frames in the “# Frames before reset” edit box.

If the “Use Pattern Matching” checkbox is not checked, the sphere position will
be calculated using blob analysis, explained in the next section. The image is binarized
using the threshold value entered in area 3 of Figure 4.9. Binarization sets all pixel
values above the threshold to 1 (white) and all values below the threshold to 0 (black).
Blob analysis finds the largest white region and calculates the center of mass. If the
“Show binary” checkbox in area 1 is checked the binarized buffer is displayed. The

threshold value is set so the largest possible area of the sphere is white while the

99

background is black. This maximizes the precision of the x and y positions while
preventing false matches to bright areas in the background.

Pattern matching analysis is performed if “Use Pattern Matching” is checked.
The model is selected and its parameters set in area 4 of Figure 4.9 before acquisition is
begun. If “Show binary" is checked in area 1 the center of the sphere is displayed on top
of the image. The model parameters are adjusted to center the model. If the model
matches to another region of the image (mismatches), the model image is replaced. This
iterative process produces a model valid over a large range of sphere positions with a
minimum analysis time.

Checking the “Beam Tracking” checkbox selects the beamtracker subroutine,
which tracks two objects displaced in the x direction, used in stability measurements of
the beam positions by pattern matching the laser reflections from the cell’s glass-water
interface (Section 4.3). The image is shifted by the number of pixels entered in the lower
edit box, first to the right then to the left. Pattern matching analysis finds a single
instance of the model in each shifted image. The sampling period is entered in the top
edit box in milliseconds (to a minimum of 10 ms). The x and y positions and the

integrated brightness (2 position) are stored to the filename in area 3 of Figure 4.9.

Data acquisition

To track the particle over long time periods without filling the computer memory,
images are analyzed and discarded in the 5 ms interval between image transfers and only
the x, y, and 2 positions recorded. At 200 frames per second, storing 40 x 40 pixel
images at 1 byte per pixel requires more than 300 kB per second, filling the computer’s

512 MB of physical memory in less than 30 minutes (or 350,000 frames). Storing the

100

position as two floating point (8 byte) and one long precision (4 byte) values requires less
than 4 kB per second, increasing the upper limit to 37 hours.

The Matrox Imaging Library has two routines for finding an object in an image:
blob analysis and pattern matching analysis. Blob analysis finds connected regions of
similar pixels in an image, in this case white pixels representing the sphere on a black
background. The x and y positions are determined from the center of mass. Pattern
matching analysis finds the position of a model in the image. Pattern matching is slower
than the blob analysis, but has greater precision. Both routines find the x and y particle
center and sum the pixel values in an 11 x 11 box centered on x and y within the 5 ms

window, therefore all data acquisition uses pattern matching analysis.

Particle Tracking by Blob Analysis

Blob analysis finds regions of similar pixels in an image and calculates features
based on the distribution of pixels. These features range from blob area, perimeter, and
position to complicated geometrical features such as roughness, convexity, and moment
of inertia.

To find the sphere center, the image is copied to a binary (1 bit) image and the
“center of gravity” determined. The 0.6 pm diameter sphere appears as a fuzzy white dot
surrounded by a dark ring (Figure 4.11a). Pixel values above a threshold value, input in
area 3 of Figure 4.9, are set to 1 and values below are set to 0. The threshold is set just
above the average background value, low enough to make the sphere’s blob as large as
possible but high enough to keep any background blobs smaller than the sphere’s blob.
Figure 4.11b shows Figure 4.11a binarized with a threshold of 169. The sphere blob is

21 pixels in area with three smaller background blobs visible.

101

 

Figure 4.11 Blob analysis. The acquired image (a) is binarized with a threshold of 169
pixel values. The largest blob in the binarized image (b) is found and the center
of mass computed (represented by the intersection of the two lines). (c) The
original image with center.

The only relevant blob analysis features to particle tracking are the center of

gravity in x and y. The values, if and y . are calculated from the first moments,

Zap, Exp,
’— and T: ’ ,
2p. y 21),

 

f:

where p,- is the pixel value of the blob at position (x,-, y,-). The center of gravity is
calculated from either the binary image (p,- = 0 or 1) or the grayscale image (p,- = 0 to
255). To take advantage of the higher resolution, WinMi12 uses the grayscale center of
gravity. Figures 4.11b and c show the position of the center of the image on the binary
and grayscale images. The threshold is adjusted so the largest blob is the sphere,
therefore the analysis is always performed on the first (largest) blob in the result list.

Even with the grayscale center of gravity, the resolution of the blob analysis is
larger than the pattern matching analysis. The dark diffi‘action ring around the sphere
limits the size of the blob to 5 x 5 pixels, less than 'A the sphere’s area. The estimated
resolution of this method is 'A pixel. compared to less than 1/8 pixels for the pattern

matching analysis.

102

Particle Tracking by Pattern Matching Analysis

Pattern matching analysis finds the instance of a user defined model in an image.
The image containing the model is captured from the camera and the model parameters
chosen using RunningCOG. Pressing the “Pattern Match” button in the user interface
(Figure 4.9, area 3) begins the real time data acquisition function. This routine continues
until the number of frames in the “# of frames” edit box are captured, during which the x,
y, and 2 positions are stored in memory. After acquisition these data are written to the
hard disk and analyzed.

The Matrox Imaging Library’s pattern matching routine scans the model across
the image, searching for the region that is most similar. This is quantified by the
normalized correlation of the image pixel values I,- and model pixel values M,- over the N

overlapping pixels,

NZIiM, (21,. )ZM,

1e—e.il~.:M.-z-w'

The routine outputs the position of the model where the score, defined as r2 x100% , is

 

 

maximized. The match center is calculated from the model center, entered by the user.

Because the model does not change, the sums 2M, and 21M,2 are calculated before

the pattern matching analysis begins, reducing the analysis time.
To achieve sub-pixel accuracy, a curve is fit to the values neighboring the position
of the maximum score and the exact peak position calculated. The highest accuracy

achievable based on the fit algorithm is 0.05 pixels. Noise in the image and the model

103

increase the uncertainty to around 0.1 pixels in x and y. At 0.071 um/pixel this
corresponds to 7 nm.

To further increase the pattern matching speed, the algorithm performs a
hierarchical search. It begins with a lower resolution version of the target image and
model to exclude regions of the image without an instance of the model. It continues at
higher and higher resolutions, narrowing down the model position. Because the position
is already known at lower resolutions, the number of correlation functions that need to be
calculated is reduced. A counter-intuitive result of this procedure is that larger models
tend to be found faster than smaller ones. Smaller models cannot be sub-sampled as
much as larger ones, so the search must start at a higher level. The lowest resolution is
chosen by the MIL software during the preprocessing step described below.

The pattern match position is not returned unless the score is greater than a user
defined minimum acceptance level. If the score is lower than this level, the model and
target image are considered not to match. If multiple objects in the target image have
scores above this value, all are returned to the results buffer in order of decreasing score.
A user defined certainty level can reduce the time spent finding the mismatched objects.
The first time the pattern matching algorithm returns a score higher than the certainty
level, the algorithm assumes it has found the object and stops searching for a better one.
Due to distortions of the sphere’s image during large 2 excursions, the acceptance level is
set around 20 %. Larger levels mean no matches at large 2, while smaller levels lead to
mismatches at large 2. The certainty level is set to 50% to speed analysis when the

sphere is in the focal plane.

104

The model selection is the most important part of the pattern matching analysis.
A model that does not match the particle during its full range of motions will cause
mismatches or fiames with no match. If the model is not centered the measured x and y
positions change with 2 position.

The model is a subset of an image of the sphere while it is in the bistable
potential, called the model image. This image is stored as a .mim file (a TIFF 6.0 file
with extra information in the comment field). The image is acquired using the “Grab
lmage” button in area 4 of Figure 4.9, then immediately smoothed. From this image the
center of the sphere is chosen. The model is defined by an upper left comer, a lower left
comer, and a center measured from the upper left comer.

To make sure the model represents the largest range of possible particle images
the model is tested. First the particle is tracked using RunningCOG with the pattern
matching checkbox checked. The image on the overlay monitor shows the trapped
particle with a white dot over the matched center. The model parameters are adjusted so
the center overlaps with the actual particle center. If the model is seen to mismatch or
have no match either the acceptance level is changed or, more frequently, a new model
image is chosen. Once the model matches by eye, a short (20,000 frame) data acquisition
run is performed. The number of no match frames, displayed in the “# no blob frames”
edit box in area 4 of Figure 4.9, should be as close to the minimum of 16 as possible. If
the number is too large (greater than 50) a new model is chosen. Finally, the xz and yz
projections of the potential energy are calculated. A tilt in either distribution shows that

the model center is not positioned correctly, probably due to the particle being between

105

pixels in the model image (e.g., the center is at 10.5 pixels instead of 11.0 pixels) and a

new model is chosen.

Real Time Pattern Matching Algorithm

The real time data acquisition and analysis takes place in a single function,
OnPatMatch(), called when the “Pattern Match” button is pressed. All buffers are
allocated and all parameters set before the time critical loop acquires images and
calculates the sphere position. The x, y, and 2 positions are each stored in a global array
until the desired number of frames is acquired. After the loop the arrays are processed to
remove skipped frames and the allocated buffers are freed.

The real-time analysis is to the highest priority to avoid being interrupted or
delayed by other programs. Visual C++ allows the program to set its priority class in
Windows 2000. At the beginning of OnPatMatch() the priority class is set to
REALTIME, telling the operating system to execute commands from WinMi12 before any
others. The priority of the current thread is set to TIME_CRITICAL, so commands from
this thread will execute before any other thread in WinMi12. These commands prevent
other applications, such as Windows Explorer or the GPIB temperature monitoring
software from causing delays in the analysis.

OnPatMatch() next reads the parameters from the user interface and sets up the
MIL buffers. The model image and parameters are loaded, the certainty and acceptance
levels are set, and the LUT and ROI values are downloaded to the digitizer. The data
arrays are allocated, with double precision for the x and y position and long integer
precision for the 2 position. Two additional arrays are created to keep track of both

mismatched or no match frames. The MIL buffers are allocated on the Genesis board and

106

the Host CPU. An additional buffer and display is allocated on the VGA system if the
“Show buffer” checkbox in area 1 of Figure 4.9 is checked, but the system resources
required to draw the image reduce the frame rate to around 27 frames per second.

The method by which the MIL application interacts with WinMi12 during image
transfer affects the maximum frame rate. The image is acquired with the MdigGrab()
command. The framegrabber waits for the current frame transfer to finish then grabs the
next image. The application can respond to this command in three ways, depending on
the grab mode. If the grab mode is set to M_SYNCHRONOUS, the application will be
synchronized with the end of each frame transfer. It will wait until the frame has been
transferred before executing any more commands.

If the grab mode is set to M_ASYNCHRONOUS, the application is not
synchronized; it will continue executing commands while the frame is transferred. To
take advantage of this, two buffers on the Genesis system are allocated to receive the
incoming image. Buffer 1 receives the frame from the camera while the program
completes the pattern matching analysis on buffer 2. When the frame transfer is finished,
analysis of buffer 1 begins and a new image is transferred to buffer 2. In asynchronous
mode the program will wait if another MdigGrab() command appears before the previous
one has finished. (The camera cannot transfer two images at once.) This double
buffering technique reduces the time between analyses to zero.

When the grab mode is set to M_ASYNCHRONOUS_QUEUED, the program
does not stop at the next MdigGrab(). It adds another grab to the framegrabber’s queue

and continues with the program. If the analysis speed is greater than the frame rate, a

107

single image could be analyzed multiple times. The uncertainty introduced is not worth
the possible slight speed increase.

The pattern matching parameters are set to maximize position resolution and
analysis speed. The accuracy is set to the maximum level, with error i1/8 pixel. The
hierarchical search is hard coded so the final search level is at the resolution of the image.
The model is preprocessed with a typical image (transferred from the camera). This
command determines the analysis shortcuts that can safely be used without affecting

resolution, speeding up subsequent analysis.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Computer
Function
Stabilizers Generators
LS-PRO he DS345 2
CCD Camera
fl 41‘— GPIB
LS-lOO PS —<—I DS345] , "1 Genesis :

 

 

 

 

 

 

 

 

Figure 4.12 Electronic component diagram. The Genesis framegrabber sends the
EXSYNC signal to the camera and function generator DS345 1, triggering an
integer number of modulation periods. DS345 1 triggers DS345 2. Both function
generators use the DS345 2 10 MHz pixel clock. The output is sent to the laser
power stabilizers to modulate the beam power. The function generator offset
voltage, modulation amplitude, and frequency are set via the general purpose
interface bus (GPIB) during batch acquisition (4.2.3.10).

For an unknown reason, the framegrabber produces a single white frame after
around 50 images are transferred. To remove this and any other startup effects, 200
frames (approximately 1 second) are transferred before analysis begins.

The Stanford Research Systems DS345 function generators used to modulate the
beam power are triggered by the EXSYNC signal from the Genesis framegrabber (Figure
4.12). The function generators execute a burst modulation, sending an integer number of

modulation periods beginning on the EXSYNC rising edge. Since the framegrabber is

108

always sending EXSYNC pulses to the camera (even while not receiving frames). the
relative phase of the modulation and data is unknown.

The modulation is synchronized by turning on the EXSYNC pulse a known
amount of time before data acquisition begins. The exposure signal active time (Section
4.2.1) is set to the maximum of one second. Five consecutive grabs are performed,
allowing previous modulation cycles of 1 Hz or higher frequency to finish. The function
generators are reset, waiting for the next EXSYNC signal. The exact phase of the
modulation at frequencies below 1 Hz is unknown but equal to 27mf , where f is the
frequency and n is an integer, 0 S n < llfi After five frames the exposure signal active

time is set to the default value in the dcf file.

 

 

 

 

 

 

 

 

 

 

Camera output 1 2 l 3 I 4 5 6 7
Frame analyzed zero 1 I 2 I 4 l 5 I 6 1
Frame transferred 1 2 l none I 4 5 6 7

 

Figure 4.13 Origin of skipped frames during data acquisition. The rows represent the
camera output (unaffected by the analysis time), the frame number being
analyzed, and the frame being transferred. As frame 1 is transferred, the program
analyzes the previous frame (frame zero). After transfer the analysis begins as the
program transfers frame 2. If the analysis takes too long, the command to transfer
frame 3 will be skipped. The Genesis board will ignore the camera signal. After
frame l’s analysis is complete, analysis of frame 2 begins. The next frame
transfer will be frame 4, but the program labels it frame 3 and mislabels all
subsequent frames. The skipped frame number (3) is stored in an array along
with the total analysis time (in case multiple frames are skipped).

The real-time analysis sometimes requires more than 5 ms, leading to skipped
frames and changing the timing. The time critical loop begins with an MdigGrab()
command, transferring the image to be analyzed in the next iteration of the loop. The
initial iteration analyzes the previous image. The amount of time required for this

analysis is recorded with the MIL timer firnction. If this time is longer than 1.2 times the

109

frame period (6 ms) the next frame is flagged as a skipped frame. Figure 4.13 illustrates
the timing involved. After frame 1 is transferred, analysis begins and frame 2 begins
transfer to the second buffer. If frame 1’s analysis takes too long, the program does not
get to the MdigGrab() command before the camera begins transferring frame 3. After the
analysis finally ends, the program begins analyzing frame 2 while waiting for frame 4’s
transfer to begin. Even though frame 1’s analysis is too long, it is frame 3 that was
skipped. To keep track of these skipped frames, two arrays record the frame number and
the delay time, just in case the analysis causes multiple frames to be skipped. After data
acquisition is complete these frames are reintegrated into the data arrays with zeros for
the positions of the skipped frames.

After the newly acquired image is transferred to the Host CPU the command
MpatFindModel() performs the pattern matching analysis, returning the x and y position
and score of the match. In the case of multiple matches the list is sorted by descending
score. The x and y positions are extracted fiom the first result and saved to the data
arrays. The 2 position is inferred from the integrated pixel values over a 13 x 13 pixel
array centered on x and y. If there are zero matches, the positions are set to zero. The
image is copied to the Genesis VGA memory for display on the overlay monitor.

The brightness of the sphere’s image is linear with z displacement over the range
of 2 positions the trapped particle explores. As the sphere moves in and out of the focal
plane it acts like a lens, focusing the microscope illumination. An 11 x 11 pixel box was
shown to have the largest dynamic range of integrated pixel values (pixelsum) vs. particle

position relative to the focal plane for a 0.6 pm diameter sphere. (Due to diffraction, the

image of the sphere can extend outside of this 11 x 11 pixel (0.78 pm square) box.) If the

110

center of the sphere moves from one integer to another (e. g., from x = 19.99 pixels to x =
20.01 pixels) the center of the 11 x 11 array will jump by one pixel, leading to

discontinuities in the 2 position.

 

Figure 4.14 13 x 13 pixel array used in the 2 position calculation. The integrated pixel
value equals the sum over pixel values in the 11 x 11 pixelsum array, centered on
the particle’s x and y position. To reduce the effect of discrete steps in array
center, the next outer row is included in the sum, weighted by the fraction of a
pixel particle center.

The solution is to sum over the 11 x 11 pixel box (pixelsum in Figure 4.14), then
sum the next set of rows (represented by P1 through P4) and weight them based on

particle position. For example, if the particle is centered at (x, y) = (19.9, 20.2),

pixelsum = 2 image + (1 — m): image + er image + (1 — yr)z image + yrz image ,
Pl P3 P2 P4

“X”
where xr = 0.9 and yr = 0.2.

After the time critical acquisition loop, the skipped frames are filled into the data
arrays with zeros for the positions. The measured frame rate is written to the “Real fps”
edit box in area 4 of Figure 4.9, and the number of skipped frames and no match frames

6“

are written to skipped’ frames” and “# no blob frames”, respectively. The MIL buffers
allocated at the beginning of the function are freed. The three data arrays were declared

globally, and thus exist after OnPatMatch() finishes. They are either analyzed to

111

calculate the intrawell dwell times and three dimensional potential energy or saved to a
file for later analysis.

The data is saved in either ASCII or binary format. The ASCII format writes the
double precision x and y positions as 6 digit numbers and the long integer precision 2
position as 5 digit numbers, for a total of 17 ASCII characters (including decimal points,
commas and carriage returns) or 17 bytes. The binary format reduces each value to 24
bits. The x and y positions are multiplied by 100,000, then converted to binary. The 2
position, ranging in value from 0 to 30,000, is directly converted to binary. These values
are written to the file without commas or endline characters, reducing the required disk
space to 9 bytes. For a 1,000,000 frame data set (1 hour, 23 minutes), this reduces the
file size from over 16 MB to less than 9 MB. Both files are written with a header
containing the model parameters and image file, the LUT and ROI, the type of analysis
(blob or pattern matching), the file type, and the number of skipped or missing frame

with the frame number and time arrays.

Z Position Calibration

The 2 position is calculated from a calibration of the integrated brightness vs. 2
position. As the sphere moves in the z direction, focusing of the microscope illumination
causes the image to change brightness. For a small range of positions, which include the
trapped particle motions, this relationship is linear. The calibration depends on
background intensity, and must be performed every time the illumination is changed.

The 2 calibration is performed with either a sphere stuck to the surface of the cell
or with a trapped sphere brought into contact with the surface. The stuck sphere will not

move, allowing a calibration over the full range of possible particle positions relative to

112

the focal plane. However, the stuck sphere may not be the same size or shape as a
subsequently trapped sphere. The trapped sphere calibration can be performed on any
sphere, but only for z displacements above the focal plane.

The 2 position of the particle is changed by moving the cell relative to the
objective lens using the microscope’s focusing knob, driven by a z translation stage for
increased resolution. The minimum step size of the focusing knob is approximately
0.3 pm, or ‘/2 of the sphere diameter. To reduce the step size a Line Tool 2 translation
stage is placed in contact with the focus knob. The focus knob travels 22.3 um per inch
of translation. The micrometer has a resolution of less than 0.001 inches, corresponding

to 22 nm. Step sizes were usually 0.005 to 0.01 inches, or 0.1 to 0.2 pm.

 

 

 

 

J
24000-
1
S 23000]
E
Q)
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D.
22000-
i '.-"--'.' """" """".
21000 I ' l ' I ' I ' I ' I ' I ' U ' I '
3 4 £5 6 7 23 9 ‘HJ 11 12
Z (um)

Figure 4.15 Z calibration. A trapped particle was lowered to the cell bottom using the
microscope’s focusing knob. The sphere contacts the cell around 2 = 5 pm. The
integrated pixel value (pixelsum) increases linearly with position with a slope of
2500 i 200 pixelsum/pm (solid line). Decreasing focus position corresponds to
the sphere moving higher in the trap.

113

RunningCOG’s beamtracker subroutine is used to measure the x, y, and 2 position
over at least 50 frames at around 100 frames per second. For calibration of a trapped
sphere this integrates over the sphere’s motions. The RunningCOG model is chosen the
same way as the pattern matching model, therefore the 2 position range over which the
model still matched would be the same. The average integrated pixel sum is plotted vs. 2
position (Figure 4.15). A trapped sphere was lowered to the bottom of the cell (2 = 5 pm)
where the brightness increased until the model no longer matched. The slope of the

linear regression fit was 2500 pixelsum/um.

Beam Power Modulation and Synchronization

The barrier height is modulated by changing the left beam power. The LS-100
stabilizer used to modulate beam power is controlled by the signal from the Stanford
Research Systems DS345 synthetic function generator (Figure 4.12). The function
generator is triggered by the framegrabber’s EXSYNC output and synchronized to the
beginning of the acquisition.

Since the camera and function generator have their own internal clocks the
camera’s frame rate relative to the function generator was measured. One period
(measured in frames) is the camera frame rate divided by the modulation frequency. The
EXSYNC signal rate was measured relative to DS345 2, controlling the LS-PRO
stabilizer. DS345 1, controlling the LS-100, is synchronized to the DS345 2 10 MHz
clock. The LeCroy 9310AM oscilloscope was set to trigger off of the falling edge of the
EXSYNC signal in channel 1, while the firnction generator sent a 200 Hz square wave to
channel 2. The fitnction generator frequency was adjusted until channel 2 was stationary

with respect to channel 1. The oscilloscope sweep rate was increased until the maximum

114

precision of the DS345 was reached. The frame rate is 201.646278 d: 0.000005 Hz, with
an uncertainty corresponding to one part in 40,000,000. Therefore, for a typical run of
100,000 frames, the phase is expected to slip by 0.0025 frames.

The phase shift is set to a constant, integer number of frames by triggering the
function generators with the framegrabber’s EXSYNC signal, eliminating all phase slip.
The EXSYNC output from the framegrabber is sent via BNC cable to the DS345 2
function generator where it triggers the Burst modulation feature. Burst modulation
sends an integer number of periods of user defined frequency, amplitude, offset and
phase shift. After these periods are sent, the function generator returns to the offset
voltage as it waits for the next trigger. During the modulation period all triggers are
ignored.

To reduce the amount of time the function generator signal is constant (i.e., the
time it is waiting for a trigger), the modulation frequency is set so one period ends just
before the next EXSYNC pulse arrives. Ideally, the correct frequency is the frame rate
(measured above) divided by the number of frames per period. The Burst modulation
requires a finite amount of time to reset, less than 200 nanoseconds. This time is not
constant with frequency, so the function generator frequency was measured for each
desired modulation frequency (Table 4.2)

The EXSYNC and function generator time traces were sent to the LeCroy
9310AM oscilloscope. The sweep was triggered by the falling edge of the function
generator’s sine output. At the end of the burst, the function generator output went to a
constant DC offset until the next EXSYNC pulse. The modulation frequency was

reduced until the constant voltage disappeared. The modulation frequency was measured

115

to within 0.000001 Hz, or a constant voltage interval of S 1 us. Table 4.2 lists the
approximate frequency, the period in number of frames, and the function generator
setting from 0.2 Hz to 2 kHz. At modulation frequencies above 10 Hz the burst count
(number of periods output by the burst modulation) is set to 10. This allows minimum
modulation periods of 0.1 frames.

The modulation, while synchronized with the images themselves, was not
synchronized with the beginning of the data acquisition. The framegrabber is continually
producing the EXSYNC signal and triggering modulation periods, even when the frame
is not transferred. To synchronize the modulation with the first recorded frame the trigger
signal is interrupted just before the time critical loop. Using the MIL commands, the
EXSYNC active time is set to the maximum of 1 second. Five single images are
acquired, producing 5 triggers at one second intervals. The EXSYNC active time is then
reset to the default value in the dcf and the data acquisition begins.

The phase was calibrated by directly measuring the beam power with the real time
pattern matching analysis routine. Due to the difference in index of refiaction, the beam
reflects off the glass-water interface at the bottom of the cell. The laser rejection filters
below the dichroic mirror filter out this reflection before it reaches the camera. These
filters were removed, and a 4 OD. neutral density filter mounted after lens L4. The beam
reflection was a dim dot on the camera, requiring amplification of around 8 times by the
lookup table. The beam power was modulated by the LS-100 stabilizer with 3 mW peak
to peak amplitude, controlled by the function generator signal. The integrated pixel sum
measurement, used to find the 2 position of the particle, measured the beam power as a

function of time. The 10,000 data points were fit to the function

116

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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117

I = Asin(27i7’(t—t0 ))+y(, with a non linear least squares fit in Origin 7.0. The phase
shift t0 is 3.0 d: 0.3 frames over frequencies from 1 to 50 Hz.

