ABSTRACT
Title of dissertation: ULTRA-FAST DYNAMICS OF
SMALL MOLECULES IN STRONG FIELDS
Kun Zhao, Doctor of Philosophy, 2006
Dissertation directed by: Professor Wendell T. Hill, III
Institute for Physical Science and Technology
& Department of Physics
Correlation detection techniques (image labeling, coincidence imaging, and
joint variance) are developed with an image spectrometer that is capable of col-
lecting charges ejected over 4? sr and a digital camera that can be synchronized
with the laser repetition rate at up to 735 Hz. With these techniques, molecular
decay channels ejecting atomic fragment ions with different momenta (energies) can
be isolated; thus the initial molecular configurations (bond lengths and/or bond
angles) and orientations as well as their distributions can be extracted. These tech-
niques are applied to study strong-field induced dynamics of diatomic and triatomic
molecules.
Specific studies included the measurement of (1) the Coulomb explosion energy
as a function of bond angle in linear (CO2) and bent (NO2) triatomics and (2) the
ejection anisotropy relative to the laser polarization axis during Coulomb explosions
in both triatomic (CO2 and NO2) and diatomic (H2, N2 and O2) systems. All the
experiments were performed with 100 fs, 800 nm laser pulses focused to intensities
of 0.1 ? 5.0?1015 W/cm2.
The explosion energy of NO2 decreases monotonically by more than 25% from
the smallest to the largest bond angles. By contrast, the CO2 explosion energies
are nearly independent of bond angle. The enhanced ionization and static screening
models in two-dimension with three charge centers were developed to simulate the
explosion energies as a function of bond angle. The predictions of both models are
consistent with the measurements of CO2 and NO2. At the same time, the observed
explosion signals as a function of bond angle for both CO2 and NO2 show large-
amplitude vibrations. The peaks of the explosion signal distributions for both CO2
and NO2 appear near the equilibrium bond angles of the neutral systems.
The ejection angular distributions in triatomic (CO2 and NO2) and diatomic
(H2, N2, and O2) Coulomb explosions were measured; the contribution made to the
ejection anisotropy by geometric and dynamic alignment was studied by comparing
the images obtained with linearly and circularly polarized fields. Different angular
distributions of the molecules are consistent with different ionization stages, induced
dipole moments and rotational constants. A narrower distribution of H2 than other
diatomics is confirmed by its nearly complete dynamic alignment in the field. The
results also show that the large dynamic alignment in the linear triatomic CO2 is
consistent with the fact that more electrons have been removed and the precursor
molecular ion spends more time in the field prior to the explosion than diatomic
systems such as N2 and O2.
ULTRA-FAST DYNAMICS OF
SMALL MOLECULES IN STRONG FIELDS
by
Kun Zhao
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2006
Advisory Commmittee:
Professor Wendell T. Hill, III, Chair/Advisor
Professor Julius Goldhar, Dean?s Representative
Professor Millard Alexander
Professor Rajarshi Roy
Professor Robert Walker
c? Copyright by
Kun Zhao
2006
DEDICATION
This dissertation is dedicated to my grandparents.
ii
ACKNOWLEDGMENTS
This work partially belongs to my wife, Xuanxuan, the best thing that ever
happened to me. She is a good wife and my best friend. She gives me all the help,
support and love in every facet of my life and research. Without her, my life would
be in vain.
First and foremost I?d like to thank my advisor, Professor Wendell Hill for
giving me such an extraordinary opportunity to challenge physics as well as myself
and have fun in the process. He is an excellent advisor and a nice person. Without
his extreme patience and constant guidance over the past years, this work would be
simply impossible. His knowledge and insight on physics has enriched me. It has
been a great pleasure to work with and learn from him.
Special thanks are extended to my defense committee members whose names
begin this dissertation. I could not ask for a more qualified group for the final step
of my graduate study. Special thanks are also extended to Mr. Tim Howard and
Mr. Lee Elberson, who read this entire manuscript and gave me enormous help on
the language.
My colleagues in our group have enriched my school life in many ways. Mr. Ilya
Arakelyan, Mr. Narupon Chattrapiban, Mr. Vishal Chintawar, Mr. Lee Elberson,
and Dr. Menkir Getahum have always helped and supported me. I really enjoy
talking about physics, philosophy, music, movies and everything else with them. I
iv
wish them a successful career.
I would also like to acknowledge help and support from professors and staff
members of Institute for Physical Science and Technology and Chemical Physics
Program, Dr. Michael Coplan, Dr. Millard Alexander, Dr. Michael Fisher, Dr.
Rajarshi Roy, Mrs. Diane Mancuso, Mrs. Patricia Wynn, and Mrs. Debbie Jenkins
in particular. Their help is highly appreciated.
