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THF‘HS

1003

This is to certify that the

dissertation entitled

Observation and Control of Intramoiecuiar Dynamics of
Gas-Phase Molecules by Ultrafast Laser Puises

presented by

. Igor Pastirk

has been accepted towards fulﬁllment
of the requirements for

Docterai degree in (:hemj SIC)!

{gm}:

Major professor

Date 07/25/2002

 

MSU i: an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

 

 

 

LIBRARY
Michigan State
University

 

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJCIRC/DateDuopes-sz

Observation and Control of Intramolecular Dynamics of Gas-Phase
Molecules by Ultrafast Laser Pulses

by

Igor Pastirk

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Chemistry

2002

ABSTRACT

Observation and Control of Intramolecular Dynamics of Gas-Phase
Molecules by Ultrafast Laser Pulses

by

Igor Pastirk

The work presented here describes experimental femtosecond three pulse four-
wave mixing techniques as a tool in observation and laser control of molecular dynamics.
The ability to control population and coherence transfer in molecular systems is of great
importance in the quest for controlling the outcome of laser-initiated chemical reactions.
The approach is based on the use of different femtosecond pulse sequences and applied
ﬁeld’s phase characteristics for providing a fundamental understanding of underlying
non-linear optical processes. It relies on a large body of experimental measurements on a
model system consisting of isolated (gas-phase) iodine molecules that have resonant X
'ng (ground) to B 3110+u (excited) electronic transition with applied laser ﬁelds. Control
over population and coherence transfer is demonstrated by selecting speciﬁc pulse

sequences.

The technique is also used to sort and measure the processes that contribute to

coherence relaxation of gas-phase 12. The observed relaxation times for vibronic

coherence are determined using photon echo and reverse transient grating measurements
at different temperatures to isolate inhomogeneous and homogeneous components.
Different pulse sequences are used to select ground or excited state vibrational
coherences. Measurements of ground and excited state wave packet spreading times due
to anharrnonicity, a process that does not involve energy dissipation or phase relaxation,

are also presented.

Femtosecond ground state dynamics of gas phase N204 and N02 are studied by
non-resonant four-save mixing (FWM) technique, applying a transient grating pulse
sequence. Both fast vibrational dynamics (< 1 ps) and long time (up to 100 ps after initial
excitation) rotational revivals are recorded for both samples. Observed fast (133 fs) beats
in N204 are assigned to stretching mode of N-N bond. The average rotational constant for

NO; in the ground state was determined from long time measurements.

Saneli

 

 

 

 

 

 

 

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Acknowledgments

During my graduate studies at Michigan State University I met a number of
people that have helped me, in one way or another. The level of cooperation and
willingness to help that is present at the Department of Chemistry would make the list

here very long. I would like to thank them all.

I would also like to thank the past and present members of the research group.
Speciﬁcally, Emily Brown, Matt Comstock, Irene Grimberg, Peter Gross, Vadim
Lozovoy, Una Marvet, Kathy Walowicz, and Qingguo Zhang — thank you for sharing

your knowledge in the lab and your advice out of the lab.

I would particularly like to thank my advisor Marcos Dantus, for introducing the
exciting ﬁeld of ultrafast spectroscopy and laser control to me, being always open to new

ideas and supporting me both professionally and personally during my graduate studies.

 

 

 

 

 

 

 

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TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... viii
LIST OF FIGURES ........................................................................................................... ix
1. INTRODUCTION ...................................................................................................... 1
1.1. Observing chemical reactions in real-time ......................................................... 3
1.2. Controlling the dynamics of molecules .............................................................. 6
1.3. Outline of the thesis .......................................................................................... 10

2. EXPERIMENTAL (ultrashort lasers and techniques) .............................................. 13
2.1. Introduction ....................................................................................................... 13
2.2. F emtosecond pulses, characterization and experimental techniques ................ 14
2.2.1. Femtosecond pulse characterization ......................................................... 18
2.2.2. One-pulse excitation ................................................................................. 25
2.2.3. Three—pulse FWM .................................................................................... 27

2.3. Colliding Pulse Mode-locked Laser System (CPM) ......................................... 28
2.4. Titanium sapphire laser system ......................................................................... 33

3. QUANTUM CONTROL OF THE YIELD OF A CHEMICAL REACTION BY

SINGLE FEMTOSECOND PULSES .............................................................................. 38
3.1. Introduction ....................................................................................................... 38
3.2. Experimental ..................................................................................................... 40
3.3. Results and discussion ...................................................................................... 42

4. PULSE SEQUENCES AND CHIRP AS A TOOL IN FEMTOSECOND
QUANTUM CONTROL OF MOLECULAR EXCITATION AND

INTRAMOLECULAR DYNAMICS ............................................................................... 48
4.1. Introduction ....................................................................................................... 48
4.2. Four wave mixing ............................................................................................. 53

4.2.1. The role of pulse sequences in controlling ultrafast intramolecular
dynamics with FWM ................................................................................................. 53
4.2.2. Formal description of nonlinear optical processes involved in sequences’
control scheme .......................................................................................................... 55

vi

 

1(1

 

 

 

 

5.

4.3.

4.2.3. Experimental ............................................................................................. 61
4.2.4. Overlapped Pulses ..................................................................................... 65
4.2.5. Pulse Sequence I: Virtual Photon Echo and Stimulated Photon Echo ...... 68
4.2.6. Pulse Sequence II : Virtual Photon Echo and Stimulated Photon Echo 72
4.2.7. Cascaded Free-Induction Decay FWM ..................................................... 75
4.2.8. Conclusions ............................................................................................... 77

Control of quantum phase with coherent light .................................................. 84
4.3.1 . Introduction ............................................................................................... 85
4.3.2. Theory ....................................................................................................... 87
4.3.3. Experimental ............................................................................................. 98
4.3.4. Spectral information in femtosecond three-pulse F WM ........................... 99
4.3.5. Two-dimensional three-pulse F WM measurements ............................... 101
4.3.6. Three-pulse F WM with chirped laser pulses .......................................... 107
4.3.7. Conclusions ............................................................................................. 1 13

FEMTOSECOND PHOTON ECHO AND VIRTUAL ECHO MEASUREMENTS
OF THE VIBRONIC AND VIBRATIONAL COHERENCE RELAXATION TIMES OF

IODINE VAPOR ............................................................................................................ 116
5. 1 . Introduction ..................................................................................................... 1 16
5.2. Experimental ................................................................................................... 1 1 9
5.3. Results and discussion .................................................................................... 119
5.4. Conclusions ..................................................................................................... 126

6. FEMTOSECOND GROUND-STATE DYNAMICS OF GAS-PHASE

DINITROGEN TETROXIDE AND NITROGEN DIOXIDE ........................................ 127
6.1. Introduction ..................................................................................................... 127
6.2. Experimental ................................................................................................... 129
6.3. Results and discussion .................................................................................... 130
6.4. Conclusion ...................................................................................................... 138

7. SUMMARY AND CONCLUSIONS ..................................................................... 140

8. APPENDIX I .......................................................................................................... 143

9. REFERENCES ....................................................................................................... 146

vii

LIST OF TABLES

Table 1. Summary of the observed phenomena for the different pulse sequences possible
with the three laser pulses when two pulses overlap in time. For the experimental
arrangement used here, there are only four distinguishable sequences (shown in bold
face). Because beams Ea and Ec are equivalent, two additional (though not unique)
sequences are possible. Sequences for which experimental data are presented have been
labeled with a star. The phenomena include transient grating (TG), reverse-transient
grating (RTG), reverse photon echo (RPE), and photon echo (PE) .................................. 63

Table 2. Summary of the observed phenomena and the different pulse sequences possible
with three time-separated laser pulses. For the experimental arrangement used here, there
are only six distinguishable sequences (shown in bold face). Because beams E3 and Ec are
equivalent, additional (though not unique) sequences are possible. Sequences for which
experimental data are presented have been labeled with a star. The phenomena include
virtual photon echo (VE), stimulated photon echo (PE), and cascaded free-induction
decay FWM (C-FID-FWM) .............................................................................................. 64

viii

LIST OF FIGURES

Figure 2.1 Schematic of the experimental setup for an intensity autocorrelation
measurement. M - mirrors, BS — 50/50 beamsplitter, HCC - hollow comer cube mounted
on a computer controlled variable delay stage, SHG - doubling crystal, L — focusing and
collimating lens, UV — ﬁlter. ............................................................................................ 19

Figure 2.2. Quasistatic interaction length (thickness) of selected crystals as a function of
the duration of transform limited pulses for Type I SHG at 800 nm. ............................... 20

Figure 2.3. Typical autocorrelation trace (triangles) obtained for ampliﬁed titanium-
sapphire fs laser pulses with a Gaussian function ﬁt. Duration of the pulses is 50 fs. ..... 21

Figure 2.4. Typical P-FROG trace obtained for transform-limited (a) and positively
chirped (b) pulses from an ampliﬁed CPM laser with central wavelength of 620 nm. 23

Figure 2.5. a) Experimental SHG FROG trace of near TL pulses from an ampliﬁed
titanium sapphire laser, obtained by CCD and spectrometer. Darker shades correspond to
the higher intensity. b) Temporal proﬁle (thin line) and phase dependence (thick line)
retrieved from data in a) by FROG software. c) Spectral (thin line) and phase (thick line)
proﬁles of the pulses retrieved by FROG software. Pulses were 55 fs with time-
bandwidth product of 0.4646. Phase variation of about 0.1 radians across most of the
pulse duration in b) is considered to be a minor deviation from TL. The spectral phase
characteristic in c) signiﬁes minor higher order chirp. ..................................................... 24

Figure 2.6. Experimental setup for one-beam experiment. P1 and P2 — double-pass
compressor prisms. P2 is mounted on an actuator. Mirrors (M) direct the beam to an SHG
crystal where the frequency of the pulses is doubled, and pulses are focused on the
sample cell. rPD — reference photodiode used for discrimination. ................................... 26

Figure 2.7. Beam arrangement for the three-pulse four-wave mixing experiment. BS -
beamsplitters; HCC — hollow comer cubes mounted on variable delay stages; L —focusing
and collimating lenses. Detail of the spatial overlap is also shown (top) with the direction
of the signal, detected in the phase-matching direction ks. ............................................... 28

Figure 2.8. Experimental setup for CPM dye laser with general detection scheme. M and
CM - mirrors and curved mirrors, respectively; P - prisms; OC — output coupler.
Nd:YAG laser and detection are triggered and synchronized to the pulses detected by
photodiode PD. ................................................................................................................. 31

Figure 2.9. Titanium-sapphire laser oscillator (seed laser). Light from a Nd:YVO4 laser is
focused by lens (L) through one of the curved mirrors (CM) on the titanium-sapphire
crystal. CM are transparent for 532 nm light, but 100 % reﬂective in the 760-850 nm
region. Emission from the crystal is collected by CM and directed through prisms P1 and
P2 to the end mirror (EM). Length of the cavity (distance from EM to output coupler
0C) is ~ 2m. Pulses shorter than 15 fs when compressed and centered around 800 nm
with FWHM of more than 50 nm were obtained. For the proper operation of the

ix

ampliﬁer, presented in the next ﬁgure, much less bandwidth is needed (25-30 nm) and the
oscillator is adjusted accordingly. For fast determination of the bandwidth, one reﬂection
of P2 is directed from the diffraction grating to an inexpensive CCD and TV monitor.
The image can be calibrated and used for rough estimation of pulse bandwidth. ............ 35

Figure 2.10. Schematic for regenerative ampliﬁcation of pulses from titanium-sapphire
laser. Beam from the seed laser is directed through the Faraday isolator (F1) to diffraction
grating G(1) in the stretcher. F I acts as ﬁlter preventing the back-reﬂected pulses from
the ampliﬁer from reentering the seed laser cavity. The beam diffracted from G] reﬂects
from silver mirror SM to long mirror LM and back. Still expanded, the beam travels to
periscope P and back for another round trip (G1, SM, LM and back). Spatially
recompressed, but chirped to ~100 ps, the beam is sent to an ampliﬁer (see text) ........... 36

Figure 3.1 (a) Experimental measurement of the yield of the molecular pathway
producing 12 from the multiphoton dissociation of CH2I2 with 624 nm laser pulses as a
function of chirp. The insert shows the relevant energetics for the reaction. 0))
Fluorescence yield as a function of chirp following the multiphoton excitation of 12
vapor .................................................................................................. 44

Figure 3.2 Experimental measurement of the yield of the molecular pathway
producing 12 from the multiphoton dissociation of CH2I2 with 312 nm laser pulses as a
function of experimentally available chirp range. The inserts show the relevant energetics
for the reaction, as well as a plot of the maximum 12 yield enhancement recorded at 2400
fs2 chirp, as a function of laser peak intensity (measured at zero chirp). ......................... 46

Figure 4.1. (a) Beam arrangement in the forward box conﬁguration for the three-pulse
FWM experiment. The three laser ﬁelds are applied in a given temporal sequence and
overlapped spatially in the sample. The signal is detected in the direction of the wave
vector ks that satisﬁes the phase-matching condition. The dashed line indicates a plane of
symmetry that makes ﬁelds Ea and Ec equivalent. (b) Temporal sequence of the three
laser pulses for a virtual echo measurement. 2'12 and 2'23 are the time delays between ﬁrst
and second or second and third pulses, respectively. (c) Temporal sequence of the three
laser pulses for a virtual photon echo measurement. Notice that the rephasing occurs at a
time t3 + ((2 — (I) for the stimulated photon echo signal and t3 — ((2 — II) for the virtual
photon echo signal. ........................................................................................................... 56

Figure 4.2. Diagrammatic representation of the processes that contribute to the virtual
echo signal when the time interval between the ﬁrst two pulses is ﬁxed. (a) The double-
sided Feynman diagrams show explicitly the time evolution of the density matrix when
the ﬁelds act on the bra (right) or ket (left) side of the diagram and the time increases
from bottom to top. The diagram on the left, R4, is responsible for the observation of the
ground-state dynamics. The one on the right, R1, is responsible for the excited state
dynamics. (b) The corresponding ladder diagrams illustrate explicitly the energy levels
involved in each electronic interaction. A vertical dashed or solid line represents
interaction on the bra or the ket respectively. In these diagrams, time evolves from left to
right. .................................................................................................................................. 60

Figure 4.3. Experimental F WM data obtained for three different sequences having two
pulses overlapped in time. The signal corresponds to: TG, RTG and PE depending on the
temporal arrangement of the electric ﬁelds. The pulse sequence and the DSFD of the
processes contributing to each phenomenon are included in each case. The dominant
oscillation period in all cases, 307 fs, corresponds to excited state vibrations. The TG and
the RTG transients in (a) and (b) show some additional contribution of ground state
vibrations. The RTG transient presents a prominent slow modulation (2 ps)
corresponding to rotational dephasing and a large background signal. Notice that the PE
data show only excited state vibrations with very little rotational dephasing. ................. 66

Figure 4.4. Experimental transients for sequences with ﬁxed time delay between the ﬁrst
two pulses, PS I, with Tab: 25,5460 fs or Tab: 258:614 fs. The VE signal is obtained when
the initial electric ﬁeld acts on the ket side of the diagram and a PE signal is obtained if
the initial electric ﬁeld acts on the bra side of the diagram. (a) When rab=460 fs, the
oscillation period of the VE signal corresponds to the vibrational motion of the excited
state molecules. (b) When rab=614 fs, the VE signal oscillates with the ground state
vibrational frequency and the rotational dynamics (slow 2 ps modulation) are clearly
observed. Notice that the predominant feature in both PE signals (c and (I) shows 307 fs
oscillations corresponding to the excited state .................................................................. 69

Figure 4.5. Top: Experimental data for VE (thick line) and PE (thin line) measurements
with Tab: 21,5460 fs. Notice that the signals, corresponding to excited state dynamics, are
exactly out of phase. The signal intensity has not been re-scaled, but the VEI signal has
been shifted vertically. Note that the photon echo signal peaks at 154 fs. The 180° phase
difference between these transients can be understood by looking at the dynamics of the
wave packet in the B state of 12 resulting from the interaction with pulse Eb. Bottom:
Simulation of the wave packet motion on the B state of 12 under the conditions of the VB
and PE signals for PS I when Tab: 153:460 fs. In both cases signal formation depends on
de-excitation of ‘Pbm (dotted curve). At time delay z=0 fs, which corresponds to the time
delay between last two pulses, ‘I’bm is in the Franck-Condon region for VB 1, maximizing
the transition probability when the third pulse is applied there. However, for case PE 1,
LPb“) is at the outer turning point of the excited state potential when 1:0 fs, minimizing
the transition probability when the third pulse is applied in the Franck—Condon region.. 71

Figure 4.6. VE transients obtained for a sequence having the time delay between the last
two pulses ﬁxed to rba=460 fs or 21,5614 fs. The vibrational excited-state motion
dominates both transients. A very small contribution from the ground state vibrational
motion is apparent when Iba=614 fs. A larger rotational dephasing is observed (2 ps
modulation) in the 253:614 fs signal and the signal intensity is higher in this case. The
DSFD of the processes that contribute to the signal are included with the pulse sequence.

........................................................................................................................................... 73

Figure 4.7. PE transients obtained for PS 11 when rac=460 fs or Tac=6l4 fs. These
transients, mostly dominated by the excited state motion, do not show the selectivity of
the molecular dynamics with the selection of the ﬁxed time delay. In addition to the pulse
sequence, the DSFD of the processes that contribute to the signal are included. ............. 74

xi

 

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Figure 4.8 Insert: Transient grating and double-sided Feynman diagrams for the case
when pulse b is the last to arrive at the sample. The grating formed by pulses a and c
(indicated by the diagonal lines) cannot diffract pulse b in the signal direction. Violation
of the RWA is identiﬁed in the double-sided Feynman diagrams by the crossed arrows.
C-FID-FWM transients for rab=460 fs (a) and rab=6l4 fs (b). When rab=460 fs, the
oscillation period of the C-FID-F WM signal corresponds to the vibrational motion of the
excited state molecules, whereas when rab=614 fs the signal oscillates with the ground
state vibrational frequency and shows a larger rotational modulation (2 ps). Notice that
the data are very similar to those observed for VB 1 (Figure 4.4a). This implies that the
dominant pathway for signal formation can be illustrated using similar DSFD for the two
ensembles. ......................................................................................................................... 76

Figure 4.9. Cascaded free-induction decay F WM transients for PS I for rac=460 fs (a)
and rac=614 fs (b). Notice that the signal intensity for these sequences is quite low,
especially after the ﬁrst picosecond. The transients do not show selectivity between the
ground and excited state molecular dynamics with the ﬁxed time delay. The pulse
sequence and the coupled DSFD are included for each case. ........................................... 78

Figure 4.10. (a) Representation of the density matrix for a multilevel system composed of
two electronic states, each one with a manifold of vibrational levels. The indices g, g'
range over the number of vibrational levels of the ground electronic state (in the present
simulation g, g' = 1, 2). The indices e, e' range over the number of vibrational levels of
the excited electronic state (in the present simulation e, e' = 3, 4). The diagonal matrix
elements, pgg and pee, represent the population of the vibrational levels (dark shade). The
off-diagonal matrix elements of the diagonal blocks, pgg' and pee, represent the
vibrational coherence in the respective electronic state (light shade). The elements of the
off-diagonal blocks, pge and peg, represent the vibronic coherence between the electronic
states. (b) Double-sided Feynman diagrams for the processes described in the text aﬁer
the ﬁrst two electric ﬁeld interactions only. ..................................................................... 88

Figure 4.11. Simulations of spectrally dispersed FWM. At the top of the ﬁgure are
diagrams of the averaged density matrix elements <|p,j(2)|2> calculated with Equations
(4.7-4.10). The corresponding spectrograms calculated with Equation (4.11) are shown
with wavelength plotted along the horizontal axis and time delay between the second and
third pulses (r) plotted along the vertical axis. The darker regions in spectrograms
correspond to higher signal intensity. For these calculations, the initial population
distribution parameter (w) in the ground state was assumed to be zero and the pulses were
unchirped. (a) The delay between ﬁrst two pulses (rah) is 400 fs, corresponding to 5/2
vibrational periods of the ground state, 18. The density matrix diagram for this pulse
sequence (top left) shows population and coherence terms only in the upper diagonal
block, corresponding to ground-state matrix elements alone. The spectrogram (left)
clearly shows 160 fs periods, reﬂecting the vibration of the ground state (rug). (b) For Tab
= 460 fs, 3/2 vibrational periods of the excited state 22,, only the matrix elements of the
excited state (lower diagonal block) are present in the diagram. In the corresponding
spectrogram (right), the excited-state vibrations of 307 fs are evident ............................. 95

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Figure 4.12. Simulations of spectrally dispersed FWM for Tab = 460 fs, w = O, and two
values of laser pulse chirp. Again, the density matrix diagrams are time independent
throughout the region shown in the spectrograms. (a) For a small value of chirp (¢" =
1000 fsz), the changes in the dynamics are obvious (compare to Fig 2b). These
differences are expected, as seen in the density matrix diagram by the appearance of
vibrational coherences in both the ground and excited states (upper and lower diagonal
blocks). (b) It is possible to calculate the appropriate chirp where ground state elements
dominate the contribution to the signal even for Tab = 460 fs where unchirped pulses
result in seeing only excited-state dynamics. With ¢" = 4000 fs2 (Figure 3b), 160 fs
oscillation periods appear in the simulation, showing primarily ground-state dynamics. 97

Figure 4.13. Experimental spectra (dots) and ﬁt (lines) of the three-pulse FWM signals
and the transform-limited laser. (a) F WM signal observed when all three beams coincide
in time ( Tab = 0 fs; 1: 0 fs). This spectrum is very broad but shows some spectroscopic
transitions. The spectrum is ﬁt with six Gaussian functions. (b) Spectrum of the
unchirped femtosecond laser pulses used for these experiments. This proﬁle is ﬁt with a
Gaussian function with width 8 nm (FWHM). (c) FWM signal observed when rag, = 614
fs and I: 614 fs. This trace shows well-resolved spectroscopic transitions which were ﬁt
by six Gaussian functions. The spacing between the peaks in spectra (a) and (c) is 4 nm,
corresponding to the vibrational spacing of the B state of iodine ................................... 100

Figure 4.14. Spectrally dispersed experimental data for three-pulse FWM with Tab = 0 fs
and transform-limited pulses. Transients taken at six different wavelengths for 3.3 ps are
represented as (a) a topographical plot and (b) a contour plot. As before, the darker areas
in the contour plot correspond to the higher signal intensity whereas the topographical
plot uses lighter shades to show higher intensity. Spectral proﬁles were approximated by
Gaussian proﬁles (see Figure 5a and c) to generate the experimental spectrograms. Both
ground- and excited-state dynamics are observed with these experimental conditions. 104

Figure 4.15. Spectrally dispersed experimental and theoretical spectrograms of three-
pulse F WM for Tab = 614 fs and ¢" = 0 fsz. (a) The spectrogram shows the experimental
transients taken at ﬁve different wavelengths where the spectral proﬁles were
approximated by Gaussian proﬁles. For all wavelengths, the dominant feature is the 160
fs vibrational motion reﬂecting the ground-state dynamics. (b) The initial population
distribution was taken to be w = 0.33 for the calculation of the theoretical spectrogram of
the experiment shown in (a). The 160 fs vibrations from the ground state dynamics are
evident in both the theoretical and experimental spectrograms. ..................................... 105

Figure 4.16. Spectrally dispersed experimental and theoretical spectrograms of three-
pulse FWM for Tab = 460 fs and (15" = 0 fsz. (a) The spectrogram generated from the
transients taken at ﬁve different wavelengths shows only 307 fs oscillations from the
excited-state dynamics. There is no evidence of ground state dynamics. (b) The
simulation of the experiment again used w = 0.33 for the calculation. The theoretical and
experimental spectrograms are in excellent agreement, with the 307 fs oscillations clearly
evident in both ................................................................................................................. 107

xiii

Figure 4.17. Three-dimensional representation of the experimental spectra taken using
differently chirped pulses for the pulse sequence Tab = 460 fs and I: 600 fs. Chirp values
ranged from —2800 fs2 to 2800 fsz. Note the change in intensity towards shorter
wavelengths as the chirp values increase from negative to positive values. .................. 109

Figure 4.18. Spectrally dispersed experimental and theoretical spectrograms of three-
pulse FWM for Tab = 460 fs and (11" = 3300 fsz. (a) The experimental spectrogram was
generated from transients obtained with ﬁve different wavelengths as above. The signal
observed for wavelengths 624 nm and shorter originates from mixed ground- and excited-
state dynamics. It is clear when compared with Figure 8a that the addition of chirp alters
the observed signal by introducing ground-state dynamics for shorter wavelengths. (b)
The theoretical simulation, where w = 0.33, shows good agreement with the experiment
both in the dynamics features and in the intensities of the beats. ................................... 110

Figure 4.19. Diagrams of averaged density matrix elements <|p,-,-(2)|2> and corresponding
theoretical spectrograms for time delay rag, = 614 fs and w = 0.33 with ¢" = 1000 fs2 and
¢" = 4000 fsz. (a) For small chirps, the dynamics at shorter wavelengths reﬂect the
ground state while higher wavelengths represent signal predominantly from the excited
state. (b) When the pulses are chirped to the critical value, (15" = (Zn/(0,22) = 4000 fsz, the
signal for all wavelengths reﬂects excited state dynamics only. Notice that the density
matrix diagram in this case shows population but no coherence in the ground state. 112

Figure 5.1. (a) Pulse sequence, corresponding DSFD, and PE measurements of the
homogeneous vibronic relaxation between the X and B electronic states of 12. The time
delay between pulses b and a(c) was scanned. The DSFD indicates that the signal
depends on the off-diagonal terms pge. The logarithm of the amplitude normalized signal
FWM emission is plotted as a function of time delay I. The solid lines are exponential
ﬁts to the experimental data obtained at different temperatures. (b) Pulse sequence,
corresponding DSFD, and RTG measurements of the inhomogeneous vibronic relaxation
between the X and B electronic states of 12. The time delay between pulses c and a(b) was
scanned. The DSFD shows that signal depends on the off-diagonal terms peg. The
logarithm of the amplitude-normalized signal F WM emission is plotted as a function of
time delay 1. The solid lines are ﬁts to a Gaussian function of the data obtained at

different temperatures. The dashed line corresponds to the exponential decay measured
for the PE data at 108 °C (in Figure 5.1a). ...................................................................... 121

Figure 5.2. (a) RTG measurements of the vibronic relaxation between the X and B
electronic states of 12. The FWM emission is plotted as a function of time delay I. The
solid line is an exponential ﬁt to the experimental data (dots). (b) The pulse sequence and
corresponding DSFD for TG measurements of vibrational coherence in the X and B
electronic states of 12. The DSF D shows that the signal depends on the elements in the
diagonal blocks pgg, pggg pee and peer. The FWM emission is shown as a function of time
delay 2'. The solid line is an exponential ﬁt to the experimental data (dots). The dashed
line indicates the long-lived component of this signal, which decays with T1. The data in
(a) and (b) were obtained in a special cell at 2000 C (see text), and include
inhomogeneous and homogenous broadening. ............................................................... 123

xiv

 

 

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Figure 5.3. (a) The sequence and corresponding DSFD for the VE pulse sequences. The
DSFD show that the signal depends on the peev and pg'g elements. (b) VE measurements of
vibrational coherence in the B (112 =460 fs) and X (112 =614 fs) electronic states of 12.
The intensity of FWM emission is shown as a function of time delay I. The solid line is
an exponential ﬁt to the experimental data (dots). The data were obtained in a special cell
at 200 0C (see text). ......................................................................................................... 124

Figure 5.4. Measurements of vibrational wave-packet spreading: (a) excited state (only
the ﬁrst 12 ps), (b) ground state with the VE pulse sequences as in Figure 3. The decay in
modulation amplitude caused by wave packet spreading was ﬁt to an exponential
function and is included here (dashed lines). The data in (a) show a wave packet revival
at 10 ps. The corresponding power FFT is shown. ........................................................ 125

Figure 6.1. Sketch of the N204 molecule. The N-O distance (a) is 1.19 A , the N-N bond
(b) is 1.776 A , and the angle at is 1346", from reference 234. ...................................... 128

Figure 6.2. 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, d) Frequency-integrated N204 data as a function of time
delay. For all scans, the signal intensity is plotted on a logarithmic scale. .................... 132

Figure 6.3. Experimental time-resolved and frequency-integrated 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. The vertical lines indicate the position of the rotational
recurrences obtained from a simulation using Bave=(B+C)/2. ........................................ 134

Figure 6.4. FFT of the data in Figure 6.3b. The observed peak at 234 cm”1 was assigned
to the N-N stretching mode. ............................................................................................ 136

Figure 6.5. Experimental data 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. .............. 138

XV

1. INTRODUCTION

Some of the main questions that preoccupied scientists and eventually led to the

development of chemistry as one of the fundamental natural sciences were:

How do substances react with each other?