Below 1 Hz, the absolute phase shift is not known exactly. By spacing the
EXSYNC triggers 1 second apart, the phase becomes an integer multiple of 27;)”, up to 271.
For example, 0.2 Hz modulation can be triggered by any of the five pulses, depending on
when the previous period ends. The phase shift is 0, 0.47:, 0.8a, 1.27:, or 1.6m ln units of

frames, the phase shifi is 0, 200, 400, 600, or 800 frames.

Batch Data Acquisition

Afier the sphere is trapped, the gimbal mounted mirrors adjusted, the model
chosen, and the cell clamp set, data acquisition requires very little input from the
operator. At the end of each run, typically 100,000 frames (500 seconds or 8.3 minutes),
the data are saved, the modulation frequency and/or amplitude are changed, and the next
analysis begun. This process is automated in WinMi12 using a batch acquisition system.

The function generator settings are controlled by a text file. When the “Batch
Acquire” button in area 3 of Figure 4.9 is pressed, WinMi12 loads the batch file specified
in the “F ilename for analysis results” in area 4. Table 4.3 shows a typical batch file. The
first number in this file is the number of individual data sets. This is followed by six
columns of numbers representing the function generator frequency setting (from Table
4.2), the DS345 2 (beam 2) and DS345 1 (beam 1) DC offset, the DS345 2 and DS345 1
modulation amplitude (in Vpp), and the number of frames. These values are stored in 6
global arrays. WinMi12 begins a loop, iterating through each setup. The batch file in

Table 4.3 acquired 152 datasets over 18 hours.

118

Table 4.3 Typical batch acquisition file. The first number (152) is the number of data
sets to be recorded. WinMi12 performs one acquisition per row. Columns
represent the frequency, LS—PRO voltage offset (V), LS-100 voltage offset, LS-
PRO voltage amplitude (Vpp), LS-100 voltage amplitude, and the number of
frames. (The table is truncated after dataset number 32).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

152

1.008243 0.360 0.125 0 0 100000
1.008243 0.360 0.105 0 0 100000
1.008243 0.360 0.085 0 0 100000
1.008243 0.360 0.145 0 0 100000
1.008243 0.360 0.165 0 0 100000
1.008243 0.360 0.125 0 0 100000
1.008243 0.360 0.125 0 0.05 100000
1.008243 0.360 0.125 0 0.1 100000
1.008243 0.360 0.125 0 0.15 100000
1.008243 0.360 0.125 0 0.2 100000
1.008243 0.360 0.125 0 0.25 50000
1.008243 0.360 0.125 0 0.3 50000
1.008243 0.360 0.125 0 0.35 50000
1.008243 0.360 0.125 0 0.4 50000
2.016486 0.360 0.125 0 0 100000
2.016486 0.360 0.125 0 0.05 100000
2.016486 0.360 0.125 0 0.1 100000
2.016486 0.360 0.125 0 0.15 100000
2.016486 0.360 0.125 0 0.2 100000
2.016486 0.360 0.125 0 0.25 50000
2.016486 0.360 0.125 0 0.3 50000
2.016486 0.360 0.125 0 0.35 50000
2.016486 0.360 0.125 0 0.4 50000
2.520655 0.360 0.125 0 0 100000
2.520655 0.360 0.125 O 0.05 100000
2.520655 0.360 0.125 0 0.1 100000
2.520655 0.360 0.125 0 0.15 100000
2.520655 0.360 0.125 0 0.2 100000
2.520655 0.360 0.125 0 0.25 50000
2.520655 0.360 0.125 0 0.3 50000
2.520655 0.360 0.125 0 0.35 50000
2.520655 0.360 0.125 O 0.4 50000

 

The frequency, offset, amplitude, and burst counts are adjusted via the GPIB
interface with the DS345 function generator before every data set. Next the
OnPatMatch() fimction is called; data acquisition is performed as explained above. After

the set number of frames are acquired, the interwell dwell times are calculated using the

119

OnAnalysis() function, explained in Appendix E. The position and dwell time data are

saved to separate files and the next iteration begins.

4. 3. Optical Stability

To measure the effect of modulation on the transition rate, the uncertainty in the
static potential is minimized through mechanical stabilization and thermal control. Slow
changes in the trap positions in x, y, and 2, caused by thermal expansion and mechanical
relaxation, are reduced by temperature controlling the optical mounts and fixing the
objective lens to cell separation. Faster fluctuations due to air motions are eliminated by
baffling the optics and microscope. The 2 position motions are almost eliminated, while

the x and y motions are reduced by a factor of 4.

4.3.1. Laser Power Stability

The intensity of the HeNe lasers fluctuates due to input power noise and thermal
fluctuations, changing the barrier height and creating a non-white noise source. The
stabilizers positioned after the beams reduce this driving by at least a factor of four.

The beam power fluctuation amplitude was calculated from the power spectrum.
The power spectrum of fluctuations was measured using a Centronic OSDlSST silicon
photodiode sampling the beam using the beam sampling optics (Section 4.1.4). A Glan
laser calcite polarizer (Newport product numberIOGLOSAR.14) attenuated the beam to
0.6 mW total power, preventing saturation of the photodiode. The beam diameter
(0.98 mm) was smaller than the photodiode active area. The photodiode integrated over
beam profile and was insensitive to beam pointing fluctuations. The photodiode signal

was sent via coaxial cable to the HP3561 A spectrum analyzer.

120

The HP3561A Dynamic Signal Analyzer performs a discrete Fourier transform
over user defined frequency ranges from 0.01 Hz to 100 kHz. The dynamic range is
280 dBV below the input range, which was set to ~15 dBV (177 Vms). Below 80 dBV
distortions and aliasing occur. (The HP3561A defines dBV = 20 log (V), where V is in
volts.)

The HP3561A output is the magnitude of the discrete fast Fourier transform

(F FT) normalized to the bandwidth |F(v] , in units of dBV/VH2. The photodiode power

spectral density is defined S (v) = |F(v)2 .
The power over a certain bandwidth is calculated from the power spectral density

by integrating the square over that bandwidth P = ISO/)dv. The FFT is discrete, so

bandwidth

instead the power is the sum P = ZS(v)Av. The power measured over the entire
bandwidth

frequency range is equal to the total power P, = ZS(V)AV The optical power is the
v=0

intensity of the beam incident on the photodiode, and is linearly proportional to the

photodiode voltage (recorded by the HP3S61A). The total power P = V 2 / 2 , therefore

the total optical power Pop, is proportional to JP, , with conversion factor from a beam

power vs. photodiode voltage calibration.

The bandwidth depends on the input window. For a flat top window the
bandwidth 0.00955 >< frequency range. To resolve narrow peaks or low frequency
features, the spectral density was recorded over many spans.

A C++ program HP356IA.cpp was written to send a series of frequency ranges

and numbers of ms averaging steps to the HP3561A via the general purpose interface

121

bus (GPIB). The program serial polls the device until the average is complete, then

copies [F(v] to a file and begins the next average with the next frequency span and

number of averages.

[F(v) was measured over four frequency spans: 0 — 100 Hz, 0 — 1000 Hz,

0 - 10 kHz, and 0 — 100 kHz. The ranges were combined by taking 0 — 100 Hz from the
0— 100 Hz range, 100 — 1000 Hz from the O — 1000 Hz range, 1 — 10 kHz from the
0 - 10 kHz range, and 10 — 100 kHz from the 0 — 100 kHz range.

The power was measured over the entire range then over the DC and AC portions
of the spectrum. (Here DC is defined as all frequencies below one bandwidth of the input
window, in this case S 1 Hz.) Table 4.4 shows the total optical power, the DC power,
and the AC power for HeNe laser 1 and laser 2. Using the voltage to beam power
conversion, laser 1’s power fluctuations (above 1 Hz) contained 0.0018 mW out of
0.60 mW, or 0.3%. Laser 2’s power fluctuations were 0.4% of the DC power.

From the static tilt measurements in Section 5.2 the beam power fluctuations were
converted into change of barrier height AUAC. The slope of the linear term in the quartic
approximation vs. beam power was 240 kBT/um V (Figure 5.14), or 7.6 kBT/mW. 0.3%
of the approximately 7 mW input power is 0.021 mW, or AUAC = 0.16 kgT.

Measured afler the LS-100 and LS-PRO beam power stabilizers the fluctuation
spectrum amplitude was below the dynamic range of the HP3561A spectrum analyzer.
Assuming the AC power was equal to -95 dBV/\IHz (input range of —15 dBV/VH2 and
-80 dBV/VH2 dynamic range), the AC optical power was 0.5 uW, corresponding to

0.04 kgT barrier height fluctuations.

122

Table 4.4 Beam power fluctuation amplitude.

 

 

 

 

Total Pop, (mW) DC Pop, (mW) AC Pop, (mW) AUAC (kBT)
Laser 1 0.60 0.60 0.0018 0.16
Laser 2 0.50 0.50 0.0018 0.19
Stabilized laser 0.60 0.60 s 0.0005 S 0.04

 

 

 

 

 

 

 

The laser power stabilizers remove at least ‘A of the beam power fluctuations,
with the measurement limited by the sensitivity of the spectrum analyzer. The expected

noise reduction is greater than 80:1 up to 100 kHz.

4.3.2. Focal Plane Position

The barrier height depends strongly on the distance between the focal plane in the
cell bottom. Simulations by F allman and Axner [90] show that the axial trapping
efficiency (the ratio of the trapping force to the input power) falls by 50% over a distance
of 2 sphere radii. A measurement of transition rates vs. height in this system showed a
40% change in rate over 3 pm. The distance changes were eliminated by securing the

sample to a clamp around the objective.

Objective Clamp

In the first design, the cell rested on the stage with the objective viewing from
below. The cell was constructed from two 22 mm #1 cover glass pieces. They were

cleaned, sealed on three sides, then baked for 6-24 hours, similar to the process described
in section 4.1.10. A temporary glass spacer left a 100 um gap between the cover glass.
The gap was filled with 0.6 pm diameter spheres in 0.000] M NaOH solution. An air

gap of 3 to 5 mm was left at the top and the final side sealed with epoxy.

123

After setting the focus height with the focus knob on the microscope, the height
changed by 108 of mm in a few hours. This effect was independent of temperature and
was attributed to the mechanical drift of the objective turret relative to the stage.

To remove this drift, the cell was mounted on a clamp which fit around the
objective lens (Figure 4.16). It was thus independent of the microscope stage. The clamp
was a hollow stainless steel cylinder with a slit along the length. A screw on the side
tightened the clamp, allowing the height (i.e., vertical position along the objective) to be
adjusted. The cell rested on the top of the clamp, held a fixed distance from the lens.

To take advantage of the clamp, a new stage was built and the cell redesigned (see
section 4.1.10). The top surface of the cell is a 3 x 1 inch glass slide. An 18 mm
diameter #1 cover glass piece is epoxied on 3%: of its circumference and attached to the
center of the slide. A temporary #1 glass spacer is used to keep the cell open. After
baking for 6 — 24 hours, the spacer is removed and the cell filled with the 0.6 pm
sphere/0.0001 M NaOH solution. The slight wedge shape created by the capillary action
of the epoxy causes the solution to wick away from the open end of the cell. The cell is

capped with epoxy, leaving a 3 — 5 mm air gap, and allowed to set for 6 hours.

‘3 . .F"

 

Cell

 

Water
Immersion Oil

   
  

 

 

 

' Objective Clamp
Objective . (D

 

 

 

Figure 4.16 The objective clamp. The clamp fits around the objective lens. It is
tightened with a screw at the bottom. The cell rests on the clamp, keeping it a
fixed distance from the objective lens. Nichrome heater wire was wrapped
around it and a thermistor attached at the top.

124

The redesigned stage has a 2.5” square opening to prevent collisions with the
objective clamp. The cell spans the opening, its ends resting in a machined groove. This
allows the cell to be moved using the stage. When a particle is trapped, the objective
clamp is raised until the cell clears the groove and rests on the clamp alone.

To prevent the cell from sliding sideways a 200 gram weight is placed on top of
the cell. The weight is a 1.5 inch diameter, '/2 inch tall cylinder with a % inch hole
through the center for illumination to pass through. The weight was positioned directly
above the walls of the objective clamp.

Despite the above modifications, the focal plane continued to move over long
time scales. Plots of 2 position vs. room temperature showed a direct correlation. The
external temperature changes caused thermal expansion of the objective/objective clamp
system, changing the focal plane height by tens of um/°C.

The objective clamp temperature was controlled with a heater and feedback
circuit. A 3 foot length of 30 Q per foot resistance wire was wrapped around the
objective clamp. A thermistor mounted near the top of the clamp measured the
temperature. A feedback circuit, capable of supplying 2 W to the resistance heater,
maintained the temperature to within 50 mK, even when the microscope was exposed to
air currents. When the microscope was baffled with cardboard (and later insulating
foam), the temperature fluctuations on the scale of seconds disappeared. The thermal
control circuit performance improved, keeping the objective clamp temperature constant
to within 10 mK.

Nonetheless, the focus height continued to change with room temperature shifts.

To make sure that the objective/objective clamp system was capable of compensating for

125

changes of this magnitude, the heater wire was driven with a 0 — 4 V square wave at
l mHz to evaluate the time-temperature response. This caused a 0.4 °C change in the
temperature measured by the thermistor. The focal plane height was inferred from the
integrated brightness of the beam reflections off of the cell’s glass-water interface. Over
less than 10 pm the focal plane height was monotonic with measured brightness.

It was found that, rather than the objective moving closer to the cell as it heats, the
cell first moves away (2900 seconds in Figure 4.17a). After 30 seconds the cell-objective
distance reaches a maximum and reverses direction. The distance decays exponentially

with a time constant of approximately 200 seconds.

 

    

 

 

 

 

 

 

 

 

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’2‘ 6000- .‘ """" - . """"" - ‘4
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.5 ‘ t a,
3 2000 . 2 g
m . . s
E 1800« 1 3
I: as
E” 1600 0 m
m ‘ ‘ I
10000 10500 1 1000 1 1500 12000

Time (seconds)

Figure 4.17 Objective to cell distance changes with objective clamp heating. The
brightness of the beam reflection from the glass water interface (solid lines)
decreases with increasing separation. Dashed lines indicate heater voltage, from 0
to 4 V. (a) with weight and (b) without weight on the cell. Vertical lines in (a)
are mismatches in the pattern matching analysis.

126

The effect was due to differential heating between the clamp and objective. At
the beginning of the heating cycle, the objective clamp heated up first because the wire
was in direct contact. It expanded, moving the cell away from the objective. Afier 30
seconds the objective began to expand while the clamp’s expansion rate decreased,
decreasing the separation. On the cooling cycle there was a similar effect. The objective
clamp was exposed to the air, cooling more quickly. The cell moved towards the
objective. As the objective cooled, it moved away from the cell.

The weight on the cell affected this process. Figure 4.17b shows that when the
weight was removed the distance increased for 30 seconds, then began to decrease. Afier
60 seconds, the distance began to increase again. 90 seconds later the distance decreased.

This was caused by thermal flexing of the cell itself. Upon heating, the bottom of
the cell expanded faster than the top. This caused the center of the cell to move away
from the objective. After approximately 150 seconds the temperature equilibrated across
the 1.25 mm thick glass slide and the cell relaxed again. The remaining distance change
came from the objective lens expansion. The weight either prevented the cell from
flexing or kept it in better thermal contact with the clamp. lln the latter case the cell
would still flex, but it would happen at the same time as the differential expansion of the
objective and objective clamp.

To increase the thermal contact a new objective clamp was built (Figure 4.18).
The cell rests on a 1/2 inch wide “shelf’, increasing the contact area by a factor of 10. The
objective clamp was wrapped in 80 Q of Nichrome resistance wire and heated with a
O - 4 V, 1 mHz square wave. The feedback thermistor is attached to the bottom of the

clamp.

127

 

Cell
I

...J " Water
Immersion Oil

()bjcctix'c I o

 

    
   

 

Heater Wire

 

'l‘hcrmi stor

 

 

 

Figure 4.18 The objective clamp with wide shelf. The shelf provides much larger
thermal contact area with the cell. The clamp is wrapped in 80 Q of Nichrome
heater wire.

Without the weight on the cell, the focal plane height change due to cell flexing
does not appear (Figure 4.1%). The cell-objective distance increased at the beginning of
the heating cycle, then leveled off after 140 seconds. The objective clamp expanded, but
the extra thermal mass made the expansion slower than the previous clamp. It took 150
seconds for heat to propagate through the cell (Figure 4.17b). Therefore if the cell was
flexing, its effects were obscured by the differential expansion.

With the weight, the time scale of the motion was reduced further. At the
beginning of the heating cycle the cell moved away from the objective for 40 seconds,
then the distance closed for the remainder of the cycle. The amplitude of this motion was
smaller than any focal plane height changes previously recorded. (The focal plane height
was measured as a function of the brightness of beam reflections off cell’s the glass-water
interface. The relationship is monotonic but not linear, so direct comparisons between
datasets are difficult.)

These results show that there is a change in focal plane height because
temperature fluctuations were changing the cell-objective separation. The change is

minimized when there is large thermal contact between the objective clamp and the cell.

128

The thermal contact is increased with the addition of a weight to hold the cell down. A

feedback control circuit isolates the objective from room temperature variations.

 

 

 

Brightness (pixelsum)

 

 

 

 

Brightness (pixelsum)

 

 

1 7900

 

48000 ' 49000 50000 51000

Time (seconds)

Figure 4.19 Objective to Cell distance changes with redesigned objective clamp. The
brightness of the beam reflection from the glass water interface (solid lines)
decreases with (a) increasing separation and (b) decreasing separation. Dashed
lines indicate heater voltage, from O to 4 V. (a) without weight and (b) with
weight on the cell.

Cell Clamp

The height of the focal plane continued to change, but on much longer time
scales. These motions correspond to relaxation of the objective-cell distance by 2 to
8 pm over more than 10 hours. The constant force of the weight is not large enough to
relax the stresses out on a short enough time scale.

A cell clamp was built to replace the weight. The cell clamp is made of three

pieces. The top piece is a 2 inch diameter, % inch thick brass cylinder with a 1 inch hole

129

for the illumination. Screws at the perimeter travel down past the objective clamp shelf
to two 2 inch long, % inch square aluminum bars. The aluminum bars contact the
underside of the shelf, clamping the cell to the objective clamp (Figure 4.20).

The application of the cell clamp caused the focal plane to move > 100 um into
the cell. This caused the trap to become unstable and the particle escaped. The clamp
was most likely distorting the objective clamp shelf, moving the cell closer to the
objective.

To prevent distortions, a series of washers was used as spacers to set a maximum
distortion of the objective clamp. The focal plane only moved 20-30 pm when the cell
clamp was tightened. Over 12 hours, the cell clamp relaxed and the focal plane moved
down by approximately 20 pm. The screws had loosened slightly, causing the force on
the cell to change.

The cell clamp was modified to hold two ball nose spring plungers, which have
constant force for a small range of distances. The spring plungers have a maximum force
of five pounds before the ball is entirely recessed into the plunger. Two pieces of 1 inch
long, 1/2 inch wide, 1/ 16 inch thick stainless steel under the plungers distribute the point
load over a larger area of the cell (Figure 4.20).

The cell was clamped and the spring plunders set to just under 5 pounds of force
each. Over 14 hours, the focus height drifted by less than 2 pm.

The cell clamp with spring plungers performs better than the simple cell clamp or
the weight. The spring plungers provide a constant force over 100 um (compared to the
cell clamp without spring plungers). If the screws holding the clamp relax there is still a

total of 10 pounds on the cell.

130

Cell clcmp Spring Plungers

  

Spring Plungers

   

Cell clamp

fiiiflfi‘afiéfié a:

   
 
   

new;

SCIBV/

Stainless Steel plates

Stainless Steel plates
Figure 4.20 Cell Clamp. The ball nose spring plungers press on two stainless steel
plates, distributing the force across the cell as it is held against the objective
clamp. The brass ring is kept in place with two aluminum bars that fit below the
shelf of the objective clamp.

The cell resting on the stage introduced drifis due to mechanical relaxation of the
microscope and temperature changes. The cell mounted on the objective clamp removes
the microscope relaxation. Thermal control of the objective removes most of the thermal
changes. Clamping the cell to the objective clamp reduces the relaxation of the
objective-cell distance to less than 2 pm in 14 hours. This corresponds to an estimated
25% change in the transition rate or 1 kBT in the barrier height. If the data acquisition is
limited to 2 hours (1.4 million frames) the barrier height changes would be 0.15 kBT or
less, depending on the size of the room temperature variations.

Measurement of the transition rate dependence on modulation frequency or
amplitude requires an unchanged potential over multiple data sets. The use of ten

100,000 frame (500 s) sets limit the minimum transition rate W0 , which limits the

minimum modulation frequency f (f > W0). For 10% uncertainty in W0 the number of

131

transitions N = 100 in each direction, for a mean transition rate of 0.4 Hz. (The number
of transitions increases with the modulation amplitude, reducing the uncertainty.) This
value is a factor of 10 smaller than the slowest intrawell relaxation time, allowing
modulation frequencies that can explore this region. Measurements over times exceeding

2 hours can be made if the room temperature remains constant.

4.3.3. Lens Mount Expansion

The x and y positions of the traps changed with temperature over many hours.
Both beams moved in the same direction during these changes. Since the objective was
temperature controlled, the problem was coming from one of the other optics.

To find the individual optics most susceptible to change, the positions of the laser
reflections off the glass-water interface of the cell were measured as the mounts were
heated. Around 80 Q of Nichrome wire was wrapped around the mounts (Figure 4.21).
A thermistor mounted on the post mount measured the temperature change.

Heating of the laser mount and the mirror mounts by 0.5 °C had no observable
effect on the beam position. Heating the lens mounts caused a large displacement in the
beam position. A linear fit of the x position vs. temperature had a slope of 20 nm/°C.
(Due to the camera orientation, x position changes correspond to lens motions
perpendicular to the table.)

The beam deflection occurs when the lens mount expands, moving the lens up

relative to the beam. The mount is made of aluminum and stainless steel, which have
coefficients of thermal expansion of 2.3 x10‘5 per °C and 1.8 ><10‘5 per °C, respectively.
A 1 °C change in the 225 mm tall mounts corresponds to an approximately 4.5 pm
change in lens height.

132

Lens

Thermistor

Aluminum Post
Holder

Aluminum
"‘“fiijj Base

LTD

L \'\\ \ \ \3 \ \

[f .. -- ..

Figure 4.21 Lens mount heater. Nichrome wire is wrapped around the post and
postholder and attached to a DC power supply. The thermistor placed at the top
of the postholder is used to measure the temperature.

The deflection of a beam travelling through the displaced lens was calculated
using geometrical optics. A ray travelling along the lens’s optic axis will not be deflected

by the lens. A ray that is off axis by a distance h will be deflected at an angle

(9 = tan‘l (h/ f ) by a lens with focal length f (Figure 4.22). For a 1 pm displacement of a

100 mm focal length lens, 6 = 1x 10‘5 rad. The beam will travel through the focal point
of the lens, which has been translated by h.

The optical system contains five lenses in each beam; the pairs L1-L2 and L3-L4
to resize the beam and the objective lens to focus the beam to a diffraction limited spot.

If all of the lenses move up the same amount, the beam will be displaced by

h. = fihi = [iéh entering the objective. If the expansion is not uniform, (Figure

ff; fsf.

133

4.23), the beam angle changes as well. When the beam enters the objective with an angle

(0]., the focal point is displaced by Ax = fEFL tan go], where fEFL is the effective focal

length of the objective lens.

Using a spreadsheet in Lenses4.xls (Table 4.1) to propagate the effects of lens
displacement on trap positions, a 1 pm relative displacement of any lens results in a 3 nm
shift in the beam position. This is nearly 1% of the typical beam spacing of 400 nm. A
full 1 °C change in a single lens mount would cause a 13.5 nm (more than 3%)

displacement of a single beam.

 

 

 

<—:-—>
O

 

l
V

Figure 4.22 Deflection of a beam through a single lens. The beam has moved upwards a
distance h relative to the beam. The beam travels through the focal point at an
angle 9.

L1
L2 L3 L4 Objective

 

>34
B

 

Figure 4.23 Diagram of the beam deflection as it propagates through the optical system.
A displacement of lens Ll by h will cause the beam entering the objective lens to
be deflected. The trap position moves a distance Ax.

134

To test these predictions, 100 Q of Nichrome wire were wrapped around
individual lens mounts. The wire was heated with a O — 10 V, 1 mHz square wave. The
temperature was measured with a calibrated thermistor mounted at the top of the
postholder, about ‘/2 of the total height (Figure 4.21).

The mount for lens L3 was heated by 1.25 °C, producing an x displacement of
each beam of 20 nm. (Due to the Dalsa CA-Dl camera orientation, a vertical motion of
the beam on the optical bench corresponds to an x displacement on the camera.) The x
position vs. temperature data (Figure 4.24a) are roughly linear with a slope of 17 nm/°C
for both beams. The direction of the motion corresponded to the predictions of the beam

propagation calculations.