I owe my deepest thanks to my mom, dad and sister who have always cared
for me, loved me and talked to me. I am so lucky to be born into such a wonderful
family. I wish my grandparents could see my graduation. May they rest in peace.
v
TABLE OF CONTENTS
List of Tables viii
List of Figures x
1 INTRODUCTION 1
1.1 Motivation and Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research Background and Contributions of This Study . . . . . . . . 4
2 BASIC CONCEPTS IN STRONG-FIELD MOLECULAR DYNAMICS 11
2.1 Kinetic Energy Release . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Enhanced Ionization and Charge Resonance . . . . . . . . . . 13
2.1.2 Electron Screening Effect . . . . . . . . . . . . . . . . . . . . . 21
2.2 Ejection Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 APPARATUS 31
3.1 Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Image Spectrometer and Vacuum System . . . . . . . . . . . . . . . . 42
3.3 Digital Camera for Imaging . . . . . . . . . . . . . . . . . . . . . . . 53
4 CORRELATION DETECTION 55
4.1 Image Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Image Labeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Coincidence Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4 Joint Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5 EXPERIMENTAL RESULTS AND ANALYSIS 73
5.1 Explosion Energy Measurement . . . . . . . . . . . . . . . . . . . . . 74
5.1.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.1.1 CO2 Results . . . . . . . . . . . . . . . . . . . . . . . 75
5.1.1.2 NO2 Results . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.1.3 Large-Amplitude Vibration . . . . . . . . . . . . . . . 89
5.1.2 Explosion Energy Analysis . . . . . . . . . . . . . . . . . . . . 92
5.1.2.1 Triatomic 2-D Field Ionization . . . . . . . . . . . . . 92
5.1.2.2 Static Screening . . . . . . . . . . . . . . . . . . . . . 105
5.1.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . 112
5.2 Ejection Anisotropy Measurement . . . . . . . . . . . . . . . . . . . . 116
5.2.1 Experiments and Results . . . . . . . . . . . . . . . . . . . . . 117
5.2.1.1 Angular Spread Measurement . . . . . . . . . . . . . . 117
5.2.1.2 Dynamic Alignment Measurement . . . . . . . . . . . 121
5.2.2 Data Analysis and Discussion . . . . . . . . . . . . . . . . . . 126
6 SUMMARY AND FUTURE WORK 135
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
vi
A Classical Pendulum Equation of Motion 139
B Charge Arrival Position and Time of Flight 143
B.1 Charge Arrival Position and Time on Image MCP . . . . . . . . . . . 143
B.2 Time of Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
C Image Processing 151
C.1 Image Compression and Recovery . . . . . . . . . . . . . . . . . . . . 151
C.2 Image Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
D Three-body Coulomb Explosion Kinematics 168
E Two Simulations for Electron Screening 176
Bibliography 183
vii
LIST OF TABLES
2.1 Molecular properties for H2, N2, O2, CO2, NO2 and I2. The equilib-
rium bond length (Re) and the ground-state rotational constant (B0)
are extracted from experimental measurements [114]. The moment
of inertia (Im) is calculated from the ground-state Re (The value of
NO2 is calculated at its ground-state bond angle ?be = 134?). The
ground-state rotational time is ?e = 1/(2cB0) because the rotational
energy at the ground state is E0 = h? = hcB0J(J +1) with ? = 1/?e
and J = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1 Polarization (POL) of the pulse and voltages on the Pockels cells
(PC1 and PC2) at each optical element in the amplifier cavity. The
labels of the elements are the same as in Fig. 3.2. The polarization
states and the voltages (0 or ?/4 voltage) are the ones AFTER the
pulse passes through the element. The pulse is running from left to
right. The upper table is the trip with both Pockels cells off. It
starts as the pulse is reflected into the cavity by the right surface of
the laser rod (Fig. 3.2). At the end, the vertically polarized pulse is
reflected out by the thin film polarizer (OP). The middle table is the
trip when the pulse is trapped. It also starts as the pulse is reflected
into the cavity by the laser rod. After this trip, the pulse stays in the
cavity as PC1 stays activated. The lower table is the last round trip
a pulse makes in the cavity. It starts as the pulse goes through the
laser rod toward the quarter-wave plate. At the end, the amplified
pulse is ejected out by the thin film polarizer. Two mirrors (M2 and
M3) are omitted since they do not affect the polarization of the pulse. 37
3.2 Arrival time in the unit of ns for atomicand molecular ions which have
zero initial momentum or are ejected parallel to the image MCP cal-
culated by Eq. 3.5 with Vint = 1500 volts. The charge states marked
as N/A either does not exist or is never detected. The arrival time
for an electron is 5 ns. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1 Ionization potentials of H and the first two ionization stages of C, N,