Why do they react as they do?

Is there a way to control and direct the reaction pathways towards a desired

outcome?

Answering these questions, chemists (and physicists) generated new ones and

developed numerous branches of chemistry. This thesis belongs to the branch of physical

 

i. Reproduced in part with permission from I. Pastirk, E. .I. Brown, Q. Zhang and M. Dantus, J.
Chem. Phys. 1998, 108, 4375. Copyright 1998 American Institute of Physics.

ii. Reproduced in part with permission from I. Pastirk, V. V. Lozovoy, B. I. Grimberg, E. J.
Brown, and M. Dantus, J. Phys. Chem. A, 1999, 103, 10226. Copyright 1999 American
Chemical Society.

iii. 1. Pastirk, E. J. Brown, I. Grimberg, V. Lozovoy, and M. Dantus, Faraday Discussion 1999,
1 13, 401. Reproduced in part by permission of The Royal Society of Chemistry.

iv. Reproduced in part with permission from V.V. Lozovoy, B. I. Grimberg, E. J. Brown, I.
Pastirk, and M. Dantus, J. Raman Spectroscopy, 2000, 31, 41. Copyright 2000 John Wiley &
Sons

v. Reproduced in part with permission from V. V. Lozovoy, I. Pastirk, E. J. Brown, B. I.
Grimberg, and M. Dantus, Int. Rev. Phy. Chem. 2000, 19, 531. Copyright 2000 Taylor &
Francis

vi. Reproduced in part with permission from I. Pastirk, Vadim V. Lozovoy, and Marcos Dantus,
Chem. Phys. Lett, 2001, 333, 76. Copyright 2001 Elsevier Science

vii. Reproduced in part with permission from V. V. Lozovoy, Igor Pastirk, M. Comstock and M.
Dantus, Chem. Phys. Lett, 2001, 266, 205. Copyright 2001 Elsevier Science

viii. Reproduced in part with permission from I. Pastirk, M. Comstock, and M. Dantus, Chem.
Phys. Lett., 2000, 349, 71. Copyright 2001 Elsevier Science

 

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chemistry known as molecular dynamics. It evolved from studies of rates of chemical
reactions and chemical kinetics rooted in van’t Hoffs and Arrhenius’s equations."2 In
essence molecular dynamics is concerned about motions of molecules before, during, and

after chemical transformations and the rates at which those changes occur.

When I started my graduate studies at Michigan State University in the spring of
1997, and shortly after joined Professor Dantus’ research group, the ﬁeld of laser control
of molecular dynamics encompassed a quite wide array of laser techniques with an even
wider spectrum of goals. The ﬁrst attempts at using lasers to drive molecular behavior
towards particular aims appeared in late 1960’s, but the investigators encountered
numerous problems, such as intramolecular vibrational redistribution (IVR), and the
attempts were abandoned. The ﬁeld of laser control was reestablished in the late 1980’s
by several new theoretical approaches, and its experimental achievements were sparked
by rapid advancement of ultrashort laser techniques in the early 1990’s. As in any
relatively new scientiﬁc ﬁeld laser control was vaguely deﬁned as an investigation of
light—matter interaction with the aim of altering the dynamics of evolution of a molecule.
The coherence of laser light and quantum interference make coherent and quantum

control of chemical reactions possible.

Over the last ﬁve years the ﬁeld of laser control of molecular dynamics has grown
and diversiﬁed rapidly, primarily as a result of the availability of commercial ultrafast
equipment, but more importantly as a result of a number of successful experimental
demonstrations of its possibilities. The motivation for research in this exciting ﬁeld is
compelling: the idea of using light to alter the outcome of natural molecular processes.

The achievements in the ﬁeld have been reviewed recently3'7 and applications are

 

 

 

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numerous, from the pioneering work dealing with selective bond breaking in selected
systems,8 through molecular alignment910 and selective excitationll'18 to recent
demonstrations of optimal control schemes”23 and possibilities in molecular
computing.24 There are two ultimate goals in laser control. The ﬁrst is to understand the
fundamental processes and light-matter interactions to the level that would allow design
of the ﬁelds required to drive a chemical reaction towards one out of many possible
pathways. The second, which comes as a natural extension of the former one, is to
‘produce a mole of product’, and make the method applicable in the chemical industry.
Although still experimentally unachieved, this goal now seems realizable, unlike a
decade ago. Before practical applications in chemistry, it is likely that applications in a
number of other ﬁelds such as micromachining or optoelectronics (control of a single

quantum dot or construction of a fast optical switch) will develop.

The work presented in this thesis deals with exploration and development of the
experimental techniques that would allow control of the outcome of molecular evolution
(an excitation or dissociation pathway) by controlling the properties of electric ﬁeld that

cause the observed process.

1.1. Observing chemical reactions in real-time

It has been established in the 1930’s that the fundamental chemical processes,
such as bond breakage and bond formation, occur on the femtosecond (fs = 10'l5 s) time
scale.25 Semi-empirical calculations showed that the molecules traveling on the potential

energy surface (PES) from their reactant to their product state pass through the transition

state of an activated complex. Formulation of the transition state theory26»27 showed that
Arrhenius’s pre-exponential factor A (corresponding to frequency of collisions) contains
the probability that the reaction will occur. In other words, it describes the probability
that the activated complex will cross from the reactant valley to the product valley. The
‘frequencies’ of these passages through the activated complex are on the order of 1013
Hz, which corresponds to the 10'13 s in the time domain. This time line represents the
approximate time resolution needed (both in calculations and experiments) in order to
follow the evolution of molecules on the reaction coordinate and also the approximate
period of molecular vibrations. This observation conﬁrmed the need for the development
of time-resolved spectroscopy. The dynamic processes of molecules involve
displacement of electrons and nuclei that are three orders of magnitude apart in energy.
Applying the Bom-Oppenheimer approximation allows investigation of molecular

motion by observing the dynamics of the nuclei.

The tools for measurement of such fast processes became available with the
discovery of lasers followed by the rapid development of pulsed laser sources. Pulsed
lasers allowed progress of time-resolved spectroscopy from the nanosecond in 1960’s to
femtosecond techniques in 1980’s. The shape of a particular electronic PES as well as
energy spacing between the energy levels can be determined from frequency-resolved
spectroscopy. High-resolution information about the investigated molecular systems in
the energy (frequency) domain can be obtained using non-coherent sources (ﬂashlamps,
white-light bulbs) or coherent sources (continuous-wave, nanosecond pulsed lasers). The
time resolution in time-domain spectroscopy depends on the inverse of the coherent

bandwidth of the excitation source used. The shorter the excitation pulse, the faster a

 

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molecular processes can be observed and also the broader the spectral bandwidth.
Therefore in order to resolve molecular vibrations or follow the evolution of an activated
complex, ultrashort ﬁelds are required. The groundbreaking experiment in femtosecond
time-resolved spectroscopy was the study of the transition state in the photodissociation
reaction of ICN performed at Caltech.28 It demonstrated that the observation of transition
states is possible and directly clocked the time duration of a bond breaking reaction. It
was soon followed by a series of experiments on several molecular systems that
established a new ﬁeld in molecular dynamics — femtosecond transition-state
spectroscopy (FTS).29'32 FTS was successfully applied to numerous systems ranging
from diatomics to proteins and to processes in both gas and condensed phases. The basic
principles33s34 and achievements of femtosecond spectroscopy in clocking chemical bond
breaking or formation, in solving the structure of molecules, or in solvation dynamics are

numerous and have been reviewed frequently.”40

Pump-probe and other multipulse methods allow one to study processes as fast as
the pulses themselves. The shortest laser pulses obtained so far are 4.8 fs.4"42 The
coherence and large spectral bandwidth of fs pulses lead to the formation of
nonstationary nuclear wave packets involving translational, vibrational and/or rotational
states. The high peak power associated with femtosecond pulses permits investigation of
multiphoton and nonlinear processes. The properties of femtosecond pulses that allow
time-resolved studies of elementary molecular processes will be described in the

experimental chapter of this thesis.

 

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1.2. Controlling the dynamics of molecules

The road to laser control of chemical reactions has been one with many important
tums.33~43 The invention of the laser gave scientists the hope of controlling chemical
reactions that went beyond photochemistry. The monochromatic quality of continuous-
wave (CW) lasers fostered the idea that chemical bonds could be selectively broken by
multiple photon excitation (MPE) at a speciﬁc frequency determined by the vibrational
frequency of the particular bond.44'48 It was soon discovered that the energy being
deposited in one vibrational mode was quickly dissipated to other intramolecular
chemical bonds by a process termed intramolecular vibrational redistribution (IVR).49'Sl
This realization meant that the MPE technique inevitably resulted in breakage of the
weakest bond. Therefore, the main use of MPE would be to deposit energy in selected
molecules in a mixture, based on spectral absorption. Based on a thorough understanding
of this method, selective bond-breakage would require excitation of a local mode in a
molecular system with extremely slow IVR. This excitation would need to be followed
by prompt bond ﬁssion achieved with UV excitation or a reactive collision. Vibrationally
mediated chemistry was ﬁrst proposed and demonstrated by Letokhov and coworkers.52
Selective bond breaking of the OH or OD bonds in HOD was proposed by Irnre and
coworkers53 and carried out experimentally by Crim and coworkers8 and Valentini and

coworkers.54

With the realization that IVR prevented the accumulation of energy in a given
chemical bond for periods longer than a picosecond, two different approaches to control
chemical reactions were developed independently. One approach was based on the

continuous and coherent excitation of two competing reactive pathways. This method

 

 

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dubbed coherent control (CC), is based on the constructive or destructive interference
among the different photochemical pathways.55~56 This approach has yielded some very
remarkable successes in controlling atomic ionization,57 controlling the Outcome of
chemical reactions,58 controlling the ﬂow of current in semiconductors,59 and controlling
the directionality of photoelectron emission in the gas phase.60 Coherence relaxation is an
important parameter that limits the efﬁciency of this technique. In general, coherence
relaxation is slower for gas-phase samples, especially samples cooled by molecular beam
expansion. On the other hand, the inhomogeneous and homogeneous broadening that
accompanies condensed phase measurements makes this the most challenging
environment for CC. To achieve coherent control in the condensed phase, one must
ensure that the sample-laser interactions occur on a time scale that is faster than IVR and

coherence relaxation.

The other approach to control chemical reactions circumventing IVR is founded
on the use of ultrafast laser pulses. Conceptually, the laser-sample interactions take place
in a time that is short compared to IVR. This concept was embodied in the pump-dump
theoryf’l’62 On the experimental front, Zewail, who had been active in the determination
of IVR rates,63 realized that shorter pulses would be critical for achieving control of
chemical reactions.50 The development of lasers with femtosecond pulse duration made
the pursuit of this work possible.‘54'67 Zewail’s group quickly incorporated these
techniques and dedicated their work to the study the ultrafast dynamics of chemical
reactions in the gas phase.36‘38~68 Ultrafast lasers have been used to make important

contributions to the study of chemical reactions in gases, liquids, solids and surfaces.35

The experimental observation of vibrational wave packet dynamics caused by

impulsive excitation of multiple vibrational levels using femtosecond pulses indicated
that pump-dump control could be used to achieve mode-selective chemistry using
ultrafast lasers.69 This is true provided that the timing between the laser pulses is
carefully controlled. Most of the femtosecond pump-probe experiments can be
considered control experiments, even though the coherence between the laser pulses does
not play a role in the observed signals. The timing between the laser pulses controls the
species that is excited with the probe laser. Limitations of these types of experiments
relate primarily to wave packet spreading and the lack of a well-deﬁned phase between
pump and probe pulses. Even for isolated molecules, wave packet spreading results from
the anharmonicity of the potential energy surface. This limitation can be addressed with
phase altered (chirped) laser pulses, whereby the laser chirp can be used to cause a
focusing of the vibrational wave packet at a particular point in time and space.“~70»7I
Single chirped laser pulses, in fact, have been used to control multiphoton excitation and
chemical reactions.'8’72'74 The main limitation of the use of ultrafast, precisely timed,

chirped pulses has been our limited knowledge of the molecular Hamiltonian.

A new approach to control of chemical reactions was the proposition that an
electric ﬁeld exists such that a speciﬁc target can be optimally achieved.75 This approach
to control depends on the search for the optimal laser pulse. The search, initially carried
out by computer algorithms43’76 has reached experimental realization in recent
years."~12-~19~23~77'80 Optimal control has the distinct advantage that the molecular
Hamiltonian does not need to be known. The search is carried out by a number of
statistical methods that operate on a trial and error basis. The solution is found by a

number of iterations as the optimal ﬁeld is found by convergence to a maximum

 

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attainment of the target. This iterative process, without a priori knowledge, implies that
the technique can be applied to very complex systems. Interestingly enough, some of the
theoretical work in the ﬁeld of optimal control has concluded that optimal solutions in
many cases converge on combinations of a small number of pulses with a speciﬁc time
delay and phase relationship.“82 This observation highlights the importance of every one
of the methods that have been used to control chemical reactions: (a) multiphoton
interactions, (b) short pulses, (c) pulse phase coherence, (d) timing between laser pulses,

and (e) chirp.

The above combination of characteristics can be achieved by a setup combining
collinear, phase-locked femtosecond laser pulses. Scherer et al. showed that phase
locking provided control over the laser-molecule interaction in a collinear pump-probe
arrangement.83 When they changed the phase of the laser pulses by it, their data showed
control over the excited state wave packets of iodine. The experimental realization of
these measurements required active phase control over the two laser pulses in order to
ensure a speciﬁc phase between the pulses in the interaction with the sample. This
requirement makes the technique difﬁcult. Bergmann and coworkers have developed a
method for laser control of rovibrational population transfer based on stimulated Raman
adiabatic passage (STIRAP). This technique uses two collinear pulses with different
wavelengths that are partially overlapped in time (such that the dump pulse precedes the
pump pulse) to achieve near 100% efﬁciency.84 The combination of degenerate
femtosecond laser pulses can be achieved using four—wave mixing methods. These
techniques, developed over the last three decades, involve nonlinear optical processes,

based on the coherent combination of laser pulses without requiring active phase

 

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stabilization.85'87 Coherent Anti Stokes Raman Scattering (CARS) measurements can be
considered examples of control over excitation pathways, as described in the recent work
of Schmitt et al.“,89 In some cases the emission of signal can be controlled in four-wave
mixing (FWM) experiments. This is achieved through the interference of different

nonlinear FWM signals.9O

1.3. Outline of the thesis

The experimental tools and basic design of the femtosecond lasers used in
experiments presented in this thesis are described in Chapter 2. Ultrafast experiments can
be performed with a number of different beam arrangements and experimental setups.
Details on the ones applied in this work are also presented along with pulse-

characterization techniques.

Use of phase-altered (chirped) pulses for controlling the outcome of the
dissociation of CH212 is demonstrated in Chapter 3. It is shown that even simple changes
in ultrashort pulse phase characteristics, such as chirp, can be used to enhance the yield of
one over several other possible channels in chemical reaction. The Dantus research group
at Michigan State University has recognized nonlinear optical techniques, which use
phase matching detection as a means to ensure coherence among a number of laser
beams, to be an ideal platform for the coherent combination of laser pulses. Work on
FWM, presented in this and in part in Dr. Emily Brown’s thesis,” resulted in number of
publications92'103 showing that pulse sequences can take advantage of the vibrational

dynamics of the sample molecules for achieving quantum control of electronic and

 

 

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vibrational processes with three degenerate laser pulses. These experimental
observations, as well as a theoretical understanding developed,'04 show how the pulse

sequences can be used to select among different excitation pathways.

It is demonstrated in Chapter 4 and used throughout this thesis that pulse
sequences can be used to control the observation of ground- or excited-state vibrational
dynamics in three-pulse FWM experiments. One of the reasons for the success of this
technique arises from the realization that laser-molecule interactions can take place with
two components of the electric ﬁeld exp[-i(kx-ax)] or exp[i(kx-at)]. Which one interacts
ﬁrst with the molecule changes the nonlinear optical processes that can be observed and
controlled. This dual interaction is typically explained in terms of bra and ket
interactions. It is described here how this selection is done in the laboratory and what
effects the different pulse sequences have on total interaction of the electric ﬁeld and the
sample. Further, Chapter 4 describes the experimental aspects of the spectrally dispersed
FWM technique, where coherently combined chirped femtosecond pulses are used to
control the intramolecular dynamics of the system.96'98 As with most laser control
methods, FWM based techniques are limited by the rate of coherence loss. Realizing that
quantum control can only be achieved while the system maintains its coherence,
coherence relaxation measurements were carried out (Chapter 5), which use some of the
control pulse sequences we have identiﬁed to provide information from ground- or
excited-state coherenceloo The measurement of the rate of coherence loss is presented in

order to illustrate differences caused by inhomogeneous broadening.

The methods in Chapters 3-5 represent basic experimental explorations of a

model system (the I2 molecule). An extension to somewhat larger molecules is presented

11

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2. EXPERIMENTAL (ultrashort lasers and techniques)

2.1. Introduction

The ﬁrst lasers capable of producing sub-picosecond pulses were developed in the
early 1980’s.64'67,105 Those original ultrafast laser systems were dye-based lasers and
were generally home-built and relatively large. Over the last two decades, the duration of
the pulses shortened and systems shrank to the size of desktop computers. The
experiments presented in this thesis were performed on two different femtosecond laser
systems in Professor Dantus’s laboratory at Michigan State University. The ﬁrst system,
the colliding-pulse mode-locked laser (CPM), is one of the few still operable
femtosecond dye lasers of the original design that sparked the development of ultrafast
science. It is capable of producing 50-60 fs pulses at 30 Hz. These pulses are typically
about 0.3 ml. The second system, a regeneratively ampliﬁed titanium-sapphire laser,
represents the latest generation of ultrafast technology. It produces sub 50 fs pulses at 1

kHz, with energy close to 1 mJ/pulse.

A substantial part of my work on this thesis involved construction, maintenance
and characterization of the laser systems and various experimental setups. Ultrashort
pulse characterization and both laser systems are described in some detail in this Chapter,

along with applied experimental setups and the corresponding detection schemes.

l3

2.2. Femtosecond pulses, characterization and experimental
techniques

The ampliﬁcation of light in the laser medium is possible if the gain exceeds the
loss inside the resonator.‘06'108 This condition depends on the laser medium properties,
and it is generally fulﬁlled only for a narrow range of wavelengths. The number of modes
with wavelengths A that can be ampliﬁed is determined by the length of the light path L
inside the resonator: L=m7t/2, where m is an integer. Wavelengths that satisfy this
relation are called longitudinal modes. Using the relation k=c/v, the difference in
frequency between two neighboring modes can be expressed as Av=c/2L, where c is the
speed of light and v is the frequency. This would mean that an average femtosecond laser
centered at 800 nm (pulse duration ~100 fs, with bandwidth of 10 nm and L~2 m) should
operate with more than 105 modes. If the laser operates simultaneously in several modes
with random phases - the resulting output would be a continuous-wave (CW) beam. In
order to obtain a stable train of ultrashort pulses the phase between the different modes

operating inside the resonator must be held constant. This process is referred to as mode-

locking.

Following the development of ultrafast laser sources a large number of
experimental ultrafast techniques have been developed. Some of the techniques applied in
the exploration of ultrafast phenomena are more suitable for condensed phase studies
while others are preferred methods in gas phase and molecular beam experiments. The
common denominator in virtually all time-resolved ultrafast experiments is the century
old pump-probe principle that was ﬁrst applied to study a chemical reaction with

femtosecond pulses in 1987.28 Its beauty lies in the simplicity of the principle; one pulse

initiates a change in a system that is investigated (pump), the system allowed to evolve in
time and then another pulse is applied to investigate the state of the system (probe). The
work presented in this thesis does not involve the use of the pump—probe technique
directly. It involves three pulse methods generically known as four-wave mixing (FWM),

which have some similarities and are much more powerful.

Starting from the Heisenberg’s relation for the uncertainty between position and

momentum for a wave packet, Avath/Z, and the velocity of that wave packet v (group

velocity), the uncertainty in the energy can be calculated as
AE=A0)2/2m) arm/m) :ApgwAp. The uncertainty in position can be viewed as approximate
width of the wave packet in space, which implies that it will be also spread in time. In

other words it will take a time AtgAx/v for a wave packet to pass through a certain region.

Rearranging these relations yields AEtAIZh/2. Given that E =ﬂ'a), we obtain a frequency-

time uncertainty relation for electromagnetic waves, AwwAtzl/Z. This is a very important
relation for the Work in this thesis, because it can be used to describe ultrafast pulses. It
suggests that for a given spectral width the pulses have a lower limit (minimum) in
duration. Such pulses, where the time-bandwidth product reaches its theoretical
minimum, are referred to as Fourier transform-limited, or TL pulses. The exact numerical
value of this product can vary because the light pulses can assume several temporal
proﬁles.107 Experimentally it is easier to determine the full width at half-maximum
(FWHM) of the spectral and temporal proﬁles. Ultrashort pulses are usually represented

by a Gaussian proﬁle

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where 2' is the pulse duration. For TL Gaussian pulses, the product AwAt=21n2/7r, where
Avis the frequency FWHM (in Hz) and At temporal FWHM (in s). This value is a
consequence of Gaussian shape of the pulse and equals 0.4413. The AVvAt product takes

generally higher values if different pulse shapes are assumed.‘07

The electric ﬁeld of a femtosecond pulse can be represented in time as'092110

E(t) = E0f(t)exp[—ia)0t — tag], (2.2)
where E; is the amplitude, ﬁt) is function describing the time dependence of the pulse
envelope, coo is the pulse central frequency and a is the phase modulation of the pulse in
frequency space (temporal linear chirp). Chirp is a natural phenomenon related to
propagation of light through the media. For a linear phase function across the spectrum,
the pulses are TL. If the function is quadratic, then the pulses are referred to as linearly
chirped. The spectral content of the pulses stays unchanged, but the time duration
lengthens. The pulse envelope ﬁt) is proportional to the square root of the pulse intensity
[(1) and can have a number of different shapes. As mentioned earlier, the shorter the laser
pulse the larger its spectral content will be, therefore the pulse duration and its spectral
width are related quantities. They are Fourier transforms of each other giving in the

frequency domain

16

(60— (1)0)2 —ia'(w —a)0)2

2F 2

 

 

 

6(a)) = .90 exp[-

1.

l: 50 exp[—

(0) - wo )2
27

(2.3)

where y is the complex frequency width deﬁned as y2 =1/rz + ia with Re[1/72]=I/F2

and Im[1/)’7]=a’. The linear chirp a’ is often labeled as ¢” (the notation with ¢”is used
throughout the thesis). The bandwidth of the pulses is given by deﬁnition as

F2 =1/r2+a 212. The temporal and frequency chirp are connected through ¢”I‘2=ar‘?.

The phase variation of a TL pulse has a linear time dependence (usually referred
to as ‘chirp-free’). It has been demonstrated in numerous experiments that knowledge of
both frequency and time proﬁles of the ultrashort pulses and one’s ability to alter them is
necessary in order to obtain correct information about dynamical response of the
molecule. One of the most important characteristics of an ultrashort pulse is the phase
relation among the different frequencies that are present in the pulse. The phase
characteristic of the pulse is caused by the fact that light of different frequencies travels
through dispersive media with different speeds (red propagates faster than blue light).
The frequency dependent phase Wm) of the pulse can be expanded in a Taylor series
around the central frequency (00. The time delay of the frequencies relative to one another

is given by the derivative of the total phase with respect to €015,107;

1 d3¢((00)

61mm): demo) +ldzczxwo)
6 dw3

dw do) 2 dwz (w-wo)2+... (2,4)

(co—coo)+

The ﬁrst term represents a constant time-delay, the second term is linear spectral

l7

chirp (a’ or ¢ ") and the third term is quadratic chirp. Under experimental conditions, the
frequency chirp is more often used (in units of fsz) given that the pulses have a ﬁxed
power spectrum with bandwidth F. The pulse duration is therefore proportional to the

extent of linear frequency chirp 12 =l/1"2+(¢”1)2,109

2.2.1. Femtosecond pulse characterization

Characterization of ultrashort laser pulses requires the determination of the
temporal and spectral proﬁle. There are several methods for ultrashort pulse
characterization: cross-correlation, interferometric autocorrelation, and frequency
resolved optical gating in all of its variations. These techniques share a common principle
in that pulses are measured against themselves. This is a consequence of the very short
time duration of pulses and the fact that no electronic device can have the response time
required. These pulse characterization methods are basically pump-probe ultrafast
experiments themselves, relying in general on the instantaneous response of non-linear
materials. Since the methods presented in this thesis emphasize the time-energy proﬁle
(‘shape’) of ultrashort ﬁelds, it is obvious that each experiment requires precise
knowledge of the character of the applied femtosecond pulse. All of the methods

described below were used.

The simplest and fastest optical method for characterization of the time proﬁle of
the pulse is known as intensity auto-correlation, also referred to as second-order
autocorrelation. The pulsed laser beam is split into two arms of equal intensity by a 1 mm
50/50 beamsplitter in a modiﬁed Michelson interferometer. Two beams are arranged to

propagate parallel to each other and focused on a thin non-linear doubling (second

18

harmonic generation-SHG) crystal (Figure 2.1). Depending on the wavelength, pulse-
duration characterized, and availability, the following crystals were used: lithium
triborate (LBO), B-barium borate (BBQ) and potassium dihydrogen phosphate (KDP).
There are two major classes of phase matching and SHG crystals. Type I is characterized
by the fundamental beams being polarized in either an ordinary (O) or extraordinary (E)
direction. The SHG formed is then either E or 0, respectively. Type 11 phase matching
requires two fundamental photons of different polarization (0+E), and can give either E

or 0 output.

 

PMT
HCC or

CCD

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1 Schematic of the experimental setup for an intensity autocorrelation
measurement. M - mirrors, BS — 50/50 beamsplitter, HCC - hollow comer cube mounted
on a computer controlled variable delay stage, SHG - doubling crystal, L — focusing and
collimating lens, UV — ﬁlter.

All crystals used in described experiments were of Type I. LBO has a wider
acceptance bandwidth but also a greater non-linear coefﬁcient than KDP.lll A larger
non-linear coefﬁcient implies a larger temporal walk-off, making the crystal less suitable
for characterization of very short pulses. The choice of crystal for a particular application

is therefore necessarily a trade-off. The ideal thickness of a SHG crystal used in

19

 

 

£11

or

at

 

autocorrelation measurements ranges from 10 to 300 um, depending on the material and
duration of the pulses measured (see Figure 2.2). For ~ 50 fs pulses, 200-300 um crystals
are successfully used, while for pulses of 30 fs and shorter, 50-100 um crystals are
necessary due to the huge spectral bandwidth of such a short pulse. For characterization
of pulses from titanium-sapphire seed laser (13 fs), 50 um LBO is used due to its large
acceptance bandwidth, while for measuring the ampliﬁed pulses (~50 fs), 300 pm BBC is

used. As the interaction length in the crystal increases, the group velocity mismatch can

greatly affect efﬁciency of SHG and the recorded signal.

 

 

 

 

 

 

 

2.6 _' e
2.4 :' —I—BBO

~22 - —A—LBO

£20 I —e—KDP

"€018 j

531.6 f

.51-4 r

g 1.2 _- e

9-»; 1.0 f

.9 0.8 _'

g O_6 : O/:

§O.4 " e/./
00 /...i....1..,.1....1.

O 50 100 150 200

duration of TL pulse [fs]

Figure 2.2. Quasistatic interaction length (thickness) of selected crystals as a function of
the duration of transform limited pulses for Type I SHG at 800 nm.

It is very important for the ﬁdelity of the measurements that the two beams are of

equal intensity and they experience an identical optical path. In order to avoid unwanted

20

non—linear processes in the SHG crystal (self phase modulation, self-focusing or
continuum generation) the beams should be attenuated. For the same reason (avoiding
strong ﬁelds), the two beams should cross at the crystal slightly before their focal point.
The time delay of one pulse with respect to the other is changed by altering its optical
path using a hollow comer cube mounted on a computer controlled actuator (See Figure
2.1). The autocorrelation signal appears between the two incident beams after summation
in the SHG crystal. It is detected after spectrally ﬁltering the fundamental frequency with
a UV-ﬁlter, and spatially ﬁltering the two doubled beams by a fast photo-diode or a
PMT. A typical autocorrelation trace is shown in Figure 2.3. The pulse duration is
determined by ﬁtting the curve and determining the FWHM of the autocorrelation

function, Ar. If the function has a Gaussian proﬁle the pulse duration is A1=AT/\/2. For a

sech2 proﬁle, At=At/1.54.