 

 

   

 

 

 

 

 

 

a) b)
0.92
30.90
x
0.881 _
1.82 "
21.0 21.5 22.0 22.5 23.0 21.0 21.5 22.0 22.5 23.0
L3 Temperature (C) L1 Temperature (C)

Figure 4.24 X position vs. lens mount temperature. Plot (a) shows the beam position
deflection due to heating of lens L3’s mount. Plot (b) shows the deflection due to
heating beam 2’s lens Ll ’s mount. The lines are the linear fits. Both have a slope
of 17 urn/°C.

The mount for lens L1 in beam 2 was heated to 1.4 °C. As expected, beam 2
moved 25 nm while beam I remained stationary. A fit to beam 2’8 x position vs.

temperature gives a slope of 17 nm/°C (Figure 4.24b).

135

The calculations show that the beam position change due to the motion of any
single lens should have the same magnitude, 14 nm/°C. The experiments show
17 rim/°C. This is explained by poor thermal contact between the thermistor and post. If
the thermistor were reading a temperature too low the slope of the x vs. temperature plot
would be artificially increased.

Room temperature variations of 0.1 0C, resulting in 2 nm beam displacements, are
common. Relative temperature changes across the table fluctuate by more than 0.2 °C
over hours. A thermal control system was designed to eliminate these slow temperature
variations.

Using the same setup as for the lens mount heating experiments, Nichrome wire
was wrapped around each of the remaining lens mounts. Two wires were needed for
L4’s cantilever mount. 10 k0 thermistors were mounted to the top of the post holder on
each mount. A resistance bridge measures the thermistor resistance. An integrating
amplifier uses this signal to control the current across the heater wire via a power
transistor, dissipating a maximum of 2 W.

The feedback circuits keep the temperature constant to within 25 mK,
corresponding to beam displacements of 0.4 nm. The feedback circuits filter out room
temperature fluctuations of up to 0.1 °C/hour.

Thermal variations have been shown to move the beam in the focal plane of the
objective. Calculations and measurements show that the motions due to lens mount
expansion are around 17 nm/°C, or 4% of the beam spacing. The thermal feedback
system controls the temperature of the lens mounts to 25 mK for room temperature

fluctuations of 0.1 °C/hour.

136

4.3.4. Pointing Fluctuations Caused by Air Motion

Moving air creates vibrations of optical mounts and induces index of refraction
changes in optical elements. The effect of air motion on the beam positions became
apparent during a set of coherence length measurements (Appendix C). The interference
fringes would undergo rapid motions on the order of 10 — 500 Hz when the laboratory
HEPA filter fans were set to maximum. Turning off the fans eliminated the fast
fluctuations but temperature changes caused a slow (<1 Hz) drifi of the fringe position.

A Plexiglas box was constructed to protect the optics from air motion. An
aluminum frame holds 0.090” thick Plexiglas panels. The frame is 8 feet long, 3.5 feet
wide and 16 inches tall at its tallest point. The panels are removable and allow easy
access to the optics. A 1 inch diameter hole was cut to allow the beam to exit the box and

enter the microscope.

 

1.03

1.02

1:; W will i i" ii

0.99

X (um)

0.98 . . . . . . . . .
100 105 110 115 120
Time (seconds)

 

Figure 4.25 X beam positions with Plexiglas box. The position fluctuates on the order of
0.1 Hz with the Plexiglas box closed (black) and 5 Hz with the box open (blue).

137

The position of the beams in the field of view was measured with the Plexiglas
panels on and off (Figure 4.25 black and blue lines, respectively). With the box open, the
beams move at frequencies above 5 Hz over distances of 10 nm. With the box closed, the
high frequency motions disappeared and slower, 0.1 Hz motions were visible. The sizes
of these fluctuations were 5 nm in the x direction and 12 nm in the y direction. Motions
in the x direction in the field of view correspond to motions perpendicular to the table.

The increased air vibration with the open box reduced size of the y fluctuations.
The turbulent air caused changes in the index of refraction along the beam path. A 1%
pressure variation across the 1 mm diameter beam over 10 cm causes deflections on the

order of 10 microradians. Amplified by the lenses, this causes the beam to move ~70 nm.

 

1.020
1.0153
1.0105

A 1.0055

=- 1.0003

0.9953

0.9903 I! _
0.985- .
0.930 , '

. ' Y 90 ' 9'5 . 1

X Position ( m

 

0.975

0.970
80 85

 

 

 

00
Time (seconds)

Figure 4.26 Beam separation vs. time. The dashed and dotted curves show the slow 10
nm position fluctuations of beams 1 (black) and 2 (blue). The green curve shows
the beam separation with amplitude 2 nm.

138

With the Plexiglas box closed, the fast air motions are eliminated, but the cause of
the slower beam motions is uncertain. The REO HeNe laser rms beam steering
fluctuations are less than 50 urad. These fluctuations, propagated through the optical
system, could account for the beam motions. However, the anisotropy suggests that the
motions of the lenses in the plane of the table are a contributing factor.

The beams exhibit correlated motion when the box is closed (Figure 4.26). The
distribution of beam separation has a width of only 2 nm in x and 3 nm in y. The
potential is much more sensitive to the change in the beam separation than the beam
positions, especially if the position fluctuations are slower than the relaxation rate of the
potential (3 Hz). Therefore the correlated motions with the box closed are preferred over
the uncorrelated motions when the box is open.

The Olympus microscope is 10 inches outside the Plexiglas box. To reduce the
air vibrations and temperature fluctuations at the microscope the region directly
underneath its stage was baffled. Two aluminum plates, layered with foam insulation,
were fit to both sides of the objective lens turret. A % inch diameter hole cut into one
plate allows the beam to enter. The stage has a 2.5 inch square hole, so the region just
above the stage was baffled with cardboard. This baffle fits around the condenser and is
lined with foam insulation. For further protection, the entire microscope except the lamp
housing is sealed with a plastic cover. One side of this cover connects to the Plexiglas
box, isolating the small volume where the beam leaves the box.

The microscope baffles reduced the beam position fluctuations, indicating that
some part of the microscope was moving. The cell was clamped to the objective, so there

should have been no high frequency motion. The other optical element, the dichroic

139

mirror, was held in a modified fluorescence beam splitter turret. This turret was designed
to allow light from a UV lamp in the rear of the microscope to be reflected up, into the
objective. Olympus produced a modified version that allows light to enter from the side.

Designed for easy switching of optical elements, the turret was weakly spring
clamped, and it was found that a small pressure on the microscope housing caused the
beams to move by tens of nanometers. Therefore, a fixed mount was built to hold the
dichroic mirror. The mount was designed to be interchangeable with the turret, using the
slide to mount it to the microscope and the original dichroic mirror holder.

However, measurements of the beam positions over an hour taken before and
after installing the dichroic mount showed no improvement. Beam position fluctuations
due to the beam splitter turret were either smaller than those observed or on a time scale
longer than one hour.

Apparent collective beam motions might originate in motions of the Dalsa
camera. The camera is attached to the microscope by a 6 inch long camera tube that
contains the 5x eyepiece. If the camera or tube should shift the position of the beams on
the CCD array would change.

To eliminate possible motion the camera was clamped to the optical table.
Aluminum plates of varying thickness were used as a base. A ‘A inch thick Teflon block
was used to make contact to the camera. The Teflon deforms slightly under pressure,
giving a more uniform clamping force than aluminum. The camera was clamped from
the top by an aluminum bar held in two ‘/2 inch stainless steel posts. A second piece of

Teflon made contact to the top of the camera.

140

The camera clamping reduced the y beam position fluctuations by 13%. Thus,
motion of the camera parallel to the table was partially responsible for the beam
displacements. The beam position fluctuations in the x direction were unchanged. While
the camera was probably also moving perpendicular to the table, the other fluctuation
sources were much larger, washing out any improvement.

The long term stability of the potential was increased using electrooptic,
mechanical, and thermal controls. The beam power fluctuations were reduced by at least
a factor of 4 using the LS-100 and LS-PRO laser power stabilizers. The depth of the
focal plane in the cell was held constant with the thermally controlled objective clamp
and cell clamp. Integral feedback heaters controlled the lens mount temperatures,
reducing in plane beam position changes to less than 1 nm for slow room temperature
variations. Baffling the optics and microscope eliminated the high frequency beam

position changes due to air motions.

4. 4. Summary

This chapter described the apparatus used to create the bistable potential, record
the particle positions, and isolate the system from external sources of noise. The 0.6 pm
diameter sphere is trapped by two focused laser beams positioned 0.4 pm apart. The
particle makes transitions between stable points over a barrier of between 0 and 10 kBT,
controlled with the trap separation. Gimbal mounted mirrors are used to change the
angles at which the beams enter the objective lens, changing the positions of the traps.
The beam power is modulated electro-optically in a single beam only, tilting the

potential.

141

The particle position is recorded in three dimensions at 200 Hz. The sphere’s
image is projected onto the CCD camera and transferred to the analysis computer. The x
and y center of the particle is found to i10 nm and the 2 position determined from the
image brightness to 3:40 nm. Image analysis takes less than 5 ms so the image is not
saved, reducing memory usage by a factor of 160.

Changes in the potential due to beam power and position fluctuations and changes
in the depth of the focal plane in the cell were reduced using electro-optic, mechanical,
and thermal stabilization. The beam power fluctuations were reduced by at least a factor
of 4 using beam power stabilizers, corresponding to barrier height changes of less than
0.04 kBT above 1 Hz. The beam positions were stabilized by thermally controlling the
lens mounts and eliminating air motions. Thermal control results in an estimated 0.4 nm
position fluctuation, or 17 nm/°C if the room temperature changes by more than
0.1 °C/hour. The Plexiglas box surrounding the optics removed the high frequency beam
fluctuations (> 1 Hz), leaving mean x changes of 5 nm and mean y changes of 12 nm.
The relative beam spacing fluctuations are only 2 nm in x and 3 nm in y. Additional
apparent beam motions due to motion of the camera itself were eliminated, reducing
measured y fluctuations by an additional 13%. Long term variations in the objective/cell
separation due to mechanical and thermal drift were reduced using a thermally controlled
objective clamp and a cell clamp (Figure 4.18). The motion was reduced to 2 pm over
14 hours, corresponding to 25% change in the transition rate or 1 kBT in the barrier
height. This limits the window of stability to approximately 2 hours, during which the
barrier changes by 0.15 kBT or less, depending on the size of the room temperature

variations.

142

In the next chapter the particle position F(t) is analyzed to extract the potential

energy and the transition statistics. The three dimensional potential energy surface is
calculated from the position distribution. The transition rate is determined from the x
position as the particle crosses the barrier. Using these results the distribution of paths
leading to a transition is measured for a static potential. The transition rate in a driven
system is measured as a function of modulation amplitude and frequency, verifying the
expected power law dependence of the activation energy on the amplitude near critical

driving.

143

Chapter 5 Experimental Results

This chapter describes the analysis of the Brownian particle dynamics and
measurements of transition rates in a bistable potential. To introduce the methods for
obtaining a quantitative measure of the trapping potential, the static single beam trap is
first considered. Introduction of a second beam creates a bistable potential, which
depends not only on beam power but also on beam spacing. By modulating one or both
of the beams a time-dependent potential barrier is created.

The calculation of the potential is described in Section 5.2. The particle position

time series F(t) is transformed into a three-dimensional probability distribution from
which the potential energy U (F) is calculated. The gradients of the potential near its

extrema (i.e., at its two minima and its single saddle point) are calculated. The behavior
of the potential under modulation is obtained from direct measurement as well as from
parametric representation using experimentally measured coefficients.

Analysis of over-barrier transition rates is described for static and periodically
modulated potentials. Spatial and temporal distribution functions are calculated from the
time series. The distribution of particle trajectories resulting from over-barrier transitions
is analyzed. Consideration of the conditional probability for escape leads to visualization
of the so-called prehistory distribution. This represents a measure of the “optimal path”,
the set of trajectories by which a particle is most likely to escape from a deep potential
minimum.

When periodic modulation of the potential occurs, the transition rates are a function

of modulation amplitude and frequency. The asymptotic dependence of the escape rate

144

as the modulation amplitude, the control parameter, approaches a critical value is
measured. Using a recent theory which predicts a power law singularity as a function of
the control parameter [44], we extract best estimates of critical exponents under two
conditions: the slow modulation regime, where the particle can follow the potential

adiabatically, and under nonadiabatic modulation.

5. 1. Measurement of Particle Dynamics in Single and Two Beam Traps

The particle position F(t) is used to calculate the potential energy and transition

statistics in the optical trap. Prior experiments with a particle in a single beam trap were
discussed in Section 3.3. The potential energy surface for a single beam trap in this
system was mapped using the method in Section 5.2. In a two beam trap the particle
makes transitions over the barrier between the two stable points. The rate of transitions is
calculated from the particle’s x position.

The following symbols are used in this and later sections.

 

 

 

 

 

 

 

Symbol Definition Equation

x Direction connecting stable points

y Direction with largest t,’I

z Direction of beam propagation, smallest t; '.

t2; Relaxation rate in the x direction (Hz) ti: = (oz/2711‘

F Viscous damping rate (Stokes damping) F = 6mm / m

W Mean transition rate (Hz) 1/(T)

(T) Mean dwell time (s) If.) = jzp(r)dr
o

 

 

 

The data in Sections 5.1 through 5.3 as the result of closely spaced measurements
conducted over a four-day period. The single beam potential is calculated from data
labeled 022804-01. The double well potential labeled 101403-03 is used to show the

three dimensional potential energy, transition rates, and the optimal path in Section 5.3.

145

It is the longest run (with little drift), containing 2x106 points. Data denoted as 012304-
01 are used for the tilted potentials in Section 5.2. The large amplitude static tilts in
Section 5.2 are labeled 021703-01. Table 5.1 lists the mean static barrier height AU,
transition rate W0 for the left and right wells, the x, y, and 2 components of the relaxation
time t,. in, and the overall acquisition time per data set.

Table 5.1 Static potential parameters for data used in Sections 5.1, 5.2, and 5.3. Units:
energies AU (kBT), rates W0 (Hz), relaxation times t, (s).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Dataset Label AUM, AUfigm W0 left W0 right t,,,- t,,, t,,: Time
022804-01 - - - - 0.002 0.002 0.2 5003
101403-03 3.4 2.8 2.5 4.4 0.007 0.002 0.3 2.8 hr
012304-01 5.2 5.7 0.7 0.5 0.005 0.002 0.3 5003
021703-01 3.3 3.3 1.5 1.4 0.010 0.004 0.5 15003

 

 

5.1.1. Single Beam Trap

The x, y, and 2 positions vs. time for a particle trapped in a single beam trap are
plotted in Figure 5.1. The particle center (solid lines) fluctuates in a 50 mm x 50 nm x
1mm volume around its equilibrium position (dashed lines). The position is recorded
every 5 ms, or 200 times in the 1 second interval shown.

The time scale of z fluctuations is much slower than the x and y fluctuations, with
larger amplitude. Prior experiments have shown the axial trapping forces (along the
direction of beam propagation, z) to be more than an order of magnitude smaller than the
radial (x and y) forces [49].

Figure 5.1 (right) shows the position distributions. In a harmonic potential the

position distribution P(x) is Gaussian with

10(sz emi— g0... Jay],

146

where B is the spring constant and A is a normalization constant. A least squares fit to
the distributions (solid lines) agrees well with expectations. A small deviation is noted

for z fluctuations in the tails of the distributions.

1 .80

 

1.78

1.76 w: '1“: ‘3‘), fl‘jiflquififint f/liiil": 'i

1.74 'l If

X (W)

1.721

1.70 r . r . .
132 . 4.0 4.2 4.4 4.6 4.8 '

 

1.301

l
”1,, ii ii 1.11 1le l l h A
ll 11. ll 111 011 ll
11' '

ll
1 #1“
1.261 v
8'6 I ' V ' Vfi

4.0 4.2 4.4 4.6 4.8

1.2

Y (um)

 

 

 

 

7.4 v ' V f T ' V ' '
4.0 4.2 4.4 4.6 4.8
Time (seconds)

 

Counts

Figure 5.1 Time series and position distributions for a single-beam trap. The x, y, and 2
position data are plotted as a function of time at 5 ms intervals for 1 second. The
plot ranges are 100 nm for x and y and 1.2 pm for z, with arbitrary position offset.
The dashed lines indicate the position of the trap center. The position
distributions from the full 500 second data set are plotted to the right of the time
series. Gaussian fits to each distribution show deviation from the harmonic
approximation. (Data labeled 022804-01, 100,000 frames. Standard deviations of
Gaussian fits 0} =16 nm, 0;, =15 nm, 0'2 =175 nm.)

The potential energy U (7) is calculated from the position distribution (Section

5.2). Energy contours, corresponding to a cross-section (slice) through the stable point of

the single beam trap are plotted in Figure 5.2. The potential is nearly ellipsoidal, with its

147

major axis along the axial (z) direction. The potential is also skewed in the direction of
beam propagation. This is the result of the scattering force (F 5cm in Section 3.1.2), which

decreases as the particle moves away from the beam waist, breaking the 2 symmetry.

14#. -v.

Z (um)

Y 0"“)
O

 

Figure 5.2 Single beam potential energy slices. The potential energy of each volume
element is plotted for the xz, yz, and xy planes running through the trap center.
Contours indicate energy steps of 1 kBT. The beam propagates in the +2 direction.
(Data labeled 022804-01, 100,000 frames, histogram bin size = 0.01 pm in x and
y, 0.02 pm in z.)

The curvature near the minimum is obtained from a least squares fit of a quadratic
function (solid line in Figure 5.3) to the one dimensional profiles extracted from the three

dimensional potential energy (see Section 5.2). (The energy offsets are arbitrary

normalization.) The curvatures a) are listed in Table 5.2. For a harmonic oscillator, the

spring constant k =mco2. The spring constants for a 0.6 pm diameter silica sphere

148

(m = 2.3 x10"'°kg) are listed in Table 5.2. The radial spring constants in this experiment
are an order of magnitude smaller than those measured by Ghislain [49] owing to lower
beam power (7 mW here vs. 150 mW), different sphere diameters (2a = 0.6 pm vs. 1
pm), and different Optics. The axial spring constant is 100 times weaker than the radial
spring constant. The relaxation rate of the overdamped particle near the potential
minimum is given by I," =w2/2nf, where l“=67777a/m=2.5x107s"l in water with

viscosity 7) = 0.001 kg/m 3.

 

   

 

 

 

 

 

 

 

a) b) C)
a . I f I 1
r: > a : ,
D

5 F b
r

0. ° 1 .

O

c _ .

DJ

E .

8 D h

(L . L

0.0 0T1 0.0 0:1 0.0 05 110
X (um) Y (um) Z (um)

Figure 5.3 One dimensional potential energy profiles of a single beam trap. The (a) x,
(b) y, and (c) 2 profiles centered on the potential minimum are plotted vs.
position. Quadratic fits at the minimum (solid lines) are extrapolations (dashed
lines). The potential energy tick marks: 1 kBT steps. The harmonic
approximation is valid to 3 kBT fiom the minimum. (Data labeled 022804-01,

100,000 frames. Fit function U(q) = U0 + 8,, /2(q — qo )2 (q = x, y, z) with
B. = 4100 kBT/umz, B, = 4400 kBT/umz, B. = 42 kBT/umz.)

Table 5.2 Single beam oscillation frequencies and damping rates.

 

w.- (s") ’25 (“2) k.- (N/m)
270000 470 1.7x10‘5
280000 490 1.8x10‘S
27000 4.7 l.7><10’7

149

 

 

 

 

 

 

 

 

N~<><~1

 

5.1.2. Two Beam Trap

The introduction of a second beam breaks radial symmetry, softening the trap
along the line separating beam centers (defined as the x direction). For 0.5 pm spacing
and about 10 mW in each beam, the system is bistable with a barrier height that depends

on beam separation and relative power. Figure 5.4 shows the x profile as the power in

the left beam Pl varies from 5.5 mW to 7.0 mW with the right beam at 7.3 mW. As a

 

 

 

 

 

 

 

 

* a) a 0 c j
g; o 0’. j
.i. ' ‘
i-r1 kBT ° . i
g .
.. a”. .4
it» .- . a
r “"1 . i
i ”._

l: i

m a

1:1 i .I .'

>. t '-

9 :— e

1:2

:3. i Svfim.‘.~q~' o.

93 ~ *5

8 F ... I -l
i 1
id)
i: fl:0 0.. .

f " 1
i '50-! 55".
t . , , - , 1
0.0 0.2 0.4 0.6 0.8 1.0
X0101)

Figure 5.4 X profile with varying beam power for a two beam trap. (a) for low left beam
power the particle is confined to the right beam. (b) as power increases the right
well curvature decreases and tilts towards the left beam. A second stable point
forms with a small barrier in (c). The potential nears symmetry about x = 0.46 urn
((1). (Data labeled 022804-03, right beam power 7.3 mW, leit beam powers (a)
5.5 mW, (b) 6.1 mW, (c) 6.7 mW, (d) 7.0 mW.)

150

 

0.8 -

Z (m)

0.4 —

 

 

 

Y (m)

 

 

 

 

‘00 0.2 0.4 0.6
X (um)

Figure 5.5 Three dimensional potential energy slices for a two-beam trap. The potential
energy of the volume elements are plotted for the xz (y = 0.13 pm), yz
(x = 0.16 pm), and xy (2 = 0.78 pm) planes through the left stable point. The
contours represent 1 kBT energy steps with local minima at the stable points. The
particle fluctuates around the stable point r.. As it moves in x to make a transition
it also moves up, in the direction of beam propagation +2 through n, to the stable
point r2. (Data labeled 101403-03, 2,000,000 frames, 16280 transitions.)

151

second stable point forms, the curvature of the right stable point decreases. At
P. = 7.0 mW the potential is symmetric about the line connecting the two minima. (The
beam powers for a symmetric potential may be slightly different owing to small
differences in the optical alignment.)

Three dimensional energy contours for slices through two stable points are plotted
in Figure 5.5. The two stable points r1 and r2 define the direction x. The top of the
barrier is the saddle point rb; unstable in x, but stable in y and z. The scattering force
breaks the 2 symmetry, creating a single saddle point displaced from the focal plane in
the +z direction. Therefore the particle makes transitions it moves out of the objective’s

focal plane in addition to moving along x.

 

 

 

 

 

 

 

 

 

\ u
it. ...... kxfkalfi's.

1.21

 

 

 

 

 

r74 ' é ' é .1.0e1.2-12-1g-1.8.2.0 4.6 ' 4T7 ' 4.8
Tlme (seconds) Tlme (seconds)

0
m.

Figure 5.6 Dynamics of over-barrier transitions in a bistable potential. The particle
undergoes small fluctuations near a stable point (dashed lines) x1 and xz;
occationally a large fluctuation pushes it over the barrier. (a) shows the particle’s
position as it makes multiple transitions. (b) shows the x positions during the
transition marked A, where the symbols represent the positions measured at 5 ms
intervals. (Data 012304-01-01, 100,000 frames. Mean transition rate W =
0.5 Hz.)

152

The time trace in Figure 5.6 shows the x coordinate as the particle makes
transitions between the stable points in a symmetric potential. The particle fluctuates
about a stable point until a large fluctuation drives it over the barrier. The transition

duration is 5 to 15 ms.

5.1.3. Determination of Transition Rates

Transition Criteria

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.8 right
- .- — well
1 61 u R N
A , +8X
E X.
X 1-4‘ B l l l . -5X
W w
l _

  

left

1.2.“ A
71' . . well

160 ' 165 ' ' '170' ' ' '175' ' ' '180
Time (seconds)

 

Figure 5.7 Determination of transition events. The x position vs. time is plotted, with
lines indicating the barrier top (X0) and the half-widths (:tSX). The “current” well
is indicated by the line above and below the data, with length equal to the dwell
time t. The particle makes a transition when it crosses the midpoint then the half-
width on the other side (event labeled A). A particle crossing only the midpoint is
not counted as a transition (event labeled B). (Data labeled 012304-01-01)

A transition is herein identified as an event in which the particle crosses the
saddle point by a distance 8X (Figure 5.7). The reason for incorporating 8X into the
definition is the following. At the saddle point the force exerted by the potential vanishes

and the particle undergoes free diffusion in x. It is possible that the particle could cross

153

the midpoint several times without entering the region of attraction of either stable point.
Since these diffusive transitions should not be counted, the constraint that the particle
must pass a line on the other side of the barrier (half-width) before a transition is

recorded.

The midpoint is taken at the top of the barrier. The half-width 5X is set to one
pixel (0.07] pm), around 40% of the distance from the midpoint to the stable point, close
to the inflection point of the potential along x. To assess the sensitivity to 5X, Figure
5.8a shows that the transition rate (calculated below) changes by less than 5% for 5X
between 0.071 pm and 0.2 pm. If the choice of midpoint is displaced by 0.02 pm (two

histogram bins) in either direction (Figure 5.8b) the rate changes by less than 2%. The

same half-width and midpoint positions are used for all data recorded in the same set.