and O atoms in eV [133], where I(q)p is the ionization potential from
X(q?1)+ ? Xq+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
viii
5.2 Molecular constants and properties for CO2, NO2, H2, N2, and O2.
The quantities are extracted from experimental measurements: Re,
the equilibrium bond length [114]; RC, the critical separation [this
study]; ?be, the equilibrium bond angle [114]; B0, the ground state
rotational constant [114]; n, the exponent of cosine distribution [this
study]; ??g, the angular spread of the distribution [this study]. . . . 118
5.3 Dynamic alignment parameters for H2, N2, O2, and CO2 obtained
in the experiments: the primary (secondary) channel and its contri-
bution (%), its n and ??g taken from Table 5.2, ??eff (??da) the
effective (apparent) angular spread for neff (nda), Re and RC taken
from Table 5.2, ? the degree of dynamic alignment (Eq. 5.18), R the
relative number of atomic ions generated under linearly and circularly
polarized fields (Eq. 5.14), ?R the number of molecules torqued in
the dynamic alignment, and Ti/Ti(O2) the interaction time relative to
that for O2 calculated at ??Rmol? = (Re +RC)/2 by the two methods
discussed in the text, where Rmol,e (Rmol,C) = 2Re (2RC) for CO2. . 127
C.1 First five nonzero pixels in a camera frame and their corresponding
pairs of bytes (a, b) in the compressed image file including the markers
(the pairs with a = 0). The position of each pixel (P) is computed
as P = x + y ?w, where x and y are the coordinates of the pixel on
the frame and w = 128 is the width of the frame. The position can
also be computed from (a, b) as Pi = bi + 256?summationtextj<i,aj=0 bj. . . . . 155
C.2 Occurrence frequency as a function of the number of pixels (x) in a
cluster measured in H2 (H+), N2 (N2+), O2 (O2+), CO2 (C2+ and O2+)
and NO2 (N2+ and O2+) images. Laser pulses are linearly polarized
with an intensity of 2?1015 W/cm2. The measured ?, ? (Eq. C.2) and
Ntot, as well as the number of frames involved, are listed in Table C.3. 158
C.3 The measured ?, ? (Eq. C.2) and Ntot from the occurrence frequency
as a function of the number of pixels (x) in a cluster for H2, N2, O2,
CO2 and NO2 images (Table C.2), as well as the number of frames
involved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
ix
LIST OF FIGURES
2.1 Classical over-the-barrier ionization model of an electron in the com-
bined atomic or molecular field and laser field for (a) atomic (Xq+),
(b) diatomic (Xq+, Xq+), and (c) linear triatomic (Xq+, Xq+, Xq+)
systems. The black, blue and red curves represent, respectively, the
electron potential energy as a function of the electron position, the
electron energy level, and the laser-electron coupling eEx in eV. The
laser electric field is pointing to the left. . . . . . . . . . . . . . . . . 15
2.2 Comparison of experimentally determined and theoretically predicted
appearance intensities using a simple classical theory. The solid line
corresponds to exact agreement. (Fig. 3 of Ref. [106]) Reprinted figure
with permission from S. Augst, D. Strickland, D. D. Meyerhofer, S. L.
Chin, and J. H. Eberly, Phys. Rev. Lett. 63, 2212 (1989). Copyright
(1989) by the American Physical Society. . . . . . . . . . . . . . . . 16
2.3 Simulated evolution of the features of the Cl2 molecule for a linearly
polarized laser pulse with a peak intensity of 1.25 ?1016 W/cm2 and
? = 610 nm ramped to maximum intensity in ten optical periods.
Main figure shows the kinetic energy of the atomic ion fragments vs
time. Note that Coulomb explosion begins near the peak intensity;
deceleration of the fragments sets in around 20 optical periods. In-
set: Distribution of net electrical charge including both nuclear and
electronic components. (Fig. 1 of Ref. [81]) Reprinted figure with
permission from M. Brewczyk and K. Rz?a?zewski, Phys. Rev. A 61,
023412 (2000). Copyright (2000) by the American Physical Society. . 23
2.4 Calculated and experimental multifragmentation channels kinetic en-
ergy releases vs total number of removed electrons in the Coulomb
explosion of CO2 induced by a linearly polarized laser pulse. (Fig. 1
of Ref. [82]) Reprinted figure with permission from Ph. Hering, M.
Brewczyk, and C. Cornaggia, Phys. Rev. Lett. 85, 2288 (2000).
Copyright (2000) by the American Physical Society. . . . . . . . . . 24
3.1 The overall experimental setup includes three parts. (1) The 100
fs, 800 nm Ti:sapphire laser system (Spectra Physics/Positive Light)
includes a mode-locked Ti:Sapphire oscillator (Tsunami) pumped by
an argon ion laser (BeamLok), a Ti:Sapphire regenerative amplifier
(Spitfire) with stretcher and compressor pumped by a Nd:YLF laser
(Merlin), and beam diagnostics; (2) 4? image and TOF spectrometer
housed in a vacuum chamber; (3) digital camera and oscilloscope with
computer and timing electronics. . . . . . . . . . . . . . . . . . . . . 32
x
3.2 Cavity of the regenerative amplifier. M1 ? M4 are broadband dielec-
tric mirrors, LR is the Ti:Sapphire laser rod, PC1 and PC2 are the
input and output Pockels cells, QWP is the quarter wave plate, and
OP is the thin film polarizer. . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Schematic of a single-shot second-harmonic autocorrelater. Mirrors
M1 and M2 are gold-coated broadband, while mirrors M3 ? M6 are
aluminum-coated broadband. BS is 50-50 beamsplitter. The distance
that the translation stage moves is measured by a micrometer with
an accuracy of 20 ?m. BBO (beta barium borate) is the doubling-
frequency crystal. All the reflected beams are kept in the same plane,
and both pulses arrive at the crystal with the same incident angle.