1.0

 

0.8

 

 

 

0.6

0.4

Intensity [arb. units]

0.2

 

 

-200 A o ’ 200
time delay [fs]

Figure 2.3. Typical autocorrelation trace (triangles) obtained for ampliﬁed titanium-
sapphire fs laser pulses with a Gaussian function ﬁt. Duration of the pulses is 50 fs.

21

This method can be used for fast determination of the extent of linear chirp ¢’ ’. A
series of experiments where the compressor prism (see following sections) is translated
along its bisector can be used. The shortest pulse duration is assumed to be transform-
limited (7,") and linear chirp (in fsz) is determined from the measured pulse duration (row)

by:112

 

 

2 2 _ 2
¢H=\/Tin(rri;f2 Tin) (25)

where F4ln2.

Another common method used for the characterization of ultrashort pulses is
frequency resolved optical gating (FROG), which relies on the instantaneous response of
a non-linear optical medium.“3" ‘7 There are numerous variations of this method. The
one presented here and used routinely in our lab for pulse characterization is based on the
polarization grating effect P-FROG. Two beams of equal path length and different
intensities are focused by a cylindrical lens on a non-linear material, usually a thin quartz
plate. The strong beam (pump) and weak beam (probe) are polarized at 45° relative to
each other. When the pulses coincide in space and time, the intensity of the probe
produces an optical Kerr effect (an anisotropic change in the refractive index) in the
quartz plate, which rotates the polarization of the probe pulse and allows it to transmit
through a polarizer. The beams are focused in a vertical line, thus producing a time delay
in the vertical axis. Temporal information (the pulse proﬁle) is then carried in the vertical
dimension. The signal is dispersed in a grating to obtain the spectrum along the

horizontal axis. The spectrally resolved signal is detected by a charge-coupled device

22

_ as.»

 

 

 

I’r‘
[er
pla

ol‘t

iter;
retri
tauls
chirr
latt

5mm

31m(

 

 

(CCD) camera. The software that captures the signal from the CCD changes the axes to
produce the image on the monitor, with spectral information encoded vertically and
temporal information encoded horizontally. A FROG trace is thus obtained in the (co, 1)
plane. Typical responses of FROG for transform limited and linearly chirped pulses
obtained on a commercially available FROG apparatus (Clark-MXR) are shown in Figure
2.4. The two-dimensional array of data is then analyzed by commercially available
iterative algorithm on the PC,1 ‘7 and the pulse duration and its phase characteristics are
retrieved. The method provides fast identiﬁcation of the sign of the linear chirp. If the
pulses are positively chirped (see Figure 2.4b), the slope is positive while for the negative
chirp the image has a negative slope. The only downsides of P—FROG are the relatively
low dynamic range (only pulses of ~ 10 u] and greater can be characterized) and
somewhat time-demanding alignment. especially if its intended for analyzing the pulses

at more than one central wavelength.

frequency

 

 

 

time

Figure 2.4. Typical P-FROG trace obtained for transform-limited (a) and positively
chirped (b) pulses from an ampliﬁed CPM laser with central wavelength of 620 nm.

23

 

 

 

 

m]
length In
wave

FigUr
lllanf
the h

rem e

 

The SHG-FROG is a more sensitive method to characterize laser pulses. It can be
used to measure very weak (< 1 n1) pulses. The setup is identical to a second-harmonic
autocorrelator, with the difference that the signal is spectrally resolved. The two-
dimensional (or, 1) signal is recorded. To obtain SHG-FROG traces with beams of
different wavelengths, the conversion requires a small change in the incidence angle of
the SHG crystal (phase-matching angle). A typical trace of SHG FROG with phase
information retrieved by soﬁware is presented in Figure 2.5. The crystal used for this

measurement was 300 pm thick BBQ, and the 1 kHz ampliﬁed titanium sapphire pulses

centered at 804 nm were attenuated to 1 14.1.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0 1.0 , . . ,
410 a) b) c)
g 405- '_.‘ .—,
._. E ,3.
E) 30.5- no.5? -
E 8 g
400
l- ? J
395 . ﬁ . 0,0 - I 0.0 . 4 1
-100 0 100 -100 0 100 780 800 820
time 115] time [fs] wavelength [nm]

Figure 2.5. a) Experimental SHG FROG trace of near TL pulses from an ampliﬁed
titanium sapphire laser, obtained by CCD and spectrometer. Darker shades correspond to
the higher intensity. b) Temporal proﬁle (thin line) and phase dependence (thick line)
retrieved from data in a) by FROG software. c) Spectral (thin line) and phase (thick line)

24

u

_--

 

 

 

P”
ham

pul~

cha

init:
con>
Prep
The

to in

 

 

proﬁles of the pulses retrieved by FROG software. Pulses were 55 fs with time-
bandwidth product of 0.4646. Phase variation of about 0.1 radians across most of the
pulse duration in b) is considered to be a minor deviation from TL. The spectral phase
characteristic in c) signiﬁes minor higher order chirp.

2.2.2. One-pulse excitation

The experiments presented in Chapter 3 use a simple well-characterized pulse to
initiate control over the yield of a known photochemical reaction. The experiment
consists of fs pulse preparation, excitation of the sample and ﬂuorescence detection.
Preparation of the pulses provides a fs pulse of known phase and spectral characteristics.
The pulses can be positively or negatively chirped by a prism-pair arrangement in order

to introduce positive or negative frequency chirp.

Femtosecond pulses at the fundamental frequency or their second harmonic (See
Figure 2.6) were focused on a gaseous sample. The extent of the chirp introduced in
pulses was controlled by a prism-pair arrangement from the pulse compression part of the
laser setup. One of the prisms was mounted on a computer-controlled actuator.
Advancing or retarding the prisms introduces more or less glass in the optical path, and
therefore introduces linear chirp. This setup provides precise control of the amount of the
dispersive medium that the fs pulse encounters and allows one to chirp the pulses
linearly. The intensity of the laser pulses was monitored by a reference photodiode (rPD)
in order to compensate for system instabilities and perform discrimination. Pulses were

characterized by FROG as described in the previous section. The ﬂuorescence produced

25

 

 

 

 

by
ﬁn:

tub

r651

Figu
C0m
|

 

\

Samr

 

by the sample was collected perpendicular to the beam propagation. The ﬂuorescence is
further spectrally ﬁltered through an imaging spectrometer, detected by a photo multiplier
tube (PMT) and sent to a boxcar averager. The analog signal is digitalized by an A/D
converter. The signal was recorded on a personal computer as a function of variable
parameter (chirp), using LabView programs. This kind of setup does not provide time

resolved data (on the ultrafast scale).

l\%__ signal
rPD ‘ ‘ detection

 

 

 

 

 

 

 

shg \ BS 36am
‘/ : cell : i
/ ~310 nm " ' ' ' ' M

 

 

 

from laser
~620 nm

 

 

 

 

Figure 2.6. Experimental setup for one-beam experiment. P1 and P2 - double-pass
compressor prisms. P2 is mounted on an actuator. Mirrors (M) direct the beam to an SHG
crystal where the frequency of the pulses is doubled, and pulses are focused on the
sample cell. rPD — reference photodiode used for discrimination.

26

2.2.3. Three—pulse FWM

Several experiments presented in this thesis are based on the three-pulse F WM
method. The beam arrangement is shown in Figure 2.7. Its principles are described in
detail in chapter 4. The main experimental requirements and considerations are discussed
here. The beam from an fs laser is split into three arms by two consecutive 1 mm thick
beamsplitters. The optical pathway of two of the beams is variable in order to allow time
ordering of the pulses. By changing the length of the path along which the pulse travels
across the optical table, one controls the delay of the pulses with respect to each other.
One of the hollow comer cubes (HCC) is mounted on a computer-controlled actuator
(stepper motor) capable of advancing 0.2 pm per step. This translates into 4/3 fs per step.
The second variable delay with HCC was mounted on a micrometer and its delay with
respect to other pulses was manually controlled. The three beams were arranged in the
forward box geometry (Figure 2.7 top) and focused by a 2 inch diameter 50 cm focal
length lens onto a sample cell. Two aluminum templates were machined and positioned
perpendicular to the axis of beam propagation, in order to ensure that the beams are
parallel before focusing and after the collimation lenses. Alignment is relatively simple;
three beams have to be parallel to each other and have to pass through three holes of the
template that is perpendicular to the beam propagation. If the beams are aligned properly
they will spatially overlap at the focal point. For temporal overlap it is necessary to ﬁnd
the ‘time zero’, the length of each of the three arms at which the pulses coincide in time.
This is performed by placing the SHG crystal or a thin (~100 um) quartz plate at the focal
point, and by retarding or advancing variable delays with hollow corner cubes until the

diffraction pattern is observed at the screen. The FWM signal appears in the fourth comer

27

of the square at the screen; it is collimated, spatially ﬁltered (to avoid any scattering

signal), and detected either by the PMT or the CCD through a spectrometer.

 

 

from laser
HCC HCC
A» /‘\$

 

ks=ka'kb+kc ,

 

, ,' signal ks
spatial ﬁlter ,’
k ' sample cell '
/ .
BS

BS\ ’ '

 

 

 

 

 

 

 

 

 

L L

Figure 2.7. Beam arrangement for the three-pulse four-wave mixing experiment. BS —
beamsplitters; HCC — hollow comer cubes mounted on variable delay stages; L —focusing

and collimating lenses. Detail of the spatial overlap is also shown (top) with the direction
of the signal, detected in the phase-matching direction ks.

2.3. Colliding Pulse Mode-locked Laser System (CPM)

The system used for the experiments described in Chapters 3-5 is a home-built

28

ampliﬁed colliding-pulse mode-locked dye laser (CPM). It is described in detail
elsewhere.64'67-~9'-‘01l 13,119 In short, the laser oscillator is a ring cavity consisting of 7
mirrors (M and CM), 4 prisms (P), and 2 dye jets. Rhodarnine tetraﬂuoroborate (R6G), a
gain dye, dissolved in ethylene glycol, ﬂows through one of the dye jets. The 532 nm line
of a continuous wave Spectra-Physics Millenia laser (typically operating at 1.8 W) is
used to pump the R6G dye optically to an excited vibrational level in the ﬁrst excited
state. The excited dye molecules emit broadband ﬂuorescence as they relax to the ground

state.

The ring-cavity arrangement (Figure 2.8) allows light to propagate in both
counter-clockwise and clockwise directions. The output coupler at the right edge of the
CPM cavity is partially transmissive (2%) and some light is allowed to exit the cavity.
The counter-clockwise propagating beam is used as a reference while the clockwise
propagating beam is sent to the four-stage ampliﬁer. This continuous-wave (CW) laser
can operate in the 580-620 nm range. In this laser cavity a large number of different
frequencies can exist, all oscillating with unrelated phases (multimode operation); thus
the laser is continuous wave. In order to produce pulses, the phase relationship between
all the different frequencies must be set and maintained constant in time. If the initial

phases of all of the modes are equal, one can obtain pulses and the laser is referred to as

c mode-locked’ .64-66, 105.120-122

Mode-locking is achieved using a saturable absorber dye. In the CPM it is 3,3’-
diethyloxadicarbocyanine iodide 2,3 (DODCI) dissolved in ethylene glycol circulating
through the other dye jet (absorber jet). The saturable absorber dye’s absorbance is

intensity-dependent, resulting in no light passing through it when the light intensity is

29

low. When a higher intensity light, caused by a ﬂuctuations in the laser cavity, encounters

30

 

 

 

computer ’4 ‘ —' detection
Tbox- system

car
sync. ' ....... ___‘_l
I experimental sample

cell

’ l set-up
Nd:YAG laser
........ j, _ _ _ _ _ _ _
|Nd:Yvo4|4x \ 532 “m I

532 nm T

PD P N
/ M P P
.. P OC . ________ ‘

r“! 1"]
L_]

r pt” U LJ pressor
’ ’>CM 620 nm .
M M ampliﬁer

absorber jet gain jet

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

! \
o
t

 

 

 

 

 

 

 

 

 

 

 

Figure 2.8. Experimental setup for CPM dye laser with general detection scheme. M and
CM - mirrors and curved mirrors, respectively; P — prisms; OC — output coupler.
Nd:YAG laser and detection are triggered and synchronized to the pulses detected by
photodiode PD.

the saturable absorber dye-jet it saturates the absorbance transition. This allows passage
of some light through the saturable absorber. This light passes through the gain dye
where it is ampliﬁed and then easily passes through the saturable absorber on the next
trip around the cavity. This regimen through the gain and saturable absorber dyes repeats
until the gain dye is saturated. The leading edge of the formed pulse is absorbed in
subsequent round-trips by the saturable absorber (because of its lower intensity) while the
trailing edge is not ampliﬁed by the gain dye (because of saturation). This allows

shortening of the time duration of the pulse resulting in femtosecond pulses. The ﬁnal

31

time-energy proﬁle of the pulse is obtained when the pulse reaches a steady-state

condition and does not change as it propagates through the cavity.

A requirement for stable and efﬁcient operation of the CPM laser is that the two
counter-propagating laser beams must collide in time and space at two points in the
cavity. The optimal situation occurs when they collide at the saturable absorber dye. In
this case there is less loss of intensity at the saturable absorber. Collision at the gain dye
would be the worst situation, as two counter-propagating pulses would be competing with
each other for ampliﬁcation in the gain jet. The maximum recovery time for the gain dye
is achieved if the pulses collide ﬁrst at the saturable absorber jet, and later as far from the
gain jet as possible. This occurs if two jets are placed at a distance apart equal to 'A of the
length of the cavity. Once mode-locked, the laser operates at the central wavelength of
about 622 nm with a bandwidth of up to 10 nm (245-300 cm"). The repetition rate is 100

MHz and the energy per pulse is on the order of 200 pJ.

The refractive indices of different media vary differently with respect to the
wavelength (frequency) of the light. This results in a naturally occurring phenomenon
known as group velocity dispersion (GVD). It arises when different wavelengths travel at
different velocities in a particular medium. It is often referred to as chirp. This
phenomenon will be used as a tool in the control of molecular dynamics as described in
Chapters 3 and 4. The prisms (P) in the laser oscillator are used to compensate for
temporal broadening of the pulses that is introduced by media (ethylene glycol and mirror
coatings) in the ring cavity, to produce the shortest pulses that thegiven bandwidth will

support (transform-limited pulses).

32

The energy of these femtosecond pulses is generally not large enough for gas
phase experiments, therefore the pulses need to be ampliﬁed. A four-stage dye-laser
ampliﬁer was built in our laboratory by Marcos Dantus and Una Marvet. Una’s thesis
contains a more detailed description of the ampliﬁcation process and construction
requirements. In short, the ampliﬁer (Figure 2.8) consists of four Bethune cells (BC)
ﬁlled with circulating gain dyes. Cells are transversely pumped by the frequency-doubled
(532 nm) output of a 30 Hz ﬂash-lamp pumped Nd:YAG laser (typically 10 W or 333 mJ
energy per pulse). The Nd:YAG laser is synchronized with the CPM by the SMl
synchronization electronics, which ensures efﬁcient ampliﬁcation of the femtosecond
pulses. The gain media is kiton red dye dissolved in water in the ﬁrst cell, and rhodamine
640 dissolved in water in the other three cells. Femtosecond pulses from the CPM are
ampliﬁed by stimulated emission from the excitation processes of these dyes. During the
ampliﬁcation process, the pulses travel through water and glass in the ampliﬁer cells and
are broadened by GVD. This is corrected after the ampliﬁcation by introducing negative
GVD, which is achieved by a double-pass prism pair (see Figure 2.5 bottom).123 The
pulses ﬁnally available for experiment were transform limited, 47-60 fs in duration, and

had energy of 0.4 ml per pulse.

2.4. Titanium sapphire laser system

Titanium doped sapphire (A1203) is a great laser material for ultrashort pulse
generation because of its broad gain cross-section. When pumped by a CW frequency
doubled laser (NszVO4) or an Ar+ laser, the TizSapphire emission band (ﬂuorescence),

centered around 800 nm with a 230 nm bandwidth, is capable of sustaining pulses as

33

short as 5 fs. That corresponds to only 2-3 optical cycles of the electric ﬁeld. The ﬁrst
sub-100 fs Ti:Sapphire lasers were built a decade ago, exploiting the material’s high
damage threshold.‘24'127 The mechanism necessary for modelocking Ti:Sapphire lasers
with a large number of longitudinal modes is called Kerr lens modelocking (KLM) or
self-modelocking. The Kerr lens effect relies on a relative index of refraction that is
intensity dependent. If pulses of relatively high intensity are present in the laser
resonator, they will cause self-focusing in the lasing medium (Ti:Sapphire rod). This
effect is present in all longitudinally pumped lasers to some extent and by careful
construction of the laser oscillator it can be made a preferred mode of operation. In order
to initiate and maintain modelocked operation and Kerr lensing, the pulses within the
Ti:Sapphire crystal must be kept short. During the round trip though the laser cavity the
pulse traverses the crystal and disperses (gains in duration). This group velocity
dispersion (GVD) is compensated by a prism-pair located in one arm of the laser cavity.

A typical Ti:Sapphire laser resonator is shown in Figure 2.9.

Pulses generated in resonators similar to the one described here have been
reported with pulses as short as 4.8 fs when compressed.“42 The one built in our
laboratory and used in the experiments described in Chapter 6 produces 13 fs pulses
when compressed. The repetition rate of the laser is 87 MHz, and the pulses are centered
at 800 nm with an energy of 6 nJ/pulse. The oscillator was constructed from a
commercially available kit (KM-labs) and is relatively easy to setup and maintain. The
titanium-sapphire rod in the oscillator is pumped by 3.5-4.5 W of frequency-doubled CW
Nd:YVO4 laser light (532 nm). Its ﬂuorescence is collimated by curved mirrors and

directed along two arms of the oscillator. The laser readily operates in CW mode, and by

34

 

 

 

cw NszVO4 laser \{VI

 

 

 

 

 

 

 

 

    

’ CCD

 

 

 

 

 

Figure 2.9. Titanium-sapphire laser oscillator (seed laser). Light from a NszVO4 laser is
focused by lens (L) through one of the curved mirrors (CM) on the titanium-sapphire
crystal. CM are transparent for 532 nm light, but 100 % reﬂective in the 760-850 nm
region. Emission from the crystal is collected by CM and directed through prisms P1 and
P2 to the end mirror (EM). Length of the cavity (distance from EM to output coupler
0C) is ~ 2m. Pulses shorter than 15 fs when compressed and centered around 800 nm
with FWHM of more than 50 nm were obtained. For the proper operation of the
ampliﬁer, presented in the next ﬁgure, much less bandwidth is needed (25-30 mm) and the
oscillator is adjusted accordingly. For fast determination of the bandwidth, one reﬂection
of P2 is directed from the diffraction grating to an inexpensive CCD and TV monitor.
The image can be calibrated and used for rough estimation of pulse bandwidth.

the mode adjustment it starts self mode-locking. The mode-locking can be achieved by
introducing a small physical perturbation to the oscillator, such as jiggling one of the
prisms. If properly aligned, the central wavelength is tunable in the 780-820 nm range
even in the mode-locked regime. Peak to peak stability is about 1%. Although potentially
very high at the experimental sample (~109 W/cmz), these energy densities are not
enough to perform non-linear experiments on gas-phase samples, and therefore need to
be ampliﬁed. One of the best ways to date to amplify ultrashort pulses generated in solid-
state media is a technique called chirped pulse ampliﬁcation.128 In the CPA technique,
pulses from a high repetition rate (~90 MHz) femtosecond oscillator (seed pulses) are
stretched in time by introducing the frequency chirp to the pulse (see Figure 2.10). This is

performed in the stretcher that consists of a prism pair (folded one-prism arrangement is

35

usually used to save space and cost of gratings). This increases the duration of the pulses
by a factor of 104 (100 ps), and allows the extraction of high energy from the gain
medium (another Ti:Sapphire crystal) in the ampliﬁer, while keeping the peak power
below the damage threshold. Aﬁer temporal broadening the pulses are injected into the 1
kHz frequency doubled NszLF (523.5 nm) pumped regenerative ampliﬁer. Two intra-
cavity pockels cells (PC) and half-waveplate (M2) regulate the number of the pulse’s

passes through the ampliﬁer resonator (Figure 2.10).

 

 

from

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T \J
from ns 1 kHz pump Hui

PCl

 

 

laser

 

Figure 2.10. Schematic for regenerative ampliﬁcation of pulses from titanium-sapphire
laser. Beam from the seed laser is directed through the Faraday isolator (F1) to diffraction
grating G(l) in the stretcher. F1 acts as ﬁlter preventing the back-reﬂected pulses from
the ampliﬁer from reentering the seed laser cavity. The beam diffracted from G1 reﬂects
from silver mirror SM to long mirror LM and back. Still expanded, the beam travels to
periscope P and back for another round trip (G1, SM, LM and back). Spatially
recompressed, but chirped to ~100 ps, the beam is sent to an ampliﬁer (see text).

After the pulse has gained energy in the resonator, it is ejected by PC2 and
directed through a pulse compressor that consists of another grating pair that compensate
the chirp introduced in the stretcher. This conﬁguration of the ampliﬁer generally allows
pulse energies up to 1 mJ at a 1 kHz repetition rate and duration of <50 fs, centered at
800 nm. In order to obtain stable TL pulses the alignment of the stretcher-ampliﬁer must
be perfect. Higher order chirps are easily introduced. The unit used in Dantus laboratory

is commercially available Spitﬁre (SP Lasers) regenerative ampliﬁer.

37

3. QUANTUM CONTROL OF THE YIELD OF A
CHEMICAL REACTION BY SINGLE FEMTOSECOND
PULSES

3.1. Introduction

The goal for active laser control is to devise electromagnetic ﬁelds that drive the
outcome of a chemical reaction in the desired direction.‘l3~‘29'l32 There are basically two
approaches to this problem. The frequency-resolved scheme (also known as coherent
control), proposed by Brumer and Shapiro, utilizes quantum interference between
different pathways to a ﬁnal state to exert control over the outcome.'33’134 One of the
most striking demonstrations of this scheme is found in the work of Gordon and
coworkers where they control autoionization versus predissociation in H1 and DI
molecules.‘35 The time-resolved scheme (also known as pump-dump), proposed by
Tannor and Rice, exploits the time-dependent motion of wave packets created by ultrafast
(usually femtosecond) laser pulses to manipulate the outcome of the reaction.62~I36
Experimental demonstrations of this control scheme are found in many pump-probe time-
resolved experiments, for example, the excitation of 12 to produce either the D (’53:) or
the F (’Zu”) states,‘37 or the production of Na’" or Na2+ as a function of time delay
between pump and probe pulses.‘38 The application of chirped pulses to shape nuclear
wave packets and enhance vibrational coherence was proposed by Ruhman and
Kosloff.‘39 Rabitz ﬁrst formalized the search for an optimal electromagnetic ﬁeld in

terms of spectral and temporal composition to control the outcome of a chemical

38

reaction,140 while Wilson, Mukarnel and coworkers generalized this analysis to obtain a
formalism that is more amenable for the study of thermal ensembles of molecules.1 L73
Boers et al. demonstrated the use of chirped pulses to enhance the population transfer in
the three-state ladder, Ss—>5p—>5d, of the rubidium atom. 1‘“ Experimental and theoretical
studies on the effect of chirped pulses on the multiphoton excitation of molecules showed
that the traditional saturation limits could be exceeded, thereby facilitating population
inversion. The groups of Shank, Wilson, Gerber, and Leone have recently shown
experimental evidence that tailored and phase-altered femtosecond pulses can be used to
modify the initial wave packet formed by the excitation laser'3~'6’l42'I46 and, in some
cases, tailored pulses can be used to enhance single and multiphoton excitation.'7,'3 The
results presented here provide evidence of the effect of chirp on the yield of a

photodissociation pathway for polyatomic molecules.72

The product state (i. e. electronic, vibrational, rotational and translational)
distribution resulting from the photodissociation of polyatomic molecules depends upon
the potential energy surfaces participating in the fragmentation process, along with their
couplings and the characteristics of the incident electromagnetic ﬁeld. The ﬁeld is
characterized by its frequency, duration, intensity and chirp. Chirp (see section 2.2) is
naturally caused by the propagation of laser pulses through matter, which leads to group
velocity variations as a function of frequency within the pulse, thus causing a frequency
sweep.l ‘2 In most cases the group velocity variation causes a positive chirp in which the
leading edge of the pulse is red-shifted and the trailing edge is blue-shifted with respect
to the central frequency of the pulse. Negative chirp corresponds to the opposite effect.

Increases in absolute chirp lead to a temporal broadening of the pulse and are usually

39

considered detrimental for the study of ultrafast phenomena, where the best time
resolution is required. Recently, chirp has been recognized as an important parameter that

can be used to control the dynamics of a system excited by ultrafast laser pulses.13’16'

l8,l 10,142-144,l47

The shape and time evolution of a quantum mechanical wave packet Mt)
produced through absorption of an ultrafast laser pulse are determined, in part, by the

phase factors in the following expression,
—‘E,, M
W) =Xane ’ ’ c0. (1)
n

where E, and (0,, denote the eigenvalue and eigenfunctions of each level n. The quantity
a" is given by the Franck-Condon overlap between the initial and each ﬁnal state n. The
initial phase factor for each eigenstate, the quantity in the exponential, is equal for all
states when excitation takes place with transform limited pulses, i. e. no chirp. However,
chirped pulse excitation opens the possibility for introducing different initial phases.
Therefore, the shape and dynamics of the wave packet can be controlled by chirped

pulses with the goal of affecting the outcome of a chemical process.

3.2. Experimental

The following experiments were carried out with a home built femtosecond laser
system consisting of a colliding pulse mode—locked dye laser (CPM) described in detail
in Chapter 2. The CPM output was ampliﬁed in a pulsed dye ampliﬁer. The ampliﬁed

pulses were compressed by a two—prism double—pass pulse compressor. The laser system

40

produced pulses with 47 fs temporal width and with the central wavelength at 624 nm.
Typical pulse energy was 0.4 mJ (30 Hz repetition rate), but was attenuated for the
experiments to 70 al. In cases where 312 nm femtosecond pulses were necessary, the 624
nm beam was frequency—doubled by a KDP crystal (0.1 mm) producing ~ 7 pl pulses.
The pulses were then focused into a quartz cell containing the gaseous sample.
Fluorescence signals originating from the cell were collected along the direction
perpendicular to the laser beam propagation through a 0.27 meter spectrometer. The laser

intensity of the 312 nm beam was continuously monitored by a reference photodiode.

One of the compression prisms was mounted on a computer-controlled actuator
(See Figure 2.5). Translation of the prism along its bisector varied the amount of glass
through which the beam propagated and permitted the introduction of linear chirp.
Because the amount of chirp was computer controlled, scans of signal intensity as a
ﬁrnction of chirp could be achieved in a convenient and fast manner. The extent of
positive or negative chirp was limited by thickness of the prisms. Characterization of the
femtosecond pulses was carried out by frequency resolved optical gating (FROG).
FROG measurements in our laboratory provided direct measurements of the frequency
components of each pulse as a function of time, from which the vale of linear chirp was
determined. A continuous chirp scale as a function of prism position was compiled using
the expression for group velocity dispersion as a function of prism position derived by

Fork et al.121 and simpliﬁed by Salin and Brun.‘23

Quartz cells containing iodine, 12, or methylene iodide, CH212 (Aldrich 99%),

were prepared on a vacuum line, and were degassed to less than 10‘6 Torr. Iodine

4l

scavenging agents (sodium thiosulphate and copper) were introduced in the CH212 cell to
ensure that the signal derived only from nascent iodine. Experiments were carried out at
room temperature (21° C) with vapor pressures of 0.25 and 1.2 Torr for 12 and CH212

respectively.

3.3. Results and discussion

Spectral proﬁles for the 624 nm and 312 nm pulses measured at zero, negative
and positive chirps were obtained in order to conﬁrm experimentally that the introduced
chirp did not affect the spectrum of the pulses. The intensity variations of the 624 nm and
312 nm pulses as a function of chirp were also measured. The intensity of the
fundamental remained essentially constant, whereas a small variation in the UV intensity
was observed. This variation can be understood in terms of the dependence of second
harmonic generation on the peak intensity of the incoming pulses. As absolute chirp

increases, the pulse width increases, resulting in a reduction of peak intensity.

The concerted elimination of 12 molecules following irradiation of CH212 is known
to occur for excitation energies larger than ~ 9.4 eV.‘48 The yield of this molecular
process has been measured to be on the order of 1%. Most other pathways produce
atomic iodine in its ground (I) and spin orbit excited states (1.). The nascent I2 molecules
are formed primarily in the D' state and have been detected following single and
multiphoton excitation.‘49‘l5| In our laboratory, we have explored the femtosecond
dynamics of the molecular detachment process and have found it to take place in less

than 50 fs.‘52 We have also found that other halogenated alkanes with formulas CX2Y2

42

for (X = H, D, F and Y = Cl, Br and I) undergo similar molecular detachment
processes.'53~'54 Detection of the molecular product is selectively carried out by
dispersing the laser induced ﬂuorescence and collecting only a spectral window that

contains the majority of the D'—>A' emission.