 

 

b

08.3). 0.70 l
‘ .. 0.69"

0.7 - - i -

E 0.6- . . ' ' ' E 0.68. i

3 . , - a, 0.67-

m 0.5 . ' ’5 1

0: . , a: .

8 . . , . g 0.66:

:1?) 0.4+ : i ran 0.65?

s ‘ a

1‘: 0-3‘ l: 0.64-
0.24 .- 0.63-

 

 

 

 

 

 

004 1302' 0.00 ' 002 ' 0.04 ' 0.06
Midpoint Position (pm)

0.00 0.05 0.10 0.15 0.20 0.25 0.30
Half-width (um)

Figure 5.8 Sensitivity of transition rate to changes in half-width 5X and midpoint X0. (a)
transition rate (W = (number of transitions)/(total dwell time)) for differed half-
widths. The region between 0.071 mm and 0.20 pm is insensitive to 8X, changing
by less than 5%. In (b) the transition rate varies by 2% for changes in midpoint
position of :l:0.02 pm from the saddle point. (Data labeled 012304-01-01, 5.5 kBT
barrier, saddle point to stable point spacing = 0.21 pm.)

154

Transition Rate in the Bistable Potential

The left well dwell time is defined as the time interval between a particle crossing
the line at -5X from the right to the time it crosses +6X. The transition analysis
calculates the dwell time I for left-to-right and right-to-lefi transitions. The mean
transition rate is calculated from the normalized dwell time probability distribution p(r).
In equilibrium, it is assumed that the particle has equal probability to escape at any time,

so that p(z')= xiexp(—/ir), with mean transition rate 2. The dwell time distribution

(Figure 5.9) for the potential in Figure 5.5 shows that this is not necessarily the case. The
first 10 ms wide bin shows a dip, which corresponds to the finite x relaxation time

(150 Hz). (This was also seen in earlier experiments [30].) This occurs because the
particle must relax to the bottom of the well at a rate tg'x before making another

transition. The data also indicates deviation from the exponential behavior at times
below 0.1 s. This may appear in deviation fiom a white noise spectrum at frequencies
from 10 to 100 Hz. At long dwell times, the distribution decays exponentially, shown by

the solid line in Figure 5.9.

The mean dwell time is defined by (r) = jzp(r)dr. For the discrete distribution
0

of dwell times t,- the mean dwell time is calculated by computing the average value

2'.
(r) = __ZIZV' = W "l , where W is the mean transition rate. The dwell times for the left and

right wells are recorded separately, leading to WL and WR, the left-to-right transition rate

and right-to-left transition rates.

155

 

    

 

 

 

 

0.04-
(13>
0.03-
? 0.02-1
Q.
1
0014
J
000 '1 'i I i I i ‘ “PL :Hl‘ I l l
0.0 0.5 1.0 1.5 2,0

1(seconds)

Figure 5.9 Dwell time distribution for a static potential. The solid line is an exponential
fit to the dwell times. The vertical line represents the value of the mean dwell
time. Inverse dwell time 1/(r) = 2.48 Hz, exponential decay rate 2. = 2.89 Hz.

(Data labeled 101403-03, 2,000,000 frames, 16280 transitions, Bin size 2 frames
(10 ms).)

5.2. Double-well Potential Calculation

The following symbols are used in this section.

Symbol Definition Equation
P(x, y,z) Particle position distribution. A discrete
3D histogram of particle positions

 

 

U (x, y, z) Potential energy distribution, calculated U (x, y, z) = "k3 T ln( P(x},)_v, 2)]

 

from P(x,y,z). 0
x position of left stable point (i = 1), right
x, stable point (i = 2), or saddle point

(i = b), measured from local fits.

 

Potential energy at left stable point

 

 

 

 

U,- (i = I), right stable point (i = 2), or saddle

point (i = b), measured from local fits.
AU Barrier height AU = Uh _Ul.2
a) Eigenfrequency (curvature) of U (x, y, z)

 

156

Symbol Definition Equation

 

 

 

 

A Potential Tilt. Linear term in quartic
0 approximation of x potential energy.
Bifurcation amplitude. Tilt A0 at which 21 3‘3 2
Ab the stable and unstable states merge, A b =
calculated from the quartic approx. 3R

 

 

5.2.1. Extraction of the Potential and Potential Surfaces

A particle in thermal equilibrium, measured on time scales longer than the mean
interwell transition time, has a Boltzmann position distribution. The distribution is
inverted to create a discrete map of the potential energy in three dimensions. The

position probability distribution is given by

___P(x,y,z) = lexp[_ U(xsy,2)] ’

P0 Z k 3T
where P0 is the total number of positions in the data set and Z is a normalization constant.
The three dimensional position distribution P(x, y, z) is computed from the position data.
The coordinate space is separated into discrete volume elements AV = AxAyAz .

P(x, y, z) is the number of positions in each volume element

0 otherwise

P(x,y,z)= Z {I (x<x(t)<x+Ax),(y<y(t)<y+Ay),(z<z(t)<z+Az).

frames

The potential energy U (x, y, z) is

U(x, y, z) = —kBT1n[£(1I’;y—’—Z—)] + kBTln(Z).

0
The normalization constant Z is ignored, creating an energy offset that changes

depending on the shape of the potential. Figure 5.5 shows slices through U (x, y, z),

centered on the left stable point.

157

The one dimensional potential profiles are extracted from P(x,,v,z). The x
profile is extracted for the entire width of the potential. The maximum P(x,,mz)
(corresponding to the minimum U (x, y,z)) in each yz plane is found. The diamonds in
Figure 5.10 follow the path of maximum P(x, y, 2) through the xz plane shown in Figure

5.5. The potential is

max[P(x, y, z)]] -

0

 

U(x) = —kBTln{

The y and 2 profiles are calculated at the stable and saddle points only. The y
profile is calculated for constant x, and 25, the x and 2 positions of the stable or saddle

point. U(z) is calculated for constant x, and y, (squares in Figure 5.10),

U(y)= —k,,Tin[:('3—’3”—ZS-)] and U(z)= -kBTln[£w].

0 P 0
If the saddle point is sparsely populated (due to low mean transition rate or short
acquisition time) the potential is also spatially averaged. This increases the size of the
volume elements without changing the spacing. In the Profiles data analysis program
(Appendix E) this is controlled with the rebin number. If rebin = 0 there is no averaging.
For nonzero rebin the position distribution is summed over an area perpendicular to the
profile direction. For example, when averaging in the x profile calculation the path of

minimum energy is found from the maximum sum, where

nzrebin mzrebin
P5 (x) = max 2 ZP(x,y + nAy, z + mAz)].

n=—rebin mz-rebin

The potential energy is calculated from P5,

158

P.(x) J

Uav x =—k Tln ——
e() B [Pox(2><rebin+l)2

 

8.81 :2 ? -
8.61 '7 7' i

1

8.41
l

8.21
8.0-1

Z (um)

7.81
7.6-
7.4-

 

7.0 - ; r 'E *r 'E I
1.2 1.4 1.6 1.8
)

 

 

 

Figure 5.10 Calculation of one dimensional profiles from the three dimensional
histogram. Contours represent potential energy in a single xz plane, where darker
areas have lower energy. The diamonds follow the path for calculation of the x
profile. The squares follow the paths for calculation of the stable point and saddle
point 2 profiles. The x and z axes begin with an arbitrary offset. (Data labeled
101403-03, 2,000,000 points, x bin size = 0.01 pm, 2 bin size = 0.02 pm.)

5.2.2. Parameterization of the Potential

Local and global approximations to the one dimensional profiles are used to
parameterize the full three dimensional potential. The curvature of the potential is
estimated for small displacements from the stable points and saddle point and the x
barrier height calculated. The tilt A0 of the potential due to changing beam power is

159

estimated from the full x profile for multiple static, tilted systems. This process was
adopted since direct measurement of the potential for very large tilts takes prohibitively

long times.

Quadratic Fits to Stable Points and Saddle Point
Each one dimensional profile is plotted with error bars (Figure 5.11) using

OriginLab’s Origin 7.0 graphing and analysis software (with three plots of the x profile).

The quadratic function

C
(5.1) U(q)=C. 710-qu

9

is fit to the stable points and saddle point, with coordinate q = x, y, or z. C1 equals the
potential offset U0, qo is the position of the minimum, and C2 = mwzin the harmonic

approximation. The fit is weighted by 1/02 (instrumental weighting), where 0' = 6U .

The uncertainty 5U is calculated from the number of counts in the volume element,

assuming Poisson statistics in the number of counts P ( 6P = JP ).

50411.3 zfoéfljfz;
P0 PPO P J?

The number of points fit with (5.1) varies depending on the potential and the size
of the volume elements. Typically 9 points are fit in the y profiles, 9 points at the stable
points of the x profile, 20 points in the z profiles, and 25 points at the barrier in the x
profile. The fit range is centered at the minimum. Figure 5.11 shows the results of these
fits (solid lines), plotted with the profiles. The dashed lines represent the extrapolations

of the fits, showing the deviation from the harmonic approximation.

160

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Left Right Barrier
16 16 16 ..
:3 144 14. 144
Y 5 12‘ 1211 12"
2 1
m 104 104 10- -
.72
cc, 8 81 ' 81
‘6
a” 6 6 6
0.00 0.05 0.10 0.15 0.20 0250.00 0.05 0.10 0.15 0.20 0.25000 0.05 0.10 0.15 0.20 0.25
16 - Ytem} . 16. Yéym) 16 . Yteth)
p 14. ..' 144 '5 .52. 14. "1'
g 12 E ,4 124 is j: 12 1°

Z 8 so ‘ "9 ‘0'“ a
g 10‘ 3 f 10. ‘9‘. g 10. W
3 ‘ ‘5 0'

El 8 \/ 8 v 8
8
6 . . . 6 . . . 6 . . .
0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5
16 2 111m) 16 Item), . 16 2 turn) -
p 14 " ' 14. ‘. ,1 14. . 1
51' -' r f
5 12 g ; 124 .- 12.
x 8 -_- .- -. ; _ . . .
E 10 if '|'o‘#~." . 104 . .‘Jfixz I: 104 . IA .
g 8 '- '.'-" " 8 . .3 V 8 ' " ‘. '
.. C '. .7 ‘ ' . o *0

13 U‘ "' “J l ‘7’, ‘1

O. 6 6 6 . .
1.21:31i41i51i61i71i819 1.21:31i41i51i61i71I819 1.21:31f41151161371161e

x (“111) X (m) X (um)

Figure 5.11 One dimensional profiles with quadratic fits. The y, z, and x profiles (rows)
for the left well, right well, and barrier (columns) are plotted with error bars

1/ Jcounts . The results of the quadratic fits (solid lines) are extrapolated to the
extents of each plot (dotted lines). (Data labeled 101403-03, 2,000,000 points, x
and y bin size = 0.01 pm, 2 bin size = 0.02 pm, rebin = 0 except for x barrier
where rebin = 2.)

Results of the fits in Figure 5.11 are displayed in Table 5.3. The conversion

factor between fit parameter C2 (units kgT/umz) and the curvature (8") is

02 = 1.77 x107 C2 at room temperature (293 K) for a 0.6 pm diameter silica particle with

mass m = 2.3x10"6 kg. The t,"_. (3 — 4 Hz) is 40 times smaller than tr“; (100 — 150 Hz),

161

and more than 100 times smaller than til, (500 Hz). The y and z relaxation rates differ

by about 20% between the stable points and saddle point.

Table 5.3 Double-well parameters from quadratic fits to one dimensional profiles.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a) (s") t." (Hz) U0 (km x0 (um
X left well 140000 125 7.0 1.36
x right well 160000 175 7.5 1.74
x barrier 59000 -22 10.8 1.56
Y left well 300000 570 7.0 0.125
Y right well 290000 540 7.6 0.125
Y barrier 270000 460 10.4 0.127
2 left well 23000 3.4 7.1 0.947
2 right well 22000 3.1 7.6 0.810
2 barrier 20000 2.5 10.3 1.26

 

 

 

 

 

 

 

The barrier height AULR is calculated from differences in potential offset U0 for
the x barrier and stable point fits. Averaging the position distribution (nonzero rebin
value) changes the constant C I in equation (5.1). Therefore only the U0 values for the
same rebin number are used. Table 5.4 shows the measured potential offset and
calculated AU for rebin 0 (no averaging), 1, and 2. The barrier height varies by 3% for
the different averages. Since the barrier often has very few points, the U0 value for
rebin = 2 is used. For the data in Figure 5.11 the barrier heights are AUlcfl = 3.39 kBT and
AUn-gm = 2.78 kBT.

Table 5.4 Barrier height measurement results with spatial averaging.

 

rebin = 0 rebin = l rebin = 2

 

 

 

 

 

 

 

U0 barrier 10.25 10.58 10.82
UO left 7.00 7.18 7.43
AUleft 3.25 3.40 3.39
U0 right 7.49 7.76 8.04
AU right 2.76 2.82 2.78

 

 

 

 

 

 

I62

Estimation of Potential Changes Under Modulation

Instead of measuring the changes in each volume element for each beam power,
the tilt is described by a single parameter A0. The full three dimensional potential is

approximated to fourth power in x,

B. 4 3,. B-
U3D(x,y,z)= A0x+—2ix‘ +—C—‘x4 + 2' y2 +——é'—z2 +C_L.xzz.

 

The y and z curvatures By and B: depend on the overall beam power. The 10% power
changes used in tilt experiments cause only small changes in curvature By and 8;
compared to B... The cross term CK depends on the positions of the stable points and thus
changes only with beam separation. Therefore the tilt A0 is calculated from the x profile

only, using
B 2 C 4
(5.2) U(x,y,z)=A0(x—x0)+3(x—x0) +Z(x—x0) +U0

The coefficients A0, B, C, x0, and U0 in equation (5.2) are calculated using a three
point fit to the x potential profile. Figure 5.12 shows the results of two methods of

parameterizing the potential. The solid lines are created from matching U(x) and

award” = 0 to the local fit results. This ensures that the quartic potential follows the

positions of the barrier and stable point as they merge at large tilts. The dashed lines are
the results of least squares fits with instrumental weighting. The weighting causes the fit
to match the shape of the lowest energy points and overestimates the barrier by more than
2 kBT in Figure 5.120. The transition rate dependence at large modulation amplitudes

depends on the potential when the barrier is low, therefore the three point fit is used.

163

 

 

 

 

17 '. 17 17 '
1 . I .
161': 1 16" _ ‘ 16‘ l “ l
1 z t i
151 ' - j 151 . r “ : 15 '_ ...l. :.
A I \ I , ‘ , J , I g ,‘ .
|—m 141 I V 1 ' 141 _, ,5 [.1 . 141 , "2",". '
x , I _ 4 I a 1. ~-I_ -' 7 ~
g 13" o ,3 0‘ .: 13‘ $3.”; : it 13': ? . r
1 ‘ ' 1 . . i i 1 . '.~ ,
3 2 .9. 3' 2 '- i‘. ‘ l 21 9 f 9t
c 11. I T ‘ 11‘ I I j 114 ' '
LIJ I ~ ‘ I. i ” r
E 10. ,0 P 101 .i ‘1 10‘ ' 1.
on I r t 1 I
5 9- r 9- j 9- . .
S . d i f C
O. 81 ' 81 ' 81 ‘
7.1 7-1 7+
1
6 . 6 6

 

 

 

 

 

 

 

 

1.1 1.2 113 1:4 1:5 1T6 77 1.8 1.1 12 1T3 174 1:5 1T6 1:7 1.8 1.1 12 1:3 1:4 1T5 1:6 1:7 1.8
X (um) X (um) X (um)

Figure 5.12 Quartic fits to tilted x profiles. The x profiles for increasing beam power
(squares) are approximated using three point fits (solid lines) or a least squares fit
to the entire potential (dashed lines). (Data labeled (a) 012304-01-01, (b) 012304-
01-02, (c) 012304-01-03, 100,000 points. Static tilt amplitudes (a) —0.9 kBT/um,
(b) 4.3 kBT/um, and (c) 9.6 kBT/um.)

The quartic parameters are calculated from the system of equations
6U 6U
U(xl):U|’ U(x2)=U2, U(xbleb’ a(xll: 0, ‘5;(le= 0’

using U(x) in equation (5.2). The positions of the left well x1, the right well x2, and the
barrier xb, and the potential energy at the left well U1, right well U2, and barrier Ub are
obtained with local quadratic fits. The quartic is constrained to have potential energy Ub
at the point xb but has an unconstrained spatial derivative, allowing freedom in the

position of the barrier top. Solving for the parameters,

C = 4(Ub(xl -x2)3 -U'(3x' —x2 —2bex2 —xb)2 -U2(x1—xb)2(xl—3x2 +2xb)2)

(x1 —x2 )3(xl —xb )2(x2 _xb )2 ,

 

_ 4Ul -—4U2 + C(rl —r2)3(rl + x2)
2C(x1 T "2)3

3

x0

164

_ 4Ul — 4U2 — C(xl — x2 )2 (6x5 + X? + 2.1r13r2 + 3x3 — 4x0 (x1 + 29(3))

B
2(1‘1 " x 2 )5

 

A : 4Ul —4U2 +(2x0 —)rl —Jr2 Xx, -x2 XZB + C(2xg +1]2 +132 — 2x0 (x1 + x3 )»

, d
0 4(1‘1 _le an

 

B o C
U0 :U1_A0(xl —x0)_3(xl_x0)- TIL“ ‘xol4-

The results from the static tilt measurements in Figure 5.12 are listed in Table 5.5. The
uncertainties are estimated by varying the positions and potentials from the local fits by
the fit error (6x = 2 nm, 5U = 0.1 kg?) and calculating the change in the parameters.

Table 5.5 Results of 3 point quartic fit to static tilt potentials in Figure 5.12. B and C are
weakly dependent on the tilt amplitude A0.

 

A0 (kBT/pm) B (kBT/umz) C (kBT/urn“) x0 (pm) U0 (k3?)
Figure 5.12a -0.9 a; 0.5 -490 r 20 10700 a 500 1.446 a 0.002 13.0 a 0.1
Figure 5.12b 4.3 a 0.5 -480 a. 20 10600 as 500 1.446 r 0.002 13.2 :r 0.1
Figure 5.12C 9.6 a 0.5 -430 r 20 9700 :r 500 1.446 as 0.002 13.5 a 0.1

For increasing tilts the stable and unstable points approach, eventually merging

 

 

 

 

 

 

 

 

 

 

 

when A0 equals the bifurcation amplitude A b. The extrema are the positions that satisfy
6U(x)/6x = 0. Solving equation (5.2) for A0 when xb =x,- yields Ab = 2|Bl3/2 /3~./ 3C .

For the values in row 1 ofTable 5.5, Ab = 38 i 3 kBT/um.

A0 vs. Beam Power Calibration, Large Two Beam T ilts

The relationship between beam power and A0 is found from several static
potentials with differing nonzero tilts. The powers in both beams were changed with
equal magnitudes but opposite direction using the SRS DS345 function generator’s offset

voltage V. The positions and potential energies of the extrema were obtained using local

165

fits and the quartic parameters calculated. The resulting calibration, A0 vs. offset voltage,
is plotted in Figure 5.13a, along with potential well parameters dependent on tilt.

These two-beam measurements were made with the modified trapping system
(see Appendix C). The LS-100 beam power stabilizer was replaced with another LS-
PRO, controlled using the offset voltage from the DS345 function generator. The power
vs. offset voltage calibration was linear, but with slope and intercept different from the
calibrations for the other experiments in this chapter. The beam power was 7.5 mW. The
potential was tilted by changing power in both beams in opposite directions, as opposed
to the single beam modulation in Section 5.4.

Figure 5.13a shows that A0 is linear with beam power from the three point fit for
tilts up 17 kBT/um. The bifurcation amplitude calculated from parameters B and C
obtained for A0 = 0 is Ab = 21 i 2 kBT/um. (This potential has a lower barrier than the
potentials listed in Table 5.5, requiring a smaller Ab to make the barrier disappear.) A0

changes linearly with offset voltage up to 80% of Ab. A0 = a,V + bI (and A0 = a2P+ b2)

where a; = -580 :t 20 kBT/um V and b. = 280 :t 4 kBT/um. The A0 = 0 voltage was
0.470 V and A = Ab when Vchanges by 0.036 V.

Figure 5.13b shows the positions x1, x2, and xb obtained from the local quadratic
fits. These show qualitatively that the stable and unstable points approach at large A0.
Poor statistics in the shallow stable point cause large uncertainties in both the stable point

and barrier positions, preventing a quantitative examination of x1 (A0) and xb (Ao ).

166

 

A0 (RT/rim)

 

 

 

 

 

 

X (11m)

 

 

 

 

 

 

 

 

11
101 ' o
' O

91 ' 0
8‘ . O
7‘ o i o

p 6- . °

£ 5- - .

D 4 ' '

<1 3‘ '-
21 /\
1- ‘ '
0 00' ‘

0.44 0.46 0.48 0.60 0.62
Beam 2 offset (V)

Figure 5.13 Measurement of tilted potential up to A = Ab. (a) the tilt parameter A0 varies
linearly with offset voltage (and beam power) up to :t 17 kBT/um (Ab = 21
kBT/um), with slope m = -590 :I: 10 kBT/um V. (b) the positions of the left well
(squares) and right well (circle) approach the barrier position (triangles) for large
tilt amplitudes. At large amplitude the barrier height decreases, reducing the
statistics and leading to poor local fits. (c) the barrier height vs. voltage for the
left well (squares) and right well (circles). A power law fit to the right barrier
height gives an exponent 1.9 i 0.2 for Ab set to 0.435 V and the left barrier gives
1.4 :t 0.3 for Ab = 0.515 V. (Data labeled 021703-01-, 100,000 to 300,000
points.)

167

For a metastable potential near the bifurcation point the barrier height varies as
(Ab — Ari/2 [91]. The barrier height obtained from the quadratic fits at the stable point
and minimum are plotted in Figure 5.130. The barrier height
AU(AO ) = U(xb(A0 ))— U(r2 (210)) was fit with AU(V) = 8(Vb — V)" for AU < 3 kBT(solid
lines). For the left barrier (squares) the bifurcation offset voltage Vb was set Vb = 0.515
V, fitting for s and v. The exponent v: 1.4 :l: 0.3 agreeing with the prediction. The right
barrier (circles), with Vb = 0.435 V returned v = 1.9 :t 0.2, two standard deviations from
the expected v = 1.5. Since the uncertainty is greater near the bifurcation point the fit
puts greater weight on the higher barrier points, on the edge of the range of validity in the
3/2 power law approximation.

At large A0 the particle spends less time in the shallow well. The only way to
reduce the uncertainty in the potential energy is to record for longer periods. The 20 data
sets in Figure 5.13 were collected over 5 hours, beyond the expected interval of stability
of the system (The stability was checked by returning to V: 0.470 V and measuring the
potential. The values of A0, )0, and U, in Figure 5.13 changed by < 10%.) Since A0 is
linear with beam power, the calibration for potentials with small tilts is extrapolated to

A=Ab.

A0 vs. Beam Power Calibration, Small Single Beam T ilts

With the system in the configuration described in Chapter 4, the potential was
tilted over a small range (A0 < Ab/4) and the A0 vs. V extrapolated to A = A1,. The beam
power in a single beam only was changed by a small amount, corresponding to a change

of 0.5 — 1 kBT in barrier height. The beam power was controlled by the DS345 function

168

generator offset voltage sent to the LS-100 stabilizer, (see Section 4.1). The beam power

P is linear with offset voltage V, A0 = alV + b1 .

 

 

 

 

0.08 ' 0.10 ' 0170.14 ' 0.16 ' 0.18
Offset Voltage (V)
Figure 5.14 A0 vs. offset voltage. Tilt A0 is calculated from the 3 point quartic fit for
varying beam power (in terms of offset voltage), with linear regression fit. Slope

is —240 :1: 10 kBT/um V, with offset 29 i 1 kBT/um. (Data labeled 012304—01-01
through 012304-01-06, 100,000 points each.)

The slope shown in Figure 5.14 was recorded using beam powers of 7.63 mW in
beam 1 and 7.26 mW in beam 2. Beam 1 power was adjusted in 0.125 mW steps over
five points (two points above A0 = 0, two below). The potentials were parameterized and
the slope of the tilt vs. offset voltage curve is -240i 10 kBT/um V. The bifurcation
amplitude is Ab = 38 kBT/um. This corresponds to a tilt modulation of 2 mW peak-to-
peak in each beam, or an amplitude that is 13% of the DC power.

These results suggest that the x profile of the three dimensional potential changes
with a variation in beam power as the quartic expression (5.2) does to a change in the

linear coefficient A0. The results in Figure 5.13 are for an antisymmetric tilt as both beam

169

powers were changed in opposite directions. In the modulation experiments studied in
Section 5.4 only one beam is modulated.

However, it is argued that, if the potential is deep enough (relative to an
untrapped, freely diffusing particle) a change in the depth of a single well is equivalent to
an antisymmetric change in both wells i.e., the potential barrier is a perturbation of the
overall potential well. For example, if the stable points have energy —20 kBT relative to a
free particle, the potential with an additional 2 kBT in the left well will have
approximately the same shape as a potential with +a kBT in the left well and —a kBT in
the right (where a is less than 1).