The second harmonic signal bisects the angle between the two fun-
damental beams and detected by a CCD camera. The transmitted
fundamental beams are filtered out by the infrared filter (Schott BG
series, more than 70% transmission at 400 nm and less than 1% at
800 nm). The crystal is mounted on a rotary holder in order to be ro-
tated to get phase-match when the polarization of the incident beam
changes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Second harmonic (SH) generation in the BBO crystal. This plot is
in the same plane as in Fig. 3.3. The SH signal is only generated
where the two pulses overlap both spatially and temporally. The
fundamental pulses propagate in the directions of vectork1 and vectork2. Their
temporal profiles are I1(t) and I2(t) with I1(t) = I2(t) = I(t). The
SH signal from the overlap of the pulses is SH(x), where x direction
is parallel to the face of the crystal. . . . . . . . . . . . . . . . . . . 40
3.5 Measured variation of the peak amplitude of the signal on the pho-
todiode, which is assumed to represent the peak intensity of the
laser pulse. The photodiode is connected to the digital oscilloscope
(LeCroy 9350AM) with 50 ? impedance. In this particular run,
13,000 laser pulses were recorded and the standard deviation of the
signal is 18%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
xi
3.6 Structure of the image-TOF spectrometer. There are six electrode
rings in the spectrometer. The two close to the TOF detector are
not being used. The other four (ER1 to ER4) are used to generate
a static electric field between the image MCP and the flight tube.
Their inner diameters are 40 mm to match the size of the active area
of the image MCP. The voltages on the electrodes are V4 = 2V3 =
4V2 = 8V1. Positive or negative voltages (normally 2000 volts) are
applied to take ion images or TOF waveforms, separately. The front
side of the image MCP is grounded (V0 = 0). The distances between
the electrodes and MCP are D4 = 2D3 = 4D2 = 8D1 = 76.2 mm
(3 inches) to ensure a uniform field. The focal point of the pulse is
between ER3 and ER4 so that the distance between the interaction
zone and MCP is l = 3D4/4 = 57.15 mm and the potential Vint =
3V4/4. The potential across MCP VMCP is up to 2000 volts. The
front side of the phosphor screen is connected to the back of MCP,
while the potential between the back side and the ground is Vphos =
4800 volts. The voltages on the front and back sides of the TOF MCP
are Vfront = ?2300 volts and Vback = ?300 volts. The 140-mm-long
flight tube is connected with ER4 on one end and the other end is 30
mm away from the front side of the TOF MCP. The diameter of the
tube is 40 mm and there is no field in the tube. . . . . . . . . . . . . 44
3.7 TOF spectra of Xe at laser intensities of 4.4 (top), 7.1 (middle) and
12?1014 W/cm2 (bottom) calculated by Eq. 3.4. The ions are Xe+
(8.6?s), Xe2+ (6.3?s), Xe3+ (5.2?s), Xe4+ (4.6?s) and Xe5+ (4.1?s).
The sharp single peaks at 4.2 and 4.5?s are N+2 and O+2 from the
background. The signal of Xe3+ (Xe4+, Xe5+) can be barely seen in
the top (middle, bottom) plot. . . . . . . . . . . . . . . . . . . . . . 47
3.8 TOF waveforms resulting from CO2 Coulomb explosions obtained
with 100 fs pulses focused to 2?1015 W/cm2 when the polarization
axis is parallel and perpendicular to the TOF axis. The upper traces
are rescaled by a factor of two. Note, the C structure is different in
the parallel and perpendicular traces while some O components are
missing altogether from the perpendicular trace. Each trace is an
average of 1000 laser shots. . . . . . . . . . . . . . . . . . . . . . . . 49
xii
3.9 Surface plots of ion images resulting from NO2 Coulomb explosions
obtained with linearly polarized 100 fs pulses focused to 2 ? 1015
W/cm2 when the polarization axis is parallel to the image MCP. The
upper plot was taken with no gate on the MCP, all the molecular and
atomic ions were collected; the large peak at the center was formed by
near zero-energy ions, the top of which had to be truncated in order
to show other structures on the image. The lower plot was taken
while the MCP was gated; only doubly- and triply-charged atomic
ions were collected. Each plot is an average of 500,000 laser shots. . 50
3.10 Gated images of the dissociative-ionization spectrum of NO2 gener-
ated with 100 fs, 2?1015 W/cm2 pulses. The times under each image
indicate when the MCP gate opens and closes and are relative to the
arrival time of the laser pulse to the focal point. Images (1) ? (13)
were taken under the same conditions (gas pressure, laser intensity
and gate width), but have been re-scaled relative to (6) by the follow-
ing factors: (1) 1.7?, (2) 10?, (3) 4.3?, (4) 4.3?, (5) 3?, (6) 1?, (7)
6?, (8) 95?, (9) 10?, (10) 4.3?, (11) 1.5?, (12) 1.5? and (13) 19?.
Few ions were detected in image (8). Image (14) is a composite image
of all the doubly and triply charged atomic ions (i.e., N2+, O2+, N3+
and O3+, same as the lower plot in Fig. 3.9) and image (15) is the
composite image taken with no gate (same as the upper plot in Fig.
