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This is to certify that the
dissertation entitled

Coherent Studies of Molecules from the Weak to
the Strong Field Limit

presented by
Matthew Comstock

has been accepted towards fulfillment
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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

COHERENT STUDIES OF MOLECULES FROM THE WEAK TO THE
STRONG FIELD LIMIT

by

Matthew Comstock

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Chemistry

2004

Abstract

COHERENT STUDIES OF MOLECULES FROM THE WEAK TO
THE STRONG FIELD LIMIT

by

Matthew Comstock

This work involves a series of experiments using the coherent properties of
ultrafast lasers. First, vibrational and rotational spectroscopy of the N204 molecule is
examined. The non-resonant four wave mixing rotational coherence spectroscopy
method, both homodyne and heterodyne detected, is used to investigate both short
time ground state vibrational dynamics and long time scale rotational dynamics.
Accurate rotational constants, vibrational frequencies and anharmonicities are
measured for the N204 molecule.

Rotational coherence spectroscopy is also used to investigate the effect of
high intensity laser pulses on sample molecules. Both alignment and deformation are
observed, in fields from 1010 W/cm2 to 1013 W/cmz. These effects are verified by
time of flight mass spectrometry. Resonant four wave mixing spectroscopy is used to
measure the decoherence cross sections for the coherence between the X(123+) and
B(3l'Io..+) states of 12 in a series of buffer gas molecules. Lennard-Jones parameters
are extracted for He, Ar, CsHs, N2, and 02. Finally, Multiphoton intrapulse
interference is examined in an effort to simplify the search space typically
investigated by pulse shapers. The concept of Binary Phase Shaping for two photon

excitation is investigated as a coherent control method.

In honor of my father

iii

Acknowledgements

I would like to thank my advisor for all his support over the past few years.

Thanks to Dr. Tamar Seideman for interesting discussions on interactions of
molecules in strong electric fields, to Dr. Jonathan Hall for discussing symmetry and
number theory as it applies to second harmonic generation, and to Dr. Piotr Piecuch

for performing some electronic structure calculations on C82 in strong fields.

I would also like to thank the chemistry department, the members of my committee
and the following past and present members of the Dantus group over the years.
Dr. Igor Pastirk
Dr. Vadim Lozovoy
Dr. Emily Brown
Dr. Irene Grimberg
Kathy Walowicz
Vahan Senekeri myan
Johanna DelaCruz

Mike Kangas

iv

Table of Contents

List of Tables ................................................................................................................ vii
List of Figures .............................................................................................................. viii
1. Coherence and Spectroscopy ....................................................................................... l
1.1 Introduction ............................................................................................................ 1
1.2 Outline of the thesis ................................................................................................ 2
1.3 experimental setup (laser systems) .......................................................................... 4
2. RC8 spectroscopy of N204 .......................................................................................... 6
2.1 Introduction ............................................................................................................ 6
2.2 Experimental .......................................................................................................... 7
2.2 Theory .................................................................................................................. 10
2.3 Results ................................................................................................................. 11
2.4 Discussion ............................................................................................................ l9
3. High Intensity effects ................................................................................................. 21
3.1 Introduction .......................................................................................................... 21
3.2 Experimental ........................................................................................................ 25
3.3 Results ................................................................................................................. 27
3.3.1. Evidence of Alignment and structural deformation from Transient Grating
Measurements ......................................................................................................... 27
3.3.2. Evidence of structural deformation for Coulomb Explosion ........................... 29
3 .3 .3. Alignment and Deformation of polyatomic molecules ................................... 33
3.4 Discussion ............................................................................................................ 36
3.5 Conclusions .......................................................................................................... 46

4. F emtosecond photon echo measurements of electronic coherence relaxation
between the X(12‘.g+) and B(3l'IOu+) states of 12 in the presence of He, Ar, N2, 02,

C3H8 ............................................................................................................................. 49
4.1 Introduction .......................................................................................................... 49
4.2 Theory .................................................................................................................. 52
4.3 Experimental ........................................................................................................ 57
4.4 Results ................................................................................................................. 60

4.4.1. Experiments on neat Iodine vapor .................................................................. 60
4.4.2. Experiments with buffer gases ....................................................................... 62
4.4.3. Photon Echo and Reverse Transient Grating Measurements .......................... 64
4.5 Discussion ............................................................................................................ 66
4.6 Conclusion ........................................................................................................... 72

5. Multiphoton Intrapulse Interference: Binary Phase Shaping ....................................... 73
5.1 Introduction .......................................................................................................... 73
5.2 Experimental section ............................................................................................ 76
5.3 Results ................................................................................................................. 77
5.4 Discussion ............................................................................................................ 81

6. Appendix I ................................................................................................................ 84

7. Appendix H ............................................................................................................... 87

References ..................................................................................................................... 93

List of Tables

Table.4.l Parameters calculated for the long range interactions responsible for
electronic coherence dephasing of 12 molecules and buffer gases (M) ................... 64

vii

List of Figures

Figure 2.1. Sketch of the N204 molecule. Distances and angles from electron diffraction
experiments" .................................................................................................................. 6

Figure 2.2. Beam geometry used for the femtosecond four-wave mixing experiments.
Homodyne detection (a) produces a background free signal, while Heterodyne Detection
(b) preserves the phase of the signal. ............................................................................... 8

Figure 2.3. Experimental time-resolved FWM signal obtained during the first 2 ps: a)
Frequency dispersed signal obtained as a function of time delay for N02, b) Frequency
integrated N02 data as a function of time delay. c) Frequency dispersed signal as a
function of time delay for N204, (1) Frequency integrated N204 data as a function of time
delay. For all scans the signal intensity is plotted on a logarithmic scale ........................ 12

Figure 2.4 Heterodyne detected FWM signal from N204 at 273K. Inset is a MEM-FT of
the transient, showing the 79cm'1 and 81cm‘l torsional frequencies, as well as the 255cm'
1 and 234cml vibrational frequencies ............................................................................ 13

Figure 2.5. Experimental time—resolved and fi'equency integrated Heterodyne FWM
signal (a) N02 and (b) N204 taken at 363 K and 294 K, respectively. Note the subtle
differences in the shape of the rotational recurrences obtained for both cases. The signal
intensity is plotted on a logarithmic scale. ..................................................................... 14

Figure 2.6. Experimental time-resolved and fi'equency-integrated signal obtained at 363
K (a) and 294 K (b). The main difference between the two transients afier the first two
picoseconds occurs at 76.265 ps and is indicated by an arrow. Both intensity axes are
plotted on a logarithmic scale. Some fine differences are observed in the modulation of
the rotational recurrences but not in their position. ........................................................ 17

Figure 2.7. Heterodyne detected Transient Grating spectroscopy of N204. Raw data at the
fiill and halfrecurrence time of N204 at two temperatures (a and b, respectively).
Subtraction of the two different temperature signals reveals a clear full recurrence for
N204 (d), but no half recurrence (c). .............................................................................. 18

Figure 3.1 Plot of the induced rotational trap depth on a highly polarizable molecule like
CS2 (dotted line, left axis) and the electric field strength (solid line, right axis) as firnction
of peak laser intensity. The predominantly observed molecular processes that occur at
any given intensity are indicated in the top. ................................................................... 22

Figure 3.2 Schematic drawing of the experimental setup for time-resolved transient

grating (t0p) and molecular beam (bottom) experiments. The transient grating signal was
detected at phase matching angle as a firnction of time delay between the first two strong-

viii

field pulses and the weak probe. Ion detection, in the molecular beam, was recorded as a
fimction of the angle that the polarization vector of the laser pulse makes with the time-
of-flight axis. ................................................................................................................ 26

Figure 3.3 First rotational recurrence of CS2 at three different laser intensities. Probe
intensity was held constant throughout at 2x10wW/cm2. (a) Low intensity data (circles)
and simulation (solid line) based on unperturbed molecules. (b) Data (circles) under
higher intensity indicating laser induced alignment. The simulation (solid line) assumes
unperturbed molecular geometry. (0) High intensity data (circles) showing a change in
the observed rotational revival, evidence for structural defornmtion of the CS2 molecule.
The solid line in (c) is a simulation based on 35% of the sample having a bond angle of
160°. ............................................................................................................................. 28

Figure 3.4 Time of flight mass spectrum obtained for CS2 molecules (a) when the laser
polarization vector is aligned with the TOF axis. The zoomed inset shows splitting of the
sulfirr fi'agment ion peaks caused by their Coulomb explosion towards and away fiom the
detector. (b) Angle dependent distribution of CS2+ ions, showing an isotropic distribution.
..................................................................................................................................... 30

Figure 3.5 Angle dependent detection of singly charged fi'agment ions following
Coulomb explosion. Anisotropic distributions are observed for C+ (a) and S+ (b) ions in
polar and Cartesian plots. Data for three independent experiments (squares, circles,
triangles) are accompanied by a guide to the eye (solid line) in the form of a cos‘(9)
function. S+ shows a maximum ion yield at 0° and 180°, while C“ shows a maximum at
90° and 270°. The anisotropy for the C" ions is less pronounced. ................................... 31

Figure 3.6 Angle dependent detection of doubly charged fi'agment ions following
Coulomb explosion. Anisotropic distributions are observed for c” (a) and s2+ (b) ions in
polar and Cartesian plots. Data for three independent experiments (squares, circles,
triangles) are accompanied by a guide to the eye (solid line) in the form ofa cos“(e)
firnction. 82+ shows a maximum ion yield at 0° and 180°, while C2+ shows a maximum at
90° and 270°. ................................................................................................................. 32

Figure 3.7 Time resolved TG experimental data centered on the first rotational recurrence
of C02, with the probe intensity held constant at 2xlo‘°WIcnr’. At low intensities ((a) and
(b)), the data (circles) matches well the simulation (solid line) based on an unperturbed
linear molecules. Near 1012 W/cm2 ((c)and(d)) the data shows evidence of bending. Solid
lines in (c) and (d) are simulations assuming 35% of the sample has a bond angle of 160°.
At higher intensity ((e), (t) and (g)), the signal appears above a constant background
signal. The simulation assumes that molecules in the intense field have been destroyed
and the signal arises from unperturbed linear molecules heterodyned by a constant
background. .................................................................................................................. 34

Figure 3.8 Time resolved TG experimental data centered on the first rotational recurrence
of acetylene, probe intensity held constant at 2x10lo W/cmz. At low intensity (a), the data
(circles) matches well the simulation (solid line) based on unperturbed molecules. As the

ix

laser intensity is increased (b and c), the first peak of the recurrence broadens, beginning
to overlap with the second peak The factor of 40 in intensity fiom (a) to (c) can be
explained by an enhanced alignment of the acetylene molecules in the laser field. ........ 35

Figure 3.9 Time resolved TG experimental data centered on the first rotational recurrence
of benzene, probe intensity held constant at 2x10lo W/cmz. When the grating intensity is
low (a) the data (circles) is in good agreement with the simulation (solid line) assuming
unperturbed molecules. At 3x10” W/cm2 (b), the first peak of the recurrence broadens,
and by 7x1012 W/cm2 there is evidence for significant deformation of the benzene
molecule. Simulation in (c) assumes a symmetric t0p with a c3 symmetry axis (ring
puckering mode). .......................................................................................................... 36

Figure 3.10 Schematic representation of laser induced molecular deformation process of
CS2 molecules based on a dressed state picture. The laser dresses the ground state and
brings it close to the excited states having a bent geometry. The wavepacket has a short
time to evolve in the dressed state potential during the laser pulse. The system returns to
the ground state with substantial bending vibrational energy, leading to a deviation fi'om
linear geometry, and producing a higher rotational constant. ......................................... 38

Figure 3.11 Simulation showing the effects of molecular deformation on the rotational
recurrences of a generic molecule. The time axis is in units of the full-width half
maximumofthepeaksoftherecurrence, withzerobeingtherecurrencetimeinthelinear
molecule. Plot (a) shows recurrences from the bent species, showing the increase in
rotational constant as a function of bond angle. Plots (b) through (i) show the recurrences
of a sample comprised of 35% bent molecules and 65% linear molecules for bond angles
from 5°(b) to 25° (t). The simulation in panel (e) is in good agreement with our
experimental findings shown in Figure 3.30. ................................................................. 40

Figure 3.12 Conceptual drawing of the Coulomb explosion of linear CS2 molecules when
the electric field polarization is parallel or perpendicular to the time-of-flight axis. The
detection of S+ fragment ions is expected to show a marked anisotropy. However, the C+
fragment ion detection is expected to be isotropic. ........................................................ 42
Figure 3.13 Conceptual drawing of the Coulomb explosion of bent CS2 molecules when
the electric field polarization is parallel or perpendicular to the time-of-flight axis. The
detection of 8+ fragment ions are expected to show a marked anisotropy similar to that
observed for the linear molecules. However, the C+ fragment ion detection is expected to
be anisotropic and perpendicular to that observed for the S+ fragment ions. ................... 44

Figure 3.14 Molecular models depicting the average bent angle observed for acetylene (a)
and benzene (b) consistent wit the experimental TG data obtained in the presence of
strong laser fields. ......................................................................................................... 46

Figure 4.1. A graph of the potential energy curves (solid line) involved in the electronic
coherence between the B and X states of iodine. Dashed lines represent the perturbation
caused by the approaching buffer gas molecule. The overall change in energy is indicated

by Aw. The phase shift Acp necessary to destroy the coherence during a long range
collision is given by the accumulated phase shift taking place during the interaction. 52

Figure 4.2. Pulse sequence and phase matching geometry for PE and RTG measurements.
The PE setup (a) with pulse b arrives before pulses a and c undergoes rephasing and has a
maximum at time t after the second two pulses. The RTG (b) with pulse a arriving before
pulses b and c, undergoes no rephasing and is thus maximum at the time of the second

two pulses. .................................................................................................................... 54

Figure 4.3. Calculated PE and RTG decays (sz rowsec" and A=2x 101°sec'l) showing

both gas and liquid phase limits. At short times, both PE and RTG approach the gas phase
limit (thin solid line), at longer times, PE follows the liquid phase limit (thin dashed line),
and RTG decays non-exponentially. .............................................................................. 57

Figure 4.4. Plot of homogeneous relaxation rate, 7; versus number density for neat iodine
vapor. Results for 37 measurements are shown. Horizontal error bars indicate uncertainty
in the absorption cross section of iodine, vertical error bars indicate uncertainty in the
exponential fitting routine. Dotted lines represent the 90% confidence interval based on a
linear least squares fit. Inset is a typical PE data trace plotted on a logarithmic scale. This
trace corresponds to a number density of 5x1023 m'3. .................................................... 60

Figure 4.5. Plots of homogeneous relaxation rate 7, of PE signal with 50 fs excitation
pulses, versus number density for (a) Helium, (b) Argon, (c) Oxygen, and (d) Nitrogen.
Error bars indicate uncertainty in the fit. Error in the number density is smaller than the
data points. The point at zero number density in each plot represents the relaxation rate of
Iodine at 5x1023 m'3 number density, uncertainty in this number is smaller than the data
point in both dimensions. .............................................................................................. 63

Figure 4.6. Logarithmic plot of (a) PE and (b) RTG signals with 1200 fs excitation
pulses, as a function of time (5x 1023m'3 Iodine and 2.5x 102‘m'3propane at 400K).
Experimental data (dots) are fit (solid line) using the theory without approximations,
Equations 4.1 and 4.2, presented in the text. The dashed line is a simulation based on a
simple exponential decay with a homogeneous decay rate determined by the full fit. 65

Figure 4.7. Plot of the coherence relaxation rate yversus pressure for propane buffer gas.
Both Photon Echo data (a) and Reverse Transient Grating data (b) are shown. The plots
contain a total of 37 independent measurements. Error bars indicate uncertainty in the fit.
Error in the number density is smaller than the data points. The point at zero number
density in each plot represents the relaxation rate of Iodine at 5x1023 111'3 number density,
uncertainty in this number is smaller than the data point in both dimensions. ................ 66

Figure 4.8. Depiction of the distances R involved in the long range interactions
responsible for electronic coherence dephasing. Molecules are depicted as 98% electron
density maps calculated using the Spartan molecular modeling program. ...................... 67

Figure 4.9. Experimental (square points with uncertainties) and calculated (round black
points) Lenrmrd Jones C5 parameters for the bufi‘er molecules in this study. Both data sets
follow very similar trends, however experiment and theory values differ by a factor of 15.
The round open points correspond to the calculated parameters multiplied by a factor of
15 to compensate for the greater excited state polarizability (see text). .......................... 70

Figure 5.1: A typical 4F pulse shaper. The SLM placed at the Fourier plane applies
different phases and amplitudes to different frequency components of the laser pulse. .. 73

Figure 5.2. Schematic representation of the problem. The broad bandwidth second
harmonic spectrum fi'om transform-limited pulses is represented by a Gaussian (thin
line). The objective is to introduce phase modulation to cause the two-photon spectrum
to be intense only inside the window defined by fi'equency 2(1)c and width W, and to
minimize the background B outside the window. The contrast ratio C is defined as the
integrated intensity inside W divided by the integrated intensity of light outside the
window. ........................................................................................................................ 74

Figure 5.3. Phase mask proposed based on the symmetry requirements of the problem,
using the quasi randomness of prime numbers. This mask is reflected about pixel 64, and
is designed to obtain a narrow second harmonic signal at the center of the spectrum. 76

Figure 5.4. Effect of spectral amplitude restriction on SHG. (a) Experimental spectrum of
the laser before (black) and alter filtering with windows of width 40 (red), 20 (blue), and

10 (green) nm. (b) Experimental (points) and simulation (continuous lines) for the second
harmonic spectrum of TL and spectrally filtered pulses as indicated in panel (a). .......... 77

Figure 5.5. Experimental results with binary phase shaping. (a) The spectrum of the laser
(dashed lines) and the binary phase mask (0 or 1:) are shown as a firnction of wavelength
(b) The second harmonic spectrum of the TL pulses (dashed line) and second harmonic
spectrum of the shaped pulses according to panel (a) ..................................................... 78

Figure 5.6. Comparison of a prime number inspired phase mask (black) with a mask that
was Optimized using a computer based learning algorithm (red). The insets on the left
depict the phase masks for each case. The inset on the right shows the improvement in
the contrast ratio, as defined in Fig. 1, as the learning algorithm frnds the best solution
starting from the prime number phase mask. ................................................................. 80

xii

1. Coherence and Spectroscopy

1.1 Introduction

Since the 1960’s, the concept of using lasers to drive chemical reactions to
particular outcomes has been a dream of physical chemistry. Processes such as
Intramolecular Vibrational Redistribution (IVR) prevented any significant success,
however until the late 1980’s when theoretical methods that circumvent IVR were
developed. 1’2 At the same time, laser technology progressed to the point of generating
pulses on the order of, and even faster than, the IVR timescale."6 This allowed a great

expansion in the techniques available to the laser control community. The late 1990’s saw

 

i. Reproduced with permission from
PasfirkLComakaaMDaMIsM“FennosxondgmnflmdynamiesofN204mdN02
investigated by degenerate four-wave mixing”, Chem. Phys. Letters, 349 (2001) 71-78.

ii. Reproduced with permission from

M. Comstock, V. V. LozovoyandM Dantus, 'Fmtosecondphotonecho
measurements ofelectronic coherence relaxationbetween the X(lS_g +)
andB(3r>"_u +)statesofl_2 inthepresenceofI-Ie, Ar, N_2 , o_2,

C_3 H_8 ," J. Chem. Phys. 119(13‘),* (2003), 6546-6553.

M. Comstock, V. SarekefimyanandMDm'Ulual’astIaserImhnedMolearlarAfigrnnerland
Deformation: Experimemalevidencefiomnamalmoleculesandfiomfiagrnemions'J.Phys Chem
107, (2003), 8271-8281.

iv. Reproduced with permission from
MatthewComstock, VadimV. Lozovoy, IgorPastirhandMarcosDamrs, Multiphotonlntrapulse
interference 6; binary phase shaping“, Optics express Vol.12, No.6, (2004), 1061-1066.

a divergence of techniques, based on the three main properties of the ultrafast laser", time,
intensity, and phase.

Early on, phase was used to cause interference on excited state potential energy
surfaces.2 Afler the invention of pulse shapers,7 genetic algorithms were introduced to
produce optimal pulse shapes“) Similarly, the short timescales available from modern
pulsed lasers allowed another coherent control scheme. Known as pump-dump coherent
control, the idea is to interact with the molecule faster than it can relax, avoiding IVR by
preempting it. 1,1143

The short timescales and coherent properties of ultrashort laser pulses allowed
more complicated nonlinear spectroscopies to be developed and used for coherent control
purposes."'" Similarly, the short timescales of ultrashort lasa’ pulses produce extremely
high intensities, and these high fields can be used to control the orientation and motion of
molecules as well. Molecular alignment and deformation,""23 tunneling ionization,”26

27'” and the molecular centrifuge30 are all examples of unique laser

Coulomb explosion,
molecule interactions which occur at the high peak intensities produced by ultrashort

laser pulses.