Multiphoton excitation of CH212 with 624 nm pulses produces the well known 12
D'—>A’ emission at 342 nm. Figure 3.1a presents variations in the relative yield of the
molecular detachment pathway as a function of linear chirp. The data have been
normalized to unity at zero chirp. Note that the maximum yield is observed for -500 fs2
and the minimum at 2,400 fsz. We observe a factor of 2.9 in the overall change in the
yield for this pathway as a function of chirp. These data are contrasted with a chirp scan
obtained under similar conditions with the 624 nm pulses on the yield from three photon
excitation of 12 vapor to yield D——>X and D'—>A' ﬂuorescence between 320 - 345 nm.
These results are presented in Figure 3.1b and show an opposite trend to that obtained for
CH212 dissociation. The minimum yield for the three photon excitation is found at -500 fs2
but increases by ~ 40 % as the magnitude of chirp increases. Wilson and coworkers have
investigated the chirp effect on the three-photon absorption yield for 12 at 550 nm, 570
nm and 600 nm.'7sl8 They found that the yields are signiﬁcantly affected by the chirp
(approximately factors of 2 and 3 for 600 nm and 570 nm excitation respectively). Our

results on 12 are in very good agreement with their ﬁndings.

We have also explored the effect of chirp on the multiphoton excitation of CH212
with 312 nm laser pulses. Figure 3.2 presents the yield of the molecular pathway

determined by detection of 12 D'—>A' ﬂuorescence intensity as a function of chirp. For

43

1.2

1.0

’5 0.8 ‘9' T INK» D
4,; _ .
,3, 342m
N 0.6 -— 624rm
“3 _l_ 5-
X
>" ' ‘ '
-1000 O

1.6 ‘ I '
E" . D'
E Z,
.0' 1.4 - 34211.“ ‘ ‘ -
.5. 624m L A, W
8 L . J
5 - X 3'
8 1.2 r , ﬂ
"5’ - ‘5 w
:s Q
a W
._,N 1.0 "

-1000 ' o 1000 2000
Linear Chirp [fsz]

Figure 3.1 (a) Experimental measurement of the yield of the molecular pathway

producing 12 from the multiphoton dissociation of CH212 with 624 nm laser pulses as a
ﬁrnction of chirp. The insert shows the relevant energetics for the reaction. (b)
Fluorescence yield as a function of chirp following the multiphoton excitation of 12 vapor.

these experiments, both the ﬂuorescence signal and 312 nm laser intensity were recorded
and averaged for 500 laser shots at each chirp value. The ﬂuorescence intensity was
normalized against the third power of the 312 nm laser intensity (because of the three
photon excitation and the inherent variation of 312 nm intensity as a function of chirp). It

is clear that increasing the chirp enhances the photodissociation yield signiﬁcantly.

 

if

 

 

 

 

 

 

44

 

 

The molecular pathway enhancement is found to be non-symmetric, favoring
positive over negative chirps. This observation implies that the observed enhancements
are not due to pulse width effects but rather depend on the magnitude and sign of the
linear chirp. The three scans shown in Figure 3.2 were obtained under identical
conditions except for the intentional changes in the pulse intensity from 0.8x1012 to
1.6x1012 W/cm2 (calculated for zero chirp pulses). The data is shown normalized to laser
pulse energy but is not corrected for variations in peak intensity caused by pulse
broadening as a function of chirp. The effect of laser intensity on these control
experiments is shown in the insert, where the molecular pathway yield is shown to
increase for positive 2400 fs2 chirp by factors of 3 to 25 as the zero-chirp intensity of the

laser pulses is increased from 0.8x10’2 to 2.4x1012 W/cmz, see Figure 3.2 insert.

We consider the differences between the dependence of the molecular pathway
yield on laser pulse chirp for 624 nm and 312 nm excitation to be of great interest. For
624 nm, the yield decreases with absolute chirp, while for 312 nm it increases. We have
not taken into account the fact that the temporal pulse width of the pulses increases with
linear chirp. Multiphoton transitions are, in general, expected to increase by factors
proportional to the peak intensity raised to the nth power (1") where n is the number of
photons. Therefore, multiphoton excitation is expected to be maximized when the chirp is
zero. For 624 nm excitation, the maximum yield was found for a chirp of -500 fsz;
however, for 312 nm the maximum yield in the available chirp range was found for a
chirp of 2400 fsz. Based on the pulse width change and the three photon excitation for the
312 nm case, the transition probability, proportional to (1":3), is expected to decrease by a

factor of 27 (because the pulse width triples at this chirp value). Therefore, the yield of 12

45

 

 

 

 

 

 

 

 

10 . 1 m I f r ' l
_ 25 L 12 Yield ‘ o— 1.61012W/cm2 J
15 - / -_ 1.1 1012 W/cm2
8 _ r V— 0.8 11112 WIcm2 _.
5 ' Mr - 1 o
.75.. - 0.5 1.0 1.512 2.0 2 2.5 ~
g Laser Intensrty 10 [W/cm ]
e 6 _ O 7
1531 \\\\\ -
4‘1 _ IL:— Dr q
.5 g . /
% 4 — 312 nm 342 nm / /- '-
5‘ l iA' o
0.; l- X I /- ..
._ 2
Dd 2 _ o ‘1
__ V—“v9\"\K~ .ég/ _,
0 . 1 . 1 . 1 . 1 r
-1000 0 1000 2000

Linear Chirp [fsz]

Figure 3.2 Experimental measurement of the yield of the molecular pathway
producing 12 from the multiphoton dissociation of CH212 with 312 nm laser pulses as a
function of experimentally available chirp range. The inserts show the relevant energetics
for the reaction, as well as a plot of the maximum 12 yield enhancement recorded at 2400
fs2 chirp, as a function of laser peak intensity (measured at zero chirp).

production should track with the transition probability instead of showing the observed

enhancement.

The effects caused by chirp in the excitation pulses reﬂect characteristics of the
potential energy surfaces and the nascent wave packet dynamics. For diatomic 12, Wilson

and coworkers have been able to explain their observed chirp effects based on quantum

46

mechanical calculations that show a “wave packet following” effect for positive chirp.
Currently, the potential energy surfaces involved for CH212, are not known, preventing us
from giving an accurate quantum mechanical description of the effect. In principle a
similar wave packet following effect could be responsible, given that the ﬁrst photon
transition is resonant, as in the 12 experiment ﬁ'om Wilson’s group. The order of
magnitude changes demonstrated give additional evidence that quantum control of the
yield of chemical reactions involving polyatomic molecules is a promising area of

research.

The chirp of femtosecond pulses can have profound effects on photophysical and
photochemical processes. For diatomic iodine, both positive and negative chirps enhance
I2 ﬂuorescence signals. These ﬁndings are in good agreement with those obtained by
Wilson and coworkers. However, for the multiphoton photodissociation of CH212 with
624 nm, the opposite chirping effects, with the maximum 12 yield near chirp zero, are
observed. Results obtained for 312 nm multiphoton excitation of CH212 exhibit a
minimum 12 yield near chirp zero. The yield of the molecular pathway is found to be
asymmetric with respect to the sign of the chirp and enhancements up to a factor of 25 for
positive chirps are observed. The amount of enhancement is also found to be highly

dependent on the pulse intensities.

47

4. PULSE SEQUENCES AND CHIRP AS A TOOL IN
F EMTOSECOND QUANTUM CONTROL OF
MOLECULAR EXCITATION AND INTRAMOLECULAR
DYNAMICS

4.1. Introduction

Work presented in this chapter demonstrates how three-pulse F WM can be used
to understand laser excitation, suggests how to predict the outcome from simple pulse
sequences, and suggests how to achieve experimental control over the population transfer
and coherences between two states. The approach presented here is to use pulse
sequences in three-pulse F WM to study closely the interaction between ultrashort laser

pulses and model molecular system.

A two-level system is the ideal starting point for discussing laser excitation. The

probability of excitation from the ground to the excited state in a two-level system is

expressed quantum mechanically asl<e|p - E(t) g>|2 = (e

 

p-E(t) g)<g|u-E(t)’|e> , where u
is the transition dipole moment and E(t) is the electric ﬁeld. This expression implies that
in order to transfer part of the population from one state to the other, two interactions
with the electric ﬁeld are required, one with E(t) and one with E(t)*. A good example of
this kind of population transfer is excitation by any light source as performed routinely in
linear spectroscopy methods. Both interactions are with the same ﬁeld but do not imply
that it is a two-photon process. Achieving the population inversion or driving the

excitation process itself into desired direction with the laser pulse requires individual

48

manipulation of both of the electric ﬁelds involved in the excitation. As will be described
in this chapter, full control over the transfer of population between the ground and the
excited states could be accomplished.94 Such a technique that allows individual
manipulation of the electric ﬁelds E(t) and E(t)’I can be obtained if non-linear optical
methods are combined with ultrashort excitation sources. The technique has a number of
modiﬁcations but is commonly referred to as three-pulse FWM. As a version of transient
grating (TG) techniques it has been used to investigate solids, liquids and gases almost
since the invention of the laser;l55"59 it has been investigated in depth‘50-l62 and
textbooks have been written on these subjects.35'37 When used in combination with
femtosecond pulses it reveals new possibilities, as it will be demonstrated in this chapter.
It was until recently most commonly applied for molecular dynamics studies in

condensed phases. 163465

Gas-phase FWM studies have a long history, starting with the observation of
photon echoes in SF6 ‘56 and the ﬁrst observation of molecular rotation by a third-order
nonlinear process.‘57 Numerous gas-phase applications of transient grating techniques
were pioneered by Fayer,‘59v‘66~I67 especially for the study of ﬂames. These novel probes
have been applied to the study of femtosecond dynamics in the gas phase. Zewail and
coworkers used degenerate four-wave mixing (DFWM) for probing reaction
dynamics.I68 Matemy and coworkers have studied iodine vapor using time-resolved
CARS and DFWM.83~‘69"72 By varying the time delay of one of the incident pulses
while maintaining the other two incident pulses overlapped in time, they showed that
vibrational and rotational dynamics can be observed for both the ground and excited

electronic states. Knopp et al. have used two-dimensional time-delay CARS to study high

49

energy ground-state vibrations. ‘73

Brown et al. used off-resonance transient grating methods to study the response of
atoms as well as polyatomic molecules in the gas phase and derived a semiclassical
expression to analyze the rotational coherences observed in the data.92 Resonance
transient grating measurements conﬁrmed the observation of ground and excited states.
Our group has explored different laser pulse sequences in three-pulse FWM to control the
observation of ground- or excited-state dynamics and has shown that the spectrally
dispersed three-pulse FWM method adds an important dimension to the study of non-

linear phenomena with transform limited and chirped pulses.”-102

Three-Pulse F WM is a nonlinear spectroscopic method that involves the
interaction of three laser pulses in a phase-matched geometry with a sample. It requires a
precisely deﬁned time sequence of the pulses. The effect of these sequences in control of
molecular dynamics is discussed here. The FWM signal develops from a third-order
polarization resulting from the interaction of the three electric ﬁelds and is itself a
coherent beam forming a fourth electromagnetic wave.87 Its similarity to the pump-probe
technique is straightforward; a preparation step is followed after a variable time delay by
a probing step. Its advantage lies in the greater degree of control over the preparation and
probing processes because various pulse sequences are possible. The technique of
quantum control by applying various pulse sequences presented here, is an ideal tool to
learn about and manipulate the quantum mechanical processes involved in laser control
of chemical reactions and also reveals which ‘knobs’ are available for controlling laser-

molecule interactions.

50

In addition to exploring the effect of different pulse sequences, the effect of
linearly chirped electric ﬁelds on three-pulse FWM measurements is also explored.
Chirped pulses affect the phase of the individual contributions from ro-vibrational levels
of the molecular system. These effects are followed in time and the data show different
dephasing dynamics depending on the magnitude and sign of the chirp. These effects are
also observed by spectrally dispersing the three-pulse FWM signal to measure the phase-
dependent amplitude of each of the different spectral components. This method,
spectrally dispersed three-pulse FWM, is ideal for characterizing and understanding

molecular excitation with chirped laser ﬁelds.95

The experimental demonstration of laser control of chemical reactions by other
methods has been advancing rapidly over last ﬁfteen years following the success of
femtosecond laser technology. Achievements in this area of research have been
thoroughly reviewed.43-63J37~ Coherent control depends on the quantum mechanical
interference between two excitation paths created by two phase-locked lasers. The
experiments are carried out by picking two lasers that can reach the same quantum
mechanical state by one-, two-, or three-photon transitions. This technique is typically
formulated in the frequency domain with phase as the controlling knob. The pump-dump
technique involves the transfer of population between two or more states; control is
achieved by taking advantage of the dynamics of the molecular system. This technique is
typically formulated in the time domain. The time between the pulses is the control knob;
therefore, most pump-probe experiments in the femtosecond time scale can be considered
examples of this technique. Mode-selective control involves using one pulse to select a

vibrational overtone of the molecule and a second pulse to cleave the bond that has been

51

selectively excited. This technique combines frequency and time resolution (usually in
nanoseconds). Quantum control usually involves a single laser pulse that is tailored in the
time and frequency domains. Typically the control knobs are linear chirp, pulse duration
(transform-limited pulses), and pulse intensity. Experimental demonstrations of this
technique often involve multiphoton transitions. Optimal control seeks to ﬁnd the optimal
electric ﬁeld to cause the desired excitation.‘74"76 Theoretically, this search can be
carried out by two approaches. The ﬁrst is to seek a direct solution of the quantum
mechanical problem for the optimal ﬁeld (a problem that is computationally feasible for
very simple systems). The second is to use an algorithm that quickly explores different
possible ﬁelds and contains a feedback on the degree of success that each ﬁeld attains.
Experimentally, groups have demonstrated equipment that can be used to search for the
‘ideal’ laser pulse to achieve a particular outcome.'9”?-"22vI77 This method has been
demonstrated recently to control a laser-initiated chemical reaction. It is important to
realize that the classiﬁcation of control approaches should be taken loosely, as the ﬁeld is

relatively new and under constant development.

It is clear from this brief overview that laser control of chemical reactions
depends on the repeated interaction between the laser ﬁelds and molecules in the system.
Although there are various formulations of the problem, the fundamental goal is to
control population transfer to different quantum states (leading to different products of

chemical reactions) by tailoring the electric ﬁelds in the time and frequency domains.

52

4.2. Four wave mixing

Four-wave mixing techniques provide a powerful platform for combining
coherently multiple laser pulses. The results presented demonstrate control over the
observed dynamics from ground- and excited-state populations and coherences. The
FWM signal results from the polarization of the sample following three different electric
ﬁeld interactions. The effectiveness of these techniques in controlling chemical reactions
is explored here by manipulating the time sequence between laser pulses. The phase
relationship between the pulses is maintained by detecting the signal in a phase-matching
direction. Several different pulse sequences were applied (transient grating (TG), reverse
transient grating (RTG), virtual and photon echo (V E and PE)) and their contributions to
dynamics of sample molecules were sorted out. For example, the virtual echo sequence is
achieved by the interactions of the sample with three consecutive electric ﬁelds
characterized by exp[i(kx-wt)], exp[-i(kx-at)] and exp[i(kx-ax)]. This sequence allows
control over the observed ground or excited state dynamics. With the PE pulse sequence,
characterized by interactions with exp[-i(kx-at)], exp[i(kx-at)], and exp[i(kx-at)], we
ﬁnd that control of ground and excited state populations is not achieved. These

differences between pulse sequences are shown experimentally.

4.2.1. The role of pulse sequences in controlling ultrafast
intramolecular dynamics with FWM

This section explores the different four-wave mixing phenomena that can be
obtained as a ﬁrnction of pulse sequences using degenerate ultrafast pulses. The goal is to

reach a more fundamental understanding of these processes by learning how the time

53

order between the pulses can be used to control the different nonlinear optical responses
that lead to signal formation. The data from this systematic approach provide the relative
intensities from the different nonlinear processes, and reveal pulse sequences that are
amenable to control electronic state excitation. The experimental data obtained for
different pulse sequences directly demonstrates the need to separate bra and ket
interactions for a complete understanding of FWM experiments with non-collinear laser
pulses. We have found that under certain conditions, e.g. gas-phase samples and high
sample concentration, a cascading process can occur, whereby the free—induction decay
emission, stimulated by one of the pulses, can mix with two other beams to produce an
additional source of signal, designated here cascaded free-induction decay four-wave
mixing (C-FID-FWM). This article provides a large body of experimental data on a
model system consisting of isolated molecular iodine that is well understood and can be

used to reﬁne present theories of FWM.

In the condensed phase, the collective polarizability of the sample contributes to
the nonlinear response and it is difﬁcult to separate the intermolecular from the
intramolecular processes that cause the observed signal. In dilute gases, it is much easier
to analyze in detail the photophysical processes that lead to signal formation.
Inhomogeneous broadening is small and the coherence decay is much longer than the
laser pulse duration. This latter characteristic leads to long-lived signals, on the order of

100 ps, which in turn allow for very accurate analysis.

Gas-phase molecular iodine was chosen as a model molecule for this study for
various reasons. (i) The visible X ’Zo.g <—) B ”110+u transition has been well characterized

by frequency and time domain spectroscopies. (ii) The vibrational periods of both the X

54

and B states are longer than the duration of our laser pulses, allowing us to impulsively
excite wave packets in each electronic state. (iii) The vibrational periods of the X and B
states are quite different, 160 fs and 307 fs respectively, making the assignment of the
signal relatively easy. (iv) At the wavelength of excitation, molecular iodine can be
treated as a two (electronic) level system. The state reached by two-photon excitation is

repulsive and does not contribute to the signal discussed in this study.

4.2.2. Formal description of nonlinear optical processes involved in
sequences’ control scheme

There are numerous publications in the literature on the nature of nonlinear
optical processes.”87 Some work involves the density matrix approach,l°4 whereas in
other cases the wave packet formalism is used to understand FWM-type processes.'7"I78
We ﬁnd that the density matrix approach provides a complete framework for interpreting
nonlinear optic phenomena. The density matrix approach naturally incorporates all of the
initially populated vibrational and rotational states necessary to express the collective
nature of a coherent process. However, for some processes the beneﬁts of the intuitive

wave packet picture were exploited.

Three-pulse FWM involves a sequential interaction of three electric ﬁelds with a
molecular ensemble. The incident pulses are typically arranged in the forward-box
geometry shown in Figure 4.1a (see detailed beam arrangement in Figure 2.6), and the
coherent signal is emitted in the direction kg with frequency (05. Both the wave vector
and the frequency of the signal are linear combinations of the wave vectors and

frequencies of the applied ﬁelds. The wave vector kg: ka— kb+ kc satisﬁes the phase-

55

 

a) Phase matching

ks zka'kb+kc

 

 

2'12 T23 2'12 PE

 

 

t1 t2 (3 tlme
c) Virtual Echo sequence

2'12 ,'
£723

 

 

 

)

t1 t2 t3 tlme

Figure 4.1. (a) Beam arrangement in the forward box conﬁguration for the three-pulse
FWM experiment. The three laser ﬁelds are applied in a given temporal sequence and
overlapped spatially in the sample. The signal is detected in the direction of the wave
vector ks that satisﬁes the phase-matching condition. The dashed line indicates a plane of
symmetry that makes ﬁelds E8 and Ec equivalent. (b) Temporal sequence of the three
laser pulses for a virtual echo measurement. 712 and 123 are the time delays between ﬁrst
and second or second and third pulses, respectively. (c) Temporal sequence of the three
laser pulses for a virtual photon echo measurement. Notice that the rephasing occurs at a
time (3 + (t2 - t,) for the stimulated photon echo signal and t3 - (t2 — II) for the virtual
photon echo signal.

matching condition, ensuring conservation of energy and momentum. The source of the
signal is the third-order polarization in the direction ks, P513 ’(t), that arises after weak

interactions with the applied ﬁelds. For homodyne detection, the intensity of the signal

i586,87

56

°° 2
[FWM oc ﬂPQ’Kt) dt. (4.1)

If the applied electric ﬁelds are ultrashort laser pulses, various rotational,
vibrational and in some cases, electronic levels can be excited impulsively due to the
large bandwidth. In those cases the signal reﬂects the coherent superposition of molecular
polarizations, showing coherent oscillations (or quantum beats). In this way, the optical
polarization contains information regarding the electronic and nuclear dynamics
manifested in the spectra. The coherent signals in a FWM process involve the entire
molecular ensemble that is interrogated by the laser electric ﬁelds. The interpretation of
the signals requires the density matrix formalism for which the statistical nature of the
quantum mechanical ensemble is explicitly taken into account. The density matrix
formalism will be just brieﬂy described here. The most detailed description of this
formalism is presented by Grimberg et. al.104 The third-order polarization of the

molecular ensemble is deﬁned a535'87

13%) = T4132?) (1)] . (4.2)

where P is the polarization operator and ﬁg” (t) is the third-order density matrix with a

spatial dependence 6”“ 'r .

The temporal evolution of the density matrix can be represented diagrammatically
using either double-sided Feynman diagrams (DSFD)35'87’I79 or ladder diagrams.180 Each
diagram is associated with a sequence of transformations or pathway in a phase space of

the density matrix. The total number of diagrams for a given direction ks is 48. However,

57

under the rotating wave approximation (RWA), the number of possible pathways is

reduced to eight. When the system includes only two electronic states, the frequency

dependence of the slowly rotating terms is exp[i(a)—a)eg )t], where a) is the carrier

frequency of the pulses resonant with the transition frequency cocg. Based on the RWA the
contribution of slowly rotating terms is much greater, therefore pulse sequences having
the ﬁrst two electric ﬁelds interacting with opposite wave vectors are primarily
responsible for the FWM signal. As a consequence, the ﬁrst two pulses of a three-pulse
FWM sequence form gratings in the direction i (kl — k2), the third pulse Bragg diffracts
(Figure 2.6) from that grating and the direction of the signal is given by kg = k3 i (k1 —
k2). The speciﬁc pulse sequence indicated by the subindices 1, 2, and 3 determines a
speciﬁc non-linear process detected (TG, RTG, VE or PE) at the direction kg and the

corresponding coherent transient effect probed.

When collected at k3 = — k1 + k2 + k3, the signal corresponds to a stimulated
photon echo (PE) process.87 This process is similar to Hahn spin echo'31 in NMR.‘32
When the signal is collected at kg = k. — k2 + k3, the signal has been named a virtual
photon echo (VE) phenomenon. Alternate names for this non-rephasing FWM signal can
be found in the literature. One of the key differences between VB and PE phenomena is
the fact that PE phenomena can be regarded as a reversible decay process for which the
polarization rephases after the interaction with the third laser pulse. For PE this
polarization reaches a maximum at a time (3 + ((2 — t1) (see Figure 4.1b) whereas for VE
the polarization reaches a maximum at a time t3 — (t2 — t1) (see Figure 4.1c), hence the

name virtual and the lack of observable rephasing. This difference is of critical

58

importance for samples where the inhomogeneous broadening is signiﬁcant such as in
liquids and solids. For PE measurements, the inhomogeneous dephasing is cancelled by
the optical rephasing. Each experimental signal obtained with a speciﬁc pulse sequence
in the direction ks is the coherent sum of the signal corresponding to several third-order
pathways in the Liouville space. Therefore, more than one diagram may be associated
with a particular three-pulse FWM phenomenon. Each diagram contributes additively to
the overall third order polarization and the signal is proportional to the square of that
polarization. Diagrams for which the ﬁrst interaction is on the ket side correspond to VE
phenomena, whereas the diagrams that initially show a bra side interaction correspond to

PE phenomena.

We have chosen the pulse sequence k1 — k2 + k3 (see Figure 4.2) to demonstrate
the different density matrix pathways involved in a VB process. In the DSFD (Figure
4.2.a), the bra (right side vertical line) and the ket (left side vertical line) interactions are
kept separately. Each electric ﬁeld interaction is indicated with a short slanted arrow. The
convention for the sign of the electric ﬁeld wave vectors is that positive wave vectors
with exp[i(kr-at)] point to the right while negative wave vectors with exp[-i(kr-ax)]
point to the left on the diagram. Its relative position and label t., t2 or t3 are used to
indicate the time ordering of the laser pulse in a given sequence. The ﬁrst interaction with
the system causes a change on only one side of the Feynman diagram (bra or ket) giving
rise to an electronic coherence, designated pmge or p")cg respectively. The second
interaction produces a population change, designated pmgg or p‘z’ce, and a ro-vibrational

coupling within each electronic state, designated p‘z’g'g or p‘z’ggv in the ground state or

59

pm“, or pm“ in the excited state. The symbols g g' and e e' represent different ro-
vibrational levels in the ground and excited states respectively. Note that the DSFD in the
left, labeled R4, proceeds through the formation of pmg'g while the one on the right, R1,
proceeds through the formation of p(2’ec., The third interaction produces the third order

electronic coherence pme'g or p‘J’cg' that gives rise to the signal according to Equations 4.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

and 4.2.
a) Double-Sided Feynman Diagrams
E
SN ES \
E3
pe'g t peg'
E3
E2 pg'g 1066'
p t2 ’2 p
eg t] EST-ﬁg. E2
E1 E1
lg) <g| lg) <g|
R4 b) Ladder Diagrams R1
1e) 5 1 1e)
0’1 “02 ‘03 0’s 0’1 ‘0’2: ‘03: “’5
lg) ‘ ' lg)
2'ab 7 Tab 7

Figure 4.2. Diagrammatic representation of the processes that contribute to the virtual
echo signal when the time interval between the ﬁrst two pulses is ﬁxed. (a) The double-
sided Feynman diagrams show explicitly the time evolution of the density matrix when
the ﬁelds act on the bra (right) or ket (left) side of the diagram and the time increases
from bottom to top. The diagram on the left, R4, is responsible for the observation of the
ground-state dynamics. The one on the right, R1, is responsible for the excited state
dynamics. (b) The corresponding ladder diagrams illustrate explicitly the energy levels
involved in each electronic interaction. A vertical dashed or solid line represents

interaction on the bra or the ket respectively. In these diagrams, time evolves from left to
right.

60

The corresponding ladder diagrams are also shown in Figure 4.2.b. Each energy
level is designated by a horizontal line with the time evolving from left to right.
Interactions on the ket side are shown as solid vertical arrows and on the bra side as
dashed vertical arrows. For FWM processes such as coherent anti-stokes Raman
scattering (CARS) involving laser pulses with different carrier frequencies, the ladder
diagrams may be preferred because they depict energy explicitly. Unfortunately, in this
case it is not as easy to keep track of the state of the bra and ket aﬁer each pulse

interaction as it is in the DSFD.

4.2.3. Experimental

The experimental setup used here has been described in Chapter 2. The
experiments were performed with a home-built ampliﬁed CPM dye laser producing 60 fs
pulses (transform-limited) centered around 620 nm. The bandwidth at FWHM is typically
about 8 nm. At the output of the laser ampliﬁer, the femtosecond pulses were
compensated for linear chirp in a double-pass prism arrangement with an average energy
of 350 111. After the compression, the beam was split by two successive beam splitters
into three beams of approximately equal intensity attenuated down to ~ 20 11.1 each. The
three beams were combined in the usual FWM forward box geometry and focused by a
500 mm lens in a quartz cell containing the iodine vapor. The amount of iodine in the cell
was limited so that the optical density reaches a maximum of 0.4 OD at 620 nm and is
independent of temperature for T > 80° C. The experiments presented here were carried
out at 140° C. Pulses in one of the beams (EC or Eb, depending on the setup) were

delayed with respect to the other two by a computer-controlled actuator (variable t

61

delay), while the ﬁxed delay (rm, where n, m=a, b, or c) was adjusted by delaying the
pulses from the second beam with respect to the ﬁxed one (beam E3 in all cases) using a

micrometer.

All the experimental data were obtained in the direction kg = ka — k, + kc. The
subindices a, b, and c can take any time order 1, 2 or 3; therefore, for the same phase-
matching geometry, different phenomena such PE or VE can be detected. The spatially
ﬁltered three-pulse FWM signal was collected by a spectrometer with wide spectral
acceptance (8 nm resolution). The transients were taken at 300-400 different time delays
(about 200 shown) and averaged for nine scans. At each time delay I, the signal was
collected for 10 laser shots. The laser intensity was monitored on a shot-to-shot basis;
laser pulses with intensity outside one standard deviation were rejected. The accumulated
laser intensity for each given time delay was stored and the ﬁnal transients were
normalized by this intensity. An effort was made to maintain constant laser intensities for
all the transients and to present all transients using a consistent intensity scale to allow

direct comparisons.