3.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1 The composite image of NO2 Coulomb explosions (lower right image)
showing the momentum distribution of Nq+ and Oq+ ions (75% q
= 2 and 25% q = 3). This distribution contains 500,000 camera
frames, i.e., 500,000 laser shots, each centered at 800 nm, with a
pulse width of 100 fs and linearly polarized (horizontal in the images)
with a peak intensity of 2 ? 1015 W/cm2. The vertical peak near
the center is composed of Nq+ ions while the distribution parallel to
the polarization axis is a mixture of both Nq+ and Oq+ ions. Four
samples of single frames (upper three and lower left) are displayed. . 58
4.2 Locations where charges ejected isotropically into 4? sr with the same
energy and a Gaussian energy spread will accumulate on a 2D image:
(a) an isotropic distribution of charges in the absence of the static
electric field form a sphere; (b) a simulated 2D image when all tra-
jectories are collected on the MCP; (c) a surface plot of a quarter
of the upper right image, the charge signal peaks at the edge of this
bowl-shaped distribution where land the trajectories initially ejected
parallel to the MCP (p? ? 0); (d) a corral-shaped distribution formed
when only the trajectories with p? ? 0 are collected. . . . . . . . . . 59
xiii
4.3 Correlation images. The concentric circles and spokes in the left panel
divide the momentum distribution image (the same as that displayed
in Fig. 4.1) into sectors. The gray arrow indicates labeled ions (i.e., a
subset of ions with a narrow momentum distribution moving down-
ward, 6 o?clock); the white arrows indicate the correlated sectors (ions
moving toward 2 and 10 o?clock). The right panel shows the corre-
lation image for the Coulomb explosion where all three atomic ions
are ejected simultaneously. The gray (white) arrows indicate the final
momenta of the labeled (correlated) charges. . . . . . . . . . . . . . 62
4.4 Two correlation images for the Coulomb explosion of CO2 taken under
the same conditions as in Fig. 4.1. Ions moving toward 3 o?clock in
the left image and those moving toward 6 o?clock in the right image
are labeled. As the correlated ions show up on the same correlation
images, linear explosion events on the left and bent events on the
right are displayed and isolated. . . . . . . . . . . . . . . . . . . . . 63
4.5 Two correlation images for NO2 taken from the same data set as in
Fig. 4.1 showing a two-body dissociation channel (left) and a three-
body channel (right). Ions moving to the left along the polarization
axis in the left image and those moving in the direction perpendicular
to the polarization axis in the right image are labeled so that two-
body and three-body events are isolated. . . . . . . . . . . . . . . . 64
4.6 Selective average. The image in the left panel and the gray and white
arrows are the same as in Fig. 4.3. The correlation image (selective
average) in the right panel is the difference between averaging only
those frames that have nonzero counts in the labeled sector and all
500,000 frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Selective average. The images in the upper row shows that a selec-
tive average is generated by taking the difference between an average
distribution composed only of frames that have nonzero counts in the
labeled sector (lower right image) and the average distribution of all
500,000 frames. The lower row shows two single frames with zero and
nonzero counts in the same labeled sector. . . . . . . . . . . . . . . . 66
xiv
4.8 Center-of-mass momentum distributions of ejected C2+ (center spots)
and O2+ (outer spots) ions associated with the symmetric CO6+2
Coulomb explosion channel, induced by linearly polarized (horizon-
tal), 100 fs pulses at 800 nm focused to 2?1015 W/cm2. (a) Average
distribution composed of 500,000 frames. (b) Selective average show-
ing only the momenta of the three charges ejected simultaneously
from a CO6+2 precursor with a bond angle of 163?. (c) A ball-and-
stick diagram of CO2 defining the molecular axis (dashed line) and
the geometric angle, ?g = 0?, which is the angle between the laser
polarization and the molecular axis (the line connecting two outer
ions). (d) The same as (c) for ?g = 10?. (e) Triple-coincidence image
(see text) corresponding to (c). (f) The same as (e) but correspond-
ing to (d). The arrows in (e) and (f) represent the final momenta
of the ejected ions (vectorpOleft, vectorpC, vectorpOright). The angle between vectorpOleft and
vectorpOright defines the center-of-mass angle ?CM. (g) Grid illustrating how
contributions to the signal strength are determined. Block 1 makes
a unit contribution while block 2 contributes only 0.6, corresponding
to the area enclosed by the circle. . . . . . . . . . . . . . . . . . . . 68
4.9 Images showing the anisotropies in the ejection angular distribution
of Coulomb explosions for selected bond angles: (1) CO6+2 at 180?,
(2) CO6+2 at 163?, (3) NO6+2 at 132? and (4) H+2 . These images were
constructed by adding a series of coincidence images similar to those
shown in Figs. 4.8e and 4.8f. The dashed lines represent the axes,
parallel and perpendicular to the laser polarization, which cross at
the center-of-mass. In (1) ? (3), the largest arcs correspond to O2+
ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1 Momentum distributions of the C2+ and O2+ ions associated with
the symmetric CO6+2 Coulomb explosions: average image composed
of 500,000 laser pulses (left); correlation images (selective averages,
Sec. 4.2) showing only the momenta of the three charges ejected si-
multaneously from a CO6+2 parent that is linear and parallel to the
polarization axis (center) and bent at a bond angle of ? 165? with
the plane of the molecule parallel to the MCP face (right). The ge-
ometric center of each image is the center of mass of the explosion
and the arrow indicates the polarization axis. The width of the MCP