1.2 Outline of the thesis

In my dissertation I will cover several different types of experiment based on
understanding the filndamental light-matter interactions used to control chemical
processes. All of these processes involve the coherence of laser pulses, and use this
coherence to either investigate or control some aspect of the sample molecule. First, I will

discuss a type of ultrafast Four Wave Mixing (FWM) spectroscopy, called Rotational

Coherence Spectroscopy (RCS).3 1'3 2 This is based on observing the reflective index of a
medium as a filnction of time after a strong non-resonant pulse passes through a sample.
The signals can be F ourier-Transformed to produce classic microwave rotational spectra,
though RCS requires no intrinsic dipole moment in the sample molecules. Resolutions of
10"’cm'l are routinely obtained, and symmetry properties of the molecules can be
examined in detail.

Secondly, using the high structural resolution of RCS, I investigate changes to the
molecular structure caused by the presence of high electric fields. By applying a
femtosecond laser pulse to a sample, peak intensities of 1016W/cm2 are achieved
routinely. The corresponding electric fields are of the same order of magnitude as the
electric field between the nucleus and the electron. This strong coupling of the laser pulse
with the molecule can produce alignment,"”2°'22 deformation23 and/or ionization.24’27’33
These effects change the signal produced by the RC8 method, and a filll analysis is given.

The third topic I will discuss is electronic decoherence, as studied in our lab by
the photon echo method. This is a type of resonant FWM spectroscopy that allows us to
measure the rate of homogeneous decoherence in a sample. Iodine was the target
molecule, and electronic coherence dephasing cross sections were measured to be
roughly 10 times greater than those expected for a number of buffer gas samples.

Finally I will address the relatively new field of pulse shaping technology,
specifically a new technique using only two phase values across 128 pixels to control
two-photon excitation. This binary phase shaping (BPS) greatly reduces the search space

for learning algorithms, leading to better convergence in shorter times. Using the

symmetry of the problem, we can design binary phase masks which will produce the

desired results, even without resorting to computer algorithms.

1.3 experimental setup (laser systems)

Three laser systems were used for the work covered by this dissertation. The first
was a Colliding Pulse Mode-locked dye laser (CPM) operating at 620nm. The
Rhodamine 590 Chloride gain jet was pumped by the output of a Spectra Physics
Millenia NszLF laser. The saturable absorber dye jet 3,3’ diethyloxadicarbocyanine
iodide (DODCI) produces 47fs pulses at 87MI-Iz. The pulses are then amplified at 30Hz
by a four stage dye cell amplifier, pumped by a Spectra Physics Quanta-Ray Nd:YAG
laser. Afier amplification the pulses are compressed to 60fs using a dual prism
arrangement. The CPM system could produce up to 30mW output power.

The second laser system used in this work is a regeneratively amplified
femtosecond titanium-sapphire laser system (Figure 2). The output fi'om a continuous
wave NszV04 laser (Millennia — Spectra Physics Laser) pumps a femtosecond titanium-
sapphire oscillator (KM Labs), capable of producing <15 fs pulses at 80 MHz. This
output is amplified by a regenerative amplifier (Spitfire — Spectra Physics Lasers)
pumped by 1 kHz NszLF laser (Evolution-X — Spectra Physics Lasers). Amplified
pulses centered at 805 nm are nearly transform limited and have a maximum energy of
0.8 mJ/pulse and time duration of 50 fs. Characterization of the pulses was performed

both with a non-collinear autocorrelator and by fi'equency resolved optical gating

(FROG). 34

The third system I used was a Titanium Sapphire oscillator (Kapteyn-Murnane
labs) system pumped by a Spectra-Physics Millenia Nd:YLF laser, capable of producing
10fs pulses tunable from 840nm to 780 nm at 87MHz repetition rate. In each chapter, the

laser system or systems used will be identified in the experimental section.

2. RC8 spectroscopy of N20.

2.1 Introduction

Nitrogen oxides (N 0;) have been the subject of numerous studies primarily for
their role in both stratospheric and tropospheric chemistry and in particular, their links to
the ozone photochemical cycles and smog formation. These relatively small molecules
are especially interesting due to their complex spectroscopy as evidenced by theoretical
and experimental studies of their ground and excited states?“6 All except NO have
multidimensional potential energy surfaces, and numerous curve crossings that result in

vibronic chaos making these molecules very interesting to dynamicists.

112.7°
134.6° ¥

 

   

   

 

 

<—1.77A m

Figure 2.1. Sketch of the N204 molecule. Distances and angles are from electron
diffraction experiments."7

The nitrogen dioxide/dinitrogen tetroxide monomer/dimer pair has received
particular attention. At one atmosphere and room temperature their equilibrium
concentration is about 3:17 for N02:N204, respectively.“48 The dimer is favored at

higher pressures and lower temperatures. Dinitrogen tetroxide is particularly intriguing

because of its unusually long N-N bond (1.75 and 1.78 A, solid and gas phase,
respectively)49’5° and the fact that the 0N0 bond length and angle are identical to those of
the monomer (see figure 2.1). Because of its (Du) symmetry N204 does not have a
permanent electric dipole moment. The dissociation energy of the N-N bond is 0.59 eV
and its first electronic resonance is at 340 nm” It has been proposed41 that the use of off-
resonance excitation should make it possible to probe the dimer directly at room
temperature. Most of the vibrational fiequencies for these molecules are established and
assigned,37'3 8'41" 1 but there is still some disagreement between experiment and theory on
several modes. The photodissociation dynamics of N02 were studied recently using
ultrafast methods. 5 2'5 3 Here we exploit the ability to detect low fi'equency Raman modes
of short-lived species and to measure high accuracy rotational constants of femtosecond
time-resolved four wave mixing (FWM) to study of the ground state dynamics of
N204/N 02.
2.2 Experimental

The amplified Titanium Sapphire laser output was split into three beams using
dielectric beam splitters and recombined in the forward-box geometry (see figure 2.1a).
The beams occupy three corners of a 25 mm square and were determined to be parallel
over one meter by using a template of the appropriate geometry. A 50 mm diameter 0.5 m
focal length lens focused the beams into the sample cell, the beams crossing at an angle
of 29°. This crossing angle produces a transient grating with a 16 um spacing. In all
experiments the output beam was attenuated to ~60 it] per pulse measured before
splitting. Two beams were overlapped in time and space to form a transient grating in the

sample while the third one was time delayed with a computer controlled actuator. This

configuration allows homodyne transient grating (TG) and reverse transient grating
(RTG) experiments, where the signal beam emerges with wave vector k,=k.-kb+kc. When
the initial two pulses are not overlapped in time, other types of FWM experiments
(photon echo and virtual echo) are possible.’4 The main advantage of the homodyne
signal is that it is spatially distinct from the other laser beams, and is therefore
background fi'ee. Another option for FWM experiments is to use only two laser beams
(see figure 2.2b) overlapped with a shallow angle to maximize the interaction area In this
case, one beam is chopped and a lock-in amplifier is used to observe changes in the
refractive index of the sample caused by the first beam This heterodyne signal has the
advantage that it reveals the phase of the signal, information that is lost in the homodyne

configuration

 

Forward BOX Horn odyne Detection Two Pulse Heterodyne detection

 

 

 

 

 

Figure 2.2. Beam geometry used for the ferrrtosecond four-wave mixing
experiments. Homodyne detection (a) produces a background fiee signal, while
heterodyne detection (b) preserves the phase of the signal.

For the homodyne configuration, the event when all three pulses overlap in time

is referred to as time-zero. The three 10 mm diameter beams were focused onto a 6

inches long custom-designed quartz cell containing the gaseous sample by a two-inch

diameter lens with focal length of 50 cm. Heating the entire cell with heating tape
controlled the temperature of the cell. The FWM signal beam was spatially filtered by a
set of irises, collimated and sent into a spectrometer (Triax 320, Jobianon). Before
reaching the entrance slit of the spectrometer, the signal was attenuated by a neutral
density filter of OD=l-2. The signal was then dispersed by the spectrometer’s grating,
and was registered by a 2000 x 800 pixel liquid nitrogen cooled CCD. The entrance slit
of the spectrometer was 100 micrometers (0.8 nm resolution). The wavelength range on
the CCD in this setup (grating dictated) was about 60 nm. In order to preserve disk
space, every 20 pixels were binned together. For the shorter scans (~1-2 ps duration) the
frequency-dispersed signal was recorded every 10 fs, while for the longer scans the time
steps were 20 fs. Signal was registered for each time delay with an integration time of
100 ms. Each data set is the average of several such scans.

For the heterodyne case one beam, the pump, was mechanically chopped at a
frequency of 318Hz, and the other beam, the probe, was delayed in time with respect to
the pump using a computer controlled actuator. Heating the entire cell with heating tape,
or submerging the cell in an ice bath controlled the temperature of the cell to within 3
degrees centigrade. The event when the two pulses overlap in time is referred to as time-
zero. After the sample, the probe beam is spatially filtered and fiequency doubled using a
150m KDP SHG crystal. This allows background fiee detection with a photodiode,
while maintaining signal linearity. This can be seen if one considers that the probe
electric field is the sum of the laser and the heterodyne FWM signal. Alter fiequency

doubling, the electric field of the probe beam is:

E 2(t) = (E ”(1) + EmUDz = (15'3,.,.,a.(t))2 + 2Epmb¢(t)EFWM(t)+ (1'3matr(t))2 (2-1)

So long as the FWM signal is small with respect to the probe field, the
(E W (t))2 term is negligible. The (E pm,” (t))2term is likewise unimportant as it only
contributes to the signal at time zero. This leaves the linear term, showing amplification
of the FWM signal by the probe beam. The FWM signal intensity was found to be 300
times smaller than the probe beam, therefore the linear signal approximation is very
good.

The sample (99.1% N204) was obtained fiom Sigma-Aldrich and used without
further purification. The sample cell was evacuated to 10” Torr on the vacuum line and
filled directly from the lecture bottle. There was no experimental evidence of
contamination. Experiments were performed at different sample pressures. All the data
presented here was obtained with a cell that was loaded to an initial pressure of 400 Torr
at room temperature.

2.2 Theory

Simulations of TG data presented here follow the quantum mechanical expression

for the time dependent ntsceptibility,”"" (see appendix for derivation)
132(1) 05 N I335 (Iv PJ [ COS( 2 “(Fez-171)!) +008( 2 71807.! -FJ-2)t)], (2.2)

where F] = B,.l(.l+l)—D.12(J’-I-l)2 for the molecule, 3., is the vibrationally dependent
rotational constant, D is the centrifugal distortion constant, ,6 is the anisotropic
polarizability, N is the number density, p1 is the normalized rotational distribution
including spin statistics (3,1) for all natural isotopes, pJ=gr(2J+ l)exp(-27th/k7), and qt.
is the normalized thermal vibrational distribution for all ground state vibrational modes.

Due to the off-resonance nature of the process, Al==0,i2, rotational recurrences are seen

10

every rm=1/(4Bc). The homodyned TG signal is given by (an2 /dt)2, while the
heterodyned TG signal is given by (sz /dt). See Appendix 1 for a derivation of these
formulae. For samples containing molecules with different rotational constants, the
homodyned signal is given by |(dP., /dt+ de /dt)|2, leading to interference terms between

the two populations58

2.3 Results

In the gaseous state the N204 and N02 molecules are in thermodynamic
equilibrium (at room temperature K = 8.8 atm'l).59 We controlled the temperature of the
sample cell fiom 294 K (80% N204) to 363 K (94% N02). A change in optical density of
the sample at 800 nm fi'om 0.045 (294 K) to 0.071 (363 K) was observed and was not
I considered substantial to qualitatively affect the measurements. Some experiments were
carried out with a cell filled to 760 Torr. The overall signal level was somewhat smaller
due to the increased 0D leading to absorption of the excitation photons. The transients
yielded qualitatively identical results, but required a higher temperature to obtain good
N02 signal (383 K), probably due to a change in the equilibrium conditions and some re-
absorption of the signal. It is necessary to note that it is impossible to have 100 % pure

gaseous N204 in the cell because of the aforementioned equilibrium.

11

 

 

 

 

 

 

 

 

g .
E 1.
a .
-—- a
E
5 8
r< 800
7
l.
320 f i '
l—I .-
E 810 ,2 1 -
'E' 800 .5: i :
r< ’: - -
790 . j 3 - ‘
o 500 100 1500 2000

Time Delay [fs]

Figure 2.3.i Experimental time-resolved FWM signal obtained during the first 2
ps: a) Frequency integrated N02 data as a filnction of time delay, b) Frequency dispersed
signal obtained as a function of time delay for N02. c) Frequency integrated N204 data as
a function of time delay, d) Frequency dispersed signal as a firnction of time delay for
N204. For all scans the signal intensity is plotted on a logarithmic scale.

In figure 2.3 we present frequency-dispersed transients obtained as a function of
time-delay between the two time-overlapped pulses and the third pulse for N02 (2.3a)
and N204 (2.3c). In the two-dimensional contour plots (a and c) the spectrum of the
signal in the region from 780 nm to 830 nm is registered versus delay time every 10 fs.
The dark current level of the CCD was subtracted from these data. The intensity scale is
logarithmic to highlight the weaker features. In the case of N02 (figure 2.3a) there is a
broad, featureless peak 160 fs after time zero as well as another at 500 fs. Figure 2.3b

shows the frequency-integrated (780-830 nm) transient signal. These features (figure 2.3a

and b) can be assigned to the initial rotational dephasing in the sample and are consistent
with results for off-resonance FWM signal.”60 Measurements for N204 were collected in
the same manner as described for N02 and are presented in figure 2.3c and 2.3d. These
rewlts show in addition to a broad rotational dephasing envelope a series of beats with a
period of 133 fs. The beats are detectable up to 3 ps after time zero but with much lower

intensity.

 

 

 

 

 

 

 

 

 

 

 

 

 

L

 

1.0L - , . , . , . , . , .
z , . , , T 1
'3 q 1.0L .
5008. n g . . .
a , n Ell-0.8: ‘
0.6-
EGG- m g ’ , -
0.4- -
a . U .. _ .
80.4r- “2102b 4 -l
t- 2 ’
a. .l o. .
o
I

160 ‘ 260 ‘ 300
Frequency (crri‘)

,0

N
r
Q

 

0 A 500 A 1000 . 1500. 2000
Delay time (fs)

1

L J

J
251) 3000

 

 

 

Figure 2.4 Heterodyne detected FWM signal fiom N204 at 273K. Inset is a
MEM-FI‘ of the transient, showing the 79cm'1 and 81cm'l torsional fi'equencies, as well
as the 25 5cm'l and 234cm'l vibrational fiequencies.

Figure 2.4 shows the first 4 ps of the heterodyne detected FWM signal at 273K.
The initial rotational dephasing observed during the first picosecond is vibrationally

modulated. The 125fs oscillations correspond to the NN stretching motion of the N204

l3

molecule. Fourier transform of the signal results in a single fi'equency at 242 cm’l, which
is in agreement with the experimentally obtained value for the N-N stretching mode fi'om
gas-phase Raman spectroscopy.“ Applying the Maximum Entrophy Method (MEM) of
analysis61 on the same set of data yielded two frequencies; 255cm‘l and 234cm“. These
two values were consistently obtained following analysis of difi’erent data sets (not
shown). The wave packet motion involves at least three vibrational levels in the ground
state and from the spacing of the two peaks one can obtain an anharmonicity constant for
the molecule, v.x.=21cm". The 79cm’1 and 8 1cm'l fi'equencies correspond to the

torsional motion of the NN bond.“62

 

Intanslterel ]
O
t»
O\
U.)
W

 

Intensity [rel]

 

 

 

 

WWI“)

Figure 2.5.i Experimental time-resolved and frequency integrated Heterodyne
FWM signal (a) N02 and (b) N204 taken at 363 K and 294 K, respectively. Note the
subtle differences in the shape of the rotational recurrences obtained for both cases. The
signal intensity is plotted on a logarithmic scale.

Long time homodyne scans were performed at 363 K (N02) or 294 K (N204). We

present data in the time-delay range fiom 9 to 42 ps in figure 2.5. The transients show the

14

frequency-integrated intensity recorded in the 785-825 nm range as a function of time
delay. Note that the vertical axis (intensity) is logarithmic. The N02 data presented in
figure 2.5a, show rotational recurrences every 9.945 ps. The transient in figure 2.5b,
taken at 294 K, exhibits similar features with the same periodicity. The recurrences
observed at 294 K are about four times lower in intensity than those observed for N02 at
363 K taking into account the attenuation filter used for the higher temperature transients.
The transients for both temperatures were taken under identical laser conditions except
for attenuation. This results in slightly different background and noise levels between the
two transients. In this range of time delays both data sets exhibit rotational recurrences
that can be assigned to N02. Under close examination some differences are observed in
the shape of the rotational recurrences obtained in the two transients. The difi’erences are
especially clear at the half recurrence observed in both transients near 29.5 ps (see figure
2.5), and are due to centrifugal distortion efi‘ects.

Nitrogen dioxide is an asymmetric top molecule, but the similarity of the B and C
constants allows us to view it as a symmetric top with B...=(B+C)/2. From the observed
rotational recurrences of 19.680 ps (9.840 ps half-recurrences) in the long time delay
transient of N02 (figure 2.5a) we can calculate the average rotational constant as
Bm=ll4crm were c is the speed of light and r... is the time between rotational
recurrences.57’60 The obtained value is 0.42343 cm'1 :1: 0.00007 cm". This value can be
compared to the experimental (0.42202 cm")63 and theoretical (0.423325 cm'l)’3 values
calculated for BM, respectively. Simulations based on the semiclassical formalism
outlined in Appendix I and the average rotational constant for N02 using the literature

values are in general agreement with the observed data The rotational recurrences

15

obtained at longer time delays (1 > 40 ps) exhibit more complex modulation that cannot
be simulated with a symmetric top approximation, even though their positions can be
reproduced exactly.

In the case of N204 the signal obtained at long time delays is more complex. The
small features at 9.840 and 19.680 ps (figure 2.5b) could be assigned to rotational
recurrences originating from the 16% N02 molecules present at room temperature. The
positions of these features coincide precisely with the rotational recurrences of N02,
however, the signal intensity is about one order of magnitude stronger than expected
based on the observed intensity at 363 K and because of these measurements’
dependence on the square of the number density.”60 Furthermore, there are some subtle
difl‘erences in the shape of the recurrences (see figure 2.5) leading us to consider
assigning a significant contribution of these features to photoproducts that result fi'om
excitation by the first two pulses. The N-N bond dissociation energy is only 0.59 eV
while the photon energy of each pulse is 1.55 eV. The principal axis of the N02 moieties
is aligned with the N-N bond of N204, therefore, the initial rotational alignment would be
preserved in the photodissociation process. The product N02 molecules formed by
photodissociation would realign and become ‘visible’ to the third pulse at the same time
delays as the thermal N02 molecules originally in the sample, but would have a different
rotational temperature. There is a possibility that multiphoton excitation yields N02
molecules in the 2B2 excited state, which has slightly different (and still not exactly

determined) rotational constants.

l6

 

 

 

 

 

 

 

 

 

E E 363 K

5. “raft Maw
10: 4]
E i 294 K \ \ 3
: All i
«J MWJJW Wm

 

00000 70000 00000
Tm Delaytts]

Figure 2.6.i Experimental time-resolved and fi'equency-integrated signal obtained
at 363 K (a) and 294 K (b). The main difference between the two transients after the first
two picoseconds occurs at 76.265 ps and is indicated by an arrow. Both intensity axes
are plotted on a logarithmic scale. Some fine differences are observed in the modulation
of the rotational recurrences but not in their position.

Only one rotational recurrence feature was observed after the first picosecond that
can be assigned to N204. At 76.265 ps (figure 2.6) a small feature is present in the room
temperature sample that is clearly absent at higher temperatures. This feature gains in
intensity compared to the one at 78.540 ps (N02 recurrence) if the temperature of the
sample is lowered to 0° C (data not shown). The lower temperature shifts the equilibrium
further toward the dimer. If a semiclassical model and available rotational constants38’40
are used, it is possible to simulate this recurrence. The experimental data showed no
evidence of a half recurrence at 38 ps. Scans taken for time delays of up to 160 ps
showed no other feature that could be distinctly assigned to N204. One possible reason

for the absence of N204 rotational recurrences at long time delays is the rapid

thermodynamic exchange 2N02 .-—‘ N204 estimated at 300 K to be r~ 0.1 ns.