To estimate the depth of the potential, we note that a single particle is typically
trapped for days or weeks. Assuming the mean rate is 1 escape/day (1><10_5 Hz) and
comparing to the interwell escape rate of 0.2 Hz for AU = 6 kBT, the “potential depth”

0.2

1x10

 

j=16k3T.

5. 3. Determination of Optimal Particle Trajectories

In this section, consideration is given to analysis of the actual trajectories
followed by a particle in an over-barrier transition. The most probable particle trajectory
during a transition is calculated from the prehistory distribution. The particle spends
most of the time near one of the stable points. Eventually a large fluctuation will drive
the particle over the barrier into the other well. These fluctuations are rare, and the
probability of making a transition sharply peaked around the most probable fluctuation
that moves the particle to a point on the other side of the barrier. The corresponding

particle trajectory is called the optimal path [2]. The optimal path is measured from F (t)

170

by considering all of the trajectories starting in one well that end up in a certain region on

the other side of the barrier.

 

8.8- // i
. 2 /
8.61 //// _
8.41 "
E 8.21
3 .
N 8.01
7.81 ,
7.61 7’

 

1.2 1.4

 

Figure 5.15 Constraints for prehistory calculation. The rules for selecting paths are
shown superimposed on a schematic of the potential (contours represent 2 kgT

steps). The paths must be in region A at time t = 0. None of the previous 20

frames can be in the right well, region B. At least one of the previous 20 frames

must be in the left well, region C.

The left-to-right transition paths are extracted from the position data using the
Optpath program. An area around the point (xf, Zf) is chosen on the right side of the
barrier (Figure 5.15 region A). Since the relaxation rate in y is faster than the sampling
rate and y is independent of x, no useful information could be extracted from y motions.
All y positions are collapsed to increase the statistics. The program steps through the
recorded positions until it finds one in this area. The particle positions prior to reaching

(Xf, 2]) are recorded. The prehistory distribution P(x,z,ti;xf,zf,0) is created from all

positions (x, z) at time t,~ that arrive at (Xf, Zf) at time t = 0, where t,- < 0. The paths are
separated into bins of width 15 nm in x, 40 nm in z, and 5 ms (1 frame) in time.

171

Two constraints limit the paths to left-to-right transitions only. Paths are
eliminated if the particle is in the right well (inside the shaded region B in Figure 5.15)
for any of the frames stored in the prehistory distribution. Also, the path must have at
least one point in the left well (region C in Figure 5.15).

Figure 5.16 shows the prehistory distribution P(x,z,t,.;xf ,2f ,0) for times leading

up to the transition. (The shaded contours are from the xz potential slice in Figure 5.5
with 2 kBT steps.) Darker red regions represent higher probability. The fluctuations in
the distribution amplitude are caused by the low numbers of counts in each histogram
bin. The distribution is smoothed using adjacent averaging.

For large times prior to t = 0 (t,- = -85 ms in Figure 5.16) the prehistory
distribution is the static distribution P(x,z) oc exp(— U (x, z)/ k 3T ) for a particle in a single
well. At t,- = -25 ms the prehistory distribution begins to move in the +z direction. At
t,- = -15 ms it distorts in the x direction, moving towards the saddle point. By t; = -5 ms
the particle positions are distributed near the saddle point, arriving in the region around
(xf, 2]) at t,- = 0. After the transition, the distribution continues moving in the x direction
(I,- = +5 ms) before settling down into the right well.

The symbols in the t,- = 0 graph represent the peak in the probability distribution
from t,- = -25 ms to t,- = 0, then to t,- = +25 ms as the particle moves into the right well,
literally the most probable escape path. Due to poor statistics the maximum may not

represent the peak of the distribution. A weighted average

ZZxP(x,z,ti;xf,zf,0)

x2 2 P(x,z,t,-;xf,zf,0)
X Z

£01):

 

172

over a 7 x 7 bin box centered on the maximum gives the mean values 7: (and through a
similar expression, 2) for each time step. The x and 2 components are plotted vs. time in

Figure 5.17 for t,- = -25 ms to t,- = +25 ms.

 

 

 

 

 

lt=-15msl

 

 

 

 

 

 

 

X(um)

Figure 5.16 Prehistory distribution and optimal path. The distribution of x and 2
positions is given for a long time before the transition (t,~ = -85 ms), the frames
leading up to the transition (t.- = -25 ms to t,- = 0 ms), and one frame after the
transition (t,- = +5 ms). The shaded contours are the xz potential energy slice from
Figure 5.5 with 2 kBT steps. The symbols in the t,- = 0 ms distribution shows the
average position from t,~ = -25 ms to t,- = +25 ms. The particle begins near the left
stable point (x = -0.21 pm) and ends up within 30 nm in x and 70 nm in z of the
point (Xf, Zf) = (0.03, 0), 30 nm to the right of the barrier top. (Data labeled
101403-03, 709 transitions tracked.)

173

There are two separate stages of motion before the particle makes a transition.
The particle begins moving in the +2 direction only at t,- = -25 ms. When it is in the same
2 plane the particle begins moving in x, increasing velocity as it climbs the barrier. After
the transition the particle moves in x for 10 ms before settling down into the right stable
point.

Theory predicts that the optimal path in one dimension xopt is the solution to [2]
(5.3) rep, = U'(x0p, ).
For the quartic in equation (5.2) with A0 = 0 the solution is

2 l4
\/ C + exp(2|B[(t — t0 ))

 

(5.4) x...(t)

 

 

 

 

 

 

 

 

a)
0.21 /.
. ./'/
A ./
E 0.01
3:
x 1—‘//
10.21
b)
0.01 ..—--I—-
./
A -/ \.\I
E
3 -0 2 / \
N /
1./
0.4 . . . . .
120 110 0 10 20

Time (ms)

Figure 5.17 X and 2 components of the optimal path vs. time. The symbols represent the
(a) 2": vs. t and (b) E vs. t shown in Figure 5.16, t,- = 0 graph. The least squares fit
to equation (5.4) (solid line in (a)) shows that the transition is not as sharp as
expected. During a transition the particle first moves in the z direction for at least
10 ms. When 2 is within 0.2 pm of Zf the particle begins moving in x, making a
transition over the barrier at x = 0 in 10 to 15 ms. The particle continues moving
in x for 10 ms before moving down into the right stable point.

174

The curve in Figure 5.17a shows the shape of equation (5.4) with the x optimal
path. The transition takes longer than expected, due to the motions in the z direction.
Equation (5.3) shows that the optimal path asymptotically approaches the banier (x = 0),
while the actual particle makes a transition in finite time.

The two dimensional optimal path shows that this three dimensional potential can
not simply be projected onto a one dimensional bistable potential. As seen in the next
Section, both the time scales in the bistable direction (x) and an orthogonal, universally

stable direction (z) must be considered.

5. 4. Transition Rate Dependence on Modulation Amplitude and Frequency

Periodic modulation of the potential at frequency f synchronizes the particle
transitions to the modulation, increasing the transition rate W (for systems with a static
rate W0 < 2]). Under slow modulation the particle follows the potential adiabatically.
With increasing frequency the particle lags behind the modulation and ultimately the
response falls off, analogous to a damped harmonic oscillator driven above the relaxation
frequency. A necessary condition for adiabaticity is ft, <<1 , or the modulation period is

much less than the intrawell relaxation time t,.

Modulation with amplitude A —> A2“, the potential curvature azU/ax2 —>0 so

that t, —-> 00. Thus, the adiabatic condition is violated at some value of A z Afd and the

system becomes nonadiabatic. In addition, adiabaticity may be broken for weak (and

slow) driving if lat, /8t| zl [45], i.e., the rate of change of the relaxation time is large.

The activation energy 6R is predicted to have power law dependence on the

.U . . . . .
distance from Afd, 6R cc Afd —A] . In the adiabatic regime the critical exponent

 

I75

21: 3/2. In the weakly nonadiabatic regime (where only [8t,./8t| <<l is broken) the

critical amplitude is the “slow driving” bifurcation point A? depends on modulation
frequency, A3." = A2.“ + Zinf/fl and 21 = 2.

The potential in the current experiment is three dimensional, bistable in the x
direction and stable in y and z, introducing additional time scales. The relaxation times

in the static potential (calculated in Section 5.2) are always ordered t,,; >t . >t . The

r,.\ r,y
condition for adiabaticity applied to x, becomes ft”. << 1 , and [dtm flit] <<1. However,

the optimal path measurements show that the particle moves in both x and 2 when

making a transition. Thus it is likely that, in addition to the conditions on t,,x, there is the

condition fin: <<1 . Since the y coordinate is not affected by perturbations in x or z and

t, . > try the condition ft”. <<1 always holds in the present experiments.

..l

The following symbols are used in this section.

 

 

 

 

 

 

 

 

Symbol Definition Equation
A Modulation Amplitude (kgT/um) A0 = A 008(247?)
f Modulation Frequency (Hz)
AUo Static barrier height (A = 0)
Minimum barrier height during one
AU min ~ 1
modulation period.
W0 Transition rate in the static potential
l/f
W Period averaged transition rate W = f [dtW
0
VI
1—9 Escape probability per period I) = [p(r)dr
0
A Critical amplitude in power-law fits 1n(W/WO ) = C IAC — Al” + y0

 

Critical exponent in power-law fits.

 

 

 

 

Aad Adiabatic critical amplitude (from
C theory)
51 Nonadiabatic critical amplitude S. _ ad
Ac under slow driving (from theory) Ac - AC + (2717/ ’6 )f

 

176

The data in this section are taken from stable runs over a three day period. The W
vs. f data in Figure 5.21 are labeled 110303-03. The low frequency W vs. A data
(frequency group A) in Figures 5.23 and 5.24 are labeled 012904-01. The higher
frequency W vs. A data (frequency group B) in Figures 5.25 and 5.26 are labeled 012404-
01. Table 5.6 lists the mean static barrier height AUO, transition rate W0 for the left and
right wells, the relaxation time t,. in x, y, and z, and acquisition time per data point for

these data sets.

Table 5.6 Static potential parameters for data used in Section 5.4. Units: energies AU
(kBT), rates W0 (Hz), relaxation times t, (s).

 

 

 

 

Dataset Label AUM, AUn-ght W0 left W0 right t,,... t,,]. t,,: Time
110303-03 4.4 4.9 1.5 0.98 0.006 0.002 0.28 5008
012404-01 5.2 4.7 0.62 0.85 0.005 0.002 0.26 5008
012904-01 6.4 5.9 0.26 0.36 0.004 0.002 0.29 5008

 

 

 

 

 

 

 

 

 

 

 

5.4.1. Transition Rate in a Periodically Modulated Potential

Periodic modulation causes a periodic change in the tilt A0 in equation (5.2).

A0 = Acos(ant), with modulation amplitude A and frequency f The barrier height
changes as AU(t)= AU0 i §U(t), (+ for the left well, - for the right well) where AUo is

the static barrier and 6U(t) is a periodic function. The probability to escape is increased
when the barrier is lowest, synchronizing the transitions with the driving field; this leads
to synchronous pulses in escape probability.

Figure 5.18a shows x(t) (solid line) for a particle in the bistable potential. The
particle is most likely to hop out of the left well when the left barrier height (dashed line)
is low. The return transition tends to occur when the right barrier is low (and AUlen is
high), an odd half-integer number of modulation periods later. Figure 5.18b shows the
probability of escape vs. modulation period. The escapes occur during 1/2 of the period.

177

Thus, if the particle does not escape during a half-cycle, it tends to wait an integer

number of full periods before escaping.

 

 

l I

%

 

 

 

 

 

 

   

Transition Probability

 

 

 

 

 

1.1 V 1 ' I ' I V I ' " Ti I “V I '
1.0 1.2 1.4 1.6 1.8 2.0 0 1 2 3 4 5

Time (seconds) Time (periods)

 

Figure 5.18 Interwell transitions under periodic modulation. (a) the particle makes
transitions from the left well (x = 1.25 um), over the barrier (x = 1.44 pm), and
into the right well (x = 1.63 pm). The dashed line represents the left barrier
height due to periodic modulation. (b) the transition probability vs. time. Peaks
indicate that the particle generally makes transitions during the portion of the
period when the barrier is lower. (Data labeled 012404-01-93, 10 Hz, 4.9 kBT
modulation of 5.2 kBT barrier, 1366 transitions.)

The period averaged transition rate W— is calculated from the dwell time
distribution p( r), the probability of the particle leaving the well a time t after entering.
Figure 5.19 shows p(z') with f = 1 Hz for two values of A. In the adiabatic regime and
for small A the distribution is p(r)= PO exp[— 212' + Acos(27y’r)] (solid lines), having a

W
decaying exponential envelope with A = W = pr where p = ]p(r)dz' is the escape
0

probability per period.

178

 

 

 

 

 

 

 

 

 

0.40
0.20- 0.35.‘
0.30: .
0.15 0.25;
gem g0”
‘ 0.15-
0.05« 0-10: U
. 0.05: \
o.oo-' . 4. - 0.00‘ . - . - . -
o 1 2 3 4 5 o 1 2 3 4 5
1 (seconds) I (seconds)

Figure 5.19 Transition rate measurement from fit to decaying peaks. The dwell time
distribution (squares) is plotted vs. dwell time for 1 Hz modulation with minimum
barrier (a) 3.2 kBT and (b) 1.5 kBT. The superimposed least squares fit (solid

lines) gives transition rate 147/ of (a) 1.36 Hz and (b) 3.3 Hz. The lack of higher

harmonics in (b) produces a worse fit, adversely affecting the measured decay
rate. (Data labeled 012904-01-83 and 012904-01-87, AUO = 6.4 kBT, 314 and 476
total transitions, respectively.)

There are several methods to extract W from data of the type shown in Figure

5.19. Ideally, each method should produce the same transition rate (using 17 = 213 p f ).
0 Calculate W f = A fi'om a least squares fit using p(Z') = C exp[— zit + A cos(27;fr)].

o Integrate the probability under the first peak, equal to the period averaged

transition probability. E, = yNi where N] is the number of transitions with dwell

time r<1/f.

 

. . 1 N
0 Calculate the mean transrtron rate W = — =

<7) Zr.-
Since the probability to escape in the first cycle approaches 1 as A —) Ab , the

uncertainty is dominated by the number of transitions that do n_ot occur in the first period.

The rate measurement needs to be sensitive to small changes in the second peak.

179

The least squares fit is preferred when there are multiple peaks. As higher order
peaks disappear the fit overestimates the decay rate 2., fitting poorly when the second
peak has fewer than 20 counts (Figure 5.1%).

The first peak area is the number of transitions with 2' < I/f, normalized by the

total number of transitions. The probability 5p approaches unity as A —) Ab. The

uncertainty is the uncertainty in both N1 and (N - NI), i.e., it depends on the number of

 

dwell times in the second peak as well. 513 = J; + 1

 

for 67Vl =,/N,. dfip is

large for high frequencies or small modulation (where NI is small) and large modulation
(where (N - N1) is small).

The mean transition rate W = 1/(1) , at complete total synchronization W = 2 f ,

and the first probability peak is centered at r = W'. A small second peak causes W to
increase with the number of points under the peak. This measure is also valid for small
modulation and in the static case. The transition rate vs. modulation amplitude

measurements use either pp or Was noted.

5.4.2. Locating the Adiabatic and Nonadiabatic Regimes

The regions of adiabatic and nonadiabatic dynamics have been identified through
measurements of the transition rate in a modulated bistable system. Figure 5.20 shows
the regions in the frequency-amplitude plane where adiabatic and nonadiabatic
characteristics have been observed. For modulation frequencies below 0.5 Hz the
transition probability is independent of frequency, implying that the particle follows the

potential, i.e., is adiabatic. For frequencies above 0.5 Hz the system enters a crossover

180

region. In the range 3.3 Hz to 20 Hz, a power-law behavior is found with critical
exponent p z 2, consistent with theoretical predictions for the weakly nonadiabatic
regime. The critical amplitude is linearly dependent on frequency, extending above the
adiabatic bifurcation amplitude Ab. For frequencies near the relaxation rate, the particle
falls strongly out of equilibrium with the driving field and the mean transition rate

decreases.

 

   

Weakly . i .
Nonadiabatic Nonadiabatic

< Reoime

Adiabatic

0.1 1 10 100
f(Hz)

Figure 5.20 Phase diagram of experimentally observed modulation regimes. The arrows
represent frequencies for which 1n(W) vs. A was measured. Frequency group
A = red, frequency group B = blue. Below 0.5 Hz the modulation is adiabatic
(blue). Between 0.5 Hz and 3.3 Hz is a crossover regime. From 3.3 to 20 Hz the
modulation is weakly nonadiabatic (cyan), with critical exponent ,u z 2. The
nonadiabatic critical amplitude extends beyond the adiabatic critical amplitude
calculated in Section 5.2.2. Above 80 Hz both adiabatic approximations fail
(green)-

181

Transition Rate vs. Frequency Measurements

Figure 5.21 shows the mean transition rate W vs. modulation frequency f for
several modulation amplitudes. Note that frequency is plotted on a log scale. By
inspection, there are three regimes with different frequency dependence of the transition
rate. At low frequencies the transition rate increases linearly with frequency and is
weakly dependent on amplitude. Above 2 — 3 Hz the rate saturates, i.e. reaches a
maximum of W = 2f at a characteristic frequency that increases with amplitude. Another
characteristic frequency scale appears near 30 Hz where W falls until it levels off near

W = W0, the static transition rate.

 

 

 

 

6.

12- -

O
.. r A 4‘

10- i 4 :E‘

. ‘ 332‘

8- ‘ ,
A , ‘ ' O2 4
N _
at, 6.. akaT)
3

 

 

 

 

0.1 1 10 100 1000
f(Hz)

Figure 5.21 Mean transition rate vs. frequency for several modulation amplitudes.
Transition rates are plotted for 2.4 kBT (squares), 3.3 kBT (circles), 4.2 kBT
(triangles), and 4.8 kBT (diamonds) modulation of the barrier height. Solid curves
are fits to equation (5.5) with parameters listed in Table 5.7. Inset: fit of
exponential equation (5.6) to comer frequency fc. (Data labeled 110303-03, static
barrier height = 4.4 kBT.)

I82

The particle is most likely to make a transition when the barrier is low. The
instantaneous barrier height AU (t): AUO +6U(t), where 5U(t)z Acos(Z/rft) and
A at A for small tilt amplitude A. (Le, the stable and unstable points do not move much
and U (xb)-U (xi): AUO + Acos(ZJrftXx-b -x,).) The transition probability depends
exponentially on the change in the barrier height A, and a maximum instantaneous
transition rate is defined by W,l = W0 exp(A/kBT). If the modulation is slow enough

(f< WA) the transition probability per period approaches unity. The transitions are totally
synchronized and the rate W = 2f.

At higher frequencies (f > WA) the transition probability per period decreases.
The barrier is not low for a long enough time for the particle to escape every period,
producing multiple peaks in the dwell time distribution (as shown in Figure 5.18b). The
transition probability is proportional to the time the barrier is low (l/f), and the total
number of transitions is proportional to the number of periods f. The transition rate is
independent of frequency.

At higher frequencies the particle cannot respond fast enough to the changes in

the potential. The effective minimum banier height becomes A17,“ = AUO — (517 (f ),

where 617 (f ) S A . This occurs when the modulation frequency approaches the intrawell

relaxation rate.

To quantitatively model W in the low and intermediate frequency regimes an
analogy is made to the driven, overdamped harmonic oscillator. The harmonic oscillator
has a characteristic time scale, the relaxation rate, which defines the rate at which the
particle returns to the stable point. For driving above this frequency the response

decreases. In this system the analogous time scale is W4, the maximum rate at which the

183

particle makes a transition. Using a functional form similar to the harmonic oscillator

 

amplitude, W or f/Jf2 + ,BZW?’ , with constant ,8. This has the correct asymptotic

behavior, W 00 f forf<< ,BW, and W independent offforf>> flW...

In the nonadiabatic regime the barrier height is averaged over some time rm,

”hive/2
AUG): —1- IAU(r)dr z AU0 + [7111: sin(7gftm,c):| cos(Zszt),

. aV'C

 

ave "’ave'z
with the integration interval limited to one period) The mean transition rate is found by

averaging the rates for a single period

I, f - ’
W : Wof chp[ A Sln(flfiave)cos(274fi)}dt 2 W01 (,[ A sm(7zftm )) ,
0 kBT Wave kBT Wave

 

 

where the zero order modified Bessel function of the first kind 10 is defined

I0 (x) = fig exp[x cos(t)]dt .

0

Combining the frequency regimes, the transition rate vs. frequency is fit with

instrumental weighting (1/6W2, where W = uncertainty in W) using

' t t 31
(5.5) W = B—[——IO[L srn(y)]+y0, WhCI'C 7 : {flaw fave

,/f2+f(,2 kBT 7 7r flave>1

Table 5.7 Results of fits of equation (5.7) to st. f data.

 

 

 

 

 

 

 

 

 

 

A (kBT/pm) 5mm) A (an B (Hz) f. (Hz) t,,. (3) yo (Hz)
12.5 2.4 1.8 i 0.2 1.0 3: 0.2 1.0 :t 0.2 9.8 :t 1.3 0.4 :t 0.2
18.7 3.4 2.44 i 0.09 1.3 :t 0.1 1.9 :t 0.2 9.6 i 0.5 0.08 :1: 0.08
25.0 4.2 2.90 i: 0.07 1.8 :t 0.1 3.8 i 0.2 9.8 i 0.3 0.06 i 0.06
31.2 4.8 3.22 :l: 0.06 2.3 a 0.1 5.8 :1: 0.3 8.7 i 0.2 0.00 i 0.05

 

 

 

 

 

184

for the W vs. f data in Figure 5.20. The fit parameters are A, B,fl., rm, and yo. Table 5.7
displays the results. The averaging times tm. are close to the x relaxation time obtained

for the static potential (9 ms).

The fit parameter/Z = ,BW4. )2. was measured at four amplitudes and plotted vs. the
change in the barrier height (3U, extrapolated from the 3 point quartic fit (Section 5.3).
The values off} in Table 5.7 are fit with
(5.6) f, = O'exp(— 2 5U/k,,T)+ yo
(inset Figure 5.20). A = 0.77 :t 0.18, indicating that the assumption fc 0c exp(dU/kBT) is

not COITCCI.

 

 

 

 

 

 

I
I
/
/
/
An A
Total Synchronization /
< /
/
I
’ I
l— -' "
High Frequency
Weak Response
0 I I ffYVVUI I I V I'TVVI I I I IIIIII'
0.1 1 10 100
f(Hz)

Figure 5.22 Phase diagram derived from W vs. f measurements. Above the dashed line
the transitions are totally synchronized with the modulation; transition rates are
constant in A. The boundary line is given by the comer frequency fc. In this
region no information about amplitude dependence near the bifurcation point is
available. Above 50 Hz the potential becomes averaged, and the transition rate
falls until W= W0. The remaining area is explored using st. A measurements.

185

Based on these experimental results only the (f, A) plane is split into three regions
(Figure 5.22). The dashed line is the total synchronization boundary, representing the
path of f, in (f, A) space. Above this boundary the transition rate is W = 2f. All
transitions are synchronized, and the second peak in the dwell time distribution
disappears (Figure 5.19). This limits the frequency range over which there is a
measurable signal near A = A b. When the relaxation time becomes a large fraction of the

modulation period (tr z TF) the particle sees a time averaged potential and the

synchronization disappears. The modulation regime in the region between boundaries is

explored by measuring the transition rate dependence on A.

Identification of the Adiabatic Regime in Frequency Group A

 

00. ,121'ri

~0.5-1 l/T
’

l l
-II-0.2 Hz
/1. -.‘- 0.25 HZ
> -‘— 0.33 Hz
/l

 

mop)

40‘ -v- 0.4 Hz
-O— 0.5 Hz
J -—<- 0.67 Hz

—>- 1.0 Hz

 

 

 

 

 

r ' I I I ' I I I ' I -
0 5 10 15 20 25 30
A (kBT/um)

Figure 5.23 Transition probability vs. modulation amplitude for low frequency
modulation. The overlapping data (up to f = 0.5 Hz) exhibit frequency
independence, a characteristic of adiabaticity. (Data labeled 012904-01, lefi-to-
right transitions, Ab = 40 :t 4 kgT/um.)

186

In the adiabatic regime the particle follows the instantaneous potential. A
consequence is that the transition probability per period is independent of frequency,

changing only with modulation amplitude. MO?” is plotted as a function of A in Figure

5.23 for modulation frequencies from 0.2 Hz to 1 Hz. Within statistical uncertainties the
transition probability is independent of frequency for f S 0.5 Hz. Above 0.5 Hz the
transition probability decreases suggesting the presence of a crossover occurs near
f = 0.5 Hz.

To examine the power law behavior in the low frequency regime, 1n(W) vs. A is
plotted and fit in Figure 5.24 for the 0.5 Hz data set. Near the bifurcation point the

activation energy is predicted to have power-law dependence on the distance from a

critical amplitude AC [44], 1n(W): C Afd — Ar + yO , where ,u is a critical exponent and

 

C and yo are constants. For critical exponents greater than one, 1n(W) approaches yo as

A —> Afd with slope zero. To properly fit this to the data the function is modified,

C(AC —A)# +de ASAC

(5.7) 1n(W): { yo A > A

and fit with free parameters Ac, y, C, and yo. The fit is applied to amplitudes above
10 kBT/um only. The weighting is instrumental (inversely proportional to the uncertainty
squared).