gate (Sec. 3.2) was 320 ns, which covers all C2+ and O2+ ions and a
small amount of C3+ and O3+ ions (less than 20% of the total signal)
due to the overlap of signals in time. . . . . . . . . . . . . . . . . . . 75
xv
5.2 Momenta of the three charges ejected simultaneously as a function
of the bond angle (?b) in the symmetric CO6+2 Coulomb explosion
channel. Each data point represents the momentum measured from
the center-of-mass of the charge distribution in a correlation image
similar to the two right images in Fig. 5.1; the error bars represent
the standard deviation in this determination over four runs. The
upper series of points (green diamonds) correspond to |pX| for O2+;
|pX| is virtually the same for the two groups of ions so only one
point is plotted for each angle. The lower trace is |pY| for C2+ (blue
stars) and 2|pY| for O2+ (red triangles); pY is also the same for the
two groups of O2+ ions. The solid curves represent the calculated
momenta assuming RC equals 0.215 nm (see App. D for details of
this calculation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Bond length (left axis) and explosion energy (right axis) as functions
of bond angle for CO6+2 symmetric explosions with qO = qC = 2
measured by image labeling(Sec. 4.2). The error barsarethe standard
deviation of four experiments under the same conditions. . . . . . . 78
5.4 Explosion signal strength as a function of bond angle for CO6+2 sym-
metric explosions initiate with qO = qC = 2 measured by coincidence
imaging (Sec. 4.3) at the momenta of the three charges determined
by selective averages. The error bars are the standard deviation of
four experiments under the same conditions. . . . . . . . . . . . . . 79
5.5 Two- and three-body explosion channels of NO2. Left: average of
N2+ and O2+ ions over 500,000 laser shots; center: selective average
of two-body correlation channel where the bright spot to the right is
the selected sector; right: selective average of three-body correlation
channel where the bright spot in the middle is the selected sector.
Polarization axis is horizontal. The width of the MCP gate (Sec. 3.2)
was 330 ns, which covers all N2+ and O2+ ions and a small amount
of N3+ and O3+ ions (less than 20% of the total signal) due to the
overlap of signals in time. . . . . . . . . . . . . . . . . . . . . . . . . 81
5.6 Momenta of the three charges ejected simultaneously as a function
of the bond angle (?b) in the symmetric NO6+2 Coulomb explosion
channel determined by image labeling. The error bars represent the
standard deviation in this determination over seven runs. The upper
series of points (green diamonds) correspond to |pX| for O2+; |pX| is
virtually the same for the two groups of ions so only one point is plot-
ted for each angle. The lower trace is |pY| for N2+ (blue stars) and
2|pY| for O2+ (red triangles); pY is also the same for the two groups
of O2+ ions. The solid curves represent the calculated momenta as-
suming RC equals 0.26 nm (see App. D). . . . . . . . . . . . . . . . . 82
xvi
5.7 Bond length (left axis) and explosion energy (right axis) as functions
of bond angle for NO6+2 symmetric explosions initiate with qO = qC
= 2 measured by image labeling (Sec. 4.2). The error bars are the
standard deviation of seven experiments under the same conditions. 83
5.8 Contour plots of average images with both selected (middle group)
and correlated (groupson bothside) momentum vectors in three-body
correlation marked. Top and center: CO2 and NO2 images with the
locations (momenta) of correlated ions determined by selective aver-
age; bottom: NO2 images with the locations (momenta) of correlated
ions determined by joint variance. The green (red) circles mark the
selected C2+ (N2+) ions while the blue ones mark the correlated O2+
ions. The green and yellow arrows in the center plot indicate that
the selected N2+ at the bottom of the middle group is correlated with
O2+ at the top of both side groups. As the selected N2+ or C2+ is
moved up along the center group, the corresponding O2+ on each side
moves down along its arc. . . . . . . . . . . . . . . . . . . . . . . . . 85
5.9 Explosion signal strength as a function of bond angle for NO6+2 sym-
metric explosions initiate with qO = qN = 2 measured by joint vari-
ance (blue square), triple-coincidence (red triangle), at the selected
N2+ momentum and correlated O2+ momenta determined by joint
variance, along with double-coincidence (green star) at the O2+ mo-
menta. The error bars are the standard deviation of seven experi-
ments under the same conditions. . . . . . . . . . . . . . . . . . . . 87
5.10 Momenta of the three charges ejected simultaneously as a function
of the bond angle (?b) in the symmetric NO6+2 Coulomb explosion
channel determined by joint variance. The error bars represent the
standard deviation in this determination over seven runs. The upper
series of points (green diamonds) correspond to |pX| for O2+; |pX| is
virtually the same for the two groups of ions so only one point is plot-
ted for each angle. The lower trace is |pY| for N2+ (blue stars) and
2|pY| for O2+ (red triangles); pY is also the same for the two groups
of O2+ ions. The solid curves represent the calculated momenta as-
suming RC equals 0.26 nm (see App. D). . . . . . . . . . . . . . . . . 89
5.11 Bond length and explosion energy as functions of bond angle for NO6+2
symmetric explosions initiate with qO = qN = 2 measured by joint
variance. The error bars are the standard deviation of seven experi-
ments under the same conditions. . . . . . . . . . . . . . . . . . . . 90
5.12 Coordinate system employed in triatomic 2-D field ionization calcu-
lation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
xvii
5.13 Two over-the-barrier ionization pathways for a triatomic ion at ?b =
134?: saddle-point ionization (upper left) and quasi-diatomic, parallel
to the polarization axis and across the O?O barrier (upper right).