17

 

 

d

 

.
‘

 

‘
'

 

 

Heterodyne FWM Signal

 

. Heterodyne FWM Signal

 

 

 

. l l - l A l A

 

 

 

       

 

 

 

 

 

 

34 ‘ as_ as 40
i DelayTlme 1 (ps) a
.5 - I - I v I ' I ' ‘ 51m”
2100 - ‘ -
3 , 1
L: m. a E w'
. . .0
5 ° 5 o
g... .g
Enos. —D2h Simulation - gm. —D2h Simulation '
. i - __ -
E Sor‘sw . gm,” 80'8”
:1: -

34 as ' ‘ ' ‘ ' -
Delay Time t (ps) 7° 1).:le3 rim€°t (pr?

Figure 2.7 . Heterodyne detected Transient Grating spectroscopy of N204. Raw
data at the filll and half recurrence time of N204 at two temperatures (a and b,
respectively). Subtraction of the two different temperature signals reveals a clear filll
recurrence for N204 (d), but no half recurrence (0).

Heterodyne detected transients around the N204 recurrence time are shown in
figure 2.6. In the Heterodyne case, the phase and linearity of the signal is preserved so
signal subtraction is a possibility. In figure 2.6a I show Heterodyne detected Transient
grating signals from 0° and 86°C at the time of the first half recurrence of N204 (38.2ps).
figure 2.6b shows the first full recurrence of N204 at the same two temperatures. There is
clearly an additional feature around 76ps in the 0°C trace, but there is no clear feature at
the half recurrence time. Figure 2.60 and 2.6d show signal subtraction at the half

recurrence and full recurrence, respectively. The gray line corresponds to the expected

half recurrence signal from N204 in the th. geometry. The small differences at the

18

recurrence time of N02 are due to centrifugal distortion effects which are not accounted
for in this simulation. Clearly the half recurrence of N204 is absent, implying that the
geometry of N204 cannot be D2h. From the subtracted data, a more accurate rotational

constant of (B+C)/2=0. 1091 54.00001 cm'1 can be determined.

2.4 Discussion

The signal observed for both molecules is primarily non-resonant and therefore
depicts ground state dynamics. Experiments were carried out at both temperatures;
photon echo (PE) and virtual echo (VE) pulse sequences“66 were performed but yielded
no significant feature other than the time zero signal. The PE and VE pulse sequences are
produced by electronic coherences, and are isolated spatially from other processes due to
the phase matching geometry. The photon echo signal, for example, is produced by the
purely homogeneous decay of the electronic coherence, which is typically hundreds of
picoseconds long (see chapter 4 for more details). If the experiments shown here did
involve a resonant one-photon transition to the 2B2 state of N02, the PE and VE signals
would have revealed it.

The absorption cross section of N02 at 800 nm is very smallm’68 The reported and
calculated values of the B and C rotational constants for this state are very dissonant,68
and fall in the range fi'om 0.458 to 0.548 cm'1 and 0.38 to 0.449 cm'l, respectively. If this
transition were involved additional rotational recurrences corresponding to these values
would have been observed in the time-resolved FWM transients. Close examination of
the data did not reveal any other features except for the aforementioned recurrences every
19.680 ps. Given that homodyne detected FWM signal depends on the transition dipole

57,60

moment to the eighth power this process is not likely to be detected by our setup.

19

In conclusion, the average rotational constant measured fiom our experiments of
0.42343 cm'l reflects the average (B+C)/2 rotational constant of N02 in the ground state.
Similarly, using heterodyne detected transient grating spectroscopy, we were able to
measure the rotational constant of the N204 molecule (B+C)/2=0.10915i:.00001 cm'1 to
high accuracy. The experiments at room temperature (N204) allowed us to measure the
frequency of the N-N stretch of 242 cm'1 and the anharmonicity of this unusual bond. The
weakness and high anharmonicity of this bond is likely due to the changes in the
electronic symmetry of the N02 required to form the N-N bond, leading to its high barrier
to rotation and long bond length. The next electronic state known is the 3B3“ is >3 eV"6
and would require two photon transitions to be reached. There is no evidence in our data
of single or multiphoton transitions to a long-lived excited state. However, excitation to a
dissociative state would be difficult to rule out. Perhaps the negative time delay shoulder
in the N204 transient in figure 2.3d and not found in the N02 transient in figure 2.3b is
the evidence of this dissociative pathway.

The transient grating method allows us to measure, with high accuracy, rotational
constants for molecules that may be inaccessible by other means due to spectral
congestion or lack of appropriate dipole moments. We can also use half recurrences to
eliminate certain symmetry properties of molecules with indistinguishable nuclei. In this
case, rotational constants for the N02 and N204 molecules were determined to 10.5 cm'1

accuracy, and D21. symmetry for the N204 molecule has been excluded.

20

3. High Intensity effects

3.1 Introduction

High-intensity off-resonance laser fields from ultrafast laser pulses can cause very
large electric-field gradients. Molecules can experience a large torque along the
polarization vector of the field due to their anisotropic polarizability. With long laser
pulses the torque is enough to cause adiabatic alignment,“”20 however, with ultrafast
pulses the torque provides an ‘instantaneous’ kick toward alignment with the electric
field polarization.” Pulses with peak intensities of 1013 W/cm2 generate electric fields of
one volt per angstrom, causing significant changes in the electronic energy levels of the
molecule. When the electronic states have different geometry, the fast strong field
excitation process leads to molecular structure deformation. In this report we study
alignment and deformation for a number of molecular species using the time resolved
transient-grating (TG) method. For CS2, we obtain angle resolved ion yields following
Coulomb explosion in the molecular beam and find evidence for the molecular
deformation.

This chapter bridges these two types of measurements and provides experimental
evidence regarding alignment and structural deformation. The results presented here shed
new light on the understanding of the behavior of molecules in the presence of intense

and ultrashort laser fields.70

21

 

Alignment Deformatidn timing Coulomb
Ionization explosion

U.)
\
\
\
\
-w-----

 

 

 

...-t
\l
\

 

p—s

 

   

Log(Electric Field Trap Depth [K])
c N
¥
\
\
1~¥
\
N- 4
Electric Field Strength [VIA]

 

 

 

 

 

l 0 1 1 12 1 3 l4
Log(Peak Intensity [W/cmzl)

Figure 3.1ii Plot of the induced rotational trap depth on a highly polarizable
molecule like CS2 (dotted line, left axis) and the electric field strength (solid line, right
axis) as function of peak laser intensity. The predominantly observed molecular processes
that occur at any given intensity are indicated in the top.

Figure 3.1 illustrates the dependence of the electric field trap depth measured in
Kelvin for molecules with relatively large anisotropic polarizability (for example, CS2).
Laser intensities of 1011 W/cm2 are enough to align cold (10 K or less) molecules, for
example fi'om molecular beams."l Also shown in Figure 3.1 is the electric field strength

as a firnction of the peak intensity of the laser. The electric field is calculated using
E = 21Z0 , where Z0 is the resistivity of vacuum (376.7 9) and I is the peak intensity of
the laser pulse. For laser intensities of 1012 W/cm2 the electric field can exert sufficient
force on a molecule to cause alignment. Higher peak intensity fields ~10l3 W/cmz,

produce an electric field strength that is comparable to that experienced by valence

electrons in molecules (~ lV/A), and structural deformation takes place. At these electric

22

field strengths ionization takes place and at higher laser intensities multiple ionization
and Coulomb explosion are observed.

The observation of molecular alignment stemming fiom high intensity
nonresonant laser excitation has been theoretically explored by Friedrich and
Hershbach."”72 The intense laser field causes an induced dipole in the molecule, which
suppresses the rotational motion and leads to aligned pendular states."”72 Friedrich and
Herschbach developed the laser-induced alignment theory based on earlier observations
of dissociative multiphoton ionization experiments on C0 and 12 with intense infrared
lasers conducted by Normand et al.73 and Dietrich et a1. 33 respectively. Demonstrations of
the laser-induced alignment technique have been conducted by Kim and Felker on large
non-polar moleculesw’20 and by Stapelfeldt and coworkers on smaller molecules."’75
Posthumus et a]. have examined the multiphoton dissociative ionization of H2, N2 and 12
with 50 fs pulses and found that only H2 and N2 showed alignment characteristics.76
Corkum and coworkers have combined intense off-resonance chirped circularly-polarized
fields to induce rotational acceleration and thus constructed a molecular centrifuge.30
Alignment with resonant intense laser fields has been examined theoretically by
Seideman. When on-resonance fields are used, the intensity does not have to be as strong
as in off-resonance experiments to generate alignment.22 She noted that as the mass of the
molecule increased (and thus the rotational period increased), longer pulse duration is
required to align the molecules. The study by Brown et al; on the resonant excitation of 12
using 50 fs pulses confirmed the propensity for multiphoton excitation rather than
alignment for a relatively heavy molecule.77 Non-resonant excitation is advantageous,

especially with larger molecules, because there is less propensity for ionization.22

23

Ortigoso et a]. calculated the conditions (laser pulse and rotational constant of the
molecule) for which recurrences of alignment can be expected with nonresonant Short
pulses.78

One of the most widely used methods for studying the behavior of molecules
under intense laser fields is the Coulomb Explosion (CE) method. In the CE method, the
ultrafast intense field causes mulitple ionization of the parent molecule. Mapping the
distrubution of the fiagment ions gives information about the molecular structure prior to
explosion.28’29’79'93 While powerfill, it is very difficult to determine a sequence of events
leading to the observed fi'agment distribution using the CE method. For example, it is
important to determine if structural deformation takes place in the neutral molecules or it
arises after ionization Similarly, it is important to determine the laser intensity that
causes alignement and deformation, an aspect that is difficult to determine by the CE
method. This specific question is addressed in this study.

Research from our group has taken advantage of the experimental measurement
of rotational recurrences following the formation of an off-resonance polarization
transient grating, a method introduced by Heritage et al.94 With femtosecond pulses one
is able to make much more accurate determinations of rotational constants and hence
better determination of molecular structures”7 Soon after we recognized that the
transient grating method would allow us to measure alignment and molecular
deformation induced by the strong off-resonance field provided the third pulse (the

probe) was kept at a low and constant intensity while the first two pulses, which coincide

in time and form the transient grating, provide the strong field. This setup was used to

24

make our first observations of alignment and structural deformation following intense

off-resonance excitation.23’6°’95

3.2 Experimental

The data presented here comes fi'om two different laser systems. Most of the data
was acquired with the CPM laser system described in the introduction. The pulse energy
used for each experiment is indicated in the text. The transient grating experiments were
obtained using the forward box geometry. The first two high-intensity laser pulses
coincided in time and were followed by a weak probe pulse delayed by a computer
controlled actuator. The signal registered corresponds to the diffraction of the probe laser
from the transient grating formed by the two intense pulses. 5 7 The degree of molecular
alignment and deformation are measured from changes in the rotational recurrence
intensity and shape observed after field-fiee evolution of the rotational wave packets. The
data for CO2 was obtained using the regeneratively amplified titanium sapphire laser

described in the introduction.

25

 

MCP
Detector

  

m plate fragmen?
on Time of Flight
. a l .

.l _ @ G—gm-d

   
  

 

 

 

 

repeller

    
 

Transient

Molecular
Grating setup Sm"

23:20-2“ pm" Beam setup

 

 

 

 

 

Figure 3.2ii Schematic drawing of the experimental setup for time-resolved
transient grating (top) and molecular beam (bottom) experiments. The transient grating
signal was detected at phase matching angle as a function of time delay between the first
two strong-field pulses and the weak probe. Ion detection, in the molecular beam, was
recorded as a function of the angle that the polarization vector of the laser pulse makes
with the time-of-flight axis.

For TG experiments (see Figure 3.2 left) the intensity of the first two laser pulses
was varied between 1010 to 1013 W/cm2 using a set of matched neutral density filters, all
with the same substrate thickness. The intensity of the third pulse, the probe, was
maintained at a fixed 101° W/cmz. The signal was collected background-free at the phase
matching geometry. The samples were introduced in a 10 cm long cylindrical quartz cell.
In some cases, for example in CS2, a cold finger in the cell was kept in ice to reduce the
sample density to 100 torr. Sample densities were 400 torr for C2H2 and C02, 100 torr for
CS2, and 150 torr for C6H6.

Ionization data was obtained using a molecular beam apparatus (see Figure 3.2,

right). The samples were introduced using He as a carrier gas, which entrained some of

the vapor pressure of the sample. The sample and carrier gas (He, 1700 torr) expanded

26

through the 1 mm diameter aperture in the pulsed valve. A 2 mm diameter skimmer
produced the molecular beam, which proceeded to the ionization chamber. The pulses
0.25 m] in energy were focused onto the molecular beam by an f = 150 mm lens.
Diffraction size of the beam in the focal plane is estimated at 20 um and experimentally
determined beam size is 26 um. At the focus, the peak power on the molecular beam then
is 4.6><10l4 W/cmz. The voltages on the extractor and repeller plates were kept at 1800
and 2500 V respectively with a distance between them being equal to 1.5 cm. The
pressure in the ionization chamber was ~ 8><10’8 Torr when the valve was closed and
~7><10’7 when the valve was operated at 30 Hz. The ions were allowed to travel in a drift
free zone of 50 cm before being detected by a micro-channel plate. Twenty time-of-flight
mass spectra (TOF-MS) were averaged before they were analyzed. The polarization
dependence of the ion intensities was obtained taking advantage of a zeroth-order half-
wave plate, which was rotated manually from 0 to 180 degrees in 15 degree increments
with respect to the time of flight axis. A calcite polarizer was used to ensure very high

polarization purity (greater than 1:10000) of the beam.

3.3 Results

3.3.1. Evidence of Alignment and structural deformation from Transient
Grating Measurements

Three-pulse transient grating experiments were carried out to explore the
occurence of alignment and structural deformation. The first two pulses, overlapping in
time, provided the strong electric field, while the third pulse with much lower intensity,

probed the molecules at their rotational recurrence. The full rotational recurrence of CS2

27

is expected at 76.5 ps.57'94 When the laser pulses are weak, this feature can be observed
and accurate rotational constants can be determined.’ 7’60”” Fig. 3.3a, shows
experimental data for the case when all laser pulses are weak. The simulation of the data
with no adjustable parameters fits well the experimental results (see Fig. 3.3a). When the
laser intensity of the first two pulses is increased by an order of magnitude, the filll
rotational recurrence observed increases in intensity. The data obtained at this intensity
can still be fitted by a room temperature distribution of linear CS2 molecules (see Fig.
3.3b). Under these conditions, signal intensity can only increase if there is a larger
anisotropic polarization of the sample. This higher polarization results fiom a greater
number of molecules being pulled into alignment by the electric field. Previous work
describing the timescales needed for alignment have considered the very low temperature
available in molecular beams?!“ At the higher temperatures here, the time scale of
molecular rotation is much closer to the pulse duration of the laser. The RMS rotational
time of CS2 at room temperature is 1.8 ps (JRMS=62). This means that during the 60 fs

laser pulse, a molecule with the RMS rotational energy can rotate 15 degrees.

 

  
   

.2 C(3)' ' - - 7 2x 10'°'W/cmzr C(b) ' - l x 10"W/cm21kc). V 2x110'2‘W7cmiI
'0 : -Simulation: : -Simulation]: -Simulation}
- “Data x 2 ° Data .

     

 

 

 

 

FWM Signal Intensity (an)
N b O D

O

LLAAA-‘AA-AA‘AAAA-lAAAA ALLAAAAIAAAA‘LAJJ .............

75.0 76.0 77.0 78.0 75.0 76.0 77.0 78.0 75.0 76:0 77:0
Time Delay 1 (ps) Time Delay “I (ps) Time Delay t (pa)

 

iio’

Figure 3.3“ First rotational recurrence of CS2 at three difl‘erent laser intensities.
Probe intensity was held constant throughout at 2x10'°W/cm2. (a) Low intensity data
(circles) and simulation (solid line) based on unperturbed molecules. (b) Data (circles)
under higher intensity indicating laser induced alignment. The simulation (solid line)
assumes unperturbed molecular geometry. (0) High intensity data (circles) showing a
change in the observed rotational revival, evidence for structural deforrmtion of the CS2

28

molecule. The solid line in (c) is a simulation based on 35% of the sample having a bond
angle of 160°.

Simulations of TG data presented here follow the quantum mechanical expression for the
time dependent susceptibility given in Chapter 2 (see equation 2.2)

When the intensity of the first two pulses is increased above 1012 W/cmz, the firll
rotational recurrence increases in intensity and changes shape (see Fig. 3.3c). Most
markedly, the first feature increases in intensity by a factor of two. The features observed
at long time delays, >77.5 ps, increase in intensity and new oscillations are observed.
Attempts to explain these changes in the rotational recurrence structure by Raman
induced changes in the rotational population in the manner of Rosca-Pruna and
Vrakking,'°°'102 have previously been shown to be inadequate to explain our experimental
results.” Simulation of the data for high intensity excitation shown in Fig. 3.3c, requires
the introduction of bent molecules. A static bond angle of 160°
(B=0.110536cm",C=0.109995cm") was used in our simulation to reproduce the
experimental data.

3.3.2. Evidence of structural deformation for Coulomb Explosion

In order to corroborate the observed molecular structure changes we studied the
effects induced by the strong field in Coulomb explosion experiments. Figure 3.4a shows
the TOF-MS resulting fi'om excitation with pulses with 4.6><10l4 W/cm2 peak intensity.
In addition to the parent ions (CS2+), (CS22+) multiple fi'agments are observed. We can
distinguish the singly ionized fragments (C and S+) from the doubly ionized fragments
(C2+ and 82*). Figure 3.4b shows the ionization yield of the parent ion as a fiinction of the

polarization vector of the excitation beam. Notice that the yield is isotropic, as expected,

29

because of the isotropic distribution of CS2 molecules in the molecular beam. The data in
Figure 3.4b serves also to confirm that there is no systematic bias in the probability of ion
detection as a filnction of the polarization vector of the excitation field with respect to the

time of flight axis. A similar isotropic distribution was observed for the CS22+ ion.

 

 

 

 

 

 

 

 

 

 

 

 

5 (a)
.—.4 Zoomed 32"
=3
5.9. s3+°
2.3
l:
.03
52
C
.9 s

1 “i.

5|
0 2:5 5
Time of Flight [us]

Figure 3.4ii Time of flight mass spectrum obtained for CS2 molecules (a) when the
laser polarization vector is aligned with the TOF axis. The zoomed inset shows splitting
of the sulfilr fiagment ion peaks caused by their Coulomb explosion toward and away

from the detector. (b) Angle dependent distribution of CS2+ ions, showing an isotropic
distribution.

The angle dependent ionization yield for 8+ and for C+ was measured, and the
results are presented in Figure 3.5 in Cartesian and polar coordinates. The experimental

data obtained for three separate runs are depicted as circles, squares and triangles. The

lines are best fitted by cosine or sine functions to the fourth power. Notice that the yield

30

for 8" follows a cosinusoidal distribution, with maxima at zero and 180 degrees (see
Figure 3.5a). The yield for C ions follows a sinusoidal distribution with maxima at 90

and 270 degrees (see Figure 3.5b).

(a) 120 9°

 

 

 

 

 

 

 

 

 

 

 

 

180
210
--.- . 270
a. a I I I '3'
EE'LO. gyro
'g 01v i;
9 309+
C +4
3 0.6‘ E ‘
r0 . 4‘0
m 3
D. 0.4' a
C - . . - ~ c0.
2 0 90 180 270 360 2

' i - 130 270'360
Angle 0 90 Angle

Figure 3.5ii Angle dependent detection of singly charged fiagrnent ions following
Coulomb explosion. Anisotropic distributions are observed for C+ (a) and S+ (b) ions in
polar and Cartesian plots. Data for three independent experiments (squares, circles,
triangles) are accompanied by a guide to the eye (solid line) in the form of a 0084(9)
function. S+ shows a maximum ion yield at 0° and 180°, while C+ shows a maximum at
90° and 270°. The anisotropy for the C“ ions is less pronounced.

The angle dependent ionization yield for S2+ and for C2+ was measured, and the
results are presented in Figure 3.6 in Cartesian and polar coordinates. The experimental
data obtained for three separate runs are depicted as circles, squares and triangles. The

lines are best fitted by cosine or sine functions to the fourth power. Notice that the yield

31

for 82+ follows a cosinusoidal distribution, with maxima at zero and 180 degrees (see
Figure 3.6a). The yield for C2+ ions follows a sinusoidal distribution with maxima at 90
and 270 degrees (see Figure 3.6b). Notice that the results for the doubly ionized
fi'agments Show greater sensitivity toward the angle of the polarization vector of the
excitation pulse compared to the singly ionized fragments in Figure 3.5. This indicates
the doubly charged particles are ejected from a multiply charged parent with more

significant bending than the singly charged ions.