The fit parameters are listed in the caption of Figure 5.24. The critical exponent is
,u = 1.5 i 1.9, consistent with the predicted ,u = 3/2 but with uncertainty too large to draw
a conclusion. However, as noted blow, there is good reason to believe the asymptotic

power law is not valid when W~f

187

 

0.0-1

 

|n(W)

-0.5-

 

 

 

r0 5-1.01,5.2,0-2.5
A(kBT/um)

Figure 5.24 1n(W) vs. A in the adiabatic regime. Modulation frequency: 0.5 Hz. The

solid line is a fit to equation (5.7), with parameters Ac = 19 :t 7 kBT/um, C = -13
i 65, ,u = 1.5 i 1.9, and yo = 0008 i 0.036. The error bars are calculated from
statistical uncertainty in the number of transitions. (Data labeled 012904-01, lefi-
to-right transitions, calculated A b = 40 d: 4 kBT/um.)

The error bars are calculated from the statistical uncertainty in the number of

transitions N. W = x/N/T and 61n(W)=o'W/W =1/JN.Forf= 0.5 Hz measured

over 500 seconds the number of transitions under total synchronization is 250 so

x/N = 16. This uncertainty propagates through the least squares fit.

The bifurcation amplitude calculated from the quartic tilt approximation is
Ab = 40¢ 4k3T/um, twice Ac. Measurements closer to the bifurcation point are not
experimentally possible; since small changes in transition rates near complete
synchronization are masked by statistical fluctuations. To increase the precision the
synchronization boundary (Figure 5.22) would need to be shifted by changing the corner

frequency. Since fc oc WO , increasing the static barrier height would move the boundary

188

to the left. Forfi < 0.5 Hz at A = Ab, the static transition rate would need to be decreased

by a factor of 10 (from 0.2 Hz to 0.02 Hz).

Identification of the Nonadiabatic Regime in Frequency Group B

Figure 5.25 shows 1n(W) vs. A for 6 data sets in frequency group B, ranging from
3.3 Hz to 20 Hz. The solid lines represent least squares fits with the power law in
equation (5.9). As A —9 AC the transitions are totally synchronized so that W= 2f, i.e., yo
can be treated as a known parameter. This reduces the number of fitting parameters

The total number of transitions N is the same for both wells. The sum of the total
dwell times T1 + T 2 = T, the total acquisition time. When the transitions are totally
synchronized,

T1T,+T,11

 

The mean transition rates for the left (WL) and right (WR) wells are not independent. In a
symmetric potential WR = WL = 2f, but in an asymmetric potential WL may be greater
than 2f. The difference in mean transition rates WL - WR nonetheless decreases as A
increases. Therefore yo is set equal to 1n(2 f)

The values of ,u in Table 5.8 yield a typical value of 2.3:t 0.3 over this frequency

range. This agrees with the predicted exponent 2 in the nonadiabatic regime. At 20 Hz,

the maximum modulation amplitude is smaller than the critical value AC.

189

 

 

3.3 Hz

 

 

 

 

 

 

 

 

 

 

2‘
g 1
s
._
0- "
0'1'0'2'0'3'0'40'5'0'60
A(kBT/um)
6.67 Hz
2.
E
0-1 I
C
0'1'0r2'0'3'0f40'50'60
A(kBT/um)
10 Hz
1
24
E
I
O.
I
0 '1'0'20'30'40'5'0'60

A (kBT/pm)

ln(W)

1n(W)

 

2-1

01

 

5H2

 

 

T'E'2'0'80'4'0'5'0'60

A (kBT/um)

 

2.

 

8H2

 

 

AII'I'T'I'I'
0102030405060

A (kBT/um)

 

 

20 Hz

 

 

0'1'0'2'0'3'0f4'0'5'0'60

A (kBT/um)

Figure 5.25 1n(W) vs. A in the nonadiabatic regime. Solid lines are fits to equation
(5.9), with parameters listed in Table 5.8. (Data labeled 012404-01, lefi-to-right

transitions, Ab = 36 i 4 kBT/um.)

190

Table 5.8 Power law fit results for the nonadiabatic regime.

 

 

 

 

 

 

 

 

 

 

 

 

f(HZ) AC (kBT/itm) C [u yo
3.3 34 i 6 -0.7 i 2.3 2.4 :t 0.9 1.93

5 48 i 4 -0.6 :t 1.0 2.3 i 0.4 2.31

6.7 48 i 3 -0.6 i 0.8 2.3 d: 0.3 2.60

8 52 d: 3 -0.4 i 0.5 2.3 :t 0.3 2.76

10 54 :t 3 -0.5 i 0.5 2.2 :t 0.2 3.00

20 8] d: 5 ~01 :t 0.1 2.4 i 0.3 3.70

 

 

The critical amplitude A derived from these fits increases with frequency. To
examine only the frequency dependence the data in Figure 5.25 were fit again (Table
5.9). AC is plotted to the same firnction in Figure 5.26 and fit by A. = c+ (if. The slope d
= 2.13 i 0.04 kBT/um Hz and c = 30.1:t 0.5 kBT/um.

Table 5.9 Power law fit results for the nonadiabatic regime, with ,u = 2 and yo = 1n(2 f )

 

 

 

 

 

 

 

 

 

 

 

 

f(Hz) Ac(kaT/11m) C .11 Yo
3.3 31 i l -2.8 i 2.3 2 1.93

5 45i0.8 -l.8i 1.0 2 2.31

6.7 45 i 0.7 -2.0 i 0.8 2 2.60

8 49 i 0.7 -l .66 :t 0.08 2 2.76

10 51i0.6-1.47i0.06 2 3.00

20 73 i 0.6 -0.80 i 0.01 2 3.70

 

 

The change in the critical amplitude is predicted to be linear in frequency [45],

with AcSI = Afd + 2727f/fl (see Section 2.3.3). For a quartic potential in the x direction

(see equation (5.2)), the coefficients yand ,6 are

 

 

fl_;and ,_ Af“ _2831’2/34/3C__1_§
mT 7 6Cx5 6C B/3C 9C°

Using the values in the first row of Table 5.5, the slope of the fit parameter A vs. f is

 

490 kBT/umz
9x10700kBT/um4

 

7 -1
27r2.5x10 S \/ =0,63kBTs/um.

2
flzbml‘,’i = —1
16 9C 1.77x107ums'2(k3T)

Therefore A? = A 3" + 063/" for A in units of kBT /um.

191

 

 

 

 

80 Ii
E 60 1
£3
Fm
£ -1
<° I
40-
, I
20 t I I I ' I ' l
0 5 10 15 20

f (Hz)

Figure 5.26 Frequency dependence of the critical amplitude. Ac from least squares fits in
the nonadiabatic regimes is linearly proportional to the modulation frequency.
Slope d = 2.11 at 0.05 kBT/um Hz and offset c = 30.6 i 0.6 kBT/um. (Data labeled
012403-04, lefi-to-right transitions, Ab = 36 i 4 kBT / 11m.)

The experimentally determined slope d in Figure 5.25 is four times larger than the
value of 2717/,6 , calculated from the parameter representation of the potential However,

the theoretical model is an expansion of the potential about the bifurcation point, trat than
about the equilibrium (A = 0) potential. This calculation assumes that the potential does

not deviate from the quartic approximation at large A. The ratio B/C from the large

amplitude tilts in Section 5.2.2 (Figure 5.13) is constant to 0.8 Afd .

Details of the Experimental Procedure

The mean transition rate was calculated for consecutive data sets with constant

modulation frequency and increasing amplitude using the batch data acquisition function

192

of WinMi12 (Section 4.2.3). The traps were positioned approximately 0.45 11m apart for
a barrier height of greater than 6 kBT and mean transition rate from 0.1 to 0.2 Hz for
frequency group A and AU = 5 kg?" and W = 0.5 Hz for frequency group B. A sample
batch progression is shown in Table 4.3. A data acquisition begins with a series of static
tilts from which the potential parameters and estimated adiabatic critical amplitude are
calculated. The next 9 data sets send a constant frequency (1.008243 Hz) and increasing
modulation amplitude (in units of Vpp) to the SRS DS345 function generators controlling
the LS-lOO beam power stabilizer. The frequency is incremented for the next run and the
amplitude progression repeated.

Even with the optical stability improvements in Section 4.3 there is a drifi due to
temperature variations, with time scales on the order of hours. To minimize this effect on
the individual amplitude progressions the data sets are as short as possible. With an
unmodulated transition rate of 0.2 Hz there are 50 i 7 transitions in each direction during
a 100,000 frame (500 second) data set (assuming Poisson statistics). With increasing
modulation amplitude the transitions become synchronized and the maximum mean
transition rate is twice the modulation frequency. The relative uncertainty decreases and
thus the data set length is decreased, as shown in Table 4.3. If the system drifts the static
tilt measurements in Section 5.2.2 are no longer valid. After every three or four
amplitude progressions another static tilt measurement is made.

To determine if the system drifted during an amplitude progression the static
potentials measured before and afier are compared. The barrier heights and stable point
positions are calculated from the x profile. Assuming the barrier varies linearly with the

beam positions (for a small range of displacements), a 5 nm change creates a 0.07 kBT

193

change in a 6 kBT barrier (1%). Since the transition rate depends exponentially on the
barrier height the rate changes by 7 %. A “stable” amplitude progression has < 5 nm
beam position change, < 0.1 kBT barrier height change, and transition rate changes within

the statistical uncertainty for the static potentials before and after the progression.

5. 5. Summary

The particle dynamics and the energy surfaces in the bistable potential have been
calculated from the position 7 (t), recorded at 5 ms intervals for 500 seconds (typically).

The overdamped particle is in thermal equilibrium, and the Boltzmann relation is used to
calculate the potential energy in the volume the particle explores. From this three
dimensional map the time scales of the system are estimated. At the stable points there
are three orthogonal relaxation rates: the slow 2 rate (3 Hz), the intermediate x rate
(150 Hz) and the fast y rate (500 Hz). The transition rate depends on the barrier height
and is usually 0.] Hz to 2 Hz.

The transitions rates are measured from x(t). The mean transition rate was
calculated from the mean dwell time in each well. The optimal path is calculated by
computing the condition probability of the particle occupying the well escaping at time I,-
later. The results indicate a process that involves two distinct steps: a preparative step in
which the particle moves in the +2 direction, then an over the barrier along x in a time t,,...

The adiabatic and nonadiabatic regimes of the periodically modulated bistable
potential were identified through measurements of transition rate vs. modulation
amplitude and frequency. The existence of the adiabatic region was inferred from the
frequency independence of the transition probability per period. Above f = 0.5 Hz the
probability decreased with increasing amplitude, indicating that the particle was not

194

following the potential. A comparison to the predicted power law dependence on
amplitude could not be performed (with reasonable uncertainties) due to limits on the

system’s long term stability. The nonadiabatic region 3.3 Hz st 20 Hz was modulated
beyond A = Afd and the function 1n(W): C IA, — A|fl + yo fit with known yo. The critical
exponents agreed with the predicted ,u = 2, and the critical amplitude Ail was linearly

dependent on frequency. The next chapter compares these results with the theoretical

predictions [44-46].

195

Chapter 6 Discussion

6. 1. Theoretical Predictions of Adiabatic Criteria

In the previous section, adiabatic and nonadiabatic regimes were tentatively
identified by examination of the 1n(W) vs. A data with some guidance from recent theory
[45]. Since the relevant relaxation times are available from knowledge of the modulated
potential, we are now able to check for internal consistency in our assignment of

adiabatic and nonadiabatic regimes.

 

ft = 1' IACS'

/
/ weakly
, Nonadiabatic

  
  
 

 

Nonadiabatic
\ dt /dt=1
\l’,X

Adiabatic

dgxnn==01

  

 

 

 

 

0 . . ....n, 4 . ....n, . . ....n.
01 1 10 100
f(Hz)

Figure 6.1 Boundaries of adiabatic and nonadiabatic regimes in the (f, A) plane. Black
lines are the adiabatic critical amplitude Afd (solid) and AS' (dashed). Red lines

01,, /6t| = 0.1 (solid) and lat” /6t| :1
(dashed). The blue line is the strongly nonadiabatic crossover, ft”. =1, and the

 

are the weakly nonadiabatic crossover,

cyan line is ftm =1, the crossover due to t,,z. The modulation frequency is

plotted on a logo scale.

196

In a periodically modulated one-dimensional potential recent theoretical work has
identified an adiabatic regime and two nonadiabatic regimes [45]. The regions are
characterized by relationships between the modulation frequency f and the intrawell

relaxation time. At low modulation frequency and amplitude the particle follows the

field; the motion is adiabatic. As frequency increases the condition létm /6tl << 1 first is

violated, leading to a weakly nonadiabatic regime. During part of the modulation period

the relaxation time t,,,- changes too quickly and the particle lags behind the modulation.

The solid and dashed red lines in Figure 6.1 represent lat” /6t| = 0.1 and [arm/821:1,
respectively. At yet higher frequency ft”. =1 and the particle falls out of equilibrium

through the entire modulation period. Above the blue ft“. =1 line in Figure 6.1 the

system is highly nonadiabatic.
The distortion of the stable and unstable states causes the bifurcation amplitude
(the amplitude at which the states merge) to increase with frequency. For relatively slow

driving, theory predicts Af' = A 3“ + 2727f/fl . This boundary is plotted in Figure 6.1 as

the dashed black line.
For multi-dimensional system additional adiabatic criteria may appear that depend

on the relaxation times in other directions. In the present system it is known that, since

t

IT"?

> tr",r , the condition ft” <<l must also be satisfied to obtain adiabaticity. The cyan
curve in Figure 6.1 represents ftm =1. (t,,: is assumed to be independent of the

modulation amplitude.) This condition adds two new regions to the (f, A) plane. The

region where ft” <<l is violated but lat” /0t| << 1 holds is not expected to be adiabatic

197

but also not weakly nonadiabatic. When both conditions are violated it is unknown if the

system is weakly nonadiabatic or enters some other frequency regime.

6. 2. Comparison to Experimentally Determined Frequency Regimes

The regions where the adiabatic approximations are violated are compared to the
results in Section 5.4 (Figure 6.2). Below f = 0.5 Hz the transition probabilities are

insensitive to frequency and thus adiabatic. From f = 3.3 Hz to 20 Hz the power-law

. A . .
Aj' — Al + y0 are consrstent wrth ,u = 2 and the

 

exponents obtained from fits to 1n(W) = C

bifurcation amplitude scales linearly with frequency. f = 1 Hz to 3.3 Hz is a crossover

region between adiabatic and weakly nonadiabatic.

 

 

 

 

 

 

I
/
/
A
’ I
A 3". "' " :_ _
Total Synchronization ‘ \
\
< \
\
Adiabatic Nonadiabatic
,_ in . inn.
0.1 1 10 100
f (Hz)

Figure 6.2 Comparison of experimental data with theoretical frequency regimes. The red
and blue arrows represent the frequencies of the 1n(W) vs. A measurements in
frequency group A and frequency group B, respectively. The experimentally
identified adiabatic region and nonadiabatic (power-law exponent ,u = 2) regions
are marked. The green line is the total synchronization boundary from Figure
5.22. (The other curves are labeled in Figure 6.1.)

198

The boundaries are estimated from our measurements as follows. The x

relaxation time is rm. = Ziz‘IT/a)2 , where mm2 = U ”(209): B + stz using the quartic

approximation U (x): on + Bx2 / 2 + Cx4 / 4 and x, is the position of the stable point. T

is the Stokes damping rate 2.5x107 s' and the conversion from kBT/um to s'2 is

a)2 = 1.77 x 107 U". The position ofthe minimum x, is found by solving U'(xs) 0 ,

' ’3 .. 73
x (A )_ 3A,, l+(,/A02.A,;+l-A0.A,,)
5 0 — . ‘3 9
23 (,lAgAf +1 — AO Ab)

3‘2 Alf .

 

with bifurcation amplitude Ab = 218

 

Under modulation, A0 = Acos(27y’t) and x, and t, are time dependent. The
adiabatic conditions are applied to the period maximum t,,,- and |ai,,_,./az|. Solving

numerically, the boundary ftm. = 1 is shown in Figure 6.2 (blue line). At low frequencies
the boundary is close to A = Ab. At f = 10 Hz it begins falling, reaching A = 0.9Ab at f =

27 Hz. All modulation violates this condition atf= 90 Hz.

lat”. /6t| is calculated and the amplitude A found such that maxlatm. /6t| = 0.1

(solid red curve in Figure 6.2) and maxlatm. /6tl = 1 (dashed red curve). The solid curve

leaves the area around A = Ab at f = 0.3 Hz and reaches A = 0.9Ab at f = 3 Hz.

The adiabatic condition from the z relaxation time is calculated using t,,: = 0.26 s,

measured from the local quadratic fits. ftr‘: = l for f = 3.8 Hz.

The f _<_ 0.5 Hz data satisfy all of the adiabatic criteria, as expected by the

frequency independent transition probability per period. Above 0.5 Hz the transition

199

probability decreases, possibly due to violation of fins <<l. This occurs when

ft” = 0.1, lending some justification to the assumption that max’o‘tm /6tl = 0.1 (solid
red line in Figure 6.2) represents the onset of the weakly nonadiabatic regime.
Alternatively, if nonadiabatic z motions do not affect the transition probability the

breakdown would have to come from violation of [firm /éit| << 1 or ft” << 1. Since the

transitions become totally synchronized and the transition probability reaches unity for
A = A2“ / 2 (i.e., the amplitude dependent nonadiabatic crossover is far away), this means
that the assumptions are violated for frequencies two orders of magnitude below
maxlatm. /c9tl =1 and ftm. =1. If this were true, the fully nonadiabatic regime would
extend to very low frequencies and the observed weakly nonadiabatic behavior would not

be seen. Since this is not the case, adiabaticity in the z direction must be considered.

The measured nonadiabatic region from 3.3 Hz to 20 Hz falls below
maxlatm. /at| = 0.1 for the majority of the modulation amplitudes. This contradicts the
critical exponent ,u = 2 obtained from the power-law fits and the frequency dependent
bifurcation amplitudes.

The three dimensional potential energy surfaces (Figure 5.5) show that the stable

point to saddle point distance is 200 nm in 2. If the stable point and saddle point

approach as A —> A 2" then there must be motion in the z direction, and another adiabatic

condition |6tm /6t| <<1 appears. Assuming the potential is a quartic of the form

2 4 2
(120(82): Acos(27gfi)+3%+£{_+ 8.2

 

+szx2z

200

the path of the stable point in x and 2 can be estimated. The coefficients B and C are
calculated in Section 5.2. The B: and C (.3 coefficients are estimated by the curvature in z

and the (x, 2) positions of the stable and saddle point when A = 0, respectively. The
stable and saddle point positions under adiabatic modulation near Afd are shown in

Figure 6.3. (The model does not exactly represent the potential in z, underestimating the
saddle point to stable point distance.) The red and blue curves show that the stable points

travel large distances in 2 as they move towards the saddle point.

 

-0.02
-0.04
-0.06

—0.08

-0.12

 

 

 

 

 

-10 —5 0 5 10

Figure 6.3 Positions of stable and saddle points in a modulated bistable potential. The 2
and x positions of the stable points (red and blue) and the saddle point (green)
approach and recede once per period. In the static potential the stable points are
(x, z) = ($0.2 pm, 0.05 pm) and saddle point (x, z) = (0, 0). The stable and saddle
points meet at (:l:0.120 urn, 0.017 um) and the other stable point is at (0.240 um,
0.066 um).

Since it does not appear in the model, the z relaxation time changes are estimated

from experimental parameters. The experimental potential is tilted by changing the laser

201

power, changing the gradient force in all three dimensions. The change in relaxation time

Arm is inversely proportional to the change in power AP, (Army or sz oc AF oc AP.

The power is modulated P = PO(1 +§Pcos(2nft)), with (P z 0.15 when A = Afd. The

maximum 6th. /6t over one period is

 

 

 

 

 

 

   
      
 

maxiétm /at| = 5sz (2nf)max|sin(2zgn) = 5P!” (24f). ..._

The amplitude at which [aim /6t = 0.1 is
_A_ _ 0.1 F.'
4““ 240.1512... ' E

/
/
/
A
l
A a“. - " ’ :
Total Synchronization
< \
\

 

Adiabatic Nonadiabatic

  

 

 

0.1 1 10 100

f (Hz)

Figure 6.4 Comparison of experimental data with adiabatic criteria on the x and z
relaxation rates. The magenta line represents lath, /6tl = 0.1 . The data with ,u = 2

are fully contained in the z weakly nonadiabatic regime while the adiabatic data
are outside.

202

The Ian‘s/84 =0.l condition (magenta line in Figure 6.4) falls between the

adiabatic and nonadiabatic data. |az,.., /8t| = 0.1 predicts that modulation below f = 0.35

Hz is adiabatic very close to A = A), and f = 3.3 Hz is weakly nonadiabatic above A =
0.1Ab, agreeing with observations.
As promising as this explanation appears, it conflicts with approximations made

in the theory [45]. The transitions are assumed to take place only when t = 0, A is closest

to Ab, t,,, is a maximum, and maxlatrv‘. /6tl occurs. Equivalent assumptions are W << f or

AUO >> kBT; transitions are rare until the barrier height is substantially reduced during

modulation. In the z direction, maxlatm /6t| occurs when t = 1/ 2 f (when P = P0 ). At

t=0,

 

at” /6t| = 0 and the condition lath: /6tl <<1 is always true.

The assumption AUO >> kBT is not satisfied in the experimental system. The
static barrier height AUO = 6 kBT. The transition probability histogram (Figure 5.18b)

shows that the transitions are distributed over approximately half of the period (including

the time lath: /6t‘ is near the maximum) with a peak at t = 0. The mean transition rate

W = 0.2 Hz, with modulation frequencyf= 3.3 Hz. The total synchronization effect is a
direct consequence of the low barrier. Even small modulation amplitudes (A < A?“ / 2 in

Figure 6.4) can increase the instantaneous escape rate WA enough so the particle makes a
transition every period. Under the AUO >> kBT assumption this only occurs very close to
the bifurcation amplitude.

It is possible that the power-law frequency scaling does not even apply under the

experimental conditions. More likely, the power-law relation 1n(W )oc AcSI — A’/1 has a

 

203

. 3:2
combination of the adiabatic 1n(W)oc Afd—Al and the weakly nonadiabatic

 

1n(W) cc Aj' — A h , depending on when the particle escapes. If the transition occurs when

 

 

létm. /6t| = 1 (close to t = 0) then ,u = 2. If lat”. /6tl << 1, then ,u= 3/2. The difference

between these effects is too small to see with the current uncertainties in W

Finally, there is the ft”. << 1 condition that is violated forfz 3.8 Hz, and should

obscure any weakly nonadiabatic effects due to the z relaxation time. If the particle can
not respond to changes in the 2 potential, it will not respond to changes in the rate of
change of the potential. In other words, the particle cannot lag behind the modulation
(weakly nonadiabatic) if it can not follow the modulation at all (nonadiabatic).

However, there is definitely some weakly nonadiabatic behavior observed at
modulation frequencies between 3.3 Hz and 20 Hz. Estimations of the x weakly
nonadiabatic boundary show that, for the lower end of the frequency range, the
modulation amplitude is too low to leave the adiabatic (in x) regime (below the red
curves in Figure 6.4). Therefore there must be some weakly nonadiabatic effect in 2.

Why it is not concealed by the nonadiabatic behavior is unknown.

6.3. Summary and Conclusions

The following phenomena have been observed and measured in the periodically

modulated bistable potential.

204

Adiabatic regime

o The adiabatic regime exists for frequencies which satisfy lat”. /8ti << I , ftm <<1,

165.; /8tl <<l, and ft” <<l. Slow modulation with respect to x alone is not

sufficient for complete adiabaticity.

The transitions may become totally synchronized for A well below A,,, obscuring
asymptotic behavior for experiments with poor sensitivity.

The 1n(W) vs. A power law fit finds p = 1.5 :t 1.9, consistent with the predicted
,u = 3/2 [45].

The modulation frequencies required are so slow that the uncertainty in the mean
transition rate is very large. To increase the number of transitions recorded the

measurements at a single amplitude and frequency requires system stability

improvement by l to 2 orders of magnitude.

Nonadiabatic regime

The region of total synchronization moves above A b.

The critical exponent is ,u = 2.3 i 0.3, agreeing with the predicted ,u = 2 for the
weakly nonadiabatic regime [45]. This represents the first measurement of a new
critical exponent for nonadiabatic critical behavior.

The critical amplitude Ac is linearly dependent on the modulation frequency. This
confirms a new prediction of the theory for nonadiabatic modulation. The slope is
larger than predicted in [45] due to uncertainties in the quartic approximation

parameters near A = A b.

205

o The nonadiabatic regime appears to result from violation of 16th. /8t| <<1, not

lat”. /6t

 

<< 1. Thus we find that, in the three dimensional systems, t,,: > t,,,- and the

relaxation in the z direction can not be neglected.