Arrows in the upper plots indicate the electron pathways. The lower
plots are the corresponding potential curves along and centered at the
O?Oaxis. The contours in theupper plots and the short line segments
in the lower plots indicate the electron energy levels in three potential
wells. The right plots correspond to 5% reduction in RC relative to
the left but at a cost of a 10% increase in ionization intensity. Other
parameters in the calculation are ? = 1.0, s = 0.01, q1 = q2 = q3 =
2, and Ip = 35.1 eV which is the ionization potential for O+. . . . . 95
5.14 Calculated RC (upper plots) and Iia (lower plots) as functions of ?b
(Eqs. 5.2 and 5.3) with ? = 0 (violet, dashed), 0.25 (green, solid), 0.5
(orange, short-dashed), 0.75 (blue, dot-dash), 1.0 (red, long-dashed)
for the quasi-diatomic (left panels) and the saddle-point (right panels)
pathways. Other parameters in the calculation are s = 0.01 nm, q1
= q2 = q3 = 2, and Ip = 35.1 eV which is the ionization potential for
O+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.15 Calculated RC (upper plots) and Iia (lower plots) as functions of ?b
(Eqs. 5.2 and 5.3) with s = 0 (violet, dashed), 0.01 (green, solid),
0.02 (orange, short-dashed), 0.03 (blue, dot-long-dash), 0.04 (red,
long-dashed), 0.05 (light blue, dot-dash) for the quasi-diatomic (left
panels) and the saddle-point (right panels) pathways. Other param-
eters in the calculation are ? = 0.5, q1 = q2 = q3 = 2, and Ip = 35.1
eV which is the ionization potential for O+. . . . . . . . . . . . . . . 98
5.16 Extreme value of the smoothing parameter. The potential structure
at the upper limit of s, 0.08 nm for Ip = 35.1 eV, R = 0.26 nm,
and ? = 1.0 at an intensity of 3.3 ? 1014 W/cm2. The short line
segments indicate the energy level of the bound electron in each well.
In this example, the bottom of the left potential well coincides with
the energy level of the electron. . . . . . . . . . . . . . . . . . . . . . 99
5.17 Experimental values of RC(?b) for CO6+2 (blue triangles) and NO6+2
(red squares) explosions taken from Figs. 5.3 and 5.11. The RC-model
curves with Ip = 35.146 eV, q = 2, and ? = 0.8 for both saddle-point
and quasi-diatomic pathways are represented by the short-dashed and
dashed curves respectively. For saddle-point ionization s = 0.074 nm
(0.055 nm) for CO2 (NO2) while for quasi-diatomic ionization s =
0.074 (0.046) nm for CO2 (NO2). . . . . . . . . . . . . . . . . . . . . 101
xviii
5.18 Experimental values of RC(?b) for CO6+2 (blue triangles) and NO6+2
(red squares) explosions taken from Figs. 5.3 and 5.11 and the saddle-
point RC ? ?(?b)-model curves with Ip = 35.146 eV, q = 2, and
? = a + bsin(?b/2) with b = 2.489 and a = -1.639 (-1.539) for CO2
(NO2). The smoothing parameter is s = 0.074 nm (0.055 nm) for
CO2 (NO2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.19 A comparison between the linear explosion energies for COZ+2 (Z =
3?9) calculated by the dynamic screening model of Ref. [82] (blue
triangles) and by qeff (red squares). The experimental value for RC
(= 0.215) nm measured for the 6-electron channel along with Re =
0.116 nm were used to determine ? and qeff (Eqs. 5.6 and 5.7). . . . 106
5.20 Energy deficit (ED) as a function of ?b calculated by SS-model (Qel
= 1). Top: x0 = 0.57, 1.27, 1.98, 2.69, 3.39 nm with y0 = 0.014 nm
and s = 0.02 nm; middle: y0 = 0.007, 0.014, 0.021, 0.042, 0.085 nm
with x0 = 1.27 nm and s = 0.02 nm; bottom: s = 0.005, 0.01, 0.02,
0.03, 0.05, 0.1 nm with x0 = 1.27 nm and y0 = 0.021 nm. . . . . . . 109
5.21 Experimental values of ?(?b) for CO6+2 (blue triangles) and NO6+2 (red
squares) explosions derived from Figs. 5.3 and 5.11 by Eq. 5.7. The
SS-model ?(?b) curves with Qel = 1.85, x0 = 1.27 nm, y0 = 0.021 nm
(0.014 nm), and s = 0.03 nm (0.01 nm) for CO2 (NO2) are represented
by the solid curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.22 Experimental values of Ek(?b) for CO6+2 (blue triangles) and NO6+2
(red squares) explosions taken from Figs. 5.3 and 5.11. The RC??(?b)
model curves (dashed) for CO2 and NO2 are derived from Fig. 5.18
by Eq. D.8 with R = RC. The SS-model curves (solid) are derived
from Fig. 5.21 by Eq. 5.6. The model parameters are the same as
those in the figure captions. . . . . . . . . . . . . . . . . . . . . . . . 113
5.23 A comparison between the relative probabilities for Coulomb explo-
sion as a function of ?g: (upper) the symmetric CO6+2 explosion chan-
nel at 180?, (middle) the symmetric NO6+2 channel at its equilibrium