 

 

 

 

?:
.2.
E‘
“c’
2
E
x
t0
0.
n.
8......5...
— 0 90 180 270 360 _ 0 90 180 270 360
Angle Angle

Figure 3.6u Angle dependent detection of doubly charged fragment ions following
Coulomb explosion. Anisotropic distributions are observed for C2+ (a) and S2+ (b) ions in
polar and Cartesian plots. Data for three independent experiments (squares, circles,
triangles) are accompanied by a guide to the eye (solid line) in the form of a cos’(0)
function. S2+ shows a maximum ion yield at 0° and 180°, while C2+ shows a maximum at
90° and 270°.

32

The ionization data shown in Figure 3.5 and 3.6 is consistent with the angular
distribution of the fragment ions obtained with a titanium sapphire laser centered at 800
um.”88 Similarly we are able to interpret our data in terms of a molecular structure
deformation that is consistent with bending. The angular dependence observed in the
molecular beam is consistent with the molecular structure deformations measured by the
TG method on the neutral molecules.

3.3.3. Alignment and Deformation of polyatomic molecules

Figure 3.7 demonstrates the behavior of C02 molecules under strong field
excitation. These experiments were carried out with a titanium sapphire laser centered at
800 nm. For low intensities (Figure 3.7a) the rotational recurrence can be fit by a
simulation of C02 molecules at room temperature with overall signal intensity as the only
adjustable parameter. As the laser intensity is increased (Figures 3.7 b-d), the rotational
revival gains in intensity and changes overall shape. Increases in intensity can be related
directly to laser-induced alignment, while changes in the shape of the revival are linked
directly to changes in the molecular structure. Based on our simulation the angle of the
deformed CO2 molecules is 1613:05 degrees. For stronger fields (Figures 3.7e-g) a
background signal is observed and the shape of the revival changes substantially. The
background is due to ionization and plasma formation. The rotational recurrences are
deformed by interference between the plasma emission and the TG signal. This aspect

will be discussed below.

33

 

Vi

a" ”WW/cw _ (b) lxlO"W/crn' ” (c) “mum”: “ (d) 2xlo'=w1cm= .

  

>213 . . .. - °
a“, -S'nnulation.. -Sinrnlatioa .. -Srmulatron . . ”Salmon
3 - Datax30 - Daren - Data

  

 

     

 

 

 

   

 

 

    

   

 

 

 

20.5 2i.o 2115 220 22.5 20.5 2110 213 2’20, 225 205 21.0 2i.5 22.0, 22.5 20.5 2i.o 21.5 22.0 2215
T'nneDelayups) T'lmeDelrryturs) 'l'rrneDelayturs) T'rmeDelayt(ps)
3.......... .....
$14 (e) s x lolIWIm: (f) l ‘ lOI’W/m: (8) 3 x IOI’W/cm:
g.” —si..ul.tian -Sirnulation -Sinrrllation '
g 10 e D.“ ' M ‘ M ‘
3
.5
a
E»
00
E A A A A A A A A A A A A A AA A
E 20.5 21.0 21.5 22.0, 225 20.5 21.0 21.5 22.0225 20.5 21.0 21.5 22.0, 225
Time Delay t (ps) Tune Delay 1: (ps) Tune Delay t (ps)

Figure 3.7“ Time resolved TG experimental data centered on the first rotational
recurrence of C02, with the probe intensity held constant at 2x101°W/cm2. At low
intensities ((a) and (b)), the data (circles) matches well the simulation (solid line) based
on an unperturbed linear molecules. Near 1012 W/cm2 ((c)and(d)) the data shows
evidence of bending. Solid lines in (c) and (d) are simulations assuming 35% of the

sample has a bond angle of 160°. At higher intensity ((e), (t) and (g)), the signal appears
above a constant background signal. The simulation assumes that molecules in the

intense field have been destroyed and the signal arises fi’om unperturbed linear molecules
heterodyned by a constant background.

'Figure 3.8 analyzes the effect of strong field excitation at 620 nm on acetylene.
As the laser intensity is increased the intensity of the rotational revivals increases. Again,
this increase is directly linked to laser induced alignment. For the higher intensity data
(Figure 3.80), the intensity has increased by a factor of almost 480 and some deformation

is observed. Interestingly, the main deformation is a broadening of the first feature of the

rotational revival.

34

 

  
 

 

 

 

 

 

 

 

A V V ' V I V V V '* I ' ' V ' ' I ' ' Vi """""""""""""""""

=2 14 (a) 2 ‘ .. (c) :

g, 10 -r 1

.3 ’ 2x l0 W/cm 4 x lollw/cmlgg 8 x 10‘2 W/cmzi

8’» 1 ~19 ~i

. 4 . . 15 . . 5

IE 5‘. : 1 -Srmulatron3; -Srmulat10n;

§,:_ -Datax 12 j; - Data

in i §E
E 2~ I et-
0" L— d:

A 3.5 A A ‘70 A A 75 0.0 ‘ ‘ A ‘65 “““ 7.0 “““ 7.5 “““ 0 .0‘ ‘

Time Delay r (ps) Time Delay r (ps)

 

Figure 3.8ii Time resolved TG experimental data centered on the first rotational
recurrence of acetylene, probe intensity held constant at 2x1010 W/cmz. At low intensity
(a), the data (circles) matches well the simulation (solid line) based on unperturbed
molecules. As the laser intensity is increased (b and c), the first peak of the recurrence
broadens, beginning to overlap with the second peak. The factor of 40 in intensity from
(a) to (c) can be explained by an enhanced alignment of the acetylene molecules in the
laser field.

Finally, Figure 3 .9 demonstrates the structural deformation of benzene as a
function of 620 nm laser intensity. The data obtained for low intensities (Figure 3.9a) can
be fit with the spectroscopic parameters of room temperature benzene. For higher
intensity excitation, there is not a large change in the amplitude of the features, indicating
that the alignment observed is not as substantial as that observed for the linear molecules.
Already for intermediate intensities (Figure 3.9b) the first feature in the rotational revival
is broadened towards early times, a feature that was observed for acetylene. For higher
excitation energies, the signal level increases by a factor of two and there is substantial

molecular deformation.

35

 

-(a) . (b) ‘(°)
2x loIOW/cmlj 3x10“ W/cm2 ’ 7x10'2W/cm2

—Simulation 7 — Simulation '
- Data x 2 ' Data

00
O

  
  

   

ox

— Simulation'
- Data x 3

  

       

N

O

 

 

 

 

FWM Signal Intensity (a. u.)
‘

 

45.5 44.0 45.5 45.0 45.5

45.5 45.0 45.5 45.0
Time Delay 1' (ps) Time Delay t (ps)

44.0

45,5 45.0
Time Delay 1 (ps)

Figure 3.9ii Time resolved TG experimental data centered on the first rotational
recurrence of benzene, probe intensity held constant at 2x10lo W/cmz. When the grating
intensity is low (a) the data (circles) is in good agreement with the simulation (solid line)
assuming unperturbed molecules. At 3x10ll W/cm2 (b), the first peak of the recurrence
broadens, and by 7x1012 W/cm2 there is evidence for significant deformation of the
benzene molecule. Simulation in (c) assumes a symmetric top with a C3 symmetry axis
(ring puckering mode).

3.4 Discussion

There are usually two views of off-resonance laser induced alignment. The first
one involves adiabatic alignment induced by a slowly increasing electric field (long laser
pulse) that creates a directional force that traps the molecules in pendular states.71 As the
electric field strength is reduced to zero, the rotational distribution in the sample is
restored unaffected, hence the adiabatic terminology is used. This picture is consistent
with experiments involving a nanosecond laser pulse as the source of the strong field.
The second view involves an instantaneous ‘kick’ toward alignment, a view that is
consistent with the interaction of an ultrashort pulse with supersonically cooled
molecules.21 The experiments presented using the TG method at room temperature are in
an intermediate regime. Molecules with low angular momentum feel a kick towards

alignment, essentially not moving while the field is on. Molecules with high angular

momentum are able to rotate on the timescale of the laser pulse and therefore their

36

alignment is more consistent with the adiabatic picture. The data shown in the results
section is consistent with alignment induced by strong off-resonance electromagnetic
fields. As shown in Figure 3.1, the trap depth as a function of peak intensity reaches
hundreds of K for lasers with peak intensity ~ 1012 W/cm2 and the resulting alignment of
the room temperature molecules becomes possible.

In addition to alignment, the data presented here is consistent with large
molecular structure deformations. In the case of linear molecules, bending can be
understood in terms of electronic state mixing. For C82 and C02 singlet and triplet
excited states are bent. In Figure 3.10 we show a schematic representation, fi'om a
‘dressed state’ point of view,"’3’104 leading to molecular deformation. Potential energy
curves in the absence of external fields are available in the literature. 105407 In Figure 3.10

'08 as dressed by two 620nm photons and overlapping

we show the electronic ground state
the bent lBz electronic state. The ground state wavepacket, originally at the equilibrium
geometry evolves within the pulse duration, ~ 60 fs, away from linear geometry. Once

the field is turned off, the molecules are left with high amplitude vibrational excitation.

Therefore, the rotational recurrence indicates a deviation fi'om the linear geometry.

37

 

'2 e a a
l
b»
t

.a
O
l

 

Potential Energy (103 cm'1)

 

l l l_ 1 l

l
140 130 220
Angle

 

Figure 3.10ii Schematic representation of laser induced molecular deformation
process of C82 molecules based on a dressed state picture. The laser dresses the ground
state and brings it close to the excited states having a bent geometry. The wavepacket has
a short time to evolve in the dressed state potential during the laser pulse. The system
returns to the ground state with substantial bending vibrational energy, leading to a
deviation from linear geometry, and producing a higher rotational constant.

One of the advantages of the TG method to assess molecular deformation
following strong field excitation is that the molecular geometry can be measured long
after the field has been turned off. The molecules rotate field free and their rotational
revival reveals the average bent angle (the deviation from linear geometry). In the case of
C82, the data is consistent with 35 % of the population achieving an average bend angle
of 20 i 0.5 degrees. This value is very close to the measurement from Iwasaki,“ who
determined the average bent angle following strong field excitation of C82 by angular
dependent ionization to be 18i3 degrees for C82”. The angle we have determined (160°)
is between the equilibrium angle of the 1132615..) and 1A2(1Au) states of csz.”9 The angle

and percentage of the population that is deformed was determined by a series of

38

simulations shown in Figure 3.11. Figure 3.11a shows that as the bend angle increases the
rotational revival shifis towards earlier times (the rotational constant B increases).
Because only a portion of the population is bent, (the 35% value arises from the fact that
72% of the intensity of a Gaussian pulse is contained in the central part of the beam), the
rotational revival from all the molecules in the sample needs to be added and then
squared to obtain the observed heterodyne detected signal.”’60 When we add 35% of bent
molecules we see changes in the resulting signal as shown in Figures 3.11b-f. The
simulation in Figure 3.11e fits our data closely, while the simulation in Figures 3.11d and
3.11f do not. Because the TG method is much more sensitive to neutral molecules than
ions, because of their greater polarizability, we can also establish that the molecular
deformation takes place in the absence of ionization. We have carried out ab initio
calculations108 to explore molecular structure changes induced by strong (0.5V/A)
electric fields. We have found that when the field is perpendicular to the molecular axis, a
bent molecular structure is preferred. The potential minimum of the polarized molecule
no longer corresponds to linear geometry but to a bent geometry. We have confirmed this
by ab initio calculations for C82 in the presence of a strong 0.5V/A field. The overall
change in the potential energy surface, however, is not large enough to explain the

experimental observations.

39

 

(b) n
. » — 35%
n bent 5°

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A
'° }
0
l
"g 3 .5 6 i 3
o (c) (d)
5 - n— 35% . —35%
b bent 10° bent 15°
.5 1 p I
c:
8
E I i I ‘
“a . U J U
E» 3 r a i i .2 -i a i i
m . - e - - a . - h .
(e) (0
I n _35% ‘ ' _ 35% '
LL; bent 20° bent 250

 

 

 

 

 

 

L
A-

U

4-5624-3-3653

Delay Time 1: (FWHM recurrence width)

A
'-

 

 

 

 

 

Figure 3.11“ Simulation showing the effects of molecular deformation on the
rotational recurrences of a generic molecule. The time axis is in units of the filll-width
half maximum of the peaks of the recurrence, with zero being the recurrence time in the
linear molecule. Plot (a) shows recurrences from the bent species, showing the increase in
rotational constant as a function of bond angle. Plots (b) through (i) show the recurrences
of a sample comprised of 35% bent molecules and 65% linear molecules for bond angles
from 5°(b) to 25° (t). The simulation in panel (e) is in good agreement with our
experimental findings shown in Figure 3.3c.

The bending angle found in the TG data is consistent with the Coulomb explosion

data taken in our laboratory with the same laser system. If the molecules remained linear,

4o

a case depicted in Figure 3.12, the signal from the S+ ions would show a marked
anisotropy. The anisotropy would result from two situations. When the electric field
vector is aligned with the TOF axis the 8+ ions are ejected towards and away from the
detector. Because of the repeller plate, the fragments that are ejected away from the
detector are deflected back towards the detector. This leads to a well known splitting in
the ion signal, evident in the zoomed insert of the mass spectrum in Figure 3.4 for 82+
ions, and used in photofiagment spectroscopy analysis.110 Note that molecules
perpendicular to the field are not affected by the field and yield no ions. When the field is
perpendicular to the TOF axis the 8+ ions are ejected at a high velocity in a direction that
is perpendicular to the detector. Despite the acceleration voltage, the Coulomb repulsion
is significant and in the time it takes those ions to reach the detector plane (~ 4 us) they
are already several centimeters apart and are not detected. For this reason the 8+ ions
show a distribution that can be simulated by a cosinusoidal function For linear
molecules, the detection of C+ ions is not affected by the angle of the polarization vector
of the electric field. The predicted detection is therefore isotropic, a fact that is not

supported by the data shown in the experimental data in Figures 3.5 and 3.6.

41

 

Electric Field Coulomb Explosion

Polarization
<—) (000+

8+ C+
@m g 00

Predicted lon Detection

 

 

 

 

 

 

Figure 3.12ii Conceptual drawing of the Coulomb explosion of linear CS2
molecules when the electric field polarization is parallel or perpendicular to the time-of-
flight axis. The detection of 8+ fiagment ions is expected to show a marked anisotropy.
However, the C+ fragment ion detection is expected to be isotropic.

Figure 3.13 considers the angle dependent detection of fragment ions resulting
from bent C82 molecules. Unless the bending angle is very large, there is almost no
change in the anisotropic dependence of the 8' ions discussed above for linear molecules,
see Figure 3.13 top. However, for bent molecules the C+ signal becomes anisotropic. The
anisotropy results from the following observations, based on the assumption that the
bending angle is small and that the polarizability of the molecule remains along the two S
atoms. When the electric field vector is aligned parallel to the TOF axis the bent C82
molecules can have the two 8 atoms along the field or perpendicular to the field. If the

molecules are oriented along the field, then the C)r ions are ejected in a direction that is

perpendicular to the detector and miss it. When the electric field vector is aligned

42

perpendicular to the TOF axis the bent CS2 molecules oriented along the field eject C+
ions toward the detector and register maximum signal. When the molecules are oriented
perpendicular to the field, signal is minimal because the perpendicular component of the
polarizability is very small for small bend angles. From this analysis so far, it is clear that
C ions are predicted to show anisotropic detection that is perpendicular to that observed
for 8+ ions if the CS2 molecules are bent. The predicted anisotropic ion signal fiom the
Coulomb explosion from bent CS2 molecules is shown in Figure 3.13. It is difficult to
make an accurate determination of the angle from this data without coincidence imaging.
A simple simulation of the parallel and perpendicular components of the polarizability as
a fimction of bend angle leads us to estimate the bend angles from the singly and doubly
ionized data sets to be 170 and 155 degrees respectively. The agreement between the
findings of Iwasaki et al using mass — resolved momentum imaging (MRMI) and 800nm
excitation86 wavelength and our angle determination is good We attempted to measure
the behavior of CS2 molecules under intense 800nm radiation using the TG method in a
static cell. Unfortunately we found that for laser intensities of 1012 W/cm2 and higher a
chemical reaction occurred in the cell and all the sample in the cell decomposed. After a

number of attempts we abandoned this experiment.

43

 

Electric Field Coulomb Explosion
Polarization
000 ->
<——> C,
0° A

90° 3"

TOF Axis

<—-> 000
0° V

 

 

 

 

 

Predicted lon Detection

 

Figure 3.13ii Conceptual drawing of the Coulomb explosion of bent CS2
molecules when the electric field polarization is parallel or perpendicular to the time-of-
flight axis. The detection of 8+ fiagment ions are expected to show a marked anisotropy
similar to that observed for the linear molecules. However, the C+ fragment ion detection
is expected to be anisotropic and perpendicular to that observed for the S+ fiagment ions.

The transient grating method is ideal to measure the signal from neutral species
with high anisotropic polarizability. For CO2 molecules the bond angle of the bent
species reaches 161 degrees; this value should be compared to the 140 degrees

determined by Hishikawa et 01.82 As the electric field is increased and ionization takes

place the TG signal decreases and gives way to a background signal caused by the
plasma. In Figures 3.7e-g, we showed CO2 molecules in the presence of an increasingly
strong electric field. Notice that the background increases and that the rotational revivals
change shape. Under these conditions, the plasma emits a spectrally shifted light pulse (a
spark) that provides a local field to heterodyne the TG signal.“l 1' Under these
conditions, the TG signal changes fiom being homodyne detected to heterodyne
detected.”’56 The weak heterodyne signal corresponds to linear CO2 molecules that are
practically unaffected by the electric field. It is very possible that this signal arises from
molecules outside the main focus of the laser field. Banerjee et al. have explored
differences observed in ionization and Coulomb explosion as a function of the electric
field intensity profile at the focus of the laser.” Their observation of single ionization in
the wings of the focus and highly charged species at the center is consistent with these
findings, that is, ionization and plasma formation in the focus with weak signal fi'om
unperturbed molecules from the wings.

Molecular deformation in larger molecules is evident in the experimental data
presented here. In the case of acetylene, the electronically excited state responsible for its
absorption in the ultraviolet region is known to have a staggered (trans) geometry.“
Based on this knowledge we can determine the deviation angle from linearity by fitting of
the experimental data. This data is consistent with bending of 60 i 10 degrees as shown
in Figure 3.14a. Because hydrogen atoms are very light we are unable to make a more
accurate determination of the angle. A similar analysis was used for benzene in the
presence of strong laser fields. The data clearly shows a distortion, however, the

rotational constant does not change significantly. This implies that the molecule is not

45

stretched in the plane; it is bent outside the plane. Based on this observation we assume
that the molecule assumes a puckered geometry that maintains the molecular symmetry
and achieves an angle of 15 :l: 3 degrees out of planarity, see Figure 3.14b. This
observation is consistent with the experiments of Shimizu who studied the molecular

deformations of benzene under strong field excitation by Coulomb explosion.”2

 

(a) acetylene if 1A,, state
120°

      
 

 

(b) benzene deformation
165°

 

 

 

Figure 3.14ii Molecular models depicting the average bent angle observed for
acetylene (a) and benzene (b) consistent wit the experimental TG data obtained in the
presence of strong laser fields.

3.5 Conclusions

In this chapter I have presented a study about the behavior of polyatomic
molecules in the presence of strong off-resonance ultrashort laser fields. The
experimental data are consistent with alignment of the molecules. This alignment results
from a ‘kick’ induced by the electric field acting on the polarizability of the molecule.

Our data provides clear evidence of significant molecular structure deformation resulting

46

from the interaction with the strong field. These observations are consistent with
Coulomb Explosion measurements from our group and fi'om other gl’oups.6(”32’112

Recent experiments on laser control of photodissociation reactions using shaped
pulses require a careful evaluation of the effect of strong off-resonance laser fields on the
electronic potentials of molecules“10 In particular, it is important to determine to what
extent these fields align and deform the neutral species. It would be especially important
to determine if the observed chemistry takes place in the ground or electronic excited
states of neutral molecules or if the chemistry takes place after ionization. The study
presented here, especially the TG measurements on the neutral molecules indicates that
the electric field can have a major influence on the molecular structure without causing
ionization. This study, and that of Banarjee et al. highlight difi‘erences caused by the
gradient in the electric field intensity of Gaussian Pulses, an aspect that has not been
addressed in strong field laser control. The temporal envelope of the excitation pulse has
not received sufi'lcient attention in studies of molecular deformation. In our case the pulse
duration (60 fs) is similar to the bending period of ground state CS2. Therefore during the
laser pulse the nuclei have time to move away from linear geometry. Once the field is
gone, the ground state potential is restored and the molecule is left vibrationally excited.
It would be interesting to explore the effects of pulse duration and excitation with
different temporal envelopes. This research could yield methodology to figure out the
design of the field to carry out a specific chemical reaction such as isomerization.