Three laser pulses can be arranged into six sequences (abc, acb, bca, bac, cab,
cba), each having a distinct pulse ordering. For each pulse sequence, we can deﬁne time
delays between the ﬁrst and second pulses (m) and the second and third pulses (223) (see
Figure 4.1). We can divide the pulse sequences into those where two pulses overlap in
time (Table 4.1) and those where the three pulses are separated (Table 4.2). We can
further subdivide the pulse sequences into those where the ﬁrst time delay is ﬁxed (pulse

sequence 1, PS I), and those for which the second time delay is ﬁxed (pulse sequence 11,

62

PS 11). In the experiments presented here, the phase-matching geometry of detection (see
Figure 4.1a) makes ﬁelds E, and Ec equivalent. The total number of unique combinations
and permutations is ten; these are shown in boldface type, with four in Table 4.1 and six
in Table 4.2. The ten combinations lead to the observation of different nonlinear optical
phenomena, such as transient grating, reverse-transient grating, virtual photon echo,

photon echo, and cascaded free-induction decay FWM.

Table l.

 

(I) Seqllﬁnm (11)

I ...... I JL ..... I

overlapped scanned scanned overlapped

 

 

“"6 * kl-kl'+k2 TransientGrating 0””

k-k +k' ReverseTG
cb,a c,ba a: ' 2 2

 

 

ac.b* k.+k,'-k2 ReversePE b,“ * -k1+k2'+k2 PhotonEcho

 

 

 

Table 1. Summary of the observed phenomena for the different pulse sequences possible
with the three laser pulses when two pulses overlap in time. For the experimental
arrangement used here, there are only four distinguishable sequences (shown in bold
face). Because beams Ea and Ec are equivalent, two additional (though not unique)
sequences are possible. Sequences for which experimental data are presented have been
labeled with a star. The phenomena include transient grating (TG), reverse-transient
grating (RTG), reverse photon echo (RPE), and photon echo (PE).

The experimental data are shown along with the schematic of the pulse sequence
and the applicable DSFD. Many of the labels for the DSFD have been omitted in order to
simplify the diagrams here. The DSF D corresponding to the formation of an excited state
population after the ﬁrst two interactions are shown on the right side while the leﬁ-side
diagram(s) correspond to formation of a ground state population after the ﬁrst two
interactions. The time at which each ﬁeld interacts with the system is indicated by a

dotted line. For the DSFD shown in ﬁgures on following pages (Figures 4.3-4.8,) the

63

distance between these dotted lines indicates the time delay, i. e., the smaller distance
depicts the ﬁxed time delay rm and the larger distance depicts the variable time delay I.
When the ﬁxed time delay is zero, only the variable time delay is reﬂected by the dotted
lines. The experimental transients show the FWM signal as a function of variable time
delay I. All FWM signals are displayed relative to each other, 1'. e., the transients have not
been re-scaled. This allows for direct comparisons of signal intensity among the different

pulse sequences.

 

 

 

 

 

Table2.
Phenomena (1) Sequences ([1)
ﬁxed - - -scanned scanned- - - ﬁxed
VirtualPhotonEcho ab,c alt a,bc
kl-kz-l-k3 Cm, (VEI) c’ba * (VEIl)
Stimulated Photon Echo 1,0,0 * b,ac a1:
-k1+k2+k3 baa “’5” b... “’E")
C-FlD-FWM ac,b a1: a,cb
“+1121, cal, (C-FIDFWMI) c,” * (C-FID-FWMII)

 

 

 

 

 

Table 2. Summary of the observed phenomena and the different pulse sequences possible
with three time-separated laser pulses. For the experimental arrangement used here, there
are only six distinguishable sequences (shown in bold face). Because beams Ea and Ec are
equivalent, additional (though not unique) sequences are possible. Sequences for which
experimental data are presented have been labeled with a star. The phenomena include
virtual photon echo (VE), stimulated photon echo (PE), and cascaded free-induction
decay FWM (C-FID-FWM).

64

4.2.4. Overlapped Pulses

Figure 4.3 displays the dynamics observed when two of the laser pulses are
overlapped in time while the third pulse precedes or follows the other two. In the
transient grating (TG) measurement (Figure 4.3a), ﬁelds Ea and Eb are overlapped in time
(rah = 0 fs) and ﬁeld Ec follows at a variable time delay 1. Three DSFD describe this
process and indicate that both ground and excited state dynamics (0(2),;g and pa’cc'
respectively) should be observed. In the ﬁrst two picoseconds of the transient, the
dominant oscillations reﬂect the 307 fs; the vibrational period of the B excited state of
iodine. The transient also clearly shows 160 fs oscillations, reﬂecting the ground state
vibrational period of 12. Ground state dynamics are more clearly observed at longer time
delays because of wave packet spreading in the more anharrnonic B excited state. The
fast Fourier transform (FFT) of this transient conﬁrmed the assignment of the dynamics

to the X (208 cm”) and B state (108 cm") vibrations (v" = 2-4 and v' = 6-11).93~94

In the reverse-transient grating (RTG) experiment (Figure 4.3b), ﬁeld Ec precedes
the overlapped ﬁelds E, and Eb. In this case, Ec excites an electronic coherence with a
relaxation lifetime of ~1 50 ps.97 When Eb and E3 arrive, the grating is formed with Eb and
the polarization generated by E... The observed vibrational dynamics are dominated by
307 fs oscillations, corresponding to the excited state of 12. A FFT of the transient
conﬁrms that the contribution of the excited state is greater than that of the ground state.
The time evolution of the coherence leads to the simultaneous observation of both ground

and excited state dynamics for this pulse sequence.

65

 

a) Transient Grating ab

 

o‘ .r--- -. .-
b) Reverse Transient Grating 0 ba

 

 

 

b
2 a
c
o ............................ °
- c) Photon Echo b

FWM Signal [arbitrary units]

 

 

 

 

Delay Time 2'[psj

Figure 4.3. Experimental FWM data obtained for three different sequences having two
pulses overlapped in time. The signal corresponds to: TG, RTG and PE depending on the
temporal arrangement of the electric ﬁelds. The pulse sequence and the DSFD of the
processes contributing to each phenomenon are included in each case. The dominant
oscillation period in all cases, 307 fs, corresponds to excited state vibrations. The TG and
the RTG transients in (a) and (b) show some additional contribution of ground state
vibrations. The RTG transient presents a prominent slow modulation (2 ps)
corresponding to rotational dephasing and a large background signal. Notice that the PE
data show only excited state vibrations with very little rotational dephasing.

66

In the photon echo (PE) measurement (Figure 4.3c), the pulse sequence is similar
to the one responsible for the RTG except that ﬁeld Eb is applied ﬁrst, followed by Ea and
Ec which are overlapped in time. The data show primarily excited state dynamics as can
be observed in the transient. The most important difference between these transients is
that ﬁeld Eb acts ﬁrst and is then followed by the overlapped ﬁelds E8 and EC. Photon
echo processes involve a rephasing of the coherence that is lost due to inhomogeneities in
the sample. A direct comparison between the reverse-transient grating and photon echo
transients reveals that the slow undulation (2 ps) in the RTG measurement as well as the
background are smaller in the PE transient. This difference is also apparent in the FFT of
the PE and the RTG transients, where the rotational contributions (peaks at frequencies <

20 cm") in the PE FFT are greatly reduced.

The predominance of the excited state dynamics in both RTG and PE transients
can be attributed to the fact that the wave packet motion in the excited state has a much
wider range of internuclear distances. This takes the wave packet in and out of the
Franck-Condon region. Therefore, the electronic polarization reﬂects predominantly
excited state dynamics. Only in pulse sequences where pathways involving excited state
population transfer can be avoided, the ground state dynamics predominates. Results
shown in Figure 4.3. are an example of observation of quantum dynamics, but also of the
selectivity. This selectivity in the observation of ground state dynamics can be achieved

by choosing the appropriate time delay between the pulses, described in next section.

67

4.2.5. Pulse Sequence 1: Virtual Photon Echo and Stimulated Photon
Echo

In Figure 4.4, we use Pulse Sequence I (ﬁxed delay followed by variable delay) to
observe the dynamics obtained as the ﬁxed delay is changed from 460 fs (Figure 4.41) to
614 fs (Figure 4.4.b) for virtual photon echo (VE I) and stimulated photon echo (PE 1)
sequences (Figures 4.4c and d). In both cases, ﬁelds Ea and E, are separated by a ﬁxed
time delay (rah or rba) and are followed by ﬁeld Ec after a variable time delay I. There are
two DSFD that apply to each of these pulse sequences; one shows pgvgm and the other
peeve). In the VB 1 measurement, when Tab is 460 fs (Figure 4a) (3/2 I, where 2', = 21t/cq,
for 12 in the B state), the dynamics show 307 fs oscillations, reﬂecting only an excited
state contribution. When Tab is 614 fs (Figure 4.4b) (twice the vibrational period of 12 in
the B state), the dynamics show 160 fs oscillations, reﬂecting predominately a ground
state contribution. By changing the ﬁxed time delay between ﬁelds E1, and Eb, the
observation of excited or ground state dynamics of 12 can be controlled. This type of
coherent control should not be conﬁrsed with control experiments without phase locking

or phase-matching.

In the PE 1 conﬁguration, when rba is 460 fs, the dynamics reﬂect an excited state
contribution with 307 fs oscillations; no ground state contribution is observed in this
transient. When rba is 614 fs, the 307 fs oscillations still dominate; however, two
picoseconds latter, 160 fs oscillations can be seen. Fourier transforms of these two PE
transients conﬁrm that for rba = 614 fs, there is some ground state contribution. The
signal contribution due to rotational dynamics (slow 2 ps modulation) that is clearly

observed in the VB 1 transients is much smaller in the PE 1 transients. Note that the

68

 

 

 

 

 

 

4 VirtualEchoI
3‘ a) tab=460 fs
2 a b c
_-z-_
1- 111 1
>
o ...........................
’ C
4‘ b) tab=6l4 fs c
' b
E" 3‘ b
g 2-1 ' ‘ I a a
S ’ R4 R1
.13 1‘
_Q .
E. o .................... . .......
€30 4; Photon Echol
'c'r‘: ‘ c)rba=460fs
3..
E ’ b a c
u. 2.
. Arr-1
1’ ;
C
d)rba=614fs c a
b a b
R3 R2

 

 

 

 

Delay Time 2' [ps]

Figure 4.4. Experimental transients for sequences with ﬁxed time delay between the ﬁrst
two pulses, PS I, with tab: zba=460 fs or Tab: zba=614 fs. The VE signal is obtained when
the initial electric ﬁeld acts on the ket side of the diagram and a PE signal is obtained if
the initial electric ﬁeld acts on the bra side of the diagram. (a) When rab=460 fs, the
oscillation period of the VE signal corresponds to the vibrational motion of the excited
state molecules. (b) When rab=614 fs, the VE signal oscillates with the ground state
vibrational frequency and the rotational dynamics (slow 2 ps modulation) are clearly
observed. Notice that the predominant feature in both PE signals (c and d) shows 307 fs
oscillations corresponding to the excited state.

69

selection between ground- or excited-state dynamics is much more efﬁcient for the
virtual echo setup (Figures 4.4a and 4.4b). The reason for this observation can be
deduced from inspecting the DSFD. For VB 1, the DSFD that leads to the observation of
ground state has three laser interactions acting on the ket. This leads to high selectivity
between the two states. For VB 1, the appearance of ground state dynamics arises from a
wave packet being prepared in the excited state, then pumped to the ground state, and
ﬁnally probed as a function of time, thereby giving a clear and intense ground state
signal. For PE 1, the DSFD that leads to ground state dynamics shows that the ﬁrst two
interactions are on the bra while the third interaction is on the ket. This action on an
unperturbed ground state by the third pulse leads to loss of selectivity. The reason for the
small ground state modulation in the PE 1 data is due to the interference between wave

packets generated from different initial states.

In Figure 4.5, enlargements of the VE I and PE I transients from Figures 4.4 a and
c, for Tab: rba=460 fs are shown. It is clear that the two transients are exactly out of phase
with each other; when one is at a maximum, the other is at a minimum. The applicable
Feynman diagram for the observation of excited state dynamics is on the right-hand-side
in each case of Figure 4.4. In both cases, ﬁeld Ec must interact with the wave packet
formed in the excited state by ﬁeld Eb. This process is depicted by the action of Ec on the
bra for both DSF D responsible for excited-state populations. In Figure 4.5 (bottom),
schematics based on calculated wave packets of the excitation process after the ﬁrst two
electric ﬁelds are shown. The wave packet resulting from excitation by ﬁeld Ea is shown

as a solid line; the wave packet resulting from excitation by ﬁeld Eb is shown as a dashed

70

 

 

 

7.3—3“ tab=460fs
§2_ VEI
El-
.971 PEI
2

o 1 2 3

Delay Time t [ps]

 

 

 

 

 

 

 

 

Figure 4.5. Top: Experimental data for VE (thick line) and PE (thin line) measurements
with rab= rba=460 fs. Notice that the signals, corresponding to excited state dynamics, are
exactly out of phase. The signal intensity has not been re-scaled, but the VEI signal has
been shifted vertically. Note that the photon echo signal peaks at 154 fs. The 180° phase
difference between these transients can be understood by looking at the dynamics of the
wave packet in the B state of 12 resulting from the interaction with pulse Eb. Bottom:
Simulation of the wave packet motion on the B state of 12 under the conditions of the VB
and PE signals for PS I when Tab: rba=460 fs. In both cases signal formation depends on
de-excitation of ‘I’bm (dotted curve). At time delay F0 fs, which corresponds to the time
delay between last two pulses, LPb“) is in the F ranck-Condon region for VB 1, maximizing
the transition probability when the third pulse is applied there. However, for case PE 1,
LPb") is at the outer turning point of the excited state potential when r=0 fs, minimizing
the transition probability when the third pulse is applied in the Franck-Condon region.

71

line. It is important to note that signal formation for both processes requires the
de-excitation of the wave packet ‘Pmb by ﬁeld EC. In the virtual echo case, ﬁeld Eb
interacts with the system second with rab=460 fs. At r=0 fs (time delay for ﬁeld Ec with
respect to the second pulse), the dashed wave packet is at the inner turning point of the
potential where the transition probability is maximized. Therefore the interaction results
in a maximum signal. However, in the photon echo, ﬁeld Eb interacts ﬁrst with the
system (still with rba=460 fs). At r=0 fs, the wave packet is at the outer turning point of
the potential with negligible transition probability upon interaction with ﬁeld EC. After
half a vibrational period, the wave packet returns to the inner turning point where the
transition probability is maximized and the interaction with Ec produces a maximum

signal. Therefore, the signals for PE 1 and VE I are exactly out-of-phase.

4.2.6. Pulse Sequence II : Virtual Photon Echo and Stimulated
Photon Echo

In Figure 4.6, the experimental signal resulting from Pulse Sequence II (variable
time delay followed by ﬁxed time delay) is observed. The ﬁxed delay is changed from
460 fs (a) to 614 fs (b). In this case, an electronic coherence is created by the ﬁrst ﬁeld
(EC) which evolves until ﬁelds E, and E2, arrive. The ﬁxed time delay can act as a “ﬁlter”
rather than a control parameter in selecting the observation of ground or excited state
dynamics. There are two applicable DFSD for this conﬁguration. The diagram on the left
has all the interactions on the ket while the diagram on the right has the last two
interactions on the bra. For rba=460 fs, the transient shows 307 fs oscillations,

corresponding to excited-state dynamics. The signal is weaker and shows only a small

72

background. When 2ha=614 fs, the transient is dominated by 307 fs oscillations. The
signal is stronger and shows a larger background. Weak 160 fs vibrations are also
observed. Fourier transforms have conﬁrmed that the rba = 614 fs transient shows a
contribution of the ground state. For a time delay of zba=6l4 fs, the observed background
arises from the process depicted by the DSFD on the right. The use of the ﬁxed delay as a
ﬁlter for the dynamics changes the ground state contributions to the signal slightly but

does not give the same degree of control as is observed in the VB 1 case (see Figures 4.4a

 

 

 

 

 

and 4.4b).
6‘ I
V1rtual Echo H
4- _
a) rba — 460 fs
E c b a
g 2- - - 2'- - _
.>.~ ' All/\AAANMAANW M
13
1‘: 0 . . . . . . a
E 1 b _ _
g 4_ b) 2ba=6l4fs a b
.9.” ' '
(I) _ C C
E 2- R4 R1
LL.
0 1—* v I v v r
0 1 2 3 4 5

Delay Time 1: [ps]

Figure 4.6. VE transients obtained for a sequence having the time delay between the last
two pulses ﬁxed to rba=460 fs or 21,5614 fs. The vibrational excited-state motion
dominates both transients. A very small contribution from the ground state vibrational
motion is apparent when 21,,=614fs. A larger rotational dephasing is observed (2 ps
modulation) in the 21,5614 fs signal and the signal intensity is higher in this case. The
DSFD of the processes that contribute to the signal are included with the pulse sequence.

73

In Figure 4.7, we use Pulse Sequence II to observe the dynamics obtained as the
ﬁxed delay is changed 460 fs to 614 fs. In these measurements, ﬁeld E, precedes ﬁelds Ea
and EC, setting up a stimulated photon echo (SPE). The coherence generated by the ﬁrst
laser pulse is probed by the second and third pulses. The rotational dynamics, as
evidenced by the slow 2 ps modulation, are signiﬁcantly reduced in the PE 11 transients as
compared to those observed in the VE II transients. The transient in Figure 4.7a shows
predominately 307 fs dynamics and a small ground state contribution. The transient
shown in Figure 4.7b reveals a predominately excited state contribution to the signal.

These assignments were conﬁrmed by FFT.

 

 

 

 

 

Photon Echo II

1 a)rac=460fs
i . 1 . 11
E >
E o ............................
Te * b)2'ac=6l4fs c
a) a
£7) 1‘ '
E b b

 

0 1 2 3 4 5
Delay Time 2'[ps]

Figure 4.7. PE transients obtained for PS 11 when rac=460 fs or 23c=614 fs. These
transients, mostly dominated by the excited state motion, do not show the selectivity of
the molecular dynamics with the selection of the ﬁxed time delay. In addition to the pulse
sequence, the DSFD of the processes that contribute to the signal are included.

74

4.2.7. Cascaded F rec-Induction Decay F WM

So far we have discussed a number of phenomena and outlined the rules for their
diagrammatic representation. The identiﬁcation of the eight different diagrams that can
contribute to the signal within the RWA has been shown in a number of publications.
When a particular sequence of resonant ﬁelds violates the RWA, the FWM signal should
not be observed (it is negligible) as denoted in upper-right comer diagrams in Figure 4.8.
An example would be the sequence k1 + k2 - k3 for a two electronic state system. This
corresponds to pulses E2, and Ec preceding pulse Eb (see Figure 4.8). We set out to test
this hypothesis and detected weak signals corresponding to an interesting FWM

phenomena arising under certain conditions described below.

In Figure 4.8a, the signal obtained with the pulse sequence Ec followed by Ea and
Eb is presented. The signal intensity is about a factor of four lower than the VB 1 data. It
is remarkable to see that when 2ab=460 fs, the transient shows 307 fs oscillations,
corresponding to excited state dynamics. When rab=614 fs, 160 fs oscillations are evident
indicating ground state dynamics. Therefore, we are able to observe ground or excited
state dynamics by changing the ﬁxed time delay as it was done in VB 1 (see Figure 4.4a
and b). Our explanation for this process is that ﬁeld (EC) creates a ﬁrst-order polarization
that decays by free-induction decay emission (FIDE) with time T2 > 50 ps. This emission
is designated as FIDEC. Beams E2, and Eb form the grating with populations pu’gg and
plz’ee', then the FIDEc Bragg diffracts from the grating to form the signal. Because the
FIDE is generated in a separate ensemble of molecules we designate this process a
cascade, hence the abbreviation C-FID-FWM. This situation is similar to the VE I

measurements except that the beam EC is replaced by FIDEc. The observation of

75

 

 

 

C-FID-FWM II
kb ’‘2:
'5? 1‘ X c
.+.:: a)2 Tab — ~460 fs "a c c a a
’51 forbidden a
b MAR/MNU‘N‘AAMM
8
E. a
E. c ....................... ﬂ . W T T b/Ef 117F113:EC
"a
,7, b)2 2ab=614fs . FIDEC
E 1- FIDEc
c
ensemble 121 ensemble2
G ............................

 

 

 

Delay Time 2'[ps]

Figure 4.8 Insert: Transient grating and double-sided Feynman diagrams for the case
when pulse b is the last to arrive at the sample. The grating formed by pulses a and c
(indicated by the diagonal lines) cannot diffract pulse b in the signal direction. Violation
of the RWA is identiﬁed in the double-sided Feynman diagrams by the crossed arrows.
C—FID-FWM transients for rab=460 fs (a) and rab=614 fs (b). When rab=460 fs, the
oscillation period of the C-FID-FWM signal corresponds to the vibrational motion of the
excited state molecules, whereas when 23b=614 fs the signal oscillates with the ground
state vibrational frequency and shows a larger rotational modulation (2 ps). Notice that
the data are very similar to those observed for VE I (Figure 4.4a). This implies that the
dominant pathway for signal formation can be illustrated using similar DSFD for the two
ensembles.

selectivity between ground- and excited-state dynamics conﬁrms the interpretation of this
phenomenon. Note that the only way to represent this ﬁrst-order cascading process
diagrammatically is by invoking two ensembles, one responsible for the ﬁrst-order
polarization and the other for the FWM process. This explanation does not involve a

violation of the RWA.

The signal identiﬁed here as C-FID-FWM arises when the sample has a relatively

76

long-lived polarization. The lasers are resonant with a transition and the sample is in
relatively high concentration. Under these conditions each pulse generates a free-
induction decay with an integrated emission intensity that can be comparable with the
ﬁeld that induces it. The FID emission under these circumstances participates in the
formation of FWM signal. This type of signal is best observed when other FWM
processes are forbidden. Observation of this type of signal can be found in the work of

Prior. 1 82

PS I used to explore other C-FID-FWM signals is shown in Figure 4.9. For both
transients in this ﬁgure, 2ac=460 fs and 2ac=614 fs, the dynamics are dominated by excited
state vibrations. F FT conﬁrms this observation. The ground state contribution is slightly
higher for the transient with 2ac=614 fs than for 23c=460 fs. These transients can be
interpreted as follows: ﬁelds Ec and Eb form the grating, then the free-induction decay
emission stimulated by Ea, F IDEa, mixes to form the signal, C-FID-FWM 1. Because the
ﬁxed time delay is now between the ﬁrst two pulses, one participating in the population
formation and the second in the generation of a F IDEa, selectivity is not expect between
ground and excited state dynamics. The time delay difference between the ﬁrst two
pulses is half the excited state vibrational period. This causes the 235460 fs transient to
be similar to the rac=6l4 fs transient except for being out-of-phase by half a vibrational

period (154 fs).

4.2.8. Conclusions

The classiﬁcation and the comparison of different phenomena that can be

observed with femtosecond pulses on gas-phase molecules using speciﬁc pulse sequences

77

 

C-FID-F WM I

a) raw = 460 fs

11:2— ,

FIDE
FIDEa
b) Z'ac =614 fs aj: c/bCDDEa
’0’” 1 I " , ensemble 1 ensemble 2
o

vvvvvvvvvvvvvvvvvvvvvvvvvv

 

FWM Signal [arbitrary units]

 

 

 

Delay Time 2'[ps]5

Figure 4.9. Cascaded free-induction decay FWM transients for PS I for 235460 fs (a)
and 2ac=614 fs (b). Notice that the signal intensity for these sequences is quite low,
especially after the ﬁrst picosecond. The transients do not show selectivity between the
ground and excited state molecular dynamics with the ﬁxed time delay. The pulse
sequence and the coupled DSFD are included for each case.

are presented. Three laser pulses can generate six different nonlinear optical phenomena:
virtual echo, photon echo, transient grating, reverse-transient grating, cascaded free-
induction decay four-wave mixing and reverse photon echo. While these phenomena are
closely related, it is important to distinguish between them. Some allow the experimenter
to control the observation of ground or excited state dynamics. Others allow the control
of the phase of the observed molecular vibrations. Finally, some of these techniques

achieve cancellation or enhancement of inhomogeneous broadening.

One of the important conclusions is the distinction between the bra and ket laser

interactions. Quantum mechanically, the transition probability from state g to state e

78

depends on <g>el21El><g l #Ele > implying a double interaction. The physical basis for

the double interaction comes from the fact that the electric ﬁeld has two components
exp[i(kor- 01)] and exp[-i(kor-at)]. Molecules may interact with either of the components
(thus forming a coherence state) or with both (thus forming a population in the original or
in the new state). Starting from the ground state and invoking the rotating wave
approximation, a ket interaction can only be achieved with exp[i(kr-ax)] and a bra
interaction can only be achieved with exp[-i(kr-at)]. When the three incident pulses have
distinguishable wave vectors, as for most non-collinear geometries, one is able to deﬁne a
phase-matching geometry that determines the sign and hence the nature of each electric
ﬁeld interaction. Therefore, a distinction between a bra or a ket interaction can be
experimentally achieved. The discussed phenomena correspond to products of four
matrix elements (three incident ﬁelds and formed signal). However, presented description
involves elements up to the second order (pg-(2’). The speciﬁc aim is solution of control
over the system dynamics. One can envision the presented method as expanded pump-
probe experiments, where the ﬁrst two interactions prepare the system and the third one
interrogates it. The time-integrated detected experimental signal is caused by third
interaction (00(3)) and carries information about the preparation of the system (with ﬁrst
two ﬁelds). Fourth order matrix element would carry the information about the system
after the signal is emitted and is not of interest here. Also, the interrogation of the state of
the system after third ﬁeld, and its temporal proﬁle would require additional time gating

of the signal.

The distinction that arises from the geometric arrangement of the lasers and the

79

possibility to have ﬁxed or variable time delay between them leads to the ten unique
pulse sequences described in Tables 1 and 2. The data presented show several
differences. For stimulated photon echo measurements, the signal does not contain
elements that can be attributed to inhomogeneous broadening while virtual echo
measurements do. This difference is best shown in Figure 4.3 by comparing the reverse-
transient grating (see Figure 4.3b) and the photon echo signals (see Figure 4.3c).
Coherence relaxation measurements on gas-phase iodine as a function of temperature
presented in Chapter 5, showed that the photon echo relaxation times were always longer
than the corresponding virtual echo ones and were also temperature dependent, as
expected. The relaxation measurements for reverse-transient grating measurements were
dominated by inhomogeneous contributions such as Doppler broadening and the
Boltzmann distribution of states prior to laser excitation. One of the strategies used here
to sort among different processes was to delay a pair of pulses by a ﬁxed time interval,
chosen to be equal to an integer or a half integer number of vibrational oscillations of the
excited state. This reveals a striking difference between VB and PE measurements
(Figure 4.4) where the VB 1 signal shows excited or ground state dynamics depending on
the delay between the ﬁrst two pulses. However, we note that for PE measurements this
control is not observed. The reason for this observation is that one of the diagrams for VE
involves all three interactions on the ket. This places a restriction on the laser-molecule
interactions, requiring that the electric ﬁeld interactions must be in concert with the
vibrational dynamics. The PE diagram that corresponds to the formation of a population
in the ground state is not as restrictive because only the third pulse acts on the ket. From

the point of view of control over ground or excited state dynamics, the sequence k1 — k2 +

80

k3 labeled VB 1 is the most efﬁcient. This phenomenon is easily observed in gas-phase
samples; however, in liquids VB is usually not observed because of inhomogeneous

broadening.

Obtained photon echo results agree with the well-known property of photon echo
signals, namely, that the effect of inhomogeneities is cancelled. A technique aimed at
suppressing excited-state dynamics was introduced in the nineties by Shank and
coworkers.183’184 The mode suppression technique requires ﬁxing the time delay between
the ﬁrst and third pulses while the second pulse is scanned in the pulse sequence kg = -
kl + k2 + k3. The method has been found to work only in cases of extremely large
inhomogeneous broadening. We have explored the mode suppression method under the
same conditions as all of the transients in this article. We found that the apparent
suppression of excited—state coherent vibrations is achieved by a combination of Liouville
pathways that allows ground- and excited-state contributions, thereby drowning the pure
excited-state dynamics that are observed when the time delay between the ﬁrst and third

pulses are out-of-phase with the excite-state oscillations.