0 Theory and simulations cannot yet go beyond one dimension: thus experiments

should stimulate more sophisticated theory and large-scale simulation.

6. 4. Suggestions for Further Work

It would be advantageous to decrease the statistical uncertainty in the

measurement of the critical exponents near the bifurcation point. There is a lower limit
on the mean transition rate uncertainty, proportional to 1/ W . The uncertainty is

reduced by increasing the data acquisition time, but at some point the optical system
drifts and systematic errors appear. The interval of stability can be extended by removing
temperature fluctuations from the lab, through a heating or cooling unit separate from the
building controls. This would reduce differential thermal expansion as a source of drift.
The beam positions could be controlled using an active beam steering system. The
gimbal mirrors in the system could be replaced with 1- or 2-axis, servo controlled
mirrors. A position sensitive detector in a plane confocal with the objective’s focal plane
would measure the change in beam position and feedback control electronics would
change the mirror angle, correcting for the motion. This would eliminate the small beam
motions observed in Section 4.3.4 and larger, long term pointing changes.

With long term stability measured over days instead of hours. W0 could be
reduced by an order of magnitude, moving the total saturation boundary in Figure 6.1 to

the left. The power law dependence in the adiabatic regime could then be measured for

206

longer times to reduce uncertainty. The amplitude dependent crossover from adiabatic to
nonadiabatic could be mapped, taking many small amplitude steps near A = Ab. The
frequency regimes based on the z relaxation time could be explored to determine the
effects of nonadiabaticity in a universally stable potential ( U " > 0) that is orthogonal yet
coupled to the bistable direction.

The two-beam optical trap is limited to a small range of potentials, based on the
beam powers, beam separation, convergence angle, and particle size. A spatial light
modulator (SLM) could be used to create a potential of arbitrary shape. The SLM
modifies the phase profile of an incident laser beam. Focused by the objective lens, the
Fourier transform of the image appears in the focal plane. The shape and size of the
desired intensity gradient would be controlled through the SLM phase profile. This also
has the advantage of constant trap separation, since the intensity distribution is set by the
phase profile and not individual optical elements. Interesting avenues of study would be
transitions in spatially extended systems and modification of the relaxation time in the y

and 2 directions.

207

Appendix A Quantitative Theory of Optical Trapping

The two laser-induced forces in optical trapping are the force of the radiation
scattering off of the dielectric sphere and the forces due to the intensity gradient. The
scattering force pushes the sphere in the direction of the beam propagation (axial
direction), and the gradient force acts towards the region of highest field gradient. In a
stable three dimensional trap the gradient force in the axial direction exceeds the
scattering force to prevent the particle from being pushed out of the focal point.

Theoretical calculations of these forces are simplified in two regimes. In the
Rayleigh regime, where the particle is much smaller than the wavelength of the laser, the
particle is approximated as a dipole induced by the trapping field. In the ray optics or
Mie regime, where the particle is much larger than the wavelength and the beam waist,
the principles of geometrical optics are used to calculate the deflection of the individual

rays.

A. I Rayleigh regime

A Rayleigh particle is much smaller than the wavelength of the trapping beam
(a << 1), and the time varying electric field is approximated as uniform over the volume
of the particle and, for a TEMOO beam, Gaussian over the beam waist.

A dielectric particle in an electric field will have an induced dipole moment of

p = aE, where a is the polarizability. In an anisotropic electric field, the force on the

particle is

(A.1) F =(p-V)E =a(E-V)E = éaVlElz —ExVxE =éaV1E12

208

 

 

The force on the dielectric particle is directed towards the region of highest electric field
gradient, independent of the direction of the field. This phenomenon is called
dielectrophoresis.

A dipole in an oscillating electric field scatters waves in all directions, changing
the magnitude and direction of the electric field. The momentum transferred to the
particle pushes it in the direction of the beam with a magnitude proportional to the square
of the dipole moment.

The polarizability of a spherical Rayleigh particle in a uniform electric field is
calculated from Maxwell’s equations [55]. In terms of the effective index of refi'action

m, the ratio of the indices of refraction of the particle (ns) and the surrounding medium

mz—l

m2+2

 

(nb) and the particle radius a, a = of J. The intensity of the focused Gaussian

beam at a point (x, y, 2) measured from the center of the beam waist is

(A2) [(x,y,z)=[ 2P 11+ 1 xp[ 2(x2 +y2)/w§]

aw: (22,112.,ng ' 1+ (2./1mg)

 

 

|(X.Y.Z)

\ WC

 

v
7&-

/ Z

 

Figure A.l Gaussian beam intensity profile near a beam waist.

209

 

where P is the total power, wo is the beam waist radius, and k is the wave number. The
beam propagates in the z direction (Figure A.1). The intensity is proportional to the

2

- n,,€0clE(x v z) .

square of the electric field [(1 y, z) — —2—
The scattering force [55] is proportional to the square of the volume and the

intensity.

Fscat (x9 ya 2 ) =

12856 2_ 2 22 2;",2
nanb[m 1][2P] 1 exp[— (x +y)140 2

’3 7 s"
214C m“ + 2 7mg 1 + 22‘ [(115 I + 22; kwg

 

2
12825216,: m2 —1 .
= 4 b 2 I(x,y,z)z
21 c m +2

From equations (A. l), the gradient force [86]

 

3 2 _
Fgmd(x9yrz)= 27m ”b m 1 V1(x2yrz)
c or2 +2

is cylindrically symmetric. The radial component Fwd] (r=,/x2 +y2) and axial

component F

gra

d.z are

 

 

3 2 _ ,2
Fgmdf (x, .v, Z) = — Zfla nb m2 1 4r/ 1440 [(x, ‘V’ 2);
c m + 2 l + (22,..r”kw(‘;a )2

 

 

F (x y z): _Zfla3m’ [m2 —1] 8r/1/(kwgf 1— 2r2 1(3‘ y zlé
grad..~ 9 9 C m2+2 1+(ZZ/kwg)2 1+(22/kw3)2 ’ ,

The maxrmum magrutude of Fmd‘, occurs at r = w0 /2, z = 0 and for Fgmdj,

r = 0 , z = ikwg / 2 J3 . For the trap to be stable in three dimensions, the gradient force

must be larger than the scattering force, or

210

(A.3) R =

 

 

Fgmd.: _ fills ["12 —1]l >1

F _1287r5a3w5 m2 + 2

SCI"
where F grad; is computed at its maximum.

For a silica sphere in water, 171 = 1.46/1 .33 = 1.1. A helium-neon laser focused to

a diffraction limited spot will have a beam waist diameter 2w0 2: xi = 633nm. The

condition (A3) is satisfied for sphere diameter 2a S 0.5 pm. (Particles this size are no
longer Rayleigh particles.)
For a total power of 100 mW, the maximum forces on a 0.1 urn diameter sphere

are

FM, (x, y, z) = 6.4 x 10''3 N

F d‘z(x,y,z)= 2.2x10'l3 N

gm

Fm,(x,y,z) = 2.1x 10"“ N

A.2 Ray optics regime

For particle diameters much larger than the wavelength (a >> 21), the wavelength
is assumed to be zero, the beam is decomposed into rays and treated with geometrical
optics. Rays reflect and refract at surfaces and diffraction is ignored. An off axis ray
enters the particle and is refracted towards the particle center. The ray is refracted again
as it exits. The photon momentum change at each point creates a force on the particle

proportional to the intensity.

In Figure A.2, a single ray with momentum an/c is incident on the sphere at an

angle 6. The ray is reflected or transmitted with probabilities R and T, the Fresnel

211

reflection and transmission coefficients. The transmitted beam travels in a straight line

until it reaches the other side of the sphere, where it will again be reflected or transmitted.

Incident on the nth interface the ray will have decreasing power PT R "'2 .

P 0

 

V

YlT

Figure A.2 A single ray incident on the sphere at angle 6 is partially transmitted with an

angle or refraction a. The beam is either transmitted or reflected at the next
surface.

The force in the direction of the initial beam, the scattering force, is

 

P
F =17 =3b—{1+Rcoszo—

z 5
C

T 2 [cos(26 — 2a)+ R cos 26]
1+R2+2Rcos2a ’

where a is the angle of refraction nb sin6 = ns sina. The force in the y direction is the

gradient force.

 

.V g C

P 2 —
F 2F an Rsin26—T [cos(262 2a)+RcosZ6] .
1+R +2Rcos2a

The total force is calculated by integrating over all incident rays in the tightly
focused trapping beam. For a particle displaced a distance S in the z direction (Figure
A.3), the ray converges to the focal point with angle ¢~ The incident angle 6 is calculated

from a sin ¢ = S sin 6. Unlike the Rayleigh regime, the force does not depend on the size

of the particle.

212

 

 

 

 

Figure A.3 Calculation of force from a single focused ray. The ray converges with angle
(15 to a point S away from the particle center. Momentum transfer causes a force in
the negative 2 direction.

The force is integrated numerically [82]. It is advantageous to write the force in

terms of the efficiency Q that depends on the particle displacement, F = anQ/c. The

maximum value of Q = 2, corresponding to total backwards reflection. With increasing
convergence angle, the efficiency rises until it reaches a maximum near 80°. The
outermost rays contribute most to the trapping force.

Due to cylindrical symmetry, the force in the transverse direction F g will be zero.
The axial efficiency increases with displacement from the focal point. It reaches a
maximum Q 2 0.4 when the particle is one sphere radius from the focal point. At this
point the outermost rays are closest to the angle of maximum efficiency.

For a displacement S ' in the transverse direction, the force is again computed

from the integral over individual rays. The incident angle 6 is asin¢ = S 'sin6. The

213

 

integration is more involved because the cylindrical symmetry is broken. The trapping
efficiency increases with particle position until it reaches a maximum of 0.3 at
displacements of one sphere radius.

The magnitude of these forces is estimated from F = anQ/c. For a particle in
water (21,, = 1.33) trapped by 1 mW total power, the force at the maximum 2 displacement
(Q = 0.4) is 1.8 x10": N z 1.8 pN . The force depends on the convergence angle and the

particle position, but not the particle size.

A.3 Intermediate Regime

In the intermediate size regime, the beam waist diameter and wavelength of the
trapping beam are on the order of the size of the particle. Neither of the prior
approximations is valid. Near the beam waist the Gaussian intensity profile in equation
(A.2) no longer satisfies Maxwell’s equations and the forces calculated from the
electromagnetic field interactions are no longer accurate. The accuracy is improved
[87,92-95] by introducing a higher order correction to the Gaussian beam in terms of

1 ii. Barton and Alexander [87] provide a fifih-order correction to the

S = =
kwo 271' w0

 

Gaussian beam. The correction becomes less accurate as s approaches one. For s = 0.2

(wO = 0.821), the percent error is 1.2%. However, for the 0.6 pm diameter sphere used in

this experiment there is no analytical formulation of the potential.

214

Appendix B Dependence of Trapping Forces on Optical
System

The experiments in this thesis measure the dynamics of an overdamped particle in
a potential with a small range of possible curvatures. These parameters can be modified
by changing the optics, particle size, laser wavelength, and surrounding medium. To
reach the underdamped regime the damping rate F can be decreased by using a larger
particle or a lower viscosity. The Kramers crossover [12] from T/co >> 1 to F/a) << 1 can
be explored by changing the surrounding medium, thus increasing the trap curvature a)
and decreasing T. This appendix estimates the effect of changing certain optical
parameters on the trapping force, calculated in the Rayleigh regime.

The trapping force is calculated from the induced dipole moment, resulting in

equation (3.2) (Section 3.1.2),

 

 

3 3 2
na m —1 2
(13.1) Fm =- b VE,
gd 2 m2+2 l

where a is the particle diameter, nb is the index of refraction of the surrounding medium,
and m is the effective index (ns/nb). The damping rate is given by the Stokes damping

67r77a _ 977
_ 2 ,

 

 

rate T = for a sphere where r] is the viscosity and p is the density.

m

The electric field is related to the intensity of the focused beam. In cgs units

nbc

I(r,z) = |E(r,z]2. The TEMOO laser has intensity profile I (r) = 10 exp(— rz/r,2 ). For

8n

215

a focused Gaussian beam propagating in the z direction with a focal point at z = 0 the

width r, = r0,/1+.~2/z,3 (Section 4.1.6) and

P

“32’ <>..()i(71

for a beam with total power P. The beam waist radius r0 and Rayleigh range 2;; are

 

 

 

functions of wavelength ,1 and convergence angle 6, r0 = 221/ 7:6 . and ZR = 42/7t62 .

Substituting (8.2) into (B. l ),

2 3 2
Fmdz47mba [m 11W

 

 

 

 

g c m2 +2
4n§a3P mz—l 1 —r2 .
= 2 V 2 21216)“) 2‘ 2/2‘
c m +2 r0‘(l+z .zR) r0(l+z [2,?)
. . . . 6 ~l 6 . 6 . .
In cylindrical coordinates, V=r—+6——+z—. The radial and axral force

6r r 66 62

components are

4n2a3P m2 -1 2r —r2
(B3) ,3 r =- b exp - , e ,and
g d‘ c m2 +2 r04(1 3,2452%?)2 r02(1+zz/z§)

8n2a3Pz m2 -1 2r2 —r2
(13.4) Fm . =- b 1— - ~ exp , - .

The potential curvature is calculated from the spring constant (the spatial

 

 

 

 

 

derivative of the force) near the trap center. The curvatures are

 

 

 

 

 

2 3 2
2 aFgradr nba P m —1
a), at: 0C 4 2
:0 r0 m + 2
2 3 2
a): ct 0C 2
62 r:0 r022§(1+ 2252??)2 m +2

 

216

The adjustable parameters in the optical system are beam power, sphere size,
index of refraction of the medium, laser wavelength, and objective lens numerical

aperture. The force depends linearly on the beam power. The sphere size affects the

trapping force and the damping rate. Fa oc a3 , and T oc a‘2 , therefore Fa /(oa oc a‘3'S .

a

Part of the index of refraction dependence is hidden in m in (B3) and (8.4),

2 mz—l
Flimnl) 2 '
m +

The wavelength and convergence angle 6 (where 6 = sin—1(NA/nm) and n,,, is the
index of the imaging medium) are implicitly included via the beam waist radius and

Rayleigh range. For small displacements from the beam waist, Fgmd‘r ocr/ro4 and

Fgmdl oc z/rozzfe . The wavelength dependence is the same for both directions, F ,1 or: [4.

The convergence angle dependence is stronger for the axial force, Fe; oc64 and

F01 at 66.
Table 8.1 compares the forces for multiple trapping systems, based on the

curvatures measured for the system in this experiment. The force is F = mwzr, where

the maximum radial force is calculated at (r, z) = (r0 /\/2 ,0) and the maximum axial
force at (0,2R /~/3). The damping rate is calculated for a silica sphere (p = 2200 kg/m3).

The relaxation rate tfl is calculated from the curvature and damping rate. Force units are

N and frequency units are S'1 or Hz.
The first row has the forces for the current experimental setup. The curvatures are

estimated a), = 200,000 s'I and a): = 20,000 s". The first two rows are the forces using

217

different objectives, 100x dry with NA 0.9 and 60x with NA 0.8. The next two rows use

the original objective but change the particle size to a = 1 11m and a = 0.1 pm. The next

three rows change laser wavelength from 633 nm to 488 nm, 532 nm, and 1064 nm. The

final rows change the medium from water to air, with larger particles. The final row

shows the reduction of the damping rate at reduced pressure (with 77 = 0.01 77mm).

Table 8.1 Trapping forces for various trapping parameters.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—l —l

F,.max (N) F: max (N) (1),.(s") (a(s") trr (HZ) tn: (HZ) Us")
original 2.2x10'l2 1.5x10'l4 200000 20000 250 2.5 2.50x107
100xdry 1.9x10’l2 1.2x10'M 180000 1700 210 1.9 2.50x107
60xdry 1.1x10'12 5.9x10’l5 120000 9800 100 0.6 2.50x107
a=lpm 8.1x10'” 5.6x10’l3 200000 20000 2800 28 2.3x106
a=0.1um 8.1x10'l4 5.6x10‘l6 200000 20000 28 0.28 2.3x108
2:488 nm 3.7x10'12 2.6x10'” 340000 34000 720 7.2 2.50x107
2=532um 4.8x10'l2 3.3x10‘l4 280000 28000 510 5.1 2.50x107
2=1064nm 4.6x10"3 3.2x10‘15 71000 7100 32 0.32 2.50x107
Air,2pm (5.4x10"0 3.5x10'“z 180000 14000 47000 290 1.04x105
Air,3pm 2.1x10'9 1.2x10'll 180000 14000 110000 660 4.60x104
Air,5pm 9.9x10‘9 5.4x10‘ll 180000 14000 300000 1800 1.7x10“
Vacuum 9.9x10'9 5.4x10°“ 180000 14000 30x107 180000 1.7x102

 

The current experiment is in the overdamped regime (1760 = 125). To reach the

underdamped regime a larger particle should be trapped in air. Removing the air with a

vacuum pump reduces the damping further.

218

 

 

Appendix C Coherence Measurements of the Millennia Laser

To increase the flexibility in the optical system the Spectra-Physics Millennia
laser was installed. It emits SW at 532 nm, for a maximum of 2.5 W per beam in the
current experiment. By comparison, the HeNe lasers are limited to a maximum of
12 mW entering the microscope (see Section 4.1). The greater power allows novel

trapping systems, such as multiple traps or scanning beam traps.

M2 plate beamsplitter M2 plate

 
 

polarizer

    

Stabilizer 2
Cell

 

beam riser

beamsplitter Microscope

Figure C.1 Single laser optical trapping apparatus. The single beam was split into two
equal intensity beams with the first beam splitter. Each beam was stabilized,
expanded by lenses L1 and L2, and positioned with the gimbal mounted mirrors
before being recombined at the second, polarizing beam splitter. Beam 1’s
polarization was rotated by 90° by a half-wave retarder. The beams were
expanded again by lenses L3 and L4 and elevated by the beam riser before
entering the microscope. A half-wave retarder and Glan calcite polarizer were
used to attenuate the beam after it exited the laser.

The single laser beam was split into two arms, each with a beam power stabilizer
and beam steering optics. The beams were recombined before entering the microscope
(Figure C.1), essentially creating a Mach-lender interferometer. To prevent interference

one arm was made 15 cm longer, multiple times the expected coherence length of 3 cm.

219

Subsequent observations showed that this did not prevent interference. The actual
coherence length was predicted to be many meters.

To create two independently steerable optical traps the single laser was split into
two beams by a nonpolarizing beam splitter (Figure C.l). A half-wave retarder and
polarizer were positioned before the beam splitter for coarse power control. Each beam
went through a LS-PRO stabilizer and the beam expanding lens pair L1-L2. Beam 1’s
polarization was rotated by 90° by a half-wave retarder. Gimbal mounted mirrors were
used to change the angle of the beams entering the objective lens (Section 4.1.7). The
beams were recombined at a polarizing beam splitter. The beam, emitted from the laser
head at a height of 4.5 inches, was elevated to 8.5 inches with a beam riser (i.e., a
periscope). A quarter-wave plate transformed the orthogonal linear polarizations into
orthogonal circular polarizations. The other elements in Figure C] are described in
Section 4.1.

The orthogonally polarized and supposedly uncorrelated beams interfered in the
focal plane of the objective. The reflections of the beams from the cell’s glass/water
interface were visible on the Sony monitor (Section 4.1.8). As expected, a circular
diffraction pattern was present for a single beam. An interference pattern appeared when
both beams were viewed, changing shape with changing beam separation. This occurred
for orthogonally polarized beams with optical path lengths differing by more than 5
coherence lengths.

Finite coherence length is derived from the spectral properties of a laser [96]. A
laser with delta function spectral output (i.e., all photons emitted have the exact same

wavelength) has an infinite coherence length. If split and recombined in an

220

interferometer the beams always interfere with maximum fringe visibility. A laser with
finite width spectral output emits multiple wavelengths. The phase between the photons
increases as the beam propagates. In an interferometer, if the optical path length
difference is large enough the phases become uncorrelated. The beams no longer
interfere and are considered incoherent.

Assuming a Gaussian spectral density with width Av the coherence length
[C = c/Av. For gas lasers the width ranges from more than 10 GHz (IC = 3 cm) to less
than 5 MHz (IC = 60 m). The quoted spectral line width of the Millennia laser is 15 GHz,
for [c = 2 cm. With the 15 cm path length difference in Figure C.l there should have been
no visible fringes.

Stationary mirror

laser

%}MUZ\4I: flammable mirror

[lllllllllllllIIIIIIIIIIIIU
ND polarizer

 

 

lens meter stick

photodiode
Figure C.2 Michelson interferometer for coherence length measurements. The laser is
attenuated using a neutral density filter (ND) and the half-wave retarder/polarizer
combination. The beams, split by the beam splitter, are sent to two mirrors. The
mirror position was changed to vary the path length, using a meter stick to
measure position. The combined beams were expanded by the 25 mm focal
length lens and the interference pattern sampled by the photodiode.
To measure the actual coherence length the Michelson interferometer in Figure
C.2 was constructed. The beam was split by the beam splitter and sent to two mirrors.

The path length was adjusted by moving one of the mirrors in the direction of beam

propagation, with a meter stick used to measure position. The path length change was

221

 

twice the change in mirror position. The beams were recombined at the beam splitter,
expanded by a 25 mm focal length lens, and measured by a photodiode (Centronic
OSD155T silicon photodiode). The distance between fiinges was 5-10 times the width of
the active area of the photodiode to reduce spatial averaging.

The coherence length appears as a reduction of visibility
V = (1

—1min )/(lmax + 1mm), defined in terms of the maximum and minimum fiinge

intensity. The maximum visibility V = 1, for minimum intensity [min = 0. The
photodiode output was recorded on a LeCroy 9310AM digital oscilloscope as the static
mirror was moved by hand. Typically 5 fiinges were cycled, corresponding to 1 pm
motion of the mirror. The minimum and maximum photodiode voltages were measured
using the cursor functions on the oscilloscope.

Figure C.3 shows the visibility vs. path length difference over nearly 1 meter.
The visibility falls rapidly for small path length differences, as expected by the quoted
2 cm coherence length. After 10 cm the visibility increases, coming to a peak near
20 cm. The peak structure repeats with period 22 cm and varying amplitude. Repeated
measurements show that the peak structure changes over time.

The quoted 15 GHz line width is actually the width of the gain profile. The gain
profile determines which longitudinal cavity modes are sustained. The large gain profile
was designed to reduce power fluctuations due to mode hopping by including more than
300 modes. Thus the line width of the individual modes is much smaller, on the order of
5 MHz. The output spectrum is a comb structure, with lines spaced approximately
100MHz apart. The visibility of such a spectrum would be a series of peaks with

spacing corresponding to the spectral line spacing, as shown in Figure C3.

222

_n
O

 

 

 

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.é‘
:3:- 0.6-
.‘L’ < I
i 041 '-
.§’ '1 \ /l\g . .
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or) . . . -. . =2 e . '

0

20

40

00

80

 

'100

Path Length Difference (cm)

Figure C.3 Visibility vs. path length difference for the Millennia laser. The visibility
falls initially, corresponding to the expected 2 cm coherence length. At larger
path length differences a periodic structure appears.

In a dual beam system the potential, proportional to the intensity gradient, gains
an additional interference induced periodic structure. Figure 0% shows a typical one
dimensional potential in x (solid line) with the intensity distribution (dotted line). Figure
C.4b introduces interference effects into the intensity, with resulting potential. The
interference pattern is very sensitive to motion of the optical mounts. A position change
of 0.1 um causes a 7r change in phase, reversing the interference pattern and changing the
calculated barrier height by ~20 %. The particle averages over the spatial changes,
reducing but not eliminating the effective barrier height changes.

The Millennia diode pumped solid state laser has many advantages over two
HeNe lasers, especially in beam power and stability. However, the changes in the

potential due to interference and the driving created by interference fluctuations introduce

extra noise. Considering these effects, the HeNe lasers are more stable.

223

 

l(x)

 

l(x)
U(x)

U(x)

 

 

 

 

 

 

 

 

Figure C.4 Change in the potential due to interference. (a) the potential U(x) (solid line)
and intensity distribution l(x) (dotted line) for the unperturbed system. (b)
interference (visibility = 0.3) added to the intensity distribution, with resulting

potential.

224

a:

 

 

Appendix D Trapping in Air and Vacuum

Kramers solved the Fokker-Planck equation in the overdamped and underdamped
regimes (with some constraints) [1]. In the intermediate regime the prefactor in equation

(2.2) changes with damping, with a peak centered around T/(o =10 (where T is the

damping rate and a) is the potential curvature) [12,13]. To explore this range a system for

trapping 11m diameter silica spheres in air was created, drawing from the prior work of
Ashkin and Dziedzic [52-54,97,98] and others [99-101].

The dual beam optical trapping system shown in Figure C.l was used to trap
2.3 pm to 10 um diameter silica spheres in air and vacuum. The particles were lifted off
the surface using ultrasonic vibration of the substrate and trapped in a single beam. The
pressure in the cell was lowered from atmosphere (760 torr) to 250 mtorr using a vacuum
pump, crossing from the overdamped to the underdamped regime.