In conclusion, the time resolved TG method is ideal to measure molecular

alignment and molecular structure deformation induced by strong laser fields. The

method provides a high degree of accuracy, and has been used to determine structural

47

deformation in C 82, CO2, acetylene, and benzene. It is important to point out that the TG
measurement is made by the time-delayed probe pulse, which has minimal influence on
the molecular system and arrives a long time alter the strong field has been turned off.
This is in contrast to other ionization-based experiments, where the strong field affects
the molecule and releases the charged particles. The TG measurements here are
consistent with alignment and bending dynamics that take place in neutral molecules, in

the absence of ionization.

48

4. Femtosecond photon echo measurements of electronic
coherence relaxation between the X(lzg+) and B(3l'In..+) states of

12 in the presence of He, Ar, N2, 02, C311,,

4.1 Introduction

There is considerable theoretical and experimental interest in controlling the
amplitude and phase of quantum mechanical states in atoms and molecules.”’6°’113 The
motivation behind these efforts comes fiom evidence that intramolecular dynamics and
chemical reactivity can be controlled using phase and amplitude shaped pulsesz’9 A
second motivation is the future construction of a computer that will operate through the
coherent manipulation of individual quantum states."‘""6 For both of these applications,
coherent interactions between the laser field and the atom or molecule are required. Here
we will explore the loss of electronic coherence in gas phase iodine molecules, a model

117,118
as a

system that has been explored for the coherent manipulation of information,
function of number density and the addition of different buffer gases. The rate of
decoherence determines how long information can be manipulated without loss of
fidelity. The loss of coherence and the nature of the long-range interactions between
ultracold atoms and Bose-Einstein condensates has become a subject of recent
interestm”120 These measurements may help to shed light on that subject as well.

This chapter is a continuation of work fi'om the Dantus laboratory, where we have

used femtosecond three-pulse four-wave mixing (F WM) methods, involving different

49

pulse sequences, to measure many of the processes that contribute to coherence
relaxation in ground and excited state iodine.""""""66’121 We make the distinction between
different measurements, such as electronic coherence decay (involving two electronic
states), from vibrational or rotational coherence decay (involving vibrations or rotational
motion in a given electronic state), and spatial coherence decay (involving the motion of
atoms or molecules in space). The focus of this chapter is on the nature of interactions
(long-range collisions) that lead to a loss of electronic coherence (decoherence). For these
measurements, we can ignore the lifetime of the upper electronic state, T1, because it is
on the microsecond time scale. These measurements involve pure iodine as well as iodine
in the presence of buffer gases that range fi'om He atoms to propane molecules.

The signal measured from coherent spectroscopic methods, such as FWM,
depends on phase coherence among all molecules emitting the field. As the phase
coherence is lost the strength of the signal diminishes. Measuring the rate of decohaence
and understanding the nature of the interactions that cause it in gas phase environments
are the goals of this work In the gas phase, decoherence typically results from long-range
interactions, which cause a phase change in the polarization of the single molecule.
Figure 4.1 shows an iodine molecule in an electronic coherence between the B(3l'Io..+) and

the X('Zg+) states, showing the potential energy curves for the relevant electronic states.
The molecule, whose polarization oscillates with fi'equency a), can be perturbed through
long range interactions with buffer gas molecules. This perturbation changes the energy
of the B and X states and result in a time dependent frequency change Aalfl). The

accumulated phase change Ad; that results from one such long-range interaction leads to

50

the measured dephasing as the emitted electric field destructively interferes with the
emitted fields fi'om other molecules in the sample.

In this study, two types of pulse sequences are used: photon echo (PE) and
reverse transient grating (RTG). 122 Both methods measure the electronic coherence decay
that results fiom the first order polarization pl" (see Figure 4.2); however, they have a
very different dependence on homogeneous and inhomogeneous contributions. In
condensed phases, the inhomogeneous contribution to coherence loss is very significant
making PE measurements the only viable method to obtain measurements of electronic
coherence decay in the condensed phase. In the gas phase, inhomogeneous contributions
are much smaller; therefore, both PE and RTG can provide valuable information. In
principle, for very low pressures and small molecules it is possible to measure the
Doppler free line width of individual spectroscopic transitions to estimate the pure
homogeneous broadening. However, for room temperature iodine in the presence of
buffer gases, time resolved measurements as performed here are clearly advantageous
and provide both homogeneous and inhomogeneous decoherence rates. The theory
section, describes the signals from the two methods and recover the isolated molecule and
condensed phase limits. The results section shows that, under certain conditions, these
measurements involve an intermediate case, between the two limits, that has not received

much attention in the literature.

51

 

 

 

Aaxt)=AUB(t)-AUX(r)
A9: =IAw(t)dt ‘

     

Energy , 104cm'1

 

 

 

2T5 3T0 3'5 4'0
Radius(l-l) ,A
Figure 4.1.iii A graph of the potential energy curves (solid line) involved in the
electronic coherence between the B and X states of iodine. Dashed lines represent the
perturbation caused by the approaching buffer gas molecule. The overall change in
energy is indicated by Am. The phase shift Ado necessary to destroy the coherence during

a long range collision is given by the accumulated phase shift taking place during the
interaction.

4.2 Theory

The intensity of the PE signal as a filnction of delay time in the liquid phase has
long been known to conform to an exponential decay with time constant ‘ATz', where T 2'
is the homogeneous relaxation time.123 Similarly, it has been shown that for gas phase
samples, the photon echo decays with rate l/2T 21124 This section explores the reason for
the factor of two difference for these two limits, and develop a means of extracting T 2'
even from the intermediate region where neither the gas phase nor the liquid phase limits
apply.

To correctly describe the evolution of the system including relaxation and obtain
an expression that bridges the gap between the liquid phase and the gas phase limits we

will need to use a density matrix approach. The formalism we use to describe PE and

52

RTG is based on this approach, which was published earlier."4 Here we apply it to derive
simple formulas for the intermediate between liquid and gas phase cases. For PE and
RTG, the first pulse arrives at time zero while the second and third coincide in time,
following after a delay 2'. In the case of PE the emission goes in the phase matching
direction kpfk3'i'k2-k] (responses R11 and R1”), while in the case of RTG the emission
goes in the phase matching direction kRm=k3-k2-hh (responses R1 and RM)” as depicted
in Figure 4.2. There are no differences in amplitude between the Liouville pathways R1
and RW and between R” and R111 for a two level system. Notice that the PE signal has a
maximum at time 27. For RTG the maximum is at zero time (before the second pair of
pulses, in the “virtual” non-physical region), as shown in Figure 4.2. The difference
between PE and RTG is very important when molecules within the ensemble have

different resonance conditions because of inhomogeneities.

53

 

 

 

  

Reverse Trandent Grating

 
 

 

 

 

Figure 4.2. Pulse sequence and phase matching geometry for PE and RTG
measurements. The PE setup (a) with pulse b arrives before pulses a and c undergoes

rephasing and has a maximum at time t after the second two pulses. The RTG (b) with
pulse a arriving before pulses b and c, undergoes no rephasing and is thus maximum at
the time of the second two pulses.

The inhomogeneous relaxation can be described by the normalized Gaussian spectral
distribution exp[-(a)-ab)2/A2] of each transition fi'equency with width A The signal
measured as a function of the delay 1 corresponds to the temporal integral of the emission
intensity. We can separate the electric field of the PE and RTG signals because they are

generated in different directions. It is possible to analytically calculate the relaxation in

54

the case of an exponential homogeneous decay and a Gaussian inhomogeneous decay.

For these cases we obtain

SPE(T)G1 Iexp(—271)exp(—(t — 2r)2 A2/2)dt =
r

exp(2;QA-2)[Lena“tA+2°-’m‘)](2/tt)°-’/A (4.1)

SRT0(T) 0C Iexp(—271)exp(—t2A2/2)dt =
T

exp(2;2A—2—4 27)[1+erlI2'o‘5rA-Zo‘syA'l)](2/1t)°‘5/A (4.2)

The first exponent for both cases is the homogeneous decay (2 7 is the decay rate which is
twice the decay rate of the elements of the density matrix). The second Gaussian function
(AZ/2 -— which is twice as fast as the Fourier image of the inhomogeneous distribution in
the frequency domain) is centered at time Zr for the PE (echo spike) and at zero time for
the RTG (inhomogeneous dephasing).

We define the homogeneous relaxation time in terms of the relaxation rate T 2'=1/7 and
the inhomogeneous relaxation time in terms of the inhomogeneous spectral width
T 2‘=1/A. In the limit 1T2. << T 2'2 there is no difference between the PE and RTG signals.

In this case, Equations 4.1 and 4.2 can be simplified to obtain
Spec) = Same) as exp(-227) = exp(-21/T2'), (4.3)

In the opposite case rT2'>>T2'2, the liquid phase limit, there is a difference between the

PE and RTG signals. We can replace the integral formula for the PE Gaussian field with

55

a delta firnction at time Zr and get a simple decay with rate 47. In this case equations 4.1

and 4.2 can be written as follows. (see appendix II)

Spec) oc exp(-4T}’) = exp(-4 tm'), (4.4)

5mm cc exp(-'/2A212) = exp(- 22/213”), (4.5)

Note that for short time delays, 1, the product 1T2, is always small. Therefore, for
small delays we expect the gas phase limit to be a good approximation. For large I, the
gas phase limit is no longer accurate even for gas phase experiments. Figure 4.3
demonstrates this transition between the two limits. Note that initially the PE has a decay
rate of 27, then goes through an intermediate region and, finally, at long delay times, has
a rate of 47. In the case of RTG, the initial exponential decay with rate 2 7 transforms to a
faster Gaussian decay. For neat iodine at low density we found the PE and RT G signal
decays are dramatically different; exponential for PE and Gaussian for RTG.64 In the case
of high density buffer gas or very short time, homogeneous dephasing is faster than
inhomogeneous dephasing, making PE and RTG decay with approximately equal rates
(27). This condition was accomplished for experiments with relatively high pressure of
buffer gas and is different fiom the previous work fi'om our group, where the

homogeneous dephasing much slower.64

56

 

 

Log10(lntensity)

 

 

 

 

 

 

 

o ‘fo ‘1‘00'150‘200'250
T , PS

Figure 4.3. Calculated PE and RTG decays (7=2x101°sec'l and A=2x1010sec'l)

showing both gas and liquid phase limits. At short times, both PE and RTG approach the

gas phase limit (thin solid line), at longer times, PE follows the liquid phase limit (thin
dashed line), and RTG decays non-exponentially.

4.3 Experimental

The experiments were carried out using the CPM laser described in the introduction For
some experiments, requiring longer pulses, the bandwidth of the CPM was purposely
narrowed to ~1 nm and shifted to 615 nm. The fourth stage of the amplifier was double-
passed, producing pulses that were 1.2 ps in duration with average pulse energy of l m].
Pulse durations were determined by autocorrelation in a second harmonic generation
crystal.

The laser was split into three beams of equal intensity, which were attenuated to less than
80 1.1.] per pulse (short pulse mode) or less than 300 uJ per pulse (long pulse mode) and
recombined at the sample in the forward box geometrym’m (see Figure 2.2). The beams

occupy three corners of a 25 mm square and were determined to be parallel over one

57

meter by using a template of the appropriate geometry. A 50 mm diameter 0.5 m focal
length lens focused the beams into the sample cell, the beams crossing at an angle of 29°.
This crossing angle produces a transient grating with a 16 um spacing at 620nm. At the
temperatures used in these measurements, the transient decays by diffusion after 60 ns.
This timescale is three orders of magnitude longer than the coherence decay times being
determined and is therefore neglected.

A computer-controlled actuator delayed the first beam, and the other two beams
were overlapped in time. The pulse sequence defines physical process emission of which
we detect in phase matching direction km= ltd-kb-l-kc (see Figure 4.2). By changing the
arrangement of the beams at the lens we controlled whether beam b or beam (1 was first.
PE measurements with ka= -k1+k2-hk3 require beam b to arrive at the sample before
beams a and c, while RTG measurements with kRm = k 1-k2+k3 require beam a to arrive
before beams b and c.“5 Time zero for both beam arrangements occurs when all three
beams are overlapped in time. The temporal overlap of the beams was found using a
removable 0.1 mm thick quartz plate.

The signal beam was collimated with a 0.5 m focal length lens, identical to the
one that initially focuses the beams, and was spatially filtered through a 50 um pinhole
before being sent to a 0.27 m monochromator for detection (homodyne) by a
photomultiplier tube. Data for the short pulse laser was collected at 620 nm, and for the
long pulse laser data was collected at 615 nm. Both detection wavelengths represent the
center of the laser’s Gaussian spectral profile in the particular laser arrangement to which
they correspond. Data were collected using a boxcar integrator, averaging 30 laser pulses.

The laser pulse intensity was monitored with a photodiode and pulses with energy outside

58

1.5 standard deviations from the mean energy were discarded. Typical data sets were
averaged for at least ten scans of 300 time delays each.

Sample cells consisted of 1 inch quartz cylindrical tubes 4 inches in length, with
optical windows. The cells were pumped to 10'5 torr while the solid iodine sample
(Kodak Chemical) was frozen with dry ice. The sample cells were then thawed, refiozen
and pumped out again. Buffer gases (AGA) were added at room temperature and buffer
gas pressure was measured with a baratlon on a sealed gas line. Optical density
measurements were made using a CW intracavity doubled Nd:YVOs laser (532nm) and a
photodiode covered by a 530i10 nm bandpass filter to exclude stray light. The number
density of iodine in the neat iodine cell was determined by transmission using established
iodine absorption cross-section data.128 The Nd:YVO4 laser was determined to be of
sufficient bandwidth to blur any fine structure on the absorption curve. This was
confirmed using an absorption spectrometer with resolution set to half the laser
bandwidth. A curve of iodine numerical density versus temperature was created. Our
measurement of the extinction coeficient, a: 813i20 Lmol'lcm'l is in excellent

agreement with that of previous measurements: e= 810 :I: 24 Lmol‘lcm".128 Using

0D =snl we calculated the number density, n. These data agreed favorably with

129”” in the temperature region considered here.

measurements fi'om two other sources
Buffer gas sample cells were all heated with a heating tape until the desired iodine
number density was reached. For pure iodine measurements, the number density was
varied from 1x1023 m'3 to 8x 1023 In3 by controlling the temperature of the sample cell.

All measurements involving buffer gases were made at a number density of iodine 5x1023

in3 and “0°C-120°C.

59

4.4 Results

4.4.1. Experiments on neat Iodine vapor

Photon echo traces obtained from pure iodine vapor were taken as a firnction of
number density, using 60 fs laser pulses. Figure 4 shows a plot of the homogeneous
relaxation rate, 7, as a function of number density. Each point on this plot represents a PE
data set at a specific temperature in the pure iodine cell. A typical data set involved the
measurement of PE signal intensity as a function of time delay between the first pulse
and the other two pulses. Inset is a typical PE data set plotted on a logarithmic scale. The
PE decays were measured from r= -10 ps to as much as r= 800 ps, with at least 300 data
points per scan. The final data sets were the result of from 10 to as many as 100 averages,

requiring approximately four hours of acquisition time.

 

T ' I ' I 1 I ' l T T Y 1 ' l ' l

p
1

o=1mrm,a’ i

  
  

 

   

l (a.u.)r'i

‘

   

 

 

 

Relaxation rate 1 , 10’ sec"

 

' PE Slgna
uh
I

3420 ‘400 .
..flPQ‘F’.
2.0 so 6.0 8.0
Number Denslty 102’m"

Figure 4.4. Plot of homogeneous relaxation rate, 7, versus number density for neat
iodine vapor. Results for 37 measurements are shown. Horizontal error bars indicate
uncertainty in the absorption cross section of iodine, vertical error bars indicate
uncertainty in the exponential fitting routine. Dotted lines represent the 90% confidence
interval based on a linear least squares fit. Inset is a typical PE data trace plotted on a
logarithmic scale. This trace corresponds to a number density of 5x1023 m'3.

Experimental decays were fit to exponential decays of the form exp(-27r) to
obtain 7. It has been verified previously that PE signals from gaseous iodine decay
exponentially“130 We use equation (3) to fit the PE data because the Doppler dephasing
time T 2' for iodine under these conditions is 720 ps, making the condition rTz' << T 2'2
valid. This was verified experimentally, as the RTG signal decays exponentially with a
strong temperature dependence. The solid black line in Figure 4.4 is a linear fit to the
data, and the dotted lines represent the 90% confidence interval for that fit. Error bars in
each dimension represent two standard deviations of the uncertainty of our
measurements. Uncertainty in the decay rate comes directly from noise in the data set.
Uncertainty in the number density is due to uncertainty in the absorption cross-section of
iodine. The uncertainty in the number density used in Figure 4.4 results fiom the
uncertainty in the absorption cross section reported previously. '28
From a plot of coherence decay rate versus numerical density, we extract a cross section,
6, for electronic phase relaxation using the equation

7 = nor": +70, (4.6)
where n is the number density of the gas sample, 9‘ is the average relative speed of the

colliding molecules i7 = (8kT)" 2(7:,u)"” 2 , and 70 is the relaxation rate at zero pressure.

A fit through the experimental data in Figure 4 allows us to determine a cross-section of
1170 i 110 A2 for neat iodine vapor, and a 1/70 of 583:4 ns. This 1/70 value is in good
agreement with our estimation of the diffusional dephasing, 60 ns. The cross section
obtained from figure 4 is in good agreement with the measurement obtained earlier in our
group, 1150 j: 150 A2.“ The electronic coherence decay cross-section for iodine obtained

by Zewail et al is 590 j: 110 A2. This value differs from our findings by a factor of 2.131 It

61

is unclear how the T 2 values were obtained from the data in this reference. The difference
could be explained if they obtained T 2' directly from the observed echo decay time
without the factor of 2.
4.4.2. Experiments with buffer gases

Electronic coherence relaxation rates were measured as a function of number density for
various buffer gases. The number density of iodine in the cell was monitored with a CW
laser beam as described in the experimental section. PE transients were obtained for
several buffer gas pressures for each of the different buffer gases. Each of the data sets
was fit to Equation 4.3 (single exponential decay with lifetime l/(2 7), and 7 values were
extracted. These data are shown in Figure 5 for PE in the presence of He, Ar, 02 and N2.
The solid lines are linear fits to the data, from which coherence relaxation cross sections
were extracted (see formula 6) and collected in Table 4.1. Error bars represent the 90%
confidence limits for each data point based on the uncertainty in the fit. Error in the
pressure of the buffer gas is smaller than the data points, about $2 torr. For all cases the
data points fall within the linear fit taking into account the uncertainty in the
measurement. Values for 7were also extracted using Equations 4.] and 4.2 (to take into
account inhomogeneous broadening), with a two-dimensional non-linear least squares
fitting method. These 7 values agree with the values extracted using Equation 4.3 to

within experimental error.

62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 . . . ,. 4. . . a
3 (a) Helium U ; ; (C) Oxygenz
‘5' 3; O=135i12A2 r 7703‘: 6:450i40A .v /_‘
8 : ’ it 33 ‘ ,1
in . $3 I
9 2 l? - o
9 . 1".
r:1: j 3
01‘ Z,” 1 I ' ‘
I F 0 . 5
0 Density? 1024 "14316 DENSITY, ‘102‘11'1'1'3
3i (b) Argon ' : (‘1) N1 01% 2 t’ :
.. ‘ cr—500_L70A2 ‘ 703: 0:30 "A -:
' ‘ . q) ‘ l.’ .
§2j i 3” i
2 . "02‘;
$3 ‘ ‘1 1 ,
£1i F1“: *
03. a ;
o . 5 a 10 ‘0 .1'0 2'0
DeHSIty , 1024f" Densrty , 1024 m‘3

Figure 4.5. Plots of homogeneous relaxation rate 7, of PE signal with 50 fs
excitation pulses, versus number density for (a) Helium, (b) Argon, (c) Oxygen, and
(d) Nitrogen. Error bars indicate uncertainty in the fit. Error in the number density is
smaller than the data points. The point at zero number density in each plot represents the
relaxation rate of Iodine at 5x10” rn 3number density, uncertainty in this number rs
smaller than the data point in both dimensions.