A new phenomenon in gas-phase FWM experiments, designated here as C—FID-
FWM has been presented. This signal is observed for pulse sequences where
conventional diagrammatic techniques fail to predict the appearance of signal. The
observed signal is not caused by a higher order nonlinear process or from a breakdown of
the RWA. As described earlier (and formulated in Appendix 1) the C-FID-FWM signal is
caused by an induced polarization that maintains the phase and the vector characteristics
of the pumping beam. Because the coherence relaxation in iodine at this temperature has

a lifetime of ~10'10 seconds, the free-induction decay emission can act at relatively long

81

times. The participation of an electric ﬁeld generated by a polarization in a nonlinear
optical process has precedents; this phenomenon is called cascading.135,186 More
recently, cascading has been implicated in the signal arising in ﬁfth-order nonlinear
processes. Upon close inspection the ﬁfth-order signal has been found to result from two
third-order processes that have been called sequential CARS or cascade. The emission

from one third-order polarization participates in a different third-order polarization.‘87"89

The data being reported here show a different type of FWM cascading. The
differences are: (i) resonance excitation leads to long-lived polarizations that can interact
with the other laser pulses at very long times; (ii) in the studies from the Albrecht and
Fleming groups, the cascading process is a combination of two third-order processes,
while in the case presented here, the C-FID-FWM signal results from a ﬁrst- and a
second-order process; (iii) only three pulses were used instead of four or ﬁve and (iv) we
have observed a C-F ID-F WM in a gas-phase sample. Thus, we have ensured that phase
matching occurs only for the third-order process of interest and that the observation of

FWM or C-FID-FWM depends on only on the pulse sequence.

The C-F ID-F WM transients presented here show two surprising effects: the signal
is measurable and the femtosecond time resolution is maintained. Given that at least one
of the electric ﬁelds is replaced by the free-induction decay emission, one could conclude
that the intensity would be much lower and that the time resolution would be lost because
of the picosecond lifetime of that emission. The intensity of the C-FID-FWM signal
derives from the integral over the long duration of the free-induction decay emission. In
fact, we have found that in addition to the concentration dependence the ratio between C—

FID-FWM and FWM depends on the ratio (T2)2/(T. 2,,), where T2 is the coherence

82

relaxation time, T1 is the emission lifetime and 2,, is the laser pulse duration.loo The
temporal resolution is maintained because the free-induction decay emission arises from
a vibronic coherence. Therefore, it is strongly coupled to the coherent vibrational motion

in the ground and excited states, being modulated on the femtosecond time scale.

Coherent control of chemical reactions depends on the design of an electric ﬁeld
capable of causing a molecular system to yield a speciﬁed product. These special ﬁelds
can be constructed out of coherently coupling multiple laser pulses. This work examined
the coherent coupling of three ﬁelds using the phase-matching condition and the third-
order nonlinear response of the sample as ﬁlters of the incoherent contributions. The
exploration of the various phenomena allows us to form a ‘toolkit’ that can be used to
achieve coherent control. Our understanding shows that with FWM the most efficient
control is achieved when the phase-matching geometry and the pulse sequence lead to all
laser interactions occurring on the ket. This implies using what we have called the VB 1

pulse sequence.

Presented experimental data for a number of different sequences stemming from
three degenerate femtosecond laser pulses highlight the need for understanding that laser-
molecule interactions take place in a complex frame that can be understood in terms of
interaction with exp[-i(kr-at)] or with exp[i(kr-at)]. In the language of quantum
mechanics, these interactions correspond to the bra or the ket respectively for the ﬁrst
pulse. Based on this understanding, diagrams can be drawn to illustrate the nature of the
measurement, whereas density matrix calculations can then be used to simulate the data.

Sorting out all the different phenomena is valuable for understanding different nonlinear

83

optical measurements from different groups. Finally, the observation of the C-FID-FWM
transients is an example of nonlinear experiments where the pulses are at least one order

of magnitude shorter than the coherence of the sample.

4.3. Control of quantum phase with coherent light

Spectrally dispersed FWM is shown to be an ideal tool for studying
intramolecular dynamics and this idea is applied to understanding the role of the phase
proﬁle of the laser pulse and chirp in controlling molecule-laser interactions.
Experimental control and characterization of intramolecular dynamics are demonstrated
using chirped femtosecond three-pulse four-wave mixing (FWM). The two-dimensional
(spectrally dispersed and time-resolved) three-pulse FWM signal is shown to contain
important information about the population and coherence of the electronic and
vibrational states of the system. The experiments are carried out on gas-phase I2 and the
degenerate laser pulses are resonant with the X (ground) to B (excited) electronic
transition. Control over population and coherence transfer in the absence of laser chirp is
demonstrated in previous section by selecting speciﬁc pulse sequences. When chirped
lasers are used to manipulate the optical phases of the pulses, the two-dimensional data
demonstrate the transfer of coherence between the ground and excited states. Positive
chirps are also shown to enhance the signal intensity, particularly for bluer wavelengths.
The calculations based on the multilevel density matrix formalism in the perturbation
limit allow prediction of particular pulse sequences that control the ﬁnal electronic state
of the population. In a similar manner, the theory allows us to ﬁnd critical chirp values

that control the transfer of vibrational coherence between the two electronic states. The

84

ability to control population and coherence transfer in molecular systems is of great

importance in the quest for controlling the outcome of laser-initiated chemical reactions.

4.3.1. Introduction

Experimental demonstrations of laser control of chemical reactions in the past
decade have solidiﬁed this ﬁeld of research and garnered great excitement. Emphasis in
this section is put on the controlled formation of ground- and excited-state wave packets
and the time-dependent polarization that is observed by employing a variation of the
femtosecond three-pulse four-wave mixing (F WM) technique, where the signal is
spectrally dispersed. This method is very useful to control and characterize the
intramolecular dynamics occurring after laser excitation. In this study, we experimentally
explore the role of different pulse sequences and optical phase manipulation (linear chirp)
in controlling the population and coherence transfer in gas-phase 12. The two-dimensional
(time and frequency) data provide phase information that cannot be obtained from similar
techniques without spectral dispersion. Presented experimental results and theoretical
simulations reveal the role that pulse sequences and laser chirp can play in controlling

intramolecular dynamics.

The goal of these experiments is to gain a deeper understanding about the role of
chirped laser pulses in controlling chemical reactions. The results from this study are
consistent with the experimental observation of wave packet modiﬁcation (for example
focusing), and enhancement in the yield of chemical reactions using chirped femtosecond
pulses. As in the previous section, gas-phase 12 was chosen as a model molecule for the

experimental demonstration of this technique. The visible X ’Zo+g <—) B 3110+u transition

85

has been well characterized by frequency-resolvedl90"95 and time-resolved
spectroscopy.l9‘5'I98 The vibrational spacing is quite different for these two states making

their identiﬁcation in the data a simple task.

An important step in the control of molecular excitation is the experimental
characterization of the system being controlled. Conventional pump probe (PP) and
FWM techniques give very high temporal resolution but give little or no spectral
information. Some variations of the PP method have been shown to provide frequency
resolution; among these are time-gated ﬂuorescence techniques'97 and PP-photoelectron
spectroscopy.‘999200 The additional frequency resolution helps one to obtain a more
complete characterization of the system. The time-gate used to obtain the spectral
information determines the spectral resolution that can be achieved, limited by the
uncertainty principle. Here we spectrally disperse the signal from the three—pulse FWM.
This signal contains a wealth of spectroscopic information. We have found that this
information allows us to characterize the system more fully following different pulse
sequences and different chirp values. It is shown that the pulse sequence in three-pulse

FWM allows one to form wave packets in the ground and/or excited states.

A brief theoretical background is given in terms of the evolution of the density
matrix. Calculations are used to predict interesting behavior that can be achieved by
particular sequence-chirp combinations. Complete density matrix treatment of non-linear
optical processes is beyond the scope of this thesis and can be found in the
Reference337~'04. Emphasis is put on the spectroscopic information that is contained in
the F WM signal and its connection to the density matrix. Results are presented for

various pulse sequences and pulse chirps. Theoretical calculations are found to be in

86

excellent agreement with the experimental data and are presented here insupport of the

experimental results.

4.3.2. Theory

The density matrix formulation is the preferred theoretical approach for the study
of coherent multi-wave mixing experiments.36~87 The diagonal blocks of the density
matrix of a multilevel system involving several vibrational levels in the ground and
excited electronic states represent properties within each state (see Figure 4.10a). In the
diagonal blocks, the diagonal elements (designated as pgg or pee in Figure 4.10a) are the
population of each vibrational level, whereas the off-diagonal elements (pgg' or pee.)
represent the coherence of the vibrational levels. The elements of the off-diagonal blocks
(age or peg) represent the vibronic coherence between the two electronic states. In the
weak interaction limit, each ﬁeld E” interacts linearly with the medium producing a
change pl") in the initial density matrix. When n is an odd number, pm contains the
changes in the probability amplitude of the electronic couplings, whereas if n is an even
number, p("’ represents the changes in the population and the coherence of the vibrational

levels within each electronic state.

The time evolution of the density matrix for molecules interacting with
consecutive applied weak electric ﬁelds can be represented by DSFD as described in
Section 4.2. The diagrams relevant to our pulse sequence and phase-matching geometry
are shown in Figure 4.10b. Only ﬁrst interactions with electric ﬁelds E, and E, separated
by a time delay Tab are shown. The interaction with the third electric ﬁeld EC and the

emitted signal E, are not shown.

87

a) Multilevel Density Matrix b) Double-Sided Feynman Diagrams

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I - P - (2) (2)
peg. pgg' pee '08? p”
' Ig> e'l
pg'g ‘. E,
O
1 a 9: -
i" ”"90”? '“ ""1 % Tab: : Tab: 130
' l ( ' |e> |e>
O pe’e A l f l
E. E.
peg pee pee lg> <81 18> (gI

 

 

 

 

 

 

 

 

I

Figure 4.10. (a) Representation of the density matrix for a multilevel system composed of
two electronic states, each one with a manifold of vibrational levels. The indices g, g'
range over the number of vibrational levels of the ground electronic state (in the present
simulation g, g' = 1, 2). The indices e, e' range over the number of vibrational levels of
the excited electronic state (in the present simulation e, e' = 3, 4). The diagonal matrix
elements, pgg and Pee, represent the population of the vibrational levels (dark shade). The
off-diagonal matrix elements of the diagonal blocks, pgg' and peev, represent the
vibrational coherence in the respective electronic state (light shade). The elements of the

off-diagonal blocks, pge and peg, represent the vibronic coherence between the electronic
states. (b) Double-sided Feynman diagrams for the processes described in the text after
the ﬁrst two electric ﬁeld interactions only.

The complex conjugate diagrams, mirror images of those shown in Figure 4.1b,
lead to FWM signal in a different phase-matching geometry and are not shown. The
signal is the sum of all possible processes and arises from the contributions of pg:g and peel
which correspond to coherence in the ground and excited states, respectively, after the
interaction with the ﬁrst two pulses. Each process depends on the molecular dynamics of
the corresponding electronic state; therefore, by enhancing one contribution with respect
to the other, it is possible to extract some information about the molecular dynamics of

each state.

In a FWM process, the ﬁrst two electric ﬁelds interfere and form transient and

88

static gratings in space. The third pulse probes these gratings, inducing a polarization in
the medium which emits coherent radiation. When (00, the carrier frequency of the pulses,
is resonant with an electronic transition frequency, the interfering ﬁelds induce two
successive dipole transitions yielding a redistribution of the vibrational populations and
coherences of the molecular ensemble, described in pm. In this way the time-dependent
matrix elements, plz’gvg (t) and pm“, (t), which oscillate with the vibrational frequencies of
the ground and excited states, cog and we respectively, describe the transient grating. The
diagonal terms, pmgg and pm“, describe the static grating which decays with T1. Note in
Figure 4.10b that the diagram on the right contributes to pmee' while the one on the left
contributes to plz’gvg. The transient FWM signal will be a maximum at the times when all

molecules radiate in phase as long as the dephasing time of the excitation coherence, T2,

is longer than the duration of the pulses.

The density matrix formulation for the interpretation of the signal obtained from
chirped three-pulse F WM experiments is applied. The theoretical analysis is based on a
model that includes two electronic states with two vibrational levels each; the states are
labeled as |1>, |2>, |3>, and |4> where g = 1, 2 and e = 3, 4. The coherence relaxation
time for 12 in the gas phase is much longer than the times considered here; therefore, no
relaxation terms are included. Resonant experiments reduce the electronic manifold to the
states that are coupled by the electronic dipole interaction. In order to control the
molecular dynamics of such states, the theory should include the parameters that

characterize their dynamics (i.e. vibrational frequencies). Therefore, at least two

vibrational levels with an energy separation of AEg = haig or AEe = ha), in each electronic

89

state will be needed. Although the effect of anharmonicity cannot be included in the
formulation with this reduced model, an analytical solution to the optimal control
problem is achieved. Unless otherwise noted, it is assumed that the initial populations are
pm)“ = p(°’22 = 0.5 and p(°’33 = p(°’44 = 0. The parameter w = pm)” - p(°’22 is used to

indicate a different initial population distribution.

The three pulses are of same frequency and designated as Ea, Eb, and EC and the
FWM signal is emitted in two directions deﬁned by the wave vectors kyg = k, -— kb + kc
and k5 = — k0 + k1, + kc. The former applies to observation of a virtual echo and the latter
to the conventional photon echo. Calculations and experimental results shown here apply

to detection in the km direction.

As described in chapter 4.2 the simultaneous control of the population and
vibrational coherence within each electronic state can be achieved by manipulating the
time delay between the ﬁrst two pulses, Tab. After the interaction with two unchirped

pulses, the matrix elements of the ground and excited states are

(0
Pg) = —A2 COS[&ZTCIP—lcos[(wo i 7g}... " (k, - k;.)' X], (4.3)

P1230) = A2 coleBTT‘QJeiiwAH’T”)cos(a)ofab - (kg - kb)' x), (4-4)

0)”!

p33) = A2 cosl-iZ—ﬂl cosllwo $—%—l2ab — (kg — k b) xl, and (4.5)

90

a) 2'
.02.?)(0 = A2 cos[ ‘2 °°

 

lei’m"(’+’r”” )cos(a)02ab — (kg — k b)- x) (4.6)

where A is the area under the pulse and the upper sign corresponds to the ﬁrst
combination of indices and the lower sign to the second. Setting 20;, = (n + l/2) 2,. where 2,.
= 2n/we, we obtain a signal that is mostly characterized by the dynamics of the excited
state and similarly for 22,), = (n + ’/z)2g with 2,, = 22t/wg. The vibrational coherence in the
ground state corresponds to the transformation of the wave packet after an excitation —
de-excitation process, whereas the vibrational coherence in the excited state corresponds
to the interference of two wave packets after excitation. The probability of each
mechanism is represented by the diagonal blocks of the density matrix after two pulses,

pm“ and p (2’29! respectively.

When the ﬁrst two pulses are equally chirped, the chirp is manifested in the
vibrational coherence matrix elements plz’gga) and pmeea) and is absent in pa)“ and
pa)“. This is because the population matrix elements are obtained after a second-order
process with two equal and opposite transition paths and the phase shiﬁ of the successive
chirped pulses cancels out. When the transition paths are different, as for the coherence
matrix elements, a net phase due to the chirped pulses arises. The modiﬁcations

introduced by the chirped pulses in the spatial average of the density matrix elements are

<lpg,’ 2> 0C [1 + cos(a)erab )l, (4.7)

91

 

<lpé§ll2>oc 1+cos(werab)cos ¢ 2" , (4.8)

(2)
pee

 

 

<

2> cc [1 + cos(a)grab )l (4.9)

n 2
2 a)
(2) >01: 1+cos(a)grab)cos ¢ 2g (4.10)

pee'

 

 

 

<

where the brackets denote the average with respect to coordinate 11. ¢" is the linear
spectral chirp of the pulses obtained from the phase shift of the chirped electric ﬁelds,
¢(a)) -.~: ¢(a)0) + (a) - (any ¢"/2 with ¢" = 6221160)sz Note that these expressions.
(Equations 4.7-4.10) are time independent. The modulus square cancels the time-

dependent terms in Equations 4.2 and 4.4.

After time delay 2', the third pulse in a FWM experiment probes the population
and the vibrational coherence generated by the ﬁrst two pulses. The resulting polarization
can be represented by the third-order density matrix [3(3) containing only the matrix
elements that oscillate with the transition frequencies (neg = (E, — Eg)/h where g = 1, 2
and e = 3, 4. Therefore, the emitted light contains useful spectroscopic information about

2)” + p(2)E, where the components that

the system. The overall signal depends on pm = p‘
lead to signal in the km; direction are separated from those that lead to signal in the k};

direction. The intensity of the emitted light corresponding to each transition in the

direction kVE as a function of 2'is given by

92

. , ,. . 2
VE 2)VE o (2)25 (2)15 — <1) (2)VE
1,,g (2) 0C pie +el pee. (2)—pgg +e ' pgrg (2) (4.11)

where (I) = ¢"a)gZ/4. Since the diagonal terms of the density matrix p‘z’VEgg and

pawn“? are time independent, the time evolution of the spectral line intensity depends on

the vibrational coherence matrix elements. Because these matrix elements oscillate with a

vibrational frequency cog or (02, Equation (4.11) can be expressed as

1§f(2) 0c 1”,.(2) + 1"E.(2) + 1”“.(2), with (4.12)
[1'5 ( (2) 2
g 2) oc lpgg'l [cos(a)g2) + l] and (4.13)
z 2
I’Ee(2) 0C <Péf" >[cos(a),,2)+1]. (4.14)

 

 

1’58. and [”58 represent the spectral intensity contributions that oscillate with the
frequencies mg and (02., respectively. IVEC represents the cross terms that oscillate with a

. . . . . . E VE . .
lrnear combination of both frequencres; 1fe1ther [V g or I e rs zero, IVEC 1s zero.

Figure 4.11 represents the simulated relationship between the amplitude of the
averaged density matrix elements, as indicated in Equations (4.7-4.10), and the spectral
intensity, as shown in Equation (4.11). The three-dimensional diagrams of the averaged
density matrix elements for the four-level system are shown at the top of the ﬁgure. The
height represents the amplitude of ( l p.112) l 2). The spectral intensities as a function of time

delay 2 are shown in the spectrograms. The spectrograms show the contributions of the

four transition lines that correspond to the four-level system (141 = 4—21, 131 = 3—>1, 2.42

93

= 4—>2, and 132 = 3——)2). Gaussian functions with a Alt/2 F WHM are used to simulate the
experimental spectral resolution where Al = 2.41—2.31 = 142—132. Based on Equations (4.7-
4.10), one can identify the time delay 20b as a useful parameter in the design of pulse
sequences that enhance or eliminate speciﬁc elements in the density matrix. Two extreme
cases with either exclusive ground- or exclusive excited-state dynamics are shown in
Figure 4.11. When 2a}, = 400 fs = 52g/2 (Figure 4.11a), the spectral intensities oscillate
with the vibrational frequency cog, with 28 = 160 fs, consistent with the vibrational
frequency of iodine in the ground state. The signal corresponds to the time evolution of
the probability amplitude in the ground state because only the matrix elements pa’gg and
plz’gvg( 2) are different from zero (see the averaged density matrix diagram at the top of
Figure 4.11a). When 20;, = 460 fs = 32e/2 (Figure 4.11b), the spectral line intensities
oscillate with vibrational frequency me, with 2., = 307 fs, consistent with the vibrational
frequency of the B 3110.u state of iodine (v' = 7-11). As can be seen from the averaged
density matrix diagram (Figure 4.1 lb top), only the elements of the excited state
contribute to the signal in this case. Therefore, by changing the time delay between the

ﬁrst two pulses, it is possible to control the population and the vibrational coherence of

the system.

The use of chirped pulses as an alternative means of controlling the
intramolecular dynamics is explored in the simulations shown in Figure 4.12. The
spectrograms along with the diagrams of the averaged density matrix elements

correspond to time delay 20;, = 460 fs with chirped pulses having a spectral chirp ¢"=

1000 rs.2 or wag/2 = 11/4 (Figure 4.12a) and 25" = 4000 fsz or (25"(082/2 = 1: (Figure 4.12b).

94

a) Tab = 400 fs b) Tab = 460 fs

Time delay 2, [ps]

 

A'41 A'31 A'42 A'32

Figure 4.11. Simulations of spectrally dispersed FWM. At the top of the ﬁgure are
diagrams of the averaged density matrix elements <|p1j(2’|2> calculated with Equations
(4.7-4.10). The corresponding spectrograms calculated with Equation (4.11) are shown
with wavelength plotted along the horizontal axis and time delay between the second and
third pulses (2) plotted along the vertical axis. The darker regions in spectrograms
correspond to higher signal intensity. For these calculations, the initial population
distribution parameter (w) in the ground state was assumed to be zero and the pulses were
unchirped. (a) The delay between ﬁrst two pulses (2,2,) is 400 fs, corresponding to 5/2
vibrational periods of the ground state, 23. The density matrix diagram for this pulse
sequence (top left) shows population and coherence terms only in the upper diagonal
block, corresponding to ground-state matrix elements alone. The spectrogram (left)
clearly shows 160 fs periods, reﬂecting the vibration of the ground state (cog). (b) For 22,1,
= 460 fs, 3/2 vibrational periods of the excited state 2,, only the matrix elements of the
excited state (lower diagonal block) are present in the diagram. In the corresponding
spectrogram (right), the excited-state vibrations of 307 fs are evident.

95

The introduction of chirped pulses produces a signal that oscillates with both
vibrational frequencies cog and are, conﬁrmed by the presence of coherence vibrational
terms belonging to both electronic states in the averaged density matrix diagram. When
more than one oscillating frequency is present in the spectrogram, the spectral line
intensity is slowly modulated in time with an approximate period of 3 ps. This effect can
be attributed to the interference, ICVE( 2), of the vibrational coherence terms from the

different electronic states according to Equation (4.12).

Comparing the averaged density matrix elements for different values of chirp for
2,), = 460 fs (Figures 4.1 lb top, 4.12a top, and 4.12b top), we can see that the chirp does
not affect the population terms but only the vibrational coherence terms, pa’ggﬂ) and
plz’eea), in agreement with previous the discussion. Thus, the time dependence of the
spectral signal can be manipulated through chirped electric ﬁelds by controlling the
coherence vibrational terms of the density matrix after two pulses. Figure 4.12 highlights
the importance of spectral dispersion; the intensity of the spectral features varies at
different wavelengths due to the different phases involved. This information would not be

available if the spectral information was integrated.

The density matrix elements after two pulses are the ensemble-averaged products
of two time-dependent amplitudes, p‘z’y-(t) = (c,-(t) cj(t)’), of the wave packet components
in each electronic state. A change in the magnitude of p‘z’gga) or pmeeﬂ) due to chirped
pulses will correspond to a change in the relative phase of the wave packet components.
This manipulation shapes the wave packet in each electronic state, allowing the focusing

or spreading of the wave packet.‘3~l4,'6~7O

96

Tab = 460 fs

a) ¢"=1ooofe:2 b) ¢"= 40002132

Time delay 2, [ps]

 

A'41 A'31 A": A:12 A'41 2‘31 A42 A‘32
Wavelength Wavelength

Figure 4.12. Simulations of spectrally dispersed FWM for 22,), = 460 fs, w = 0, and two
values of laser pulse chirp. Again, the density matrix diagrams are time independent
throughout the region shown in the spectrograms. (a) For a small value of chirp (¢" =
1000 fsz), the changes in the dynamics are obvious (compare to Fig 2b). These
differences are expected, as seen in the density matrix diagram by the appearance of
vibrational coherences in both the ground and excited states (upper and lower diagonal
blocks). (b) It is possible to calculate the appropriate chirp where ground state elements
dominate the contribution to the signal even for 22,), = 460 fs where unchirped pulses
result in seeing only excited-state dynamics. With ¢" = 4000 fs2 (Figure 3b), 160 fs
oscillation periods appear in the simulation, showing primarily ground-state dynamics.

97

When the initial population distribution is not limited to a single vibrational state,
the density matrix elements represent the evolution of all levels in the system after
successive laser pulses. A coherent coupling arises within each electronic state as a
consequence of interaction with phase-matched electric ﬁelds. This results in additional
coherence terms introduced, for example, by the thermally populated vibrational levels in
the sample prior to laser excitation. The density matrix formulation treats these
coherences naturally without having to artiﬁcially add these contributions using phase
coherent or incoherent sums. By taking only four levels into account, analytical
expressions are obtained for the spectrally dispersed F WM signal as a function of the
control parameters 22,), and (21" using the density matrix formalism. Simulations based on

these expressions are in close agreement with the experimental data.

4.3.3. Experimental

The experimental setup is identical to one previously described in section 4.2.
These experiments were conducted using 60 fs pulses (FWHM when transform-limited)
centered at 620 nm with 8 nm (FWHM) spectral width. The chirp was measured by
analyzing the pulses with a frequency-resolved optical gating (FROG). The laser was
split into three beams, each with energy of ~ 20 11.1, which were arranged in the forward
box conﬁguration before being focused in the quartz sample cell containing neat iodine
vapor at 140 °C. The time delay between the ﬁrst two pulses, 20),, was controlled by a
manual translator and the time delay between the second and third pulse, 2, was scanned
to yield the FWM signal transients. The three-pulse FWM signal was collected by a

spectrometer placed at the phase-matching condition km; for different wavelengths within

98

the 605 — 635 nm spectral range. The spectral acceptance of the spectrometer was set to 2
nm. Narrower spectrometer slits did not seem to sharpen the observed spectral features.
Both FWM spectra and transients were measured with these same conditions. The spectra
were obtained between 600 —- 640 nm and averaged for 10 scans with 200 points per scan
and 10 laser shots per point. Each transient was taken at 150 different time delays and
averaged for 20 scans. At each time delay 2, the signal was collected for 10 laser shots,
where the energy of each laser pulse was required to be within one standard deviation
from the mean. These transients, collected at different wavelengths, were then combined

to generate the spectrograms shown in the following section.

4.3.4. Spectral information in femtosecond three-pulse FWM

Most four-wave mixing experiments measure the intensity of the signal either as a
function of excitation wavelength (frequency-resolved FWM) or as a function of time
delay between laser pulses (time-resolved FWM). For FWM measurements involving
resonant excitation, the polarization can be sufﬁciently long-lived that it carries relevant
spectroscopic information."3 This is particularly the case for gas-phase samples where the
polarization can persist for tens to hundreds of picoseconds.100 Spectrally dispersed
FWM measurements are not reserved only for long-lived polarization systems; for a
radiative decay of 1 ps, one can expect to get 5 cm'1 spectral resolution. The spectral
resolution in this case is almost two orders of magnitude better than the bandwidth of the

femtosecond lasers used for the experiment.

In Figure 4.13, the spectrum of the transform-limited femtosecond laser used for

these experiments (b) with the spectra of three-pulse FWM signal for two cases, (a) and

99

 

 

 

 

      
 

Intensity [arb. units]
I

 

 

. —l

Tab= 614 fs -
T: 614 fs

 

 

609 613 617 621 625 629 633
Wavelength 2» [nm]

Figure 4.13. Experimental spectra (dots) and ﬁt (lines) of the three-pulse FWM signals
and the transform-limited laser. (a) FWM signal observed when all three beams coincide

in time (22,), = 0 fs; 2= 0 fs). This spectrum is very broad but shows some spectroscopic
transitions. The spectrum is ﬁt with six Gaussian functions. (b) Spectrum of the
unchirped femtosecond laser pulses used for these experiments. This proﬁle is ﬁt with a
Gaussian function with width 8 nm (FWHM). (c) FWM signal observed when 20,, = 614
fs and 2= 614 fs. This trace shows well-resolved spectroscopic transitions which were ﬁt
by six Gaussian ﬁrnctions. The spacing between the peaks in spectra (a) and (c) is 4 nm,
corresponding to the vibrational spacing of the B state of iodine.

(c) is shown. The experimental data (dots) were ﬁt by Gaussian functions (black lines).
The laser has a bandwidth of 210 cm". Trace (a) corresponds to the signal observed when
all three beams coincide in time. This signal spectrum is very broad but shows some
spectroscopic features with an average width of 80 cm". Trace (c) shows well-resolved
spectroscopic transitions and corresponds to a three-pulse FWM in which 201, = 614 fs and
2: 614 fs. The average width of these features is 50 cm". The main difference between

traces (a) and (c) is that the signal in the former case includes multiple FWM pathways,

100

many involving non-resonant processes that are short-lived,92 and the latter case arises
from a long-lived polarization that decays in hundreds of picoseconds.'00 The spectral
features are much narrower than the laser but are signiﬁcantly broadened by the
rotational temperature of the sample (spectral congestion) and the spectral resolution of
the spectrometer. No signiﬁcant improvement in the spectral resolution was obtained

when much narrower slits were used.

The spacing between the features in Figure 4.13c is A?» z 4 nm and corresponds to
the vibrational spacing in the B 3110+u state?“ The transitions arise from a polarization
involving vibrational levels v' = 7-11 in the B state and vibrational levels v" = 2-4 in the
X ’Zo+g state. The vibrational levels with the highest probability of excitation are v' = 9
and 10. Given the very small rotational constant of iodine molecules, there is a great deal
of spectral congestion. Furthermore, the coincidental factor of two between the
vibrational frequency of the ground-state and that of the B state results in multiple band
overlap. The data in Figure 4.13 make it evident that spectral dispersion of the three-pulse
FWM signal provides an additional dimension with a wealth of information about the
system. In the following section the changes in the spectra as a function of time delay of
the pulses involved in the three-pulse FWM process are explored, as well as the effects of

chirped-pulse excitation.