Attempting to create a bistable trap similar to the one made in water, two beams
were focused approximately one sphere radius apart by an objective lens. In air (and
vacuum) the particle was only confined in the focal plane for a small range of powers.
Larger powers “ejected” the particle into an upper stable point, where the laser scattering
forces were opposed by gravity instead of the gradient force (see Section 3.1.1). In
addition to those at the centers of the beams there were also stable points of unknown
origin in the focal plane. (These effects were also observed in a HeNe trapping system,
ruling out the Millennia interference (Appendix C) as a cause.)

The particle was stably trapped in a single beam for extended periods. The

potential curvature was measured as a function of pressure. The crossover from

225

 

 

 

overdamped to underdamped was observed in the position fluctuation spectrum,
measured as described in Section 3.2.2 [49]. With decreasing pressure the Lorentzian
spectrum created by the overdamped particle transformed into a series of peaks,

corresponding to the natural frequencies of the potential.

D. 1 Apparatus

The dual beam optical trapping system is shown in Figure CI. The 5 W
Millennia laser was used so the beam power could be varied by a factor of 10, or a factor
of 3 in potential curvature. The beam was focused by a 40x dry objective lens into the
vacuum cell, constructed from a piezoelectric cylinder. The spheres were deposited on
the bottom surface and lifted off by 70 — 80 kHz vibration. The cell was moved using the
microscope sample stage until a particle was drawn into the trap and the vibration turned
off. A mechanical pump slowly removed the air from the cell, to pressures below

100 mtorr.

D.1.1. Trapping Optics

The dual beam optical trap was created by the Spectra-Physics Millennia 5js laser,
split into two arms with a beam splitter. Each beam was stabilized and power controlled
using an LS-PRO beam power stabilizer. The beams were recombined with a polarizing
beam splitter and sent into the microscope. A 40x objective lens (NA = 0.75) with a
working distance of 1 mm focused the beams into the cell.

The particle position was tracked using the system outlined in Section 4.2. The
0.071 urn/pixel conversion for a 100x objective becomes 0.27 urn/pixel for a 40x

objective after the microscope’s internal 1.5x magnifier was removed. Typically the

226

 

1mg. '.'. ‘ v. ' 7 vi.

 

particles trapped were 4.87 um (18 pixels) in diameter. The 2 position determination
integrated over a 15 x 15 pixel box with dithered edges (Figure 4.14). The power and
angle of illumination from the condenser changed with the stage position due to masking
by the cell. Therefore a universal calibration of particle 2 position vs. integrated

brightness was could not be used.

D.1.2. Trapping Cell and Particles

The cell was constructed from a hollow piezoelectric cylinder 20 mm tall, 16 mm
inner diameter and 19 mm outer diameter (APC International, product number 42-1020)
(Figure D.l). The bottom surface was a 22 mm square piece of #2 cover glass attached
with Stycast 1266 epoxy resin. Under vacuum the 0.25 mm thick #2 cover glass flexed
10 11m in the center, compared to the 90 pm for the thinner #1 cover glass needed for the
100x oil immersion objective lens. The top surface was a 1” x 3” glass slide with an
o-ring fixture attached with Stycast epoxy. The fixture consisted of two aluminum rings
which fit the inner and outer diameters of the cylinder. Between was a 1/8” thick o-ring,
sealing to the top of the cylinder. The position of the condenser lens for proper
illumination (Section 4.1.8) restricted the height of the cell to 25 mm. A 16 gauge
stainless steel pumping line (OD 0.065 inches, ID 0.047 inches) entered through a hole in
the side, secured with Huntington’s PVC Vacuum epoxy.

The piezoelectric cylinder’s vibration was controlled through electrodes coating
the inner and outer walls. The inner wall was placed in electrical contact with the
stainless steel pumping tube using silver paint. The outer surface was connected to a 24

gauge copper wire, held in place by an o-ring. The cylinder was driven by a SRS DS345

227

 

function generator amplified by the HP 463A Precision AC Amplifier, supplying a

maximum square wave amplitude of 40 Vpp at 100 kHz.

Glass Slides,

171‘” ”1F"*‘"‘

Fill Tube ‘ P1620 Cylinder

 

ll
:1
i ,‘
l

Ami..— __c.

‘42 Cover Glass

 

Figure D.1 Vacuum cell photograph and schematic. The cell was constructed from a
piezoelectric cylinder with a #2 cover glass epoxied to the bottom. The
removable top was a 1 inch by 3 inch glass slide, sealed with an o-ring held in
place by two aluminum rings. The air was pumped out through a stainless steel
tube entering through a hole in the side.

The trapped particles were 4.87 pm diameter silica spheres (Bangs Labs
SSO4N/4745). The particles were deposited on the bottom surface of the cell with a
spatula. Few enough particles were introduced so regions of the bottom surface were
empty. Afier a sphere was trapped it was moved to an empty region to prevent additional
distortion of the beam profile by the other particles.

The piezoelectric cylinder was driven with a 2 Vpp square wave at ~76 kHz, near
its resonance frequency. A small fraction of the spheres stuck to the surface overcame
the electrostatic attraction and lifted off the surface. The moving spheres fell into two
categories: “bouncers” and “floaters”. The bouncers immediately flew out of the field of
view of the microscope. Floaters levitated 10 to 20 um above the surface at the

vibrational nodes of the cell bottom. The cell was moved with the sample stage until a

single particle was trapped in a single beam and the vibration shut off.

228

Larger spheres were easier to dislodge but required more beam power to trap.
Also, they were more likely to move to the trap’s higher stable points. Smaller spheres
were more difficult to remove from the surface and required a smaller trapping power
(making a shallower trap). Adding 10 um spheres to a cell containing smaller spheres
created more floating smaller spheres through collisions. The smallest sphere captured,
2.35 pm diameter, was trapped for 5 minutes before escaping. The 3.5 pm and 4.87 pm
spheres were found to be the best compromise. However, no bistable potential with both

stable points in the objective’s focal plane was created.

D.1.3. Trapping in a Vacuum

The cell was evacuated using a mechanical pump, connected to the cell through
Tygon and copper tubing. The cell’s 16 gauge pumping line was connected to 6 inches
of 1/ 16 inch inner diameter Tygon tubing. The tubing ended at a 16 gauge tube hard
soldered to a % inch copper adapter and connected to a compression fit tee. The other
ends of the tee led to the pressure measurement devices and the pump (Figure D2).

The pressure was measured using a 1 atmosphere gauge and a 1 torr capacitance
manometer (MKS Baritron 220). The gauges were connected to the tee with 40 feet of
'A inch copper tubing, matching the flow impedance of the 1/ 16 inch ID Tygon tubing.
The pumping end of the tee was connected to a roughing valve and a needle valve in
parallel, both mounted to the optical bench. The Franklin Electric mechanical pump was
sitting on the floor, connected to the valves through a 4 inch diameter, 16 inch long helix
of % inch copper tubing. The helix helped prevent pump vibrations from reaching the

optical bench.

229

 

 

Rough Gauge

J-

Baritron
Valves
.. -I
Microscope ‘ ’ " "
‘t' " "t Pumping

Station

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.~ I v" ‘- ‘ I"

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Cell
Trapping Optics

1

Figure D.2 Schematic of the pumping system. (top view) The cell, positioned on the
microscope stage, was connected to the pumping line and pressure measurement
line with Tygon tubing. The pumping line was separated from the cell by a
roughing valve and a needle valve for fine control of the pumping rate. The
pressure in the cell was measured at the end of 40 feet of copper tubing, with flow
impedance equivalent to the 6 inches of Tygon tubing. The pressure was
measured by a O to 1 atmosphere gauge and a capacitance manometer (Baritron).
The trapping optics were below the microscope (not pictured).

The particle was trapped before the cell was pumped down. Immediately after
spheres were added to the cell the top was replaced. After trapping a sphere the vibration
was turned off and the needle valve was opened slowly until the pressure began to drop,
typically limited to 20 torr/second. Constant pressure was achieved by limiting the flow

with the needle valve.

0.2 Results

The position fluctuation spectrum was measured for a 4.87 pm diameter sphere in
a single beam trap as a function of pressure and beam power. At approximately 10 torr
peaks appeared in the overdamped Lorentzian spectrum, representing the natural

oscillation frequencies. The peak frequencies increased with increasing beam power.

230

 

 

In a related experiment the potential curvature was calculated from the position
distribution as a function of pressure. The curvature relative to kBT increased due to the

reduction in damping.

D.2.1. Spectrum vs. Power and Pressure

The fluctuation spectrum was measured from the forward scattered trapping beam
using the technique described in [49]. A Centronic OSD155T silicon photodiode was
mounted above the condenser lens. The photodiode was moved roughly one photodiode
radius from the beam center in the x direction (see Section 3.2.2). Due to space
constraints the axial position was not optimized. The photodiode operated in photovoltaic
mode with a 10 k0 resistor in parallel. Coaxial cable connected it to the HP3561A
Dynamic Signal Analyzer (see Section 4.3.1), used to record the amplitude of intercepted
power fluctuations from 10 Hz to 4 kHz. The amplitude was measured in arbitrary
logarithmic units (proportional to dBV) that were consistent between data sets.

Figure D.3 shows the fluctuation spectrum as a function of pressure. A 4.87 pm
diameter silica sphere was trapped at atmospheric pressure. The spectrum was measured
with the pressure held constant using the needle valve. Above 1.3 torr the pressure was
measured using the 1 atmosphere gauge. The pressure was lowered until the sphere
escaped from the trap at 278 mtorr. The spectrum was averaged over 1000 sweeps using
the HP3561A’s rms averaging function. The data were transferred over the general

purpose interface bus (GPIB) to the computer.

231

 

 

 

 

 

 

 

 

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Frequency (Hz)

 

Figure D.3 Fluctuation spectrum for changing pressure. The forward scattered light
from a 4.87 pm particle trapped in a single beam was measured with a spectrum
analyzer. As the pressure was reduced the spectrum transformed from the
overdamped Lorentzian to a series of peaks representing the natural oscillation
frequencies in the underdamped system. The 2 frequency appeared at 190 Hz.
The x frequency appeared at 670 Hz. The frequency independent region of the
Lorentzian decreaseed in magnitude with decreasing pressure. The peak at 170
DDDHD-D+DI/IDDD[IDI]DI]Ell]DDDDUDDDDDDDDDDDDDDADHDd

110902, 5.6 mW beam power.)
At 760 torr and 350 torr the spectra show the expected Lorentzian shape. As the

pressure was lowered a peak formed near 1 kHz (14 torr). Somewhere between 14 torr
and 1 torr the system made the transition from overdamped to underdamped. A peak due
to the 2 potential curvature arose at 190 Hz. (This is different fiom the 170 Hz peak in
the P 2 14 torr curves, caused by a measurement artifact.) The peak at 670 Hz represents
the x curvature, with a peak near 1300 Hz representing y fluctuations (frequency doubled
due to photodiode position). As the pressure was reduced the peaks narrowed and the
background in the 50 Hz to 500 Hz range fell.

The fluctuation spectrum was measured vs. power with pressure held constant at

1.14 torr (Figure D4). The beam power was changed from 6.3 mW to 3.98 mW using
232

the LS-PRO laser power stabilizer. At higher powers the particle was ejected to the
upper stable point. The power was reduced until the particle escaped. The x and y
curvature peaks moved to lower frequency with decreasing beam power. The positions
of the second and third peaks are plotted vs. beam power in Figure D5. The third peak
positions were multiplied by 0.53, so the second and third peaks overlapped for 3.98 mW.

The positions of the peaks overlap for the entire range. The expected dependence of trap

curvature on beam power is a) at JP (see Appendix B), but Figure D.5 indicates a linear

relationship.

 

 

 

 

 

 

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Frequency (Hz)

Figure D.4 Fluctuation spectrum of the underdamped particle with changing beam
power. The x peak position (near 700 Hz) decreased with decreasing beam
power, as did the harmonic (near 1500 Hz). The 2 peak should occur near 200 Hz
and have similar power dependence. (Data labeled 110902, pressure = 1.14 torr.)

These measurements show that the system moved from overdamped to

underdamped at a pressure near 10 torr. The particle motions changed from pure

relaxation with a rate (oz/T to oscillations about the trap center with frequency a). The

233

 

 

oscillation frequency can be adjusted using the beam power, within limits. If the power
is too large the particle will be ejected into the upper stable point (or out of the trap

altogether). If the power is too small the particle will escape.

 

 

 

 

 

 

 

 

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4.0 ' 4:5 ' 5:0 ' 5:5 ' 6.0 ' 6.5
Beam Power (mW)

Figure D.5 Position of the second and third spectral peaks vs. beam power. The second
and third peaks in Figure D.4 increased frequency with increasing beam power.
The third peak position was multiplied by 560/1050, the ratio of the second peak
position to third peak position for P = 3.98 mW. (Data labeled 110902,

pressure = 1.14 torr.)

D.2.2. Curvature vs. Pressure

The position of the trapped particle was also measured, using the system
described in Section 4.2. The three dimensional potential energy was calculated and the
one dimensional profiles extracted (see Section 5.2). The z calibration changed with the
position of the trap in the cell, therefore the z curvatures could not be accurately
determined from the 2 profiles.

Figure D.6 shows the one dimensional potential energy profiles in the x direction

as the pressure was reduced. The increase in the curvature was due to the change in the

234

damping. The F luctuation-Dissipation theorem states that the amplitude of the noise (in
this case Brownian motion) is proportional to the damping. These changes are not

considered when calculating the potential energy.

 

    

 

 

 

 

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0.00 0.05 0.10 0.15

X(um)

Figure D.6 X potential energy profiles with changing pressure. The curvature increases
as the pressure decreases due to the change in the noise intensity. (Data labeled
1 11202, beam power 5.6 mW.)

The change in the x and y curvatures vs. pressure are plotted in Figure D7 The
curvatures decreased with increasing pressures up to 100 torr. Since the system was
overdamped above 10 torr it is unknown whether the peak in the fluctuation spectrum
changed.

These measurements show that the overdamped to underdamped crossover in the
Kramers rate could be measured if a bistable potential could be formed. In the water
system particles larger than the beam waist have an additional stable point between the

beams. The smallest particle trapped in air (2.35 pm with 0.6 mW) was twice the size of

the beam waist (1.2 pm). If a 1.2 pm diameter sphere could be trapped it is possible that

235

 

it would be localized in the focal plane with a bistable potential as seen in Figure 5.5.
Techniques for lifting smaller particles and keeping them stably trapped need to be

developed.

 

240001 e l - Xcurvature
22000: '35P, . . Ycurvature
200001 3.1,

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16000:
14000:
120003 I
10000; . ‘

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0:1 1 10 100

Pressure (torr)

0) (8'1)

 

 

 

Figure D.7 X and y potential curvatures vs. pressure. The curvature decreases with
increasing pressure due to the change in the damping rate. The damping does not
change above 100 Hz. (Data labeled 111202, beam power 5.6 mW.)

D.3 Summary

An attempt was made to measure the Kramers rate in the crossover regime. A
silica sphere was trapped in an air filled cell, then the air was removed using a vacuum
pump. Trapped in a single beam, the particle position fluctuations indicated a transition
from overdamped to underdamped with decreasing pressure. The peaks in the fluctuation
spectrum decreased in frequency with decreasing beam power. The power was limited to
a small range (from 6.3 mW to 3.98 mW for a 4.87 pm sphere) outside of which the

particle was either lifted to a higher stable point (gravitationally opposed levitation) or

236

 

the particle escaped from the trap. The pressure was also limited to 250 torr, below
which the particle was more likely to escape.

The system would be sufficient to study the crossover if a bistable potential could
be created. The scattering forces of two overlapping beams tend to move one or both of
the stable points out of the focal plane. Additional stable points of unknown origin also
appear. These effects may disappear if a small enough particle can be lifted from the
cell’s bottom and trapped, with a beam convergence angle large enough so the axial

gradient force always offsets the scattering forces of the combined beams.

237

 

 

Appendix E Calculation of the Potential Energy and

Transition Statistics Using Profiles

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure B] Profiles user interface. Profiles was created from WinMiIZ’s analysis
sections, areas 5 and 6 in Figure 4.8. Area 1 controls the data input, length
calibration, and analysis output. Areas 2 and 3 create the three dimensional
potential energy histogram and extract the one dimensional profiles. Area 4
controls the transition analysis. Area 5 is used to analyze multiple files with the
same filenarne prefix.

Calculation of the potential energy occurs in the Visual C++ program Profiles.
The Matrox Imaging Libraries (MIL) hardware copy protection prevents WinMi12 from
running on another computer. Profiles was created from the analysis portion of WinMi12

(see Section 4.2) with some additional code. Figure E.1 shows the Profiles user interface,

238

separated into 5 subsections of functionality, numbered in the order in which they are
typically used. The data file is loaded in area 1, applying the length scale conversion.
Results of analysis performed on the data are output to the analysis file. Area 2 controls
the three dimensional potential energy and one dimensional profile analysis. Area 3
performs the same function as area 2, and is used for comparisons between histogram
parameters. The transition analysis is controlled in area 4. Area 5 allows the user to
automate the analysis of multiple, consecutive files with the same suffix and prefix, for

use with the batch data acquisition routine in WinMi12

E. 1 Three dimensional potential energy surface

The position data are loaded from a file in area 1 of Figure B]. The x and y
values are converted from pixels to um using the conversion 0.071 urn/pixel, and the 2
values are converted from the integrated particle intensity using the most recent
calibration (see Section 4.2.3.8), modified in the “micron/Z-value” edit box.

The histogram parameters set the total analysis volume and bin sizes Ax, Ay, and
A2. These parameters are input into the top half of area 2 in Figure E1 The three
parameters in each direction are Start, # bins, and binsize (x and y use the same bin size).
The ith axis has length L1 =#bins,. xbinsizei and extends from Start; to Starti+L,-. To
verify that the parameters contain the desired range of particle motions, a series of two
dimensional projections of the three dimensional histogram (with given parameters) are
displayed using the “2-d Hist” button (Figure E2). The histogram parameters are
modified to create 10 values in the y direction at the stable points (for the local fits in

Section 5.2.2), with a typical xy bin size of 0.1 pm. The 2 bin size is set to 0.2 pm.

239

 

 

 

Figure E.2 Two dimensional potential energy projections. The “2-d Hist” button creates
the (a) xy projection, (b) xz projection, and (c) yz projection. The numbers on the
perimeter indicate the length scale (in pm), with positive 2 pointing down. The
numbers in the upper left of each window are the position of a single pixel in the
window, chosen by a left mouse click and used for estimating the stable point and
barrier positions. (Data labeled 101403-03, 2,000,000 points, xy bin size = 0.01
11m, 2 bin size = 0.02 pm.)

Pressing the “3-d Hist” button in area 2 of Figure B] calls the OnHist3d()
function, which creates the three dimensional and one dimensional potentials. The
discrete position distribution is calculated and stored in a global array named Hist[][][].
The histogram parameters define the number of bins and the size of each bin. This array
is passed to the functions that extract the one dimensional profiles.

OnHist3d() begins by reading the user inputs for the position distribution and the
estimated stable point and barrier positions. The Hist array is initialized to zero. If the
“rotate 3d hist” button is checked, the data are rotated in three dimensions until the line

connecting the left and right points is parallel to the x axis. (Typically the traps are

positioned so this is unnecessary.)

240

 

 

The position distribution is filled by stepping through every point in the data set.
If the position is inside the box defined by the histogram parameters, the Hist array
element corresponding to the position’s volume element is incremented. The array index

for each dimension is calculated

Xbin = floor[m] ,

binsize
where x,- is the current position, x0 is the Start value (Figure E.1). The floor function
rounds the fraction to the next lower integer. The total number of points is incremented
each time a Hist element is incremented.
The equation (5.2) is used to calculate the potential energy in each volume

element,

PE : _ln[ Hist[nyIzU,

total
in units of kBT. Volume elements with zero counts are set to 20 kBT. This value is used
only for file output or comparisons; it is not saved. Recalculating PE has little effect on

processing time and conserves memory.

The position distribution is averaged over (2 x rebin +1)2 volume elements, where

the rebin number is set in area 2 of Figure E. l. The direction of averaging depends on the
calculation. For the one dimensional potential energy profiles, the averaging is

perpendicular to the direction of the profile. For example, in calculating the x profile,

6y=rebin 62=rebin

sum = Z Z Hist[ny + aylz + 62]

6y=—rebin fizz-rebin

and the potential energy is

241

 

'1 Ind-'1 -.' .I' 1

 

(5.2) PE = —In sum , .
total x (2 x rebin + 1)‘

The OnHist3d() function calculates the one dimensional potential energy profiles,
from which the barrier height and potential curvatures are measured. After a header
describing the histogram parameters, the first data saved to the analysis output file are the
one dimensional profile in the x direction.

OnHist3d() calls a subroutine named StepPlanes() to step along a given direction
and find the minimum energy. If rebin is greater than zero the Hist array is averaged in y
and 2 before calculating potential energy. Figure 5.10 shows a single xz plane at the
center of the potential. Darker regions represent lower energy. The diamonds show the

path of minimum energy calculated by StepPlanes() with rebin = 2.

E.2 One dimensional potential energy profiles

Extraction of the y and 2 one dimensional profiles from the three dimensional
potential energy takes as input the three dimensional coordinates of the stable points and
saddle point. These values are either input into the left, right, and barrier edit boxes in
area 2 of Figure E. l , or, if the “Find” box is checked, are found from the x profile. When
input by hand the values are estimated from the two dimensional projections (Figure E.2).
First the x and y positions are estimated from the x-y projection, then the 2 positions from
the x-z projection.

To use the program to find the saddle and stable point positions the potential is
split at the x position in the “X Midpoint” edit box. After finishing the StepPlanes()
function for x, OnHist3d() steps through the volume elements with x less than the

midpoint value and determines the coordinates with the largest number of counts (or

242

 

average number of counts if rebin is nonzero). This process repeats for the volume
elements on the other side of the midpoint. The barrier is a saddle point, a local
minimum for y and 2 but a local maximum for x. To find the barrier position the x range
is limited to the midpoint :t 2% of the x bins. For example, for Figure E.1’s values of 100
x bins and midpoint = 1.45 the x range is 1.43 to 1.47 pm. The position with minimum
energy in this volume is considered the barrier position. The x, y, and 2 values for all
three extrema are written to the edit boxes.

Originally the StepPlanes() function was repeated for y and z. This either missed
one of the stable points, or caused the position of the minimum to jump between stable
points while stepping in y or z. A new subroutine was written to correct this problem,
Profiles(), which outputs the potential energy for lines in y and 2 running through the
stable points and barrier. The squares in Figure 5.10 represent the 2 profile paths.

Profiles() is called once for each extremum with a checked box, outputting the
profiles in the x and 2 directions. The minimum energy at the saddle point is found

within a 7 x 7 bin area in y and 2 (but not x, for which the barrier is a local maximum). If

rebin is nonzero, the position distribution is averaged over a volume of (2x rebin +1)3

bins. For the stable points the minimum energy is found within a 7 x 7 x 7 bin volume,

averaging over the area (2 x rebin +1)2. The potential for the y profile then the 2 profile

is output to the analysis results file.

The output of StepPlanes() and Profiles() contains five columns: the position of
the volume element in x, y, and z, the potential energy calculated from equation (5.2),
and the number of counts for that element. The final column is used to estimate

uncertainties. After the y and 2 profiles are output the file is closed and OnHist3d() ends.

243

 

E.3 Dwell time and transition time

The dwell times are calculated in the OnAnalyze() function in Profiles when the
“Analyze” button is pressed. The function steps through the particle positions, counting
the amount of time spent in each well. When the particle makes a transition (when it
crosses the half-width) the counter is reset and the dwell time and frame number are
stored.

If there is a mismatched frame the measured position may be on the other side of
the midpoint and half-width. If a frame is missing the position is automatically set zero.
To prevent these events from being counted as transitions, a bounding box is created
using the three dimensional histogram parameters in area 2 of Figure E]. If a frame has
a position outside of this box the counter is incremented and the particle is defined to be
in the same well in as the previous frame. Since mismatched and missing frames tend to
occur for large 2 excursions from the stable points, careful choice of the model image
limits these events to when the particle is low in the traps (-z) and unlikely to make a
transition, reducing the possibility of missing an actual transition.

The “Save Results” button saves the dwell times and transition times (in units of
frames) for each well to the analysis results file. This file is opened in Origin and a
histogram of dwell times created (Figure 5.9).

After acquisition the data are transferred to Komodo.pa.msu.edu for analysis with
the Profiles program. A batch analysis routine, controlled in area 5 of Figure 5.8, outputs
the dwell time and transition time results and the one dimensional profiles for the
sequentially numbered data files. These files are imported into Origin to calculate the

mean dwell time (Section 5.1) and the fits to the static potentials (Section 5.2). The

244

 

 

quartic fits to the static tilt data sets are used to estimate the adiabatic critical amplitude
Afd . The potential energy calculated for modulated data sets does not represent the static

potential so the one dimensional profiles fits are not performed.

 

 

245

10.

ll.

12.

13.

14.

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