63

Table.4.l Parameters calculated for the long range interactions responsible for electronic
coherence dephasing of 12 molecules and buffer gases (M).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M o, A2 R=crml7rv2 vrris'l e I): 2’; 1E , eV (.i6m6 g 36:53

He 135112 6.5 3: 1.5 1435 0.6 021 a 24,619 a 4.9x10’77 0.306x10'7"
Ar 500170 12 i: 3.5 485 3.4 163 a 15.78] “ 43.6x10'77 2.06lx10’77
N2 300:50 10 i 1.6 567 2.3 1.76 b 15.581 ° 14.2x10'77 2.215x10'77
02 4503:40 12 3: 1.1 534 3.0 1,60 b 12.0697 ° 36.9x10'77 1.816x10'77
CsHs 500155 12 :1: 1.3 465 3.6 6.31 a 10.962 " 41.3x10'77 6.861x10'77
12 f “(7)0 i 19 3: 1.4 274 10.1 9 d 9.307 ° 190x107" 9.0465(10'77

 

a. D. A McQuarrie, J.D.Simon, Physial Chemistry, p.668, University Science Books,

Sauslito, California,1997.

b. D. A McQuarrie, Statistical Mechanics, p.481, University Science Books, Sausalito

California, 1997.

c. W. G. Mallard, NIST chemistry webbook, http://webbook.nist.govlchemistry/ (2001).

d. B. Friedrich, Phys Rev A, 2000, 61 025403

e - at 383 K

f- o=1150i150 A 1‘ 5902110 A 26

g — calculation with r'6 potential and experimental data.

h- calculation with formula for the dispersive force for parameters for Iodine in X state.
4.4.3. Photon Echo and Reverse Transient Grating Measurements

For all the gaseous samples we measured both PE and RTG transients. RTG

measurements are helpfill in determining the inhomogeneous contribution to the

relaxation rate, A (see theory section). Figure 4.6 presents data obtained with 5x1023

iodine molecules per cubic meter and 2.5x1024 propane molecules per cubic meter. Data

were taken using 1.2 ps pulses to blur the vibrational dynamics of the coherence. This

allowed us to obtain a smooth decay across the entire temporal range. The dots are

experimental signal intensities plotted as a function of delay time t. The solid lines are

fits using Equations 4.1 and 4.2, the dashed lines are exponential decays (pure

homogeneous dephasing) using the 7determined by the same fit. The difference between

64

PE and RTG decay times is small and within the experimental error. Similarly, the fits
obtained from the theory (Equations 4.1 and 4.2 for the PE and RTG) and the simulations

obtained from the single exponential decay (Equation 4.3) are within experimental error.

 

 

 
  

 

       
   

 

 

 

 

0.1 . . . . . . . . 0.1 . . . .
. (I) PE , L(It) RTG
=_ - y- 913x109 see-1 3. _ t, 7- 7t2x109 see“
“i “I A = 3x109 sec"
2‘ 2‘
'6 0-01 7, 0.01 l
s s
g ..-- e 2T: :- _-- a 2T:
— Theory . a
0.001 ' EXP“. "'9.” . 0.001 E.“ ""1"“ . .
0 so 100 150 0 50 100 150 200
T , ps 1: . ps

Figure 4.6. Logarithmic plot of (a) PE and (b) RTG signals with 1200 fs
excitation pulses, as a function of time (5x1023‘m'3 Iodine and 2.5x102‘m'3propane at
400K). Experimental data (dots) are fit (solid line) using the theory without
approximations, Equations 4.1 and 4.2, presented in the text. The dashed line is a
simulation based on a simple exponential decay with a homogeneous decay rate
determined by the full fit.

Figure 4.7 shows data points obtained from 37 independent transients for PE and RTG in
the presence of propane at various pressures. The values for the decoherence cross
section obtained (see formula 6) from these two plots are in agreement to within
experimental error. From this we can conclude that inhomogeneous broadening plays a
minor role, with an upper limit of 3x109 3". This observation indicates that under these

conditions, T 2'2>> T 2'r, confirming that the gas phase limit applies for delay times 1 less

than 1 us for the I2-propane case presented.

65

 

 

 

 

 

 

v.- 4 .

8 1a)PE

3” : Propane

é 3"; 0:500:55A2

2<

L“ 1

s 1%

'5 :

g 0e

m I I I I l l

a: 0 5 10 O 5 10
Density , 1024 m'3 Density, 1024 m-3

Figure 4.7. Plot of the coherence relaxation rate 7 versus pressure for propane
buffer gas. Both Photon Echo data (a) and Reverse Transient Grating data (b) are shown.
The plots contain a total of 37 independent measurements. Error bars indicate uncertainty
in the fit. Error in the number density is smaller tlmn the data points. The point at zero
number density in each plot represents the relaxation rate of Iodine at 5x10” m'3 number
density, uncertainty in this number is smaller than the data point in both dimensions.

4.5 Discussion

The measurements of electronic coherence relaxation allow us to calculate cross sections.
Assuming collisions occur at an average distance, R, determined by 6 = 1ER2, we can then
calculate the average radius of interaction. The decoherence cross-section for iodine-
iodine collisions was measured to be 1170 A2, giving us an interaction radius of
R = 19 A Notice that this value for the interaction radius is more than ten Angstroms
greater than the Van der Waals radius (see Figure 4.8). Since we are dealing with the
relaxation of electronic coherence, no energy needs to be exchanged by this interaction.
Here we will use a hard sphere model to describe these interactions, drawing an

“interaction region” around the molecules in the coherence. We define this interaction

66

region such that any molecule passing through it will produce at least a 7: shift in the

phase of the electronic wavefunction of the iodine molecule.

 

25-—
.l

R (A)

01

 

1111111111

0

 

 

 

 

Figure 4.8. Depiction of the distances R involved in the long range interactions
responsible for electronic coherence dephasing. Molecules are depicted as 98% electron
density maps calculated using the Spartan molecular modeling program.

The colliding partner, moving with a velocity in center of mass coordinates l7 ,
interacts with an impact parameter b, the shortest distance between the two molecules
during the interaction. The interaction potential AU(r) is a function of the distance
between the two molecules, r, and it reflects the perturbation in the energy diference

between the B and X electronic states of iodine caused by the buffer molecule (see

Figure 4.1).

67

If we picture the interaction region as a sphere around the iodine molecule, using the
experimentally determined R (see Table 4.1), we can estimate how long these dephasing

interactions take, based on the average speed of the molecules, 17. The average time of
each interaction assuming an average interaction length, i =(4/3)R, is then given

by; = 1717. The values obtained using our measurements are compiled in Table 1 and
range from 0.6 ps to 10 ps for helium and iodine, respectively. The intermolecular
distances R, determined in this report, are graphically depicted in Figure 4.8, while the
molecules are represented as 98% electron density plots calculated using the Spartan
molecular modeling program. 132 The above analysis of the experimental data shows that
the interactions are long range when compared to the Van der Waals radii, and short lived
when compared to the lifetime of the electronic coherence.

As a plausible approximation, we assume the interactions to be dispersive and to follow
an r45 dependence, as obtained from the second term in the Lennard Jones potential. ‘33
Based on the distance and weakness of the interaction we assume the molecules follow a
linear path through the interaction region. The change in phase of the iodine

wavefunction is then134
|A¢(b)| = (2m)! AU(t)dt = 3nc6 /(8h17b5) (4.7)
0

We can then calculate C6 parameters for the 12 + M interactions, given in Table 1 using

the formula
C,5 = (8/3)17h(a /7t)7‘5 (4.8)
The values obtained are summarized in Table 1 and range from 4.9><10'77 to

190x 10'77 J m6 for helium and iodine respectively.

68

In order to learn about the nature of the long range dephasing collisions, we assume the
intermolecular interaction is purely due to dispersion forces as a first order
approximation. We can calculate the dispersion parameter using and the expressionm
C6“,=(3/2)a,a21112(11+12)", (4.9)
where a, and a; are the ground state polarizabilities of the two molecules, and I 1 and 12
are the ionization energies of the two molecules. The resulting values are summarized in
Table 4.1 and are plotted in Figure 4.9. Errors in the experimental numbers in Figure 4.9
represent 2 standard deviations. Comparing the experimental to the calculated C 6
parameters in Figure 4.9, we note that both follow a similar trend. There is about one
order of magnitude difference between the two sets of parameters. This difference is
primarily caused by the greater polarizability of the B state, a parameter that is not

presently known, but is expected to be greater due to the B state’s triplet character.

69

1091 o( Ce/ABOV )

I Experiment
0 Theory
0 15x Theory

 

1 He 02 N2 Ar C3H8 '2

Figure 4.9. Experimental (square points with uncertainties) and calculated (round
black points) Lennard Jones C6 parameters for the buffer molecules in this study. Both
data sets follow very similar trends, however experiment and theory values differ by a
factor of 15. The round open points correspond to the calculated parameters multiplied by
a factor of 15 to compensate for the greater excited state polarizability (see text).
Multiplying the calculated points by 15, using the values obtained fi'om the noble gases
as guidelines (see open dots), the simulation reproduces closely the observed trend and
close agreement is observed for He, 02, Ar, and 12. Only the Lennard-J ones parameters
for nitrogen and propane are a factor of three smaller. These smaller differences may be
related to the absence of non-bonding electrons in these molecules. The factor of 15
increase in the polarizability of the excited state can be justified based on calculations of
the polarizability of the ground and excited states of a number of small molecules.‘” In
this study it was determined that the polarizabilities of the excited states of many

diatomic molecules were between 12 and 20 times the polarizabilities of the ground states

of those molecules.

70

For this analysis, a number of processes that take place during collisions have
been considered and ruled out. Collision induced predissociation in iodine was studied
previously. ‘36 The cross-section for Iodine — Argon collision induced predissociation of
the B state was measured to be 11.143. This is 1/100 of the cross-section of pure
electronic dephasing, measured here, and for pressures less 1 atrn can be assumed to be
negligible. Collision-induced electronic state mixing for Iodine - Argon collision has also

been studied.137

The cross section for this process is of the same order of magnitude as
the hard sphere (Van der Waals) radius which is two orders of magnitude smaller than the
dephasing cross section measured in our paper. Therefore this process is assumed to be
negligible. Rotational-vibrational dephasing of iodine was also studied, for example. 138’!”
It was shown that all these relaxations (actually intramolecular dephasing of rotational-
vibrational wave packets) take place during the first several picoseconds (less than 15ps
for our spectral range of excitation) after excitation of the B state, an order of magnitude
shorter time scale than our measurements (up to 200ps). This type of intramolecular pulse
relaxation was discussed in our previous work“

The long-range interactions described in this report may involve other effects
(quadrupole-induced dipole, quadrupole-quadrupole, etc.). Here we do not speculate
beyond the fact that the experimentally derived values can be simulated by a model that
only takes into account pure dispersion interaction estimates. Perhaps these

measurements will encourage further theoretical analysis about the nature of these long

range interactions.

71

4.6 Conclusion

I have presented a number of photon echo measurements aimed at determining the
electronic coherence dephasing time of gas phase iodine vapor, neat and in the presence
of buffer gases. We have written the explicit dependence of the dephasing rate on the
homogeneous and inhomogeneous contributions. With the firll expression, one is able to
calculate the dephasing time T 2’ without assuming a gas or liquid phase limit. From the
experimental decay rates, dephasing cross sections have been extracted, and these are in
good agreement with previous measurements in neat iodine.

The gas phase limit applies to the systems studied here, therefore only
homogeneous relaxation plays a role and rap = 1/(2 7). The Dantus group’s earlier work
on iodine at lower densities had shown some intermediate behavior where
inhomogeneous relaxation was overwhelming and the liquid phase limit applied.64 It is
interesting to note that the differences between quenching rates for the various buffer
gases seem to have no dependence on the number of degrees of freedom available in the
molecule, and only depends on the relative polarizabilities of the quenching species. The
similarity between (for example argon and propane) could be accounted for by the large
distance of interaction and the modest amount of perturbation required to cause electronic
dephasing.

In summary, the electronic coherence dephasing rates and cross sections have been
determined for iodine molecules in buffer gases that range fiom low density to the
condensed phase limit. These cross sections were determined using a simple hard-sphere
model, but seem to describe long range interactions between the iodine and buffer gas

molecules.

72

5. Multiphoton Intrapulse Interference: Binary Phase Shaping

5.1 Introduction

Amplitude and phase shaped laser pulses are often used to control chemical reactions
and other physical phenomena. A typical pulse shaper is shown in figure 5.1. The laser
pulse is spectrally dispersed and focused through what is known as the 4F geometry. The
pulse shaping device, usually an LCD Spatial Light Modulator (SLM),7 is placed at the
Fourier plane. Voltages applied to the SLM produce a change in the retardance of the
individual SLM pixels. In this way different phases can be applied to each fiequency
component of a laser pulse. As some shapers control both amplitude and phase for at least
128 pixels across the laser spectrum there is a very large space of possible pulse shapes.
For a system with P different phases and A amplitudes across N pixels, there will be
(PXA)” possible pulse shapes. Because the field needed to optimize a given chemical
process is not known, the use of evolutionary learning algorithms (ELA) guided by
experimental results in a closed feedback loop was proposeds. Experimental
implementation of this method and successes in control of chemical reactivity have been

reported and reviewed.9’”°'141

 

   

r

flom “Sapphire Oscillator

 

 

 

Pulse Shaper

 

Figure 5.1: A typical 4F pulse shaper. The SLM placed at the Fourier plane
applies different phases and amplitudes to different fi'equency components of the laser
pulse.

73

In the low intensity regime, laser control is dominated by interference of different
nonlinear optical pathways connecting the initial and final states.142 The challenge is
. finding the proper phase for each frequency within the pulse to achieve constructive
interference at the desired pathway and destructive interference elsewhere. Here we
consider two-photon excitation of fluorescent probes, as used in two-photon microscopy,
as the target for optimization. In particular we envision two chromophores with different
two-photon absorption spectra and our goal is to achieve selective excitation by

‘focusing’ the energy available at a specific region of the two-photon spectrum, as shown

   
 

  
    
     
  

   

in Fig. 5.2.
1,
C = SIB
e' '0 ---------- 3
'00 I 1
c y r I
2 r I
t: .i I l
(D s 1,1
I J 1":
‘0 lit
- r
0.110 ........ 5
o T r . A r

 

 

 

 

' ' 1
Frequency (3°

Figure 5.2.iv Schematic representation of the problem. The broad bandwidth
second harmonic spectrum fi'om transform-limited pulses is represented by a Gaussian
(thin line). The objective is to introduce phase modulation to cause the two-photon

spectrum to be intense only inside the window defined by frequency Zinc and width W,
and to minimize the background B outside the window. The contrast ratio C is defined as
the integrated intensity inside W divided by the integrated intensity of light outside the
window.

Experiments from our group have already shown that selective two-photon excitation
is possible when one is able to tune the narrowed non-linear power spectrum to optimize

excitation of one chromophore versus another. ”3"“ Here, however, we realize that the

74

phase difference between different fi'equencies should take only two values, 0 or rt, to
maximize or minimize a given pathway.

In order to control two-photon excitation, one needs to control the non-linear power
spectrum of the laser Ez)((0),l45’146 which can be measured by obtaining the second
harmonic spectrum generated using a thin second harmonic generation (SHG) crystal.
The experimental and theoretical application of this new paradigm, which we call binary
pulse shaping (BPS), is presented herein. The effect of spectral phase modulation on

ad147-149 and

SHG has been studied by a number of groups and can be divided into bro
nsnewm"51 phase matching bandwidth.

Conceptually, the problem depends on the intrapulse interference between pairs of
frequencies. If we only consider two phases, zero and 7:, we can analyze the problem in
terms of symmetry. To maximize the SHG intensity at a frequency 2 are the spectral
phase needs to be symmetric or antisymmetric about at, such that the fi'equencies
interfere constructively. To minimize the background intensity at all other frequencies,
the spectral phase must be asymmetric with respect to all other frequencies away from
we, so that destructive interference is maximized. Here we propose the use of prime

numbers to help us generate the quasi-random phase changes required. The mask to be

used to modulate the pulses is shown in Fig. 5.3, and is designed for a 128-pixel SLM.

75

<13 LLL- “JJM_

1235 7 1113 1719 23 2931. 37 4143 47 53 64
SLM plxel

Figure 5.3.“ Phase mask proposed based on the symmetry requirements of the
problem, using the quasi randomness of prime numbers. This mask is reflected about

pixel 64, and is designed to obtain a narrow second harmonic signal at the center of the
spectrum.

5.2 Experimental section

The experiments are carried out using the titanium-sapphire oscillator laser system
described in chapter 1, capable of generating pulses as short as 10 fs after a double pass
prism compressor and a pulse shaper.143 The spectral phase of the pulse is tailored using a
computer—controlled pulse shaper of a design similar to those described by Weiner,7 and

”3"“ (see figure 5.1). For the experiments shown here, the pulses

described elsewhere
were centered near 800 nm. The spectral phase of the laser pulses was corrected using the
multi-photon intrapulse interference phase-scan (MIIPS) method, which compensates
phase distortions to obtain transform-limited (TL) pulsesm’144 The binary phase was
introduced as an addition to the compensation phase.

The shaped laser pulses, with an energy of ~ 0.5 11] per pulse and 87 MHz repetition
rate, were focused mildly, to a spot size of ~ 100 microns in diameter, on a 20 micron

thin BBBO SHG crystal. The fi'equency doubled light was collected with an optical fiber

and dispersed on a compact spectrometer.

76

5.3 Results

First, consider spectral amplitude restriction to solve the problem. By narrowing the
spectrum of the laser using a slit as shown in Fig. 5.4a, the SHG intensity obtained after
amplitude restriction is shown in Fig. 5.4b. For a spectral width of 10% of the available
bandwidth, amplitude restriction produces 100 times less second harmonic intensity than
transform limited pulses. Although the contrast ratio for this case is favorable, the low

intensity obtained at the desired wavelength makes this method experimentally

 

 

   
    

 

 

 

 

 

 

 

 

 

 

unpractical.
r r r I r r r r r T l l
1 o - _ - 1.0 -
_ -_- Experirnental_ _ T.» '" _F.xperiment:
— 5m :Th
as i. - 0.8 i- ‘_ 000’ "‘
i— -1 g '- 3. ‘ "‘
E , a.
8 0.6 — - 5 0.6 — ° : ‘
3 5 n ' _
, _ - 1.
c (D
_ 0.4 F J 5 0.4L- / ..
0 2 — - 0 2
.. | ,
7 o 740 80 820 =91 900 350 370 390 410 430 450

WWB'GV'Q“~l (nm) Wavelength (nrn)

Figure 5.4.iv Effect of spectral amplitude restriction on SHG. (a) Experimental
spectrum of the laser before (black) and after filtering with windows of width 40 (red), 20
(blue), and 10 (green) nm. (b) Experimental (points) and simulation (continuous lines) for
the second harmonic spectrum of TL and spectrally filtered pulses as indicated in panel
(a)-

The phase mask shown in Fig. 5.3 was programmed on the SLM and resulted in
dramatic narrowing of the SHG spectrum as shown in Fig. 5.5. The contrast ratio as
defined in Figure 5.1 for this mask is 2.5 when the SHG peak is centered. By shifting the

position of the mask on the SLM, hence tuning the center of symmetry, one can tune the

SHG spectrum across approximately 35nm in the SHG spectrum. Sinusoidal phase

77

modulatiorr,”3'”6'148 can not produce contrast ratios greater than 0.5 in theory, and as the

window is tuned away from the central frequency the contrast ratio drops below 0.1.

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

V T I I I l l l T I l I I F I I l I

a 1.0 i ’i“ T gnu :1- _

EU 4 b _ 4 _

.8- " u‘ H ‘0. II _
t — I ...

:5 r I“ g I. ‘1

ca 0.5 '- ‘, $.05 :

3‘ ' 'U I

'51 . g

.6 .' e;

5 E

0.0 . . 2° 0 o
700 74c 780 820 860 900 '3 370 390 410 4 450
Wavelength (nm) Wavelength (nm)

Figure 5.5.iv Experimental results with binary phase shaping. (a) The spectrum of
the laser (dashed lines) and the binary phase mask (0 or it) are shown as a function of
wavelength. (b) The second harmonic spectrum of the TL pulses (dashed line) and
second harmonic spectrum of the shaped pulses according to panel (a).

The gray line in Fig. 5.5 corresponds to theoretical calculations. The absolute value of
the spectral amplitude of the electric field was calculated from the experimental power
spectrum of the fundamental pulse [((a) using lE(a))| = 1(0))”. To simulate the
experimental results a double Fourier transform method was used. The electric field in

the time domain E(l) was calculated as the Fourier image of the complex spectral

amplitude in the frequency domain, with the formula
EU) = ilE(w)|eXPli¢(w)]eXp(-iat)dw, (5.1)

where the spectral phase ¢(a)) is the function that is introduced by the SLM. The power

spectrum of the SHG was calculated using

Isno(w) = |1E(t)2exp(iat)dt|2. (5.2)

78

The SHG amplitude was normalized using the maximum of the SHG intensity calculated
for TL pulses, ¢(ro) s O.