4.3.5. Two-dimensional three-pulse FWM measurements
As discussed in section 4.3.2, the spectroscopic transitions that are detected here
are determined by the system and their wavelengths are time independent. The time-

dependent intramolecular dynamics manifest themselves as changes in the intensity of the

101

individual spectroscopic features. Therefore, the modulation of the signal is a
manifestation of a time-dependent transition probability of the system. Based on these
principles, it is not necessary to take a high-resolution spectrum at each time delay. The
two-dimensional data were obtained by taking transients at 5 to 6 different wavelengths
and interpolating the rest of the spectroscopic information using multiple Gaussian
functions in the same way that they were used in Figure 4.13 to ﬁt the spectroscopic data.
In all cases, spectra were taken at different time delays 2 to check the two-dimensional
data for wavelength and bandwidth consistency. Given that the data contain temporal as
well as spectral information, it should be possible to invert and obtain the real space

dynamics of the system in both the ground and the excited states.

Figure 4.14 shows spectrally dispersed three-pulse F WM measurements carried
out with 20;, = 0 fs. Both plots present the two-dimensional experimental data. Figure
4.14a shows the signal intensity in the third dimension giving a topographic
representation. Figure 4.14b shows the same signal in a contour form, apparently more
useﬁil for quantitative analysis. In the contour plot, darker shades correspond to higher
intensity. The contour lines are spaced according to the square root of the signal
amplitude. One of the important points to notice is that the spectroscopic features remain
ﬁxed in wavelength and one observes their change in intensity as a function of time delay
2. This makes this technique different from wave packet imaging techniques such as
those based on photoionization19 or Coulomb explosion.202~203 With those, one obtains
the shape of the wave packet and its time evolution. In our measurements, the dynamics
of the system are obtained from the change in intensity of every spectral region. The

spectroscopy as well as the dynamics reveal the states involved; in the experiments

102

discussed here these would be the X and B states with cog = 208 cm'1 and a), = 108 cm'l

(2g = 160 fs and 22 = 307 fs).

In Figure 4.14 the observed dynamics for wavelengths longer than 620 nm are a
mixture of ground- and excited-state vibrational motion; however, the features with a
period of 307 fs, corresponding to the excited state dynamics, are more prevalent
especially in the 628 — 632 nm region and after the ﬁrst 1.5 ps. For wavelengths shorter
than 620 nm, the dynamics show a vibrational period of 160 fs that corresponds to the
ground state. The observation of ground and excited state dynamics for FWM
experiments with 20;, = 0 fs was reported ﬁrst by Schmitt et al.38,169 Basically, the ﬁrst
two pulses create populations in both the ground and excited states. The third laser pulse
probes the dynamics of both populations and the signal contains a mixture of each. Due
to the Franck-Condon factors between both of the electronic states involved, probing of
the ground state will be enhanced for short wavelengths while probing of the excited state
is enhanced for longer wavelengths. These enhancements result in the observed
differences in the signal as seen in the experimental spectrograms. With three-pulse four-
wave mixing, one is able to prepare different pulse sequences that can be used to control
the coherence and population, and therefore the observed dynamics of the system. Pulse
sequences have been used in liquid phase FWM experiments to suppress vibrational
mode contributions183 and to control the emission of signal by allowing the echo and
virtual echo components to interfere.”204 Experimental data and simulation of results for

the case when 2a), = 614 fs are shown in Figure 4.15 (a) and Figure 4.15 (b), respectively.

103

Tab=ors ¢"=0fs’

a) Experiment b) Experiment

we

Time delay 2, [ps]

 

 

- “ﬂu-rﬂﬁwﬁdﬁeﬁ‘eﬂ!

5"/i‘fﬁ’/!!§f§};3; t

  
 

I
D ID
‘—

8

612 '

Wavelength [nm] Wavelength [nm]

Figure 4.14. Spectrally dispersed experimental data for three-pulse FWM with 202 = 0 fs
and transform-limited pulses. Transients taken at six different wavelengths for 3.3 ps are
represented as (a) a topographical plot and (b) a contour plot. As before, the darker areas
in the contour plot correspond to the higher signal intensity whereas the topographical
plot uses lighter shades to show higher intensity. Spectral proﬁles were approximated by
Gaussian proﬁles (see Figure 5a and c) to generate the experimental spectrograms. Both
ground- and excited-state dynamics are observed with these experimental conditions.

104

Tab=s1us ¢=0st

a) Experiment b) Theory
, 3.3

 

3.0

2.7

Time delay 2, [ps]

 

 

 

Wavelength [nm] Wavelength [nm]

Figure 4.15. Spectrally dispersed experimental and theoretical spectrograms of three-
pulse FWM for 201, = 614 fs and ¢" = 0 fsz. (a) The spectrogram shows the experimental
transients taken at ﬁve different wavelengths where the spectral proﬁles were
approximated by Gaussian proﬁles. For all wavelengths, the dominant feature is the 160
fs vibrational motion reﬂecting the ground-state dynamics. (b) The initial population
distribution was taken to be w = 0.33 for the calculation of the theoretical spectrogram of
the experiment shown in (a). The 160 fs vibrations from the ground state dynamics are
evident in both the theoretical and experimental spectrograms.

This time delay maximizes the contribution from the ground state. The
experimental data show fast oscillations with a period that corresponds to the vibrational
frequency of the ground state. There is a small contribution from a population in the
excited state that is apparent in the 628 nm region. The data show a phase shift of about
80 fs when comparing the maxima of the spectra features for 612 nm to those at 628 nm.

This delay is probably due to the anharmonicity of the B state. Notice that the intensity

105

remains high for the bluer wavelengths. The calculations, carried out for w = 0.33, which
corresponds to the expected Boltzmann distribution between the two vibrational levels in
the ground state at 140 °C, reproduce the observed dynamics showing higher intensities
for shorter wavelengths and showing that some of the excited state dynamics are
observable for this pulse sequence. The calculations, based on a four-level density matrix,
do not have a sufﬁcient number of levels to reproduce the observed phase shift. The wave
packet calculations described in the Section 4.2 (Figure 4.5) demonstrate that the
observed ground state dynamics arise from a wave packet that evolves in the B state for a
time period 201,. During this evolution, the anharmonicity of the B state causes the wave

packet to spread.

When a pulse sequence is chosen with 20,, = 460 fs, only the excited state is
populated (see Figure 4.11). The experimental data, shown in Figure 4.16a, show very
clear 307 fs oscillations that correspond to the vibrational period in the excited state for v'
= 7-11. In this case there is no evidence of ground state dynamics. This data can be
contrasted to the data in Figure 4.15 where ground state dynamics are dominant. The
theoretical calculations, shown in Figure 4.16b and carried out for w = 0.33, are in
excellent agreement with the experimental data. For this sequence, the intensity
distribution across the spectral range is quite symmetric and shows no phase shift. Note
that for this 2gb, the calculations are independent of the initial population (w) in the

ground state.

106

Tab = 460 fs ¢"= o fsz

       

 

b) Theory
.3 \Jv\,_.4\r/\I
3.0
2.1
2.4
'3‘
2.1 ,2;
1..“
1.2
>
. 2
- e
_ 1.5 U
- 0
I 1.2 .E
_ 1-
: 0.9
; : 0.5
: ' ' : 0.3
'. l 0
D D
2 225232 =34:
Wavelength [nm] Wavelength [nm]

Figure 4.16. Spectrally dispersed experimental and theoretical spectrograms of three-
pulse FWM for 2a,, = 460 fs and (2" = 0 fsz. (a) The spectrogram generated from the
transients taken at ﬁve different wavelengths shows only 307 fs oscillations from the
excited-state dynamics. There is no evidence of ground state dynamics. (b) The
simulation of the experiment again used w = 0.33 for the calculation. The theoretical and
experimental spectrograms are in excellent agreement, with the 307 fs oscillations clearly
evident in both.

4.3.6. Three-pulse FWM with chirped laser pulses
Based on the experimental data shown so far, it is clear that the spectrally
dispersed three-pulse FWM data provide a wealth of information about the

intramolecular dynamics of a system including information about the population

107

distribution among the active electronic states. The role that chirped laser pulses have on
the excitation of molecular systems is explored in this section. Recent experiments where
chirped laser pulses have been used to control the excitation of molecules and the yield of

chemical reactions provide the motivation for this study. '5'13270273-74

Using the pulse sequence with 20;, = 460 fs and ﬁxing the time delay of the third
pulse to 600 fs provides a snapshot of the system in time. While keeping these two
parameters ﬁxed, chirp was varied. The experimental data for this measurement are
presented in Figure 4.17; data are interpolated between the spectra obtained at the
different chirp values. Notice that for zero chirp the spectrum is centered and is quite
symmetrical, as discussed earlier. For positive chirps, we notice that the intensity shifts
towards shorter wavelengths, while for negative chirps, the intensity shifts towards longer
wavelengths. These shifts and their time dependence are discussed below. It is interesting
to note that the highest signal intensities are observed for positive chirps. This
observation is consistent with the fact that optimal pumping of the B state of iodine

requires a positively chirped pulse.‘7’73,74,l ‘0

The experimental and theoretical data for the two-dimensional three-pulse F WM
where 2a,, = 460 fs and the laser pulses are chirped to 3300 fs2 are presented in Figure
4.18 (a) and (b) respectively. Comparing the experimental data in Figure 4.18a with the
experimental data in Figure 4.16a obtained for the same pulse sequence, it is clear that
the chirp has introduced ground state dynamics into the observed signal. While the

excited state dynamics are still clearly observable, there are some regions showing

108

 

 

600 ' " r , _
620 , 2800

Wavelength ,1 [11m]

 

Figure 4.17. Three-dimensional representation of the experimental spectra taken using

differently chirped pulses for the pulse sequence 2“,, = 460 fs and 2= 600 fs. Chirp values
ranged from —2800 fs2 to 2800 fsz. Note the change in intensity towards shorter

wavelengths as the chirp values increase from negative to positive values.

oscillations with the 160 fs characteristic of the ground state. There are spectral regions,
for example 620 nm, where there is a clear doubling of the frequency with every other
beat having higher intensity. The theoretical calculations in Figure 4.18b, carried out with
w = 0.33 and 25" = 3300 fsz, are in excellent agreement with the experimental data. The

calculations reproduce the shift of spectral intensity towards shorter wavelengths and the

alternating intensity between the vibrational features.

109

Tab = 460 fs ¢"= 3300 fsz

      

 

 

a) Experiment b) Theory
3.0 s: 9 O
2.7
2.4
2.1
3'
1.3 3
1..“
1.5
>1
2
1.2 g
d)
0.9 _§
.—
0.6
7 . I 0.3
’ ‘ 2 3 § 3
o u- 2 N N a g g ‘9 on 9 '9
10 GD ‘9 O O O (D 0
Wavelength [nm] Wavelength [nm]

Figure 4.18. Spectrally dispersed experimental and theoretical spectrograms of three-
pulse FWM for 2.2, = 460 fs and ¢" = 3300 fsz. (a) The experimental spectrogram was
generated from transients obtained with ﬁve different wavelengths as above. The signal
observed for wavelengths 624 nm and shorter originates from mixed ground- and excited-
state dynamics. It is clear when compared with Figure 8a that the addition of chirp alters
the observed signal by introducing ground-state dynamics for shorter wavelengths. (b)
The theoretical simulation, where w = 0.33, shows good agreement with the experiment
both in the dynamics features and in the intensities of the beats.

It is possible to ﬁnd ‘critical chirp’ values that accomplish certain tasks, for
example, eliminateing the ground- or the excited-state dynamics in the signal. Here we
would like to focus on two interesting cases. In Figure 4.19 we show simulations for the

pulse sequence with 2,, = 614 fs. When a relatively small chirp of 1000 fs2 is used (a), we

110

see that the ground-state contributions to the signal decrease (compare to Figure 4.15b).
The data show an apparent slope in which the spectroscopic features on the short
wavelength side seem to reach a maximum earlier than those on the long wavelength
side. This slope depends on the sign of the laser chirp. For negative chirps, the opposite

slope would be observed.

The second interesting case, shown in Figure 4.1%, corresponds to the same
pulse sequence but with a critical chirp ¢" = (222/20,?) = 4000 fs2 (see Equations 6 and 8).
In this case, only 307 fs dynamics are observed. The spectral intensity is still shifted
towards shorter wavelengths as expected for w = 0.33 and 20,, = 614 fs. For w < 0, a red
shift in spectral intensity would be observed. However, most notable is the phase of the
observed oscillations across the different wavelengths. The average density matrix
diagrams for both cases are also shown in Figure 4.19 (top). Notice that the laser chirp
does not affect the population, but affects the coherence between the states. For the case
shown in Figure 4.1%, the ground state coherence is no longer present and therefore not
observed. As can be seen in Figure 4.15b, the spectrogram for 20), = 614 fs is dominated
by ground-state dynamics. When the critical chirp of 4000 fs2 is introduced for 20), = 614
fs, the ground-state dynamics are completely eliminated (see Figure 4.1%). This result is
similar to the one presented in the spectrogram for 20b = 460 fs and no chirp (Figure
4.16b) where only excited-state dynamics is observed. When the critical chirp of 4000 fs2
is introduced, the dynamics are dominated by the ground state (see Figure 4.12b). From a
theoretical point of view, changing the chirp (¢") is as effective as changing the pulse
sequence (20),) to control the observed molecular dynamics. It is much more complex task

experimentally.

lll

Tab = 6141‘s

a) 2521000242 b) (152400011:2

©(‘)

1:)" '3 o 0
> ”st/7

C2; (ﬁlo?
(9‘97

Time delay 2, [ps]

 

 

A11 A'31 A'42 A'32
Wavelength Wavelength

Figure 4.19. Diagrams of averaged density matrix elements <|,0y(2)12> and corresponding
theoretical spectrograms for time delay 2a), = 614 fs and w = 0.33 with ¢" = 1000 fs2 and
(15" = 4000 fsz. (a) For small chirps, the dynamics at shorter wavelengths reﬂect the
ground state while higher wavelengths represent signal predominantly from the excited
state. (b) When the pulses are chirped to the critical value, ¢" = (212/082) = 4000 fsz, the
signal for all wavelengths reﬂects excited state dynamics only. Notice that the density
matrix diagram in this case shows population but no coherence in the ground state.

4.3.7. Conclusions

The signal arising from time-resolved FWM measurements carries valuable
information about the intramolecular dynamics of the system. The time delay between the
ﬁrst two pulses is a valuable parameter for controlling the population and coherence
transfer. In all cases the theoretical simulations were in good agreement with the

experimental observations.

The chirped pulse excitation measurements are of current interest as phase-altered
pulses are being used to control the excitation process of molecules and the yield of laser-
initiated chemical reactions. The data in Figure 4.18, obtained for (15" = 3300 fsz, show a
contribution from the ground state that was absent in the dynamics obtained with
unchirped pulses in Figure 4.16. It is possible to ﬁnd ‘critical chirp’ values that can
accomplish certain tasks, for example, introduce or eliminate ground state dynamics in
the signal. Calculations that include multiple vibrational levels in the excited state would
help to reproduce some of the ﬁner features observed, such as the phase delay evident in
the data for 22,), = 614 fs in Figure 4.15. From Equations (4.8 and 4.10) in the Theory
section, it is clear that the laser chirp (¢") can be used as a parameter to control the
coherence, and in a similar way, the time delay (201,) can be used to control the coherence

and population transfer.

The experimental data shown here illustrate the advantages of spectrally dispersed
three-pulse F WM because it yields the following results that could not be obtained from

spectrally integrated data:

1. For the case where 20;, = 0 fs, one can see that longer wavelengths are

113

dominated by excited-state dynamics while shorter wavelengths are

dominated by ground-state dynamics (see Figure 6).

2. For the case 20,, =614 fs, the data show a phase shift of about 80 fs across
the spectrum (see Figure 4.15). This shift is caused by the anharmonicity
in the excited state. Therefore, the data contain phase information that is

important in understanding the underlying intramolecular dynamics.

3. When chirped pulses are used, important spectral shifts occur. Figure 4.17
shows that for positive chirps the intensity shifts to shorter wavelengths
and the signal intensity increases. Negatively chirped pulses give a

broader spectrum that is shifted towards longer wavelengths.

The above examples demonstrate both experimentally and theoretically that the
chirp controls the coherence but not the population transfer in this system. It is important
to note that all the laser pulses are equally chirped and only two electronic states
participate in the observed dynamics. When the ﬁrst two laser pulses are chirped
differently (magnitude and/or sign), one can control the population transfer. When the
system contains a reactive pathway, the coherence can be used to enhance or decrease the
probability for the system to follow that pathway. The coherence determines the
dynamics of the system. In multidimensional systems, one may want to control the yield
of different pathways or affect the rate of intramolecular vibrational relaxation. This type
of control, where the phases of the quantum mechanical superpositions are manipulated
by the optical phases of the pulses to determine the outcome of chemical reactions,

results in true control of ﬁnal population transfer from reactant to product. Alternatively,

114

the ﬁrst two pulses can be used to prepare the system in a speciﬁc superposition of states

and the third pulse can be used to complete the chemical reaction.

Work presented here has only involved pulse sequences of linearly chirped pulses.
With phase masks one can achieve quadratic and higher order chirps. One can also tailor
the pulses into very complex electric ﬁelds, which have been used to achieve different
product yields in chemical reactions. Based on these observations, we consider three-
pulse FWM to be a powerful tool for probing intramolecular dynamics and for achieving

laser control of chemical reactions.

115

5. FEMTOSECOND PHOTON ECHO AND VIRTUAL
ECHO MEASUREMENTS OF THE VIBRONIC AND
VIBRATIONAL COHERENCE RELAXATION TIMES OF
IODINE VAPOR

5.1. Introduction

Previous chapters described control of intramolecular dynamics by manipulation
of vibrational and vibronic coherences as well as pOpulation transfer between two
electronic states of gas state molecular iodine. This chapter explores the three-pulse four-
wave mixing technique as a tool to sort and measure the processes that contribute to
coherence relaxation. All coherent control methods are affected by the rate of
decoherence of the sample. Here we show how these rates are measured with FWM
techniques. The measurements presented here illustrate how photon echo (PE)
measurements yield the homogeneous relaxation rate while the virtual echo (V E)

measurements yield the sum of the homogeneous and inhomogeneous relaxation rates.

The observed relaxation times for vibronic coherence using photon echo and
reverse transient grating measurements are compared at different temperatures to isolate
inhomogeneous and homogeneous components. Different pulse sequences are used to
select ground- or excited-state vibrational coherences. Measurements of ground- and
excited-state wave packet spreading times due to anharmonicity, a process that does not

involve energy dissipation or phase relaxation, are also presented.

The loss of quantum mechanical coherence, a T2 process, in gas phase samples is

116

caused by collisions and inhomogeneous broadening (for example Doppler broadening).
The cross section of these processes depends on the type of coherence. In a molecular
system, one can identify electronic, vibrational and rotational coherences. Vibrational and
rotational coherences can be found in a selected electronic state (ground or excited), and
electronic coherences can include rotational and vibrational components as well. In the

present study, three-pulse four-wave mixing (FWM) techniques are applied to measure
experimentally the homogeneous, T,’, and inhomogeneous, T; , vibronic coherence

relaxation times of iodine vapor.

Starting with the ﬁrst observation of photon echo (PE) in the condensed and gas
phase,‘55~156 it was clear that this nonlinear optical method, analogous to spin echo
techniques in NMR, was capable of providing coherence relaxation measurements free of
inhomogeneous broadening. In the last decade, these measurements have been combined
with ultrafast lasers to measure coherence relaxation times in large organic molecules in
solution.‘64~205‘207 This method has not been as popular for the study of gas phase

samples, and this work illustrates its advantages.

There are many studies on the relaxation time of iodine in the gas phase. Among
them are measurements of T1 relaxation (loss of population in the excited state) due to
self-quenching,208~209 cross section measurements of phase relaxation in the presence of
noble gases with femtosecond pump-probe methodsmu“ and T2 measurements by PE
techniques on the nanosecond time scale.212 The goal of our measurements is to
determine the rate of vibrational coherence relaxation of the B and X states independently

as well as the homogenous and inhomogeneous vibronic coherence involving the two

117

electronic states.

The theoretical foundation of these measurements is best expressed in terms of the
time evolution of the density matrix in Liouville space.“87 This theory as it applies to
presented measurements was brieﬂy described in sections 4.2 and 4.3, based on an
established formalism.379'0492‘3 In the presence of collisions, two types of relaxation can

be observed: T1 processes, which involve the loss of amplitude in the population terms

p,-,-. and T 2' processes, which involve the loss of amplitude in the coherence terms pi).

Inhomogeneous broadening, T; , does not depend on collisions but also leads to a loss of

coherence. Wave packet spreading due to anharmonicity does not involve coherence
relaxation, but it leads to dephasing (and rephasing) of the observed vibrational
coherence, which depends on the terms pggrand peel. This intramolecular dephasing time

is deﬁned here as a T3 process.

A set of measurements using three-pulse FWM techniques, presented in this
Chapter, sorts out all the contributions to quantum mechanical coherence dephasing in a
molecular gas. Four different pulse sequences are used to sort out the different relaxation
processes involving the (X)’>:g+ and (B)31'Io,,Jr states of molecular iodine. Homogeneous
vibronic coherence relaxation rates between the ground and the excited state were
measured using the PE pulse sequence. The full vibronic coherence lifetime was
measured with the reverse transient grating (RTG) pulse sequence. The vibrational
coherence relaxation in each electronic state was measured with VB pulse sequences as
described in Chapter 4. For this method, the timing between the ﬁrst two pulses was used

to separate the contributions from ground and excited states.

118

5.2. Experimental

The experimental setup used to carry out the measurements has been described in
previous chapters. Brieﬂy, 60 fs transform-limited pulses (FWHM) centered at 620 nm
were used. The output was split into three separate arms with ~20 1.11 per pulse, and then
combined non-collinearly in a phase matching geometry. The pulses were focused into a
45 mm long quartz cell. Temperature dependence was measured for PE and RTG in a cell
that maintained a saturated iodine vapor pressure. The rest of the measurements were
made at 200°C in a different cell where the optical density was kept constant at ~ 0.3, a
value that is well below the saturated vapor pressure. The two diagonally opposed beams
(a and c) and the beam between them (b) deﬁne the phase-matched emission in the
direction k, = k, - k2, + k. This arrangement predeterrnines the sign for each wave vector.
The time ordering of each pulse (the pulse sequence) is changed for each experiment
using optical delay lines. The homodyne-detected signal was time (10 ns) and spectrally
(16 nm) integrated. For all the pulse sequences, indicated at the top of each ﬁgure, the
laser intensities were kept constant. The measurements presented here were taken with
the same setup and under similar conditions. The only change was in the pulse sequence

that selects the process to be measured.

5.3. Results and discussion

Homogeneous vibronic coherence relaxation measurements on molecular iodine
using the PE sequence are presented in Figure 5.1a as a function of temperature. The ﬁrst
pulse in the sequence generates the vibronic coherence pge, and its dephasing is measured

as a function of the time delay 2. The double-sided Feynman diagrams (DSFD) are

119

presented to indicate the nonlinear process involved. In the range from 77 to 108 °C, the
signal intensity decays exponentially. Because the signal is homodyne detected, it is

proportional to the square of the polarization. Hence, we use T2 = 2 tcxp to calculate the
homogeneous relaxation times measured, ranging from 7222:1304 to 310 ps for the

lowest and highest temperatures (see Figure 5.1a). In this temperature range, the vapor
pressure increases exponentially with temperature?'4 Homogeneous relaxation is caused

by collisions; therefore, it is proportional to the iodine concentration in this temperature

range, 1/ T,“ = now, where u is the average molecular velocity, n the number density

and 0' the phenomenological cross section. Numerical calculation of the homogenous
cross section based on the experimental data gives 0' = 11501150 AZ. This implies that
long-range interactions are responsible for the collision-induced homogeneous vibronic
relaxation. Our value for the homogeneous vibronic dephasing cross section is larger than
the vibrational dephasing cross section measurements by Zewail’s research group in the
presence of noble gases?‘ 1,212 This is to be expected when comparing self-quenching of
iodine molecules to quenching by noble gas atoms such as helium, neon and argon.215
The cross section for coherence relaxation is also larger than those for T, relaxation due

to self quenching, as expected.

When the RTG pulse sequence is used (see Figure 5.1b top), the vibronic
coherence relaxation measurements include both homogeneous and inhomogeneous
contributions. The experiments were carried out under the same conditions as the PE
experiments. In this case, however, the data no longer follow an exponential decay.

Inhomogeneous broadening gives rise to a Gaussian line shape in the frequency domain

120

a) Homogeneous vibronic relaxation

 

Inhomogeneous vibronic relaxation

b

 

   

 

 

 

 

 

 

a
a
b ca signal __ c badgna] \ -_ -_
a peg peg b
p“ -c- -—
2’ time .. b 1' (in: c
P110100 EChO Reverse Transient Grating
1.01'"; " " 1.0-rs; - - .4 - - - 2-- - - r- ~r -
1 n" . : x
i V," -' : ‘ .' 3
. " 1 I
_ D
s K. , _ . .
a . :i
-—': 0.1i o‘oooo 'l '2'. 0.1.1 ':
£3 ‘ ' yo 1 a i 1
2. ° a. ., ..- M... a» : z
= O O 1 "' ‘
3 0 ° 0 3
= , . o’ 0 ‘ a 2 ‘
'-' IP13”) 0C exPl-217'Tz) o E lRTdT) 0C CHM-(17'1" 2) )
e 0..., ., 1 5 am. ..
: a 78°C,T'2=l302145ps 0' [_ 3o 78°C,T'2 = 229i8ps a 3
'e 88°C,T2= 778il6ps o . 9‘ 1088°C,T'2=221:t5ps o I
, .98°C,T§= 460i10ps , ‘498°C,T'2=214tsps ‘
olO8°C, T'2 = 310i 9ps °lO8°C, T’ = l99i3ps
0.001 .................................... 0.002-.-” ,_ n, ......
0.0 0.1 0.2 0.3 040506 0.70.8 0,0 01 vino: 0,3 0,4

Time Delay 2, [11s] Time Delay 2, [us]

Figure 5.1. (a) Pulse sequence, corresponding DSFD, and PE measurements of the
homogeneous vibronic relaxation between the X and B electronic states of 12. The time
delay between pulses b and a(c) was scanned. The DSFD indicates that the signal
depends on the off-diagonal terms pg... The logarithm of the amplitude normalized signal
FWM emission is plotted as a function of time delay 2. The solid lines are exponential
ﬁts to the experimental data obtained at different temperatures. (b) Pulse sequence,
corresponding DSFD, and RTG measurements of the inhomogeneous vibronic relaxation
between the X and B electronic states of 12. The time delay between pulses c and a(b) was
scanned. The DSFD shows that signal depends on the off-diagonal terms peg. The
logarithm of the amplitude-normalized signal FWM emission is plotted as a function of
time delay 2. The solid lines are ﬁts to a Gaussian function of the data obtained at

different temperatures. The dashed line corresponds to the exponential decay measured
for the PE data at 108 °C (in Figure 5.1a).

and hence in the time domain. The data in Figure 5.1b, were ﬁt with the formula
S = Ae'w '2’ ,where A is an adjustable amplitude. '05 The experimental relaxation times,

in the temperature range from 77 to 108 °C, go from T22, = 229 i 8 to 199 i 3 ps (see

Figure 1b). Notice that there is only a 13% difference in the relaxation rate, although the

number density increased from 3x1023 1113 to 1.7x1024 m'3. In Figure 5.1b, we have

121

included the exponential decay used to ﬁt the PE data at 108° C. Notice that after the ﬁrst
200 ps, inhomogeneous relaxation is faster than the homogeneous relaxation. The weak
dependence on number density and the Gaussian decay pattern indicate that
inhomogeneous contributions play a very important role in RTG measurements.

Inhomogeneous contributions include Doppler broadening. Using formulas from

Siegmartm6 Awnﬂlﬂffiﬂw and T,’=——‘4”’°2, where A019 is the FWHM
C

A200
linewidth of the Gaussian distribution, M is the molecular weight and a) is the central
frequency of the transition, we calculate T; = 1.1 ns for T: 381 K. The value measured
is of the same order of magnitude but is ﬁve times shorter. One reason for this
discrepancy is that the RTG data contain both homogeneous and inhomogeneous

relaxation. This explains why the measured rates exceed the expected 7% dependence for

inhomogeneous decay.