The agreement between experiment and theory in Fig. 5.5 is satisfactory given that
there are no adjustable parameters. Differences between experiment and theory are due to
the resolution of the pulse shaper at the Fourier plane and errors in the applied phase
mask due to cross-talk between adjacent pixels. These resolution and calibration issues
cause the experimental SHG spectrum to be slightly broader than expected, though the
contrast ratio remains 2.1, comparable to the theoretical contrast ratio of 2.7 .

To determine if a better solution exists, one can then use an evolutionary learning
algorithm (ELA), assuming a Gaussian electric field corresponding to a 10 fs pulse

centered at 800 nm. The SHG intensity was calculated according to
Isne(2at) = IiE(aL-w)E(ar,+ «2)de (5.3)

normalized to the maximum SHG amplitude for TL pulses.

BPS simplifies the calculations, especially if we assume the amplitude of the electric
field to be a constant, that is, the spectral power is equal to 1 in the allowed spectral
region. Each spectral component of the electric field, linearly dispersed in the fi'equency
domain, can be represented as a binary value (i 1) determined by bk = exp(ifi), for
d1, = 0 or 7:, respectively. The intensity of the SHG signal measured at fiequency 2a), can

be calculated with the formula

Sk=l>31bkdbk+j|2 (5.4)

79

where the integral in Eq. (5.3) is now replaced by a discrete sum. The problem of spectral
selectivity can now be formulated as finding a vector bk such that St = 1 for a), = a1; and

Sr is minimized at all other frequencies.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I ' 1 ‘ l ' l ' T ' T ' l ' l ' l
6 ...........
1.0 b I " \ ’ 1-
' V ' /\ ' ' " \ 5. .0" .
/ y I
'éfi E r \ .3 ’ .’ I
E’ ' \ a" .
I g \ E ’ .0
- so“ . . a . 1 1' ‘ fish.
:1 anwxfwe C'-
s i , x \ a" .
- f . i . \ 1;
2." W \ .
.5 E0 \ o ...........
c 05 L1t"tt-:224‘ll_e‘41io156112121: __ 0 56101532431
313 Pue' ’ Inltlalx10 °"°" "
c I
— b . d
I Flna|x10
g I
(I) ' lnitialSIHG Spectrum

Final SHG Spectrum
I
" I
I
0.0 - ‘

4.428 4.728 5.028
SHG Frequency , 1015 s"1

 

 

 

 

 

 

 

 

Figure 5.6.iv Comparison of a prime number inspired phase mask (black) with a
mask that was optimized using a computer based learning algorithm (gray). The insets on
the left depict the phase masks for each case. The inset on the right shows the
improvement in the contrast ratio, as defined in Fig. 1, as the learning algorithm finds the
best solution starting fi'om the prime number phase mask.

Figure 5.6 compares theoretical results from the phase described in Fig. 5.2 (black)
and the result of the ELA (gray), which was initiated with the proposed phase. The ELA
used 100 individuals and one parent through 30 generations using single double and triple

symmetric bit flips. The result from the learning algorithm shows ~2.5 greater contrast.

80

Experimentally contrast ratios greater than 5 are difficult to obtain when the spectrum is

152 and

not linearly dispersed over the SLM pixels. Using an SLM with more pixels,
grouping those pixels such that each group covers the same frequency range, would
improve the implementation and result in greater contrast, an observation based on these
calculations.

5.4 Discussion

The goal of narrowing the SHG spectrum was motivated by the possibility of selective
two-photon microscopy. To achieve this goal two conditions had to be satisfied,
maximizing the nonlinear power spectrum at 26%, and minimizing it elsewhere.
Maximization has been investigated previously by Silberberg and coworkers, who
identified the condition «waquafi w) for continuous functionsm’”4 Zheng and
Weiner explored the narrow phase matching bandwidth SHG output at a single
frequency, a problem that is analogous to two-photon excitation of an atom, using binary
encoded pulses as used in communications.‘”’151 Zheng and Weiner found that limiting
the phase to values of zero and 7: led to the condition flak-(0):“ ak'i' 0)). Under both
shaping conditions the maximum signal approached that obtained with TL pulses.

The problem of spectral narrowing by pulse shaping under broad phase matching SHG
conditions, or selective excitation of large organic dyes and chromophores requires
minimization of the nonlinear power spectrum away fi'om 2612. Upon study of the
problem, as expressed in Eq 5.4, there is an analogy with convolution. In this case, some
binary firnctions have the property of giving a sharp ‘delta firnction’ signal in one

location and very low background elsewhere.

81

In principle, the solutions found in this study are members of the set of solutions
obtained by arbitrary phase and amplitude pulse shaping. For a pulse shaper with N
pixels, one can generate (P*A)N shaped pulses, where P and A are the number of different
phases and amplitudes a pixel can take. Ifwe assume 100 pixels, each taking 10 different
amplitude values and 100 difi‘erent phase values, the number of different pulses is of
order of magnitude 10300. This number is extremely large, therefore, while in principle,
the solution exists to achieve the desired photonic transformation or excitation, finding it
is very time consuming except for a convex problem with a clear maximum.

Experimentalists have reduced the search space of an ELA by reducing the number of
active pixels by grouping, the type of phase functions that are allowed, modulating phase
and not amplitude, and introducing phase firnctions with a reduced numbers of
parameters. ”"157 We tried these approaches with limited success.

For one pulse experiments, the periodic nature of electromagnetic waves results in a
great deal of redundancy in pulse shaping because nonlinear optical processes do not
depend on the absolute phase or a linear variation of the spectral phase. This equivalence
can be expressed by ¢(w) <:> ¢(a)) + a + ba) , where a and b are constants. Programming
an ELA that acts on the second derivative of the phase can filter out this redundancy. The
actual phase is obtained by integrating setting a=b=0. We used such an ELA to optimize
smooth phase functions for spectral narrowing, but could not obtain a contrast ratio
greater than unity.

The advantage of BPS is that it eliminates most redundancies. For BPS and 128

2128

active pixels the search space is reduced to . If the problem has two-fold symmetry,

for example two-photon excitation, then the search space is reduced to 10", a number

82

that is at least 281 orders of magnitude smaller than would be considered for arbitrary
phase and amplitude pulse shaping as discussed above. An ELA can quickly converge in
this small search space without resorting to grouping pixels.

BPS may have significant technological advantages. A retardation equivalent to 7: is
easy and fast to obtain and calibrate. Permanemly etched masks can be made in advance
and used for specific applications such as selective two-photon microscopy. Scanning the
mask’ 3 position can yield two-photon excitation spectra.

In summary, the use of BPS for selective two-photon excitation has been
demonstrated. The contrast ratio using BPS was 6 times greater than that obtained using
sinusoidal phase masks. The amplitude of the signal is 100 times greater than would have
been obtained by amplitude modulation. BPS makes it simple to analyze a laser control
problem and to propose rational solutions, as demonstrated here with a phase mask
inspired by the quasi-random gaps between prime numbers. An ELA can be used to

improve on the proposed solution efficiently because of the greatly reduced search space.

83

6. Appendix I

This appendix shows the derivation of the rotational component of the
non-resonant homodyne and heterodyne FWM signal (see figure 2.2). Both signals
involve the same physics in the sample, where an initial field sets up a polarization in
a sample, which is then interrogated by another laser field. The difference is only in
the signal detection, which is described in the experimental section of chapter 2.

Initially, the sample consists of rotators with the uncorrelated wavefunction

i¢. —' Hr
r(°)(0,t)=2bje J (2.1+1)|J)e th (6.1)

J

where the phase ¢j is random, and 0 is the angle between the moment of inertia (I) and
the electric field polarization of the laser. J is the angular momentum, and IJ) are

ortho-normalized eigen-functions with energy levels a, = 17,011 , where FJ=BJ(J+ 1)-

DJ‘?(J+ 1)2, B=h/(8nzcl), and D is the centrifugal distortion. The initial amplitudes of

the states are defined by the Boltzmann distribution

 

b]. = Jrr—J = J(2J +l)exp(—s, ”(7'). After the first two pulses, applying the Raman
selection rule A! = 0, i2 we have a wave packet in the sample where each rotational

state has produced a wave packet consisting of three states with amplitudes aj" . This

wavepacket will evolve until the third pulse arrives.

i¢.
(2) _ J +2 . o .
‘I’ (9,t)—§bje (aJ exp(—16'J+2t/h)|J+2)+aJexp(—rsJ_2t/h)|J)

+2152 exp(—i£J_zt/h)|J-2)) (6.2)

84

From this point there are two ways we can proceed. Mukamel tells us that the
Homodyne FWM signal is proportional to the square of the absolute value of the

imaginary part of the third order time dependent susceptibility of the sample

          

, which is proportional to the time derivative of the full operator

         

‘25 For a more

       

(fluctuation dissipation theorem) '7

        

mlkT—l dtl

intuitive description, we can instead think of the anisotropic polarization produced by

the laser passing through the samme. The polarization is proportional to the average
transient alignment of the molecules in the sample P2(cos 0) oc cos2 0 —l/ 3 after

interacting with the laser. This polarization will afl‘ect the refractive index of the
medium, and temporal changes in the refractive index of the medium give us the

Heterodyne FWM signal. In this way we can write

S W (t)oc7(3)(t)ocF—T§t-Z(3)(t)oc— —,<‘I’(2)(01)'132,(cos0)“1’(2)(0t)> (6 3)

 

Heterodyne

and

S W(t)ocl 23,200). eel—W 1" 12(3)“)! «26%| <142)(t9,t)lP2,2(eosa)‘r(2)(t9t)>l (6 4)

PM
Since (T(Z)(0,t)|P2(cos 0)|‘I’(2)(19,t)) is an oscillating function that depends on J, and

wavepackets produced from difl‘erent J are uncorrelated, we can write
<a42)(9, 1)le (combine, 0):»: Jdao aJ (is+22

where (232 = 22413,, —F,), and of = 2mm, —F,_,).

 

(6.5)

 

     

 

     

 

 

85

So long as the temperature is high and the molecule is heavy (J ave >>1), the
transition probabilities (a,) are near equal. The macroscopic polarization will depend

linearly on the number density (N) of the sample, and anisotropic polarizability
,6 = la" — all] of the molecules that make up the sample. Additionally, the rotational

constant of the molecule, B, depends on the vibrational energy of the molecule, 80 we
get a final expression for the time dependent polarizability of the sample caused by
rotational Raman excitation ,
P2(t) 0: N B 2V2; qv pt cos(27tc(FJ+2-Fr)t)+cos(27tc(Fr -F 1.2)t) (6.6)
where F, = BJ(J+l)-DJZ(J+1)2 for the molecule”, 3.. is the vibrationally
dependent rotational constant, p; is the normalized rotational distribution including
spin statistics (3;) for all natural isotopes,”9 pJ=g;(2J+1)exp(-27zchFJ/k]), and qv is
the normalized thermal vibrational distribution for all ground state vibrational modes.

The FWM signals can then be written as

 

S ”2
Heterocbme
and
2
dP
2
S FWM 0: dt (6.8)
Homodyne

For samples containing multiple molecules with different rotational constants, the
homodyned signal is given by | (dPa ldt+ dPt, /dt)|2, leading to interference terms

between the two populations.”

86

7. Appendix II

This appendix describes the FWM signal decay for the photon echo (PE) and the
reverse transient grating (RTG) pulse sequences. The intermediate case is derived, and
both the liquid phase (inhomogeneous) and gas phase (homogeneous) limits are applied.

To correctly describe the evolution of the system including relaxation we have to
use the density matrix (DM) presentation. In this case the ensemble of molecules can be
described by the nth orda density matrix. Because we use a non-collinear three pulse
Four Wave Mixing (F WM) experiment and detect the 3rd order nonlinear polarization that
depends on all three pulses, each interaction increases the order of the DM by one. The
pulses are weak and the rotating wave approximation is valid, and the system has two
electronic states, ground (g) and excited (e). Non-trivial DM elements of the nth order

after interaction with the nth pulse can be calculated though the recurrence relation:54

ethanerHexplriws-tXu.l-t.)l>:..rv‘“'m(x)p‘°'"ms(x.t.rp‘“‘"a(x.t.)V‘"'ss(x)l. (7.1)

where t" is the center of the nth very short pulse (the time duration of the pulse is very
small compared to the delay times and relaxation times). In the case of homogeneous
relaxation the time dependence of the pulses is described by the Green fiinction exp[(-ia)-
”(bu 1-t,.)], where a2= (tag is the electronic resonance frequency and 7 is the relaxation

rate. The time integrated interaction operators with the nth pulse are

V‘“‘s<x)=i(2ttmwm)explik.xl. (7.2)

where ,u is the dipole moment of the transition, In, is the wave vector of the nth pulse, and

x is the special coordinate of the molecule. If all pulses are equivalent (DFWM), for the

87

interaction elements where AM) is the spectral amplitude for the nth transform limited

pulse at the electronic transition we can write

A[“'(co)= op Egn] (t)exp[i(w—a)0)t]dt (7.3)

—00

where (an is the central frequency of the degenerate pulses with amplitude Eo(t).
The third order density matrix given by the master solution has four components

(Liouville pathways)

flaw) = p13”..(x.t)+ d3”’.g(x.t)+ d””’.g(x.t)+pf””.g(x.t) (7.4)

In the case of Photon Echo (PE) and Reverse Transient Grating (RTG), the first pulse
arrives at time t1=0 and the second and third follow after a delay I and coincide in time
t2=t3=t. The common spatial phase determines the emission directions. In the case of PE
the emission goes in the direction kpg=k3+k2-kt (Ru and R111), while in the case of RTG
the emission goes in the direction kgm=k3-k2+kt (R; and Rtv). There are no differences
in amplitudes between the Liouville pathways R1 and R4 and between Ru and Rm for a
two level system. If only the ground state is populated initially, only the term p(°)gg is not

zero and using equation (7.1) we can write

13‘3”, Ees(xst) °C eXP[i(krl-:x-(0(t-21))]eXP(-Yt)A“]'(0>)A[2](03)A[3](60) (75)

names» cc expli<knrex-mt)1exp(-tt)A‘"(<o)A""(m)A‘3‘(co) (7.6)

The PE DM has zero phase at 21, but RTG is maximum at t3-t2=0 (before the last pulse in

the “virtual” unphysical region).

88

The third order dipole polarization P“) of the medium is proportional to Tr( [28(3) ). We

can easily show that

P‘3)(x,t)=u[p(3’es(x,t)+ p‘3"a(x.t)l (7.7)
In the gas phase, the electric field generated is proportional to the third order polarization,

and has two components: PE and RTG.

Epg(x,t) 9:: exp[i(kpgx-oa(t-21)+¢pg)]exp(-yt)A2(co)A'(m) + co (7.8)

Enro(l,t) 0‘3 eXP[i(karat-COIWRTGHCXP('Yt)A2(0)A.(03) + co. (7.9)

Where we introduce additional phase shits of the pulse’s fields (PM (A["](a)) =

eXP(i¢.JA’"’(ail) and me = WWW—J” and 11m = 731-721.. 7”
The difference between PE and RTG is very important in the case of an ensemble of
molecules with (for example) different resonance conditions. The inhomogeneous

relaxation described by the normalized spectral distribution of each transition around

some frequency 030 with width A is given by
n(n,no)=exp[-(n-no)’/A2]/(n“2A). (7.10)

The full field is a sum (integral) over all inhomogeneous components. Term A3(03)
defines the full intensity of the signal, and we set it to unity, because we suppose the
spectral width of the pulse is more than the inhomogeneous broadening (any relaxation

time is bigger than the pulse duration).

Epg(x,t) oc 8Xp[1(¢pg+kPE)X]eXp("Yt)]o exp[- i0)(t - 2r) — (c0 — (no )2 /A2 Liar + co. (7. 1 1)

89

ERTG(XJ)°CeXP[i(¢RrG+kRTG)X]eXP('7t)ICXPL1°” —(a> moi/421m +c.c. (7.12)

After simple integration over the spectrum
Epg(x,t) oc exp[-i(coo(t-2‘t)-kpgx-¢pg)]exp[-yt—(t-Zt)2A2/4] + c.c. (7.13)
ERTG(x,t) or: exp[-i(raot-kpgx-¢pg)]exp[-yt-t2A2/4] + c.c. (7.14)

We can write the time integrated homodyne signal

2
E[4](x,t) dt (7.15)

sm e i
T

 

 

Note that S is scalar, as the detector integrates the signal over both space and
time.
We can separate the electric field of the PE and RTG signals because they are generated
in different directions and for the time integrated signal the common phases [aha-2 r)-
kpex-¢PE] (P1330r [W-kRmx-fitm] (RTG) disappear-

It’s possible to analytically calculate the relaxation in the case of an exponential

homogeneous decay and a Gaussian inhomogeneous decay

 

 

(I) oo 2
Spri('c) at i exp(-27r)dr I exp[-iw(r—Zr)—(w—wo)2/A2] dt (7.16)
r O
2
Sarah) oc [exp(-27r)dr jexp[-rax—(a)-wo) /A ] dt dt (7.17)
r 0

 

 

The spectral integrals are simple and gave us signal in integral or analytical forms

89150:) at Qjoexp(-27t —(t — 22‘)2 A2/2)dt =
r

exp(272A'2)[ 1 -er1(2*’-’rn+2°-’yn")](2/tt)°-5/A (7. 18)

oo
SRTG(‘C) (X? I exp(-27t—12A2/2)d1 =
T

exp(212A'2-4ry)[1+erf(2°'stA-2°‘57A'l)](2/7t)°'5/A (7. 19)

We define the homogeneous relaxation time in terms of the relaxation rate T h=1/7 and the
inhomogeneous relaxation time in terms of the inhomogeneous spectral width 7}=I/A. In

the limit 1T), << 71-2 there is no difference between the PE and RTG signals. In this case,
equations 7 . 18 and 7.19 can be simplified to obtain

SPEC!) = 8310(1) 0C exp(-2‘ry) = exp(-2t/Th), (7.20)

In the opposite case rTh >> T3, the condensed phase limit, there is a difference between
the PE and RTG signals. We can replace the integral formula for the PE Gaussian field
with a delta function at time 21 and get a simple decay with rate 47. In this case equations

7.18 and 7.19 can be written as follows.

SPE(T) oc exp(-4ry) = exp(-4t/1‘t.), (7.21)

8mm 0: exp(-‘/2A2t2) = exp(-t2/2T.2), (7.22)

Note that even for this approximation, for small time delays, t, the product 1:Ttl is
still small. Therefore, for small delays we might expect the “gas phase” limit to apply.

For long 1: the “liquid phase” limit gives an accurate description. Figure 1 demonstrates

91

this observation. Note that initially the PE has a decay rate of 27, then goes through an
intermediate region and at long delay times has a rate of 47. In the case of RTG, the

initial experimental decay with rate 27 transform to a faster Gaussian decay.

92

 

10

11

12

13

14

15

l6

17

References

Tannor, D. J.; Kosloff, R.; Rice, 8. A. Journal of Chemical Physics 1986, 85,
5805.

Brumer, P.; Shapiro, M. Accounts of Chemical Resew'ch 1989, 22, 407.

Felker, P. M.; Lambert, W. R.; Zewail, A. H. Journal of Chemical Physics 1985,
82, 3003.

Felker, P. M; Zewail, A H. Journal of Chemical Physics 1985, 82, 2961.
F elker, P. M.; Zewail, A. H. Journal of Chemical Physics 1985, 82, 2975.
Felker, P. M.; Zewail, A H. Journal of Chemical Physics 1985, 82, 2994.
Weiner, A. M. Review of Scientific Instruments 2000, 71, 1929.

Judson, R. 8.; Rabitz, H. Physical Review Letters 1992, 68, 1500.

Assion, A; Baumert, T.; Bergt, M; Brixner, T.; Kiefer, B.; Seyfiied, V.; Strehle,
M.; Gerber, G. Science 1998, 282, 919.

Levis, R J .; Menkir, G. M.; Rabitz, H. Science 2001, 292, 709.
Rice, 8. A. Nature 2001, 409, 422.

Rice, 8. A; Tannor, D. J .; Koslofi‘, R Journal of the Chemical Socieor-Faraday
Transactions Ii 1986, 82, 2423.

Tannor, D. J .; Rice, 8. A Advances in Chemical Physics 1988, 70, 441.

Pshenichnikov, M. 8.; deBoeij, W. P.; Wiersma, D. A Physical Review Letters
1996, 76, 4701.

Voss, T.; Breunig, H. G.; Ruckmanrr, 1.; Gutowski, J.; Axt, V. M.; Kuhn, T.
Physical Review B 2002, 66.