For PE and RTG measurements, the signal drops to zero for long time delays. In
Figure 5.2a, we have included an RTG measurement obtained with a cell kept at 200 °C
with a low number density of iodine molecules (see experimental section). The signal
approaches zero at long times with an exponential decay time of 160 ps. For vibrational
relaxation measurements, the transient grating (TG) and VE sequences were used. In both
cases, the signals include the diagonal density matrix elements pee and/or pgg, which relax
with T1 lifetimes. Under the experimental conditions, T 1 is very long and is manifested
experimentally as a constant signal level at long times (see Figure 5.2b). The exponential
decay has lifetime 29,”, = 130 ps. The time dependent component of the TG signal

includes both p22 and pggr vibrational coherence.

122

8) Vibronic relaxation b) Vibrational relaxation

 

 

 

 

 

 

 

 

 

 

b a __ C
c be signal -- '2 ab 2' signal c "
a peg peg b m I ’ g’2 b pll's pcc'
Lb? :1," c _- c _- 2' lime r a a 0
Reverse Transient Grating ’ ' Transient Grating
? t 5.... [exp = 16012 ps > rexp = 130:1:8ps
i ’ . n?" 3:
a o a
g 1 ~ T=200°C .1 - T=200°C
a E‘
8 m
.5 1’:
[L3 .5.
LD
9‘ 1— .
o " 1 l J a L 1 1 0 1' 1 I 2 2 1 1 L
0-0 0-1 0-2 0-3 0-4 0-5 0-6 0.0 0.1 0.2 0.3 0.4 0.5 0.6
Time Delay 2, [nsl Time Delay 2, Ins]

Figure 5.2. (a) RTG measurements of the vibronic relaxation between the X and B
electronic states of 12. The FWM emission is plotted as a function of time delay 2. The
solid line is an exponential ﬁt to the experimental data (dots). (b) The pulse sequence and
corresponding DSFD for TG measurements of vibrational coherence in the X and B
electronic states of 12. The DSF D shows that the signal depends on the elements in the
diagonal blocks pgg, pggv, pee and pan. The FWM emission is shown as a function of time
delay 2. The solid line is an exponential ﬁt to the experimental data (dots). The dashed
line indicates the long-lived component of this signal, which decays with T1. The data in
(a) and (b) were obtained in a special cell at 200° C (see text), and include
inhomogeneous and homogenous broadening.

In order to separate the ground- and excited-states contributions to the signal, the
VE pulse sequences was applied as described by Brown et al.94 (see Figure 5.3 top).
When the delay between ﬁrst two pulses 212 was 460 fs, we measured a coherence
lifetime corresponding to the excited state of Texp = 152 ps. When the delay 2, 2 was 614 fs
the observed lifetime, corresponding to the ground state, was rcxp = 165 ps (see Figure
5.3). The ﬁrst tens of picoseconds of these two signals are shown together with their

Fourier transforms in Figure 5.4.

123

 

C
Vibrational relaxation " c ‘-
a b ’ c signal pf: b ‘31:}
lL—‘A , ’ -- b __

T12 T time a a

71209: 21m or 21t(n+%)

 

 

3 - o
1 Virtual Echo
:2 I . 2,,=460fs.2,,p= 152:1:8 ps
—: 2 '- 0 T12 =614fs .Taxp=16519 p5
3‘ ' is.
'5 . g ..
a _
3 ’ ’.
.5: 1 .
l: i
, o
o'. J

 

 

 

oio 0:1 0:2 0:3 0:4 0:5 0.6
Time Delay 2, Ins]

Figure 5.3. (a) The sequence and corresponding DSFD for the VE pulse sequences. The
DSFD show that the signal depends on the pee' and pgrg elements. (b) VE measurements of
vibrational coherence in the B (2,2 =460 fs) and X (2,2 =614 fs) electronic states of 12.
The intensity of FWM emission is shown as a function of time delay 2. The solid line is

an exponential ﬁt to the experimental data (dots). The data were obtained in a special cell
at 200 °C (see text).

The excited-state vibrational modulation (307 fs oscillations, 108 cm") is observed for
2,2 = 460 fs, while the ground-state vibrational modulation (160 fs oscillations, 208 cm")
is observed for 2, 2 = 614 fs. These vibrational frequencies correspond to wave packet
motion in the B state with v' z 7-10 and in the X ground electronic state with v” z 3-4,
respectively. The initial wave packet dephasing, dashed line in Figures 5.4a and b, was
obtained by eliminating the slow modulation caused by rotational dephasing and then
ﬁtting the vibrational modulation amplitude to an exponential. The data in Figure 5.4a
show a vibrational revival at 10 ps that is very similar to that observed in pump-probe
measurements with a similar preparation pulse.‘96 Because no coherence is lost during

wave packet spreading, revivals are typically expected. The wave packet spreading time

124

depends on intramolecular parameters such as anharmonicity and external parameters of
the preparation pulse such as pulse duration and chirp. The value measured for wave
packet spreading in the B state, 4 ps, is consistent with other femtosecond pump-probe
measurements. 2‘0?” The time associated with wave packet spreading of the X state, ~10
ps, is longer indicating a much smaller anharmonicity and a smaller number of
vibrational levels involved?“5»217 The measurements on wave packet spreading
presented here are in agreement with FWM experiments on gaseous iodine from other

groups173218.219

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Wave packet dephasing
a) T12=460 IS F. b) 712:614fs
31.0 lug” Tarp: 4:1 ps 3°25 Wl llllﬂll [WI l“. ‘1' Tap: 10:2 ps
.2 lllll 111111111111gull‘llllllmlllllHHIUllll’l' l 1,111
g lll'E ’ lull c2.0 ll ‘ Mlll’l’zl will.“ i
:30le lllllll‘l’ 8111711,.” llllll g llllllli I'lllllll llllllllll l llll‘i'lli llllllilllilll Wiwillsllmﬂ
E(iv'''55"1'1'ov'n1'5'h'2'0”"2'5 6----é----1,o----1,5--..io.---2,5
Time Delay 2, [ps] ,__, Time Delay 2', [pa]
=3 a
30.3 vcxcited state 550.5 ’ V round state
§°1l : 1 g
80.1 E .
E k t ’G‘TAIA‘ I I I I t I r I I t I
“0.0 100 110120130 140150 1601701'180190 200210 220 “0.0 100110120130140 150150 170100190 200 210 220
Wavenumber [cm ’1 Wavennrnber,[cm"]

Figure 5.4. Measurements of vibrational wave-packet spreading: (a) excited state (only
the ﬁrst 12 ps), (b) ground state with the VE pulse sequences as in Figure 3. The decay in
modulation amplitude caused by wave packet spreading was ﬁt to an exponential
function and is included here (dashed lines). The data in (a) show a wave packet revival
at 10 ps. The corresponding power FFT is shown.

125

5.4. Conclusions

This chapter described the use of different three-pulse F WM sequences that can
be effectively used to measure different types of coherence relaxation processes in gas-
phase molecules. Experimental results lead to the conclusion that the vibronic coherence
relaxation times in iodine vapor are strongly inﬂuenced by inhomogeneous broadening.
We obtained the homogeneous relaxation time and corresponding cross section using
photon echo measurements. The size of the cross section indicates that decoherence is
caused by long-range ‘soft’ collisions. In addition, we measured the inhomogeneous
vibronic coherence relaxation and vibrational relaxation times following selective
coherent excitation in the ground or in the excited state. Times for wave packet spreading
in each state are also reported. Using different pulse sequences to select among the
relaxation processes, these powerful methods can measure T1 times (typically 10'8 5); T2
times (typically 10'lo s) and T3 times (typically 10'12 5). Future applications will explore
decoherence processes in polyatomic molecules, an aspect that is of great interest for
efforts in coherent control of molecular dynamics. Of particular interest will be to explore
long-range dephasing collisions and wave packet spreading rates in multidimensional

systems.

These studies have been followed by a series of systematic measurements as a
function of density and in the presence of buffer gases.220 The results, obtained by Matt
Comstock and Evgeny Sudachenko in our group, are in very good agreement with the
dephasing cross section for pure iodine obtained by the initial measurements presented

here.

126

6. FEMTOSECOND GROUND-STATE DYNAMICS OF
GAS-PHASE DINITROGEN TETROXIDE AND
NITROGEN DIOXIDE

6.1. Introduction

Nitrogen oxides (NOX) have been the subjects of numerous studies primarily for
their roles in both stratospheric and tropospheric chemistry and in particular, their links to
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 statesm'232 All except NO have
multidimensional potential energy surfaces and numerous curve crossings that result in

vibronic chaos making these molecules very interesting to dynamicists.

The nitrogen dioxide/dinitrogen tetroxide monomer/dimer pair has received
particular attention. The dimer is favored at higher pressures and lower temperatures. At
one atmosphere and room temperature the equilibrium concentration of NO; is about
16%.226s233 Dinitrogen tetroxide is particularly intriguing because of its unusually long
N-N bond (1.776 A, gas phase)23‘lv235 and the fact that the ONO bond length and angle are
identical to those of the monomer (see Figure 6.1). Because of its (D2h) 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.221 It has been proposed226
that the use of off-resonance excitation should make it possible to directly probe the

dimer at room temperature. Most of the vibrational frequencies for these molecules are

127

O 0 Figure 6.1. Sketch of the N204 molecule. The
% N-O distance (a) is 1.19 A , the N-N bond (b)

\ is 1.776 A, and the angle a is 134.6°, from

(X /N __—— N \ reference 234.
O b 0

established and assigned,222~2241225 but there is still some disagreement between
experiment and theory on several modes. Time-resolved rotational dynamics of NO; were
measured by Sarkisov et. al.236 using femtosecond Raman-induced polarization
spectroscopy. The photodissociation dynamics of NO; were recently studied using
ultrafast methods.237.~238 This chapter exploits the ability to detect low frequency Raman
modes of short-lived species and to measure high accuracy rotational constants by
femtosecond time-resolved FWM to study the ground-state dynamics of NzO4/NOZ. Non—
resonant femtosecond time-resolved four-wave mixing data on gas-phase N02 and N204
are presented. The initial rotational dephasing is observed during the ﬁrst picosecond
after excitation. Fast vibrational dynamics (average value 133 i 1 fs beats) are observed
for dinitrogen tetroxide and are assigned to the V3 N-N stretching mode in the ground
state ('Ag). Rotational revivals on the picosecond time scale are recorded for both
samples. No evidence of a long-lived excited-state participation was found using PE or
VE pulse sequences with 800 nm laser pulses, however, NO; photoproducts were

observed following N204 excitation.

128

6.2. Experimental

The experiments were performed using a regeneratively ampliﬁed femtosecond
titanium-sapphire laser system (Figures 2.8 and 2.9) that was described in detail in
Chapter 2. In short, the output from a continuous wave NszVO4 laser (Millennia —
Spectra Physics Laser) pumps the femtosecond titanium-sapphire oscillator (KM Labs),
capable of producing <15 fs pulses at 90 MHz. This output was ampliﬁed by a
regenerative ampliﬁer (Spitﬁre — Spectra Physics Lasers) pumped by a 1 kHz NszLF
laser (Evolution-X — Spectra Physics Lasers). Ampliﬁed pulses centered at 805 nm were
close to transform limited and had a maximum energy of 0.8 mJ/pulse and time duration
of 50 fs. Characterization of the pulses was performed with both non-collinear
autocorrelation and frequency resolved optical gating. Both methods conﬁrmed that the
pulses were shorter than 50 fs measured at full-width half-maximum and had no residual
chirp. The output was split into three beams using dielectric beam splitters and
recombined in a forward-box geometry (see Figure 2.6). In all experiments the output
beam was attenuated to ~60 u] 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 TG and
RTG experiments described in detail in previous chapters. When the two initial pulses are

not overlapped in time, other types of FWM experiments (PE and VE) are possible.104

The event when all three pulses overlap in time is referred to as time-zero. The
three 10 mm diameter beams were focused into a 6 inch-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 F WM

129

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=1-2. 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.

The sample (99.1% N204) was obtained from Sigma-Aldrich and used without
further puriﬁcation. The sample cell was pumped out to 10'5 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 were obtained with a cell that was loaded to an initial pressure of 400 Torr

at room temperature .

6.3. Results and discussion

In the gaseous state, the N204 and N02 molecules are in thermodynamic
equilibrium (at room temperature K = 8.8 atm").239 The temperature and pressure of the

sample cell from 294 K (80% N204) to 363 K (94% N02) were controlled.233 A change

130

in optical density of the sample at 805 nm from 0.045 (294 K) to 0.071 (363 K) was
observed and was not considered large enough to affect the measurements qualitatively.
Some experiments were carried out with a cell ﬁlled to 760 Torr. The overall signal level
was somewhat smaller due to the increased OD leading to absorption of the excitation
photons. The transients yielded qualitatively identical results, but required a higher
temperature to obtain a good NO; 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 N;O4 in the cell because of the aforementioned

equilibrium.

In Figure 6.2 frequency-dispersed transients are shown that were obtained as a
function of time-delay between the two time-overlapped pulses and the third pulse for
NO; (6.2a) and N;O4 (6.2c). 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 NO; (Fig. 6.2a) there
is a broad, featureless peak 160 fs after time zero as well as another at 500 fs. Figure 6.2b
shows the frequency—integrated (780-830 nm) transient signal. These features (Figure
6.2a and b) can be assigned to the initial rotational dephasing in the sample and are
consistent with results for the off-resonance FWM signal. Measurements for N;O4 were
collected in the same manner as described for NO; and are presented in Figure 6.2c and
d. These results show, in addition to a broad rotational dephasing envelope, a series of
beats with an average period of 133 fs. The beats are detectable up to 3 ps after time zero

but with much lower intensity.

l3l

 

 

 

 

 

 

 

 

 

_ a)
E
E...
(<
b)
1045- 1
103
820:— 3 i . ' C)
E 810— -t‘ - 2 i
E“ 800* 1?: K g
(< " - Q O
790_‘ : E. - ‘
: d)
104; '
103 . . _
o 500 1000 1500 2000

Time Delay [fs]

Figure 6.2. 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 NO; data as a function of time delay. c) Frequency-dispersed signal as a
function of time delay for N;O4, d) Frequency—integrated N;O4 data as a function of time
delay. For all scans, the signal intensity is plotted on a logarithmic scale.

132

Long time scans were performed at 363 K (NO;) or 294 K (N;O4). We present
data in the time-delay range from 9 to 42 ps in Figure 6.3. The transients show the
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 NO; data presented in
Figure 4a show rotational recurrences every 19.680 ps with half recurrences in between
(9.840 ps). The transient in Figure 6.3b, taken at 294 K, exhibits similar features with the
same periodicity. The recurrences observed at 294 K are about six times lower in
intensity than those observed for NO; 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 NO;. Under
close examination some differences are observed in the shape of the rotational
recurrences obtained in the two transients. The differences are especially clear at the half

recurrence observed in both transients near 29.5 ps (see Figure 6.3).

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 Bave=(B+C)/2. From the observed
rotational recurrences of 19.680 ps (9.840 ps half-recurrences) in the long time delay
transient of NO; (Figure 6.3a) we can calculate the average rotational constant as
Bave=1/4c rm, where c is the speed of light and Tree is the time between rotational
recurrences. The obtained value is 0.42343 cm'l i 0.00007 cm". This value compares
favorably to the reported experimental240 (0.42202 cm") and theoretical227 (0.423325

cm") values calculated for Bave, respectively. Simulations based on the semiclassical

133

forrnalism,9z1'32 and the average rotational constant for NO; using the literature values
are in general agreement with the observed data (see vertical lines in Figure 6.3). The
rotational recurrences obtained at longer time delays (r> 40 ps) exhibit a more complex
modulation that cannot be simulated with a symmetric top approximation, even though

their positions can be exactly reproduced.

 

104 r l ‘

 

 

 

 

 

 

A
Y Y

N 9680 fs

Intensity [rel.]

   

103— 1 ._

Fl 1 ll 1 I n :L 1 Id

10000 20000 30000 40000 50000
Time Delay [fs]

 

 

 

Figure 6.3. Experimental time-resolved and frequency-integrated FWM signal (a) NO;
and (b) N;O4 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. The vertical lines indicate the position of the rotational
recurrences obtained from a simulation using Bave=(B+C)/2.

I34

In the case of N;O4, the signal obtained at long time delays is more complex. The
small features at 9.840 and 19.680 ps (Figure 6.3b) could be assigned to rotational
recurrences originating from the 16% N0; molecules present at room temperature. The
positions of these features coincide precisely with the rotational recurrences of NO;,
however, the signal intensity is about one order of magnitude stronger than expected
based on the observed intensity at 363 K, given the square dependence on number density
for these measurements. The signal in TO measurement is proportional to the square of
the induced polarization, which depends linearly on susceptibility and therefore on
number density.92 If the only source of rotational recurrences at 294 K (Figure 6.3b) were
N0; molecules that were present in the sample before the ﬁrst interaction with laser pulse
took place, the signal should be an order of magnitude lower. Furthermore, there are
some subtle differences in the shape of the recurrences; for example the half recurrence at
29.5 ps in Figure 6.3 lead us to consider assigning a signiﬁcant contribution to these
features from photoproducts that result from 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 1.54
eV. The principal axis of the NO; moieties is aligned with the N-N bond of N;O4;
therefore, the initial rotational alignment would be preserved in the photodissociation
process. The product NO; molecules formed by photodissociation would realign and
become ‘visible’ to the third pulse at the same time delays as the thermal N0; molecules
originally in the sample, but would have a different rotational temperature. There is a
possibility that multiphoton excitation yields NO; molecules in the 2B; excited state,

which has slightly different (and still not exactly determined) rotational constants.

The initial rotational dephasing observed during the ﬁrst picosecond is

135

vibrationally modulated. The times between ﬁrst ﬁve beats are 158, 125, 131 and 135 fs.
Fourier transformation of the signal presented in Figures 6.2 c and d results in a single
broad feature centered at 234 cm"(See Figure 6.4). The experimentally obtained value for
the N-N stretching mode from gas-phase Raman spectroscopy225 is 254 cm", and from
high resolution FTIR241 and Raman spectra on solid242 N;O4 it is 281.3 cm". The only
other frequency that could be implicated is the NO; rocking mode v10 = 265.5 cm'l;

however, this mode is not Raman active and therefore not observed in our measurements.

 

FFT

 

 

l‘\ /— -"/'
' ‘-— 1‘ A A n A \T“'r~—l__l—_A ‘h— Irri—

1100‘ A ‘150 ‘ A 200 A A .250 I 300 350 400

 

wavenumber [cm1 ]

Figure 6.4. FFT of the data in Figure 6.3b. The observed peak at 234 cm'I was assigned
to the N-N stretching mode.

Applying the Maximum Entropy Method (MEM)243 of analysis on the same set of
data yielded two frequencies: 238.5 cm'1 and 251.8 cm". These two values were
consistently obtained following analysis of different data sets (not shown). The observed
wave packet motion involves at least three vibrational levels in the ground state. The
difference between the two frequencies obtained by MEM gives the anharmonicity of this

Vibrational mode. Presumably, the 251.8 cm'1 value corresponds to the lowest level(s),

136

consistent with the reported gas phase value, and the 238.5 cm'1 value to the higher

levels. These ﬁndings are consistent with the extremely large anharmonicity of N;O4.244

Only one rotational recurrence feature was observed after the ﬁrst picosecond that
can be assigned to N;O4. At 76.265 ps (Figure 6.5) 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 (NO; recurrence) if the temperature of the
sample is lowered to 00 C (data not shown). The lower temperature shifts the equilibrium
further towards the dimer. If a semiclassical model and available rotational constants 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 N;O4. One possible reason for the absence of
N;O4 rotational recurrences at long time delays (> 100 ps) is the rate of collision; which

we estimate to be r< 0.1 ns at 300 K.

137

 

 

 

 

 

 

10“ :— l\590401s. 78720 fs/ ‘5
'7 : M l a) I
B _

m -
C
9 l- d
E

 

104 r 1
: l 76265 fs l -

Intensity [rel.]

 

 

 

103 _

I
1 L

I . I . 1'
60000 70000 80000
Time Delay[fs]

 

 

 

 

 

 

Figure 6.5. Experimental data 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.

6.4. Conclusion

The signal observed for both molecules (N O; and N;O4) 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 were performed
but yielded no signiﬁcant feature other than the time zero signal. If the experiments
shown here did involve a resonant one-photon transition to the 2B; state of NO;, the PE

and VE signals would have revealed it.

The measurements suggest that only ground state NO; contributes to the observed

138

 

signal based on the following argument. The absorption cross section of NO; at 800 nm is
very small.245»246 The reported and calculated values of the B and C rotational constants
for the 2B; state are very dissonant, and fall in the range from 0.458 to 0.548 cm'1 and
0.38 to 0.449 cm", respectively. If these transitions were involved, additional rotational
recurrences corresponding to these values would have been observed in the time-resolved
F WM transients. Close examination of the data did not reveal any other features except
for the aforementioned recurrences every 19.680 ps. Given that the homodyne detected
FWM signal depends on the transition dipole moment to the eighth power, this process is
not likely to be detected by our setup. The average rotational constant measured from our
experiments of 0.42343 cm'l reﬂects the average (B+C)/2 rotational constant of NO; in

the ground state.

The experiments at room temperature (N ;O4) allowed us to measure the frequency
of the N-N stretch of 234 cm'1 and perhaps the anharmonicity of this unusual bond. The
next electronic state known is the 3B3u which is reached with photon energies >3 eV and
would require two photon absorption to be detected. 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
found in the N;O4 transient in Figure 6.2d but not found in the NO; transient in Figure

6.2b is evidence of a dissociative pathway.

139

7. SUMMARY AND CONCLUSIONS

This work has contributed to the evaluation of different methods of using
femtosecond lasers for studying laser control of chemical reactions. The number of
ultrafast pulses, the timing between them (sequence), their phase, and chirp are shown to
be effective control parameters for the molecular system. All of these parameters play a
role in the search for optimal laser ﬁelds for control of molecular motion. The
experimental demonstration of laser control presented in the previous pages includes the
enhancement of the yield of the photodissociation reaction by manipulation of a phase
characteristic (chirp) of the applied pulses. Speciﬁcally, the formation of I; that followed
multiphoton excitation of CH;I; at 312 nm exhibited an asymmetric dependence on the
sign of chirp. Enhancements up to a factor of 25 were observed for positive chirps
compared to transform-limited pulses and were found to be dependent on pulse
intensities. Over last two years this method that involves only one control ‘knob’ has lost
popularity despite its success,‘3~‘54844512471248 and has surrendered its place to control
schemes using more sophisticated optimal control},'93032’7‘31‘77,249 where the number of
control parameters can be virtually unlimited. Very recently, it has been demonstrated

that a limited number of parameters is indeed more advantageous.23~250

The work presented in this thesis also demonstrated that femtosecond F WM
techniques could be applied in coherent control by combining a discrete number of laser

pulses. This also allows us to learn about the role of one or more control parameters in

I40

controlling a process. Theoretical as well as some experimental studies in coherent,
quantum, and optimal control have found that the optimal ﬁelds in many cases
correspond to two or three pulses with a particular phase relationship and chirp. Three-
pulse FWM methods are shown here to be very useful in the characterization of
intramolecular dynamics. Presented measurements show the dynamics of a gas-phase
model system (1;) prepared by the ﬁrst two laser pulses and its control. For the VE
sequence, when the pulses are transform limited and the delay between them is 460 fs,
only excited-state dynamics are observed, while time delay of 614 fs yields ground-state
dynamics. With the VE setup controlling the time between the ﬁrst two pulses allows to
control the observation of ground or excited state dynamics. This type of control is not
available with the PE setup. Presumably, under large inhomogeneous broadening, the
vibrational coherence is lost leaving only the signal that is independent of the coherence.

However, when the laser pulses are chirped, the observed dynamics are quite different.

One of the clearest applications of FWM methods to coherent control is in
measuring coherence relaxation. It is demonstrated here how the methods work for gas-
phase samples and a comparison has been made between photon echo and virtual echo
type measurements. The main difference is that for PE the inhomogeneous broadening
cancels, giving a more accurate homogeneous coherence relaxation time. The data
presented can be used to extrapolate a homogeneous coherence time for iodine in the
nanosecond scale, giving the possibility of performing a very large number of coherent
interactions with this gas-phase sample at room temperature. Differences between the
various phenomena (PE, VE, TG, RTG) have been demonstrated. The results reveal the

control schemes of the molecular system within the framework of phase-matching

l4l

 

detection. This implies that only the subset of molecules forming a transient grating in the
sample is controlled and yields the coherent signal observed. It is also demonstrated how
FWM methods can be used to measure coherence life times and to characterize the
intramolecular dynamics of the system that is being controlled. Future applications of this
form of control could lead to a more general molecular control of quantum mechanical
states in the gas phase and perhaps even to realization of molecular computing.24 The
non-resonant time resolved FWM technique has been applied to dinitrogen tetroxide and
nitrogen oxide in order to study their dynamics. This allowed the precise determination of
the average rotational constant and vibrational frequency of the N-N stretch mode. The
results suggest that the planarity of N;O4 could be explored with this technique. More
detailed exploration of this interesting molecular system requires tunable femtosecond
pulses that are resonant with electronic transitions of the molecule. Further explorations

would be feasible because such pulses are widely available at the present.

142

 

8. APPENDIX I

Below is a brief theoretical description of C-FID-FWM based on the solution of
the Liouville equation. Microscopic expressions for cascading are given in chapter 16 of
reference 87. Evolution of the density-matrix elements p1,- in the absence of an external
ﬁeld can be described with the free propagator, under interaction with the vibronic matrix

elements V1,.

v5“ = Alf'e'ka", where All“ = ‘fislf‘m (8.1)

where the label aindicates the interacting pulse a, b or c; i and j are indices for

vibrational levels in the excited and ground state respectively; 0),]- and 11,-,- are the

transition frequencies and dipole moment on the ijth transition; Sl-a] is the spectral

intensity of the corresponding transition induced by the transform-limited pulse a with

carrier frequency a)" and wave vector ka; Asa] are the macroscopic space-independent (x)
0]

components of V2 .

The 1n1t1al condition of the densrty matrix p19) = 11,- 1S glven by a Boltzmann

distribution. After pulse c interacts with the sample at time t], the induced polarization

causes FID with the wave vector knp = kc,

143

 

EFID(I) 0C :14 21- “ij[vi[jc]e-i(oij(t-ti)_Vljcl*ei0)ij(t-ti)]nj (82)

. . . . no ,
This ﬁeld interacts With medium, Vlj / oc ,uleF/D, after pulses a and b at times t;

and I 3 causing the third order polarization

(3) -' .. - (2)
pij (Um-e 1(1),)“ t3)Zj" Pj"j(t3)14ij"zry “no,

* f dt'eimij”(t"t3)[e-i[wi'j'(t'-tl)-kFle]-ei[wi'j'(t'-tI)-kFle]]A!f!nj'

_' .. - (2)
+e “U“ “)2," pii" (101/0021111 Pi’j'

[c

* ﬂoodt'eimi"j(t'-t3)[e'i[mi'j'(t"tl)‘kFIDXI-eilmi'j'(t"tl)‘kFlenA'v'Inj' (83)

Only resonance terms can stimulate optical transitions, therefore, terms with

wy'zwij in the ﬁrst sum and a),-~j=a),~j1 in the second sum are kept. The second order
. . 2 . . . .

denSity matrix elements, pg) in Equation (3), can be calculated With the following

formula applicable for well-separated pulses
plin)(t) ____ Ie-iwija-t")

1n] (n-l) (n-l) [n]

[n] (n-l) ("l
*{Zy [Vii'pi'j (tn)'pii' (thi'J n

1+2, [Vij' pj'j (id-Pi)" (tn)Vl3]]}- (8.4)

To calculate the C-FID-FWM signal, only non-zero density matrix elements of

p33) proportional to exp[-i(ant-(ka-kb+kc)x)] are included, because only these components

emit light in the phase-matching direction. The intensity of the time-integrated signal is

144

given by

2 (3)
l = Zij 111, where Iij oc uijﬁdtlpij (t)|2 (5)

An expression to calculate the spectral components Ig-FlDlof the C-FID-FWM
signal observed for negative rd, = tc-tb can be obtained as follows. The Equation (8.4) is

used to calculate psi), and introduce these elements into Equation (8.3). Finally, each

element is calculated with Equation (8.5) using rba = tb-ta, to obtain

[FID] 2| 10,... 2 [c1 -10” [bl* [al
Iij °Cllij 211C " ‘bllij'Aij'nj'Zive ”'"Ai'j' Ai'jn'+

- 2 [c] -° [bl‘ [8| 2
X11 e'm"T°bH1'in'j anj'elm”Tb'A 'Aij'“i’ - (6)

i’j

The time delay rba is the control parameter that allows us to select between ground and
excited state dynamics in both cases. Similarly to VE sequence, the signal is obtained as a

function of the time delay 1d,.

145

 

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