Pastirk, 1.; Lozovoy, V. V.; Grimberg, B. I.; Brown, E. J.; Dantus, M. Journal of
Physical Chemistry A 1999, 103, 10226.

Pastirk, I.; Brown, E. J.; Grimberg, B. 1.; Lozovoy, V. V.; Dantus, M. Faraday
Discussions 1999, 401.

93

 

18

19

20

21

22

23

24

25

26

27

28

29

3O

31

32

33

34

Friedrich, B.; Herschbach, D. Physical Review Letters 1995, 74, 4623.
Kim, W. S.; Felker, P. M. Journal of Chemical Physics 1997, 107, 2193.
Kim, W.; Felker, P. M. Journal of Chemical Physics 1996, 104, 1147.
Seideman, T. Physical Review Letters 1999, 83, 4971.

Seideman, T. Journal of Chemical Physics 1995, 103, 7887.

Comstock, M.; Senekerimyan, V.; Dantus, M. Journal of Physical Chemistry A
2003, 107, 8271.

Hankin, S. M.; Villeneuve, D. M.; Corkum, P. B.; Rayner, D. M. Physical Review
Letters 2000, 84, 5082.

Mohideen, U.; Sher, M. H.; Tom, H. W. K.; Aumiller, G. D.; Wood, 0. R;
Freeman, R R; Bokor, J.; Bucksbaum, P. H. Physical Review Letters 1993, 71 ,
509.

Resch, K.; Blanchet, V.; Stolow, A; Seideman, T. Journal of Physical Chemistry
A 2001, 105, 2756.

Chelkowski, 8.; Corkum, P. B.; Bandrauk, A D. Physical Review Letters 1999,
82, 3416.

Hishikawa, A; Iwamae, A; Hoshina, K.; Kono, M.; Yamanouchi, K. Chemical
Physics 1998, 231, 315.

Iwamae, A; Hishikawa, A; Yamanouchi, K. Journal of Physics B-A tomic
Molecular and Optical Physics 2000, 33, 223.

Villeneuve, D. M.; Aseyev, S. A; Dietrich, P.; Spanner, M.; Ivanov, M. Y.;
Corkum, P. B. Physical Review Letters 2000, 85, 542.

Joireman, P. W.; Connell, L. L.; Ohline, S. M.; Felker, P. M. Journal of Chemical
Physics 1992, 96, 4118.

Felker, P. M. Journal of Physical Chemistry 1992, 96, 7844.

Dietrich, P.; Strickland, D. T.; Laberge, M.; Corkum, P. B. Physical Review A
1993, 47, 2305.

Trebino, R; Kane, D. J. Journal of the Optical Society of America a-Optics Image
Science and Vision 1993, 10, 1101.

94

 

35

36

37

38

39

40

41

42

43

45

46

47

48

49

50

51

52

Hall, T. C.; Blacet, J. J. Chem. Phys. 1952, 20, 1745.

Weiner, N. R; Nixon, E. R. J. Chem. Phys. 1957, 26, 906.

Bibart, C. H.; Ewing, G. B. Journal of Chemical Physics 1974, 61, 1284.
Luckhaus, D.; Quack, M. Chemical Physics Letters 1992, 199, 293.

Stirling, A; Papai, I.; Mink, J.; Salahub, D. R. Journal of Chemical Physics 1994,
100, 2910.

Domenech, J. L.; Andrews, A. M.; Belov, S. P.; Fraser, G. T.; Lafi‘erty, W. J.
Journal of Chemical Physics 1994, 100, 6993.

Dyer, C.; Hendra, P. J. Chemical Physics Letters 1995, 233, 461.

Wesolowski, S. S.; Fermann, J. T.; Crawford, T. D.; Schaefer, H. F. Journal of
Chemical Physics 1997, 106, 7178.

Kato, T. Chem. Phys Lett. 1996.
Kirmse, B.; Delon, A; Jost, R Journal of Chemical Physics 1998, 108, 6638.

Parsons, B. F.; Curry, 8. L.; Mueller, .1. A; Ray, P. 0; Butler, L. J. Journal of
Chemical Physics 1999, 111, 8486.

Mueller, J. A; Morton, M. L.; Curry, S. L.; Abbatt, J. P. D.; Butler, L. J. Journal
of Physical ChemistryA 2000, 104, 4825.

Borisenko, K. B.; Kolonits, M.; Rozsondai, B.; Hargittai, 1. Journal ofMolecular
Structure 1997, 413, 121.

Streuli, C.; Averell, P. The analytical chemistry of nitrogen and its compounds;
John Wiley & Sons: New York, 1970.

McClellan; GunderseG; Hedberg, K. Journal of Chemical Physics 1972, 56,
4541.

Kvick, A; McMullan, R K.; Newton, M. D. Journal of Chemical Physics 1982,
76, 3754.

Melen, F .; Herman, M. Journal of Physical and Chemical Reference Data 1992,
21 , 831.

Davies, J. A; Continetti, R E.; Chandler, D. W.; Hayden, C. C. Physical Review
Letters 2000, 84, 5983.

95

 

53

54

55

56

57

58

59

6O

61

62

63

65

66

67

68

Davies, J. A; LeClaire, J. E.; Continetti, R E.; Hayden, C. C. Journal of
Chemical Physics 1999, 111, 1.

Grimberg, B. 1.; Lozovoy, V. V.; Dantus, M.; Mukamel, S. Journal of Physical
ChemistryA 2002, 106, 697.

Morgen, M.; Price, W.; Hunziker, L.; Ludowise, P.; Blackwell, M.; Chen, Y.
Chemical Physics Letters 1993, 209, 1.

Morgen, M.; Price, W.; Ludowise, P.; Chen, Y. Journal of Chemical Physics
1995, 102, 8780.

Brown, E. J .; Zhang, Q. G.; Dantus, M. Journal of Chemical Physics 1999, 110,
5772.

Lozovoy, V. V.; Grimberg, B. 1.; Pastirk, 1.; Dantus, M. Chemical Physics 2001,
267, 99.

Jolly, W. L. The inorganic chemistry of nitrogen; W.A Benjamin Inc.: New
York, 1964.

Dantus, M. Annual Review of Physical Chemistry 2001, 52, 639.

Press, W. H. Numerical Recipes in C; Cambridge University Press: New York,
1992.

Hepp, M.; Georges, R; Herman, M.; Flaud, J. M.; Lafi‘erty, W. J. Journal of
Molecular Structure 2000, 517, 171.

Herzberg, G. Molecular Spectra and Molecular Structure Vol. 111 Electronic
Spectra and Electronic Structure of Polyatomic Molecules; Krieger Publishing
Company: Malabar, Florida, 1991; Vol. 3.

Pastirk, 1.; Lozovoy, V. V.; Dantus, M Chemical Plorsics Letters 2001, 333, 76.

Lozovoy, V. V.; Grimberg, B. 1.; Brown, E. J.; Pastirk, 1.; Dantus, M. Journal of
Rarnan Spectroscopy 2000, 31, 41.

Lozovoy, V. V.; Pastirk, I.; Brown, E. J.; Grimberg, B. 1.; Dantus, M.
International Reviews in Physical Chemistry 2000, 19, 531.

Lievin, J .; Delon, A; Jost, R Journal of Chemical Physics 1998, 108, 8931.

Gillispie, G. D.; Khan, A U. Journal of Chemical Physics 1976, 65, 1624.

96

69

7O

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

Dion, C. M.; Keller, A; Atabek, 0.; Bandrauk, A. D. Physical ReviewA 1999, 59,
1382.

Yamanouchi, K. Science 2002, 295, 1659.
Friedrich, B. Physical Review A 2000, 6102, art. no.
Friedrich, B.; Herschbach, D. Journal of Physical Chemistry 1995, 99, 15686.

Normand, D.; Lompre, L. A; Cornaggia, C. Journal of Physics B-A tomic
Molecular and Optical Physics 1992, 25, L497.

Larsen, J. J .; Wendt-Larsen, 1.; Stapelfeldt, H Physical Review Letters 1999, 83,
1123.

Sakai, H.; Safvan, C. P.; Larsen, J. J.; Hilligsoe, K. M.; Hald, K.; Stapelfeldt, H.
Journal of Chemical Physics 1999, 110, 10235.

Posthumus, J. H.; Plumridge, J.; Thomas, M. K.; Codling, K.; Frasinski, L. J.;
Langley, A J .; Taday, P. F. Journal of Physics B-A tomic Molecular and Optical
Physics 1998, 31, L553.

Brown, E. J .; Pastirk, 1.; Dantus, M. Journal of Physical Chemistry A 2001, 105,
8004.

Ortigoso, J .; Rodriguez, M.; Gupta, M.; Friedrich, B. Journal of Chemical Physics
1999, 110, 3870.

Cornaggia, C.; Salin, F.; LeBlanc, C. Journal of Physics B-A tomic Molecular and
Optical Physics 1996, 29, L749.

Cornaggia, C. Physical ReviewA 1996, 54, R2555.

Hishikawa, A; Iwamae, A; Hoshina, K.; Kono, M.; Yamanouchi, K. Chemical
Physics Letters 1998, 282, 283.

Hishikawa, A; Iwamae, A; Yamanouchi, K. Physical Review Letters 1999, 83,
1 127.

Hishikawa, A; Iwamae, A; Yamanouchi, K. Journal of Chemical Physics 1999,
111, 8871.

Banerjee, S.; Kumar, G. R; Mathur, D. Physical Review A 1999, 60, R3369.

Banerjee, S.; Kumar, G. R; Mathur, D. Journal of Physics B-A tomic Molecular
and Optical Physics 1999, 32, 4277.

97

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

Iwasaki, A; Hishikawa, A; Yamanouchi, K. Chemical Physics Letters 2001, 346,
379.

Graham, P.; Ledingham, K. W. D.; Singhal, R R; Mchy, T.; Hankin, S. M.;

Fang, X.; Smith, D. J .; Kosmidis, C.; Tzallas, P.; Langley, A. J .; Taday, P. F.
Journal of Physics B-A tomic Molecular and Optical Physics 1999, 32, 5557.

Couris, S.; Koudoumas, E.; Leach, S.; Fotakis, C. Journal of Physics B-Atomic
Molecular and Optical Physics 1999, 32, L439.

Frasinski, L. J.; Codling, K.; Hatherly, P. A Physics LettersA 1989, 142, 499.

Hasegawa, H.; Hishikawa, A; Yamanouchi, K. Chemical Physics Letters 2001,
349, 57.

Sanderson, J. H.; El-Zein, A; Bryan, W. A; Newell, W. R; Langley, A. J .;
Taday, P. F. Physical ReviewA 1999, 59, R2567.

Bryant, W. A; Sanderson, J. H.; El-Zein, A; NewelL W. R; Taday, P. F.;
Langley, A. J. Journal of Physics B-A tomic Molecular and Optical Physics 2000,
33, 745.

Wu, C. Y.; Ren, H. Z.; Liu, T. T.; Ma, R; Yang, H.; Jiang, H. B.; Gong, Q. H.
International Journal of Mass Spectrometry 2002, 216, 249.

Heritage, J. P.; Gustafson, T. K.; Lin, C. H. Physical Review Letters 1975, 34,
1299.

Comstock, M.; Pastirk, 1.; Dantus, M. “Ultrafast transient-grating study of
molecules after high intensity excitation”; Ultrafast Phenomena XII, 2000.

Joireman, P. W.; Ohline, S. M.; Felker, P. M. Journal of Physical Chemistry A
1998, 102, 4481.

Frey, H. M.; Beaud, P.; Gerber, T.; Mischler, B.; Radi, P. P.; Tzannis, A. P.
Journal of Raman Spectroscopy 2000, 31 , 71.

Jarzeba, W.; Matylitsky, V. V.; Weichert, A; Riehn, C. Physical Chemistry
Chemical Physics 2002, 4, 451.

Lang, T.; Motzkus, M.; Frey, H. M.; Beaud, P. Journal of Chemical Physics 2001,
115, 5418.

Sarkisov, O. M.; Tovbin, D. G.; Lozovoy, V. V.; Gostev, F. E.; Titov, A. A;
Antipin, S. A; Umanskiy, S. Y. Chemical Physics Letters 1999, 303, 458.

98

 

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

Rosca-Pruna, F.; Vrakking, M. J. J. Journal of Chemical Physics 2002, 116, 6567.
Rosca-Pruna, F.; Vrakking, M. J. J. Journal of Chemical Physics 2002, 116, 6579.

Bandrauk, A Molecules in Intense Laser Fields; M. Dekker Publishing: New
York, 1993.

Giustisuzor, A; Mies, F. H.; Dimauro, L. F.; Charron, E.; Yang, B. Journal of
Physics B-A tomic Molecular and Optical Physics 1995, 28, 309.

Zhang, Q. G.; Vaccaro, P. H. Journal of Physical Chemistry 1995, 99, 1799.
Tseng, D. C.; Poshusta, R. D. Journal of Chemical Physics 1994, 100, 7481.

Brown, S. T.; Van Huis, T. J.; Hoffman, B. C.; Schaefer, H. F. Molecular Physics
1999, 96, 693.

Calculation performed using the CCSD(T) method with the aug-cc-PVDZ basis
set. The lowest two orbitals(correlating with the Is orbitals in C and S) were
frozen in the calculations. We also calculated the states in the presence of a .5V/A
field parallel or perpendicular to the molecular axis.

Brasen, G.; Leidecker, M.; Demtroder, W.; Shimamoto, T.; Kato, H. Journal of
Chemical Physics 1998, 109, 2779.

Sander, R. K.; Wilson, K. R Journal of Chemical Physics 1975, 63, 4242.
Pastirk, 1.; Comstock, M.; Walowicz, K. A; Lozovoy, V. V.; Dantus, M. “2D
(time-frequency) femtosecond four-wave mixing at 1014 W/cm2: Molecular and
electronic response”; Symposium on Optical Pulse and Beam Propagation 11],
2001.

Shimizu, S.; Zhakhovskii, V.; Sato, F.; Okihara, S.; Sakabe, S.; Nishihara, K.;
Izawa, Y.; Yatsuhashi, T.; Nakashima, N. Journal of Chemical Physics 2002, 11 7,
3 180.

Henriksen, N. B. Chemical Society Reviews 2002, 31 , 37.

Deutsch, D.; Ekert, A. Physics World 1998, 11, 47.

Bennett, C. H; Divincenzo, D. P. Nature 1995, 377, 389.

Chuang, I. L.; Laflamme, R; Shor, P. W.; Zurek, W. H. Science 1995, 270, 1633.

Lozovoy, V. V.; Dantus, M. Chemical Physics Letters 2002, 351, 213.

99

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

Zadoyan, R; Kohen, D.; Lidar, D. A; Apkarian, V. A. Chemical Physics 2001,
266, 323.

Greiner, M.; Mandel, 0.; Esslinger, T.; Hansch, T. W.; Bloch, 1. Nature 2002,
415, 39.

Goldstein, E. V.; Zobay, O.; Meystre, P. Physical Review A 1998, 58, 2373.

Lozovoy, V. V.; Pastirk, 1.; Comstock, M. G.; Dantus, M. Chemical Physics 2001,
266, 205.

Zhang, W. M.; Chemyak, V.; Mukamel, S. Journal of Chemical Physics 1999,
110, 5011.

Kurnit, N. A; I.D., A; Hartmann, S. R. Phys. Rev. Lett. 1964, 20, 1087.
Patel, C. K. N.; Slusher, R E. Phys. Rev. Lett. 1964, 20, 1087.

Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford University
Press: New York, 1995.

Prior, Y. Applied Optics 1980, 19, 1741.
Shirley, J. A; Hall, R J.; Eckbreth, A C. Optics Letters 1980, 5, 380.
Tellingh.J. Journal of Chemical Physics 1973, 58, 2821.

Lide, D. R CRC Hancfiook of Chemistry and Physics; Chemical Rubber
Company Press, 1996-7.

Sleva, E. T.; ZewaiL A H. Chemical Physics Letters 1984, 110, 582.

ZewaiL A H.; Orlowski, T. E.; Jones, K. E.; Godar, D. B. Chemical Physics
Letters 1977, 48, 256.

Spartan Version 5.1.3; 6-31G* electron density calculation

Simon, J. D.; MacQuanie, D. A Physical Chemistry: A Molecular Approach;
University Science Books: Sausalito, Ca, 1997.

Kubo, R In Fluctuation Relaxation and Resonance in Magnetic Systems; Oliver
and Boyd: London, 1961.

F uentealba, P.; Simon-Manso, Y.; Chattaraj, P. K. Journal of Physical Chemistry
A 2000, 104, 3185.

100

 

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

Lienau, C.;Zewai1, A H. Journal of Physical Chemistry 1996, 100, 18629.

Liu, Q. L.; Wan, C. Z.; Zewail, A. H. Journal of Physical Chemistry 1996, 100,
18666.

Zewail, A H.; Dantus, M.; Bowman, R M.; Mokhtari, A. Journal of
Photochemistry and Photobiolog a-Chemistry 1992, 62, 301.

Gruebele, M.; Zewail, A. H. Journal of Chemical Physics 1993, 98, 883.

Bardeen, C. J.; Yakovlev, V. V.; Wilson, K. R; Carpenter, S. D.; Weber, P. M.;
Warren, W. S. Chemical Physics Letters 1997, 280, 151.

'n . "‘ '"fl"

Goswami, D. Physics Reports—Review Section of Physics Letters 2003, 3 7 4, 3 85.

Shapiro, M; Brumer, P. Journal of Physical ChemistryA 2001, 105, 2897.

 

Pastirk, 1.; Dela Cruz, J. M.; Walowicz, K. A; Lozovoy, V. V.; Dantus, M. Optics
Express 2003, 11, 1695.

Dela Cruz, J. M.; Pastirk, I.; Lozovoy, V. V.; Walowicz, K. A; Dantus, M.
Journal of Physical Chemistry A 2004, 108, 53.

Lozovoy, V. V.; Pastirk, I.; Walowicz, K. A; Dantus, M. Journal of Chemical
Physics 2003, 118, 3187.

Walowicz, K. A; Pastirk, 1.; Lozovoy, V. V.; Dantus, M. Journal of Physical
Chemistry A 2002, 106, 9369.

Lozovoy, V. V.; Pastirk, 1.; Dantus, M. Optics Letters 2004, 29, 775.

Hacker, M. ,Netz, R; Roth, M. ,Stobrawa, G.; F eurer T. ,Sauerbrey, R. Applied
Physics B-Lasers and Optics 2001, 73, 273.

Broers, B.; Noordam, L. D.; Vandenheuvell, H. B. V. Physical Review A 1992,
46, 2749.

Zheng, Z.; Weiner, A M. Chemical Physics 2001, 267, 161.
Zheng, Z.; Weiner, A. M. Optics Letters 2000, 25, 984.
Wang, L.; Weiner, A M. Optics Communications 1999, 167, 211.

Meshulach, D.; Silberberg, Y. Physical ReviewA 1999, 60, 1287.

101

154

155

156

157

158

159

Dudovich, N.; Dayan, B.; Faeder, S. M. G.; Silberberg, Y. Physical Review
Letters 2001, 86, 47.

Zeidler, D.; Frey, S.; Kompa, K. L.; Motzkus, M. Physical Review A 2001, 6402.

Weinacht, T. C.; Bartels, R; Backus, S.; Bucksbaum, P. H.; Pearson, B.;
Geremia, J. M.; Rabitz, H.; Kapteyn, H. C.; Murnane, M. M. Chemical Physics
Letters 2001, 344, 333.

Herek, J. L.; Wohlleben, W.; Cogdell, R. J .; Zeidler, D.; Motzkus, M. Nature
2002, 417, 533.

Equation (6.6) also holds true for the DK=0 Raman transition of a symmetric top
molecule, and can be applied to nearly symmetric top molecules provided the
effective B constant Befl’=(B+C)/2 is used. G. Herzberg, Molecular Spectra and
Structure: Volume [[1 Electronic Spectra and Electronic Structure of Polyatomic
Molecules (Krieger Publishing Company, Malabar, Florida, 1991), pp106-108.

For electronic states of gerade symmetry, Bosonic nuclei have gJ=(21+1)(I+1) if J
is even, and g] =(21+1)I if J is odd. Fermionic nuclei in gerade electronic states
have gJ=(21+1)I if I is even, and g] =(21+1)(I+1) if J is odd. For ungerade
electronic states, the ratio of odd to even states is reversed. Here I is the nuclear
spin of indistinguishable particles exchanged during rotation. For heteronuclear
diatomics, interchange symmetry does not apply and all states are equally
weighted. For more information see C. H. Townes, A L. Schawlow, Microwave
Spectroscopy (Dover Publications, Inc.,New York, 1975), pp102-105.

102

   

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