bond angle (134?), and (lower) H2. The error bars correspond to the
standard deviation for four different runs. The solid curves are cosn ?g
fits where n = 39, 25 and 19, respectively for CO2, NO2 and H2. The
FWHM, ??g, of the distributions are respectively 22?, 27? and 31?. . 119
5.24 The widths (??g) of the orientation distributions vary as a function
of bond angle for the symmetric CO6+2 (upper) and NO6+2 (lower)
Coulomb explosion channels. The data are extracted from distribu-
tions similar to those shown in Fig. 5.23. The error bars are the
standard deviation over four different runs. The solid lines are the
equilibrium ??g values (22? for CO2 and 27? for NO2). . . . . . . . . 120
xix
5.25 Coordinate systems in the focal and detector planes, separated by a
distance l, with the x and y axes are parallel to the X and Y axes.
The laser field (vectorE) is polarized in the xz-plane at an angle ? and the
molecular axis ( vectorM) is oriented at an angle ? relative to vectorE. . . . . . . 123
5.26 Momentum distributions of Coulomb explosions obtained with a lin-
ear polarization at I0 = 2?1015 W/cm2 and ? = 0? (top), circular
polarization at 2I0 (middle) and average image composed of 18 linear
polarizations at I0 with ? ranging from 0? to 170? in steps of 10? (bot-
tom). Beneath each image is its relative ion yield normalized to the
second row in each column. The images in the top and middle rows
are composed of 200,000 laser shots and the bottom 18 ? 200,000
shots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.27 Ion yields, R vs neff for H2 (green triangle, 0.92), N2 (blue star,
0.35), O2 (red circle, 0.35) and CO2 (magenta square, 0.38). The
upper (lower) solid curve is R = 1 (R = RG). Values for ? (%) are
also given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B.1 Coordinate systems used for calculating the ion trajectory on the
image side. The laser is focused at ?o? and the wave vector vectork is in
the x-direction. The center of mass of the dynamics is ?O?, which is
the center of the image. The initial velocity of the ion is vectorv0, which is
pushed toward the detector by the static field vectorE. The field strength
is E = Vint/l (Fig. 3.6). The detector, XY?plane, is parallel to
xz?plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
B.2 Momenta of protons ejected from H2 under the influence of a linearly
polarized laser field at 1.5 ? 1015 W/cm2 as functions of the static
field in the spectrometer (V4, see Fig. 3.6). Since the image calibra-
tion, charge qe, mass m and distance l (Fig. B.1) are the same, the
momentum p is expressed by Rpeak??V4 (Eq. B.8), where Rpeak is the
radius of the proton peak in the unit of pixel. The data points were
taken with V4 varying from 500 to 2500 volts with 500 volts at each
step. The dashed lines are the average of the five points in each plot.
The error bars correspond to ?1 pixel which is the uncertainty of the
determination of the peak positions. The upper plot is for the inner
peak from the low-energy dissociation, while the lower one for the
outer peak from Coulomb explosion. The insets (blue rectangles) are
the image taken at V4 = 2000 volts to show where Rpeak was measured. 148
xx
B.3 Structure and coordinate system of TOF spectrometer. The laser is
focused at ?o? and propagates in or out of the paper. The x?axis
is parallel to the external fields vectorE1 and vectorE3 and the flight tube, while
z-axis is parallel to the detector, which is a 50-mm-diameter MCP
with 40-mm active area. The initial velocity of the ion is vectorv0. The
distances and potentials are, according to Fig. 3.6, l1 = 19.05 mm, l2
= 140 mm, l3 = 30 mm, and Va1 = V4/4 = 500 volts, Va3 = 300 volts. 149
C.1 Flowchart of the image compression process. Flag and Shift are used
to compute the shift which will be placed in the marker. Counter is
used to count the number of nonzero pixels in this frame. . . . . . . 153
C.2 Flowchart of the process of image recovery fromthe compressed image
file. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
C.3 Comparison of two images before (left) and after (right) the back-
ground elimination. Both images are composed by 10,000 single
frames, each of which corresponds to one laser pulse. . . . . . . . . . 154
C.4 Flowchart of the procedure to determine the centroid of a cluster on
an image. The adjacent pixels p1(i1,j1) and p2(i2,j2) are determined
when |i1 ?i2| ? 1 and |j1 ?j2| ? 1. In the special case of |i1 ?i2|
= |j1 ?j2| = 1, they are called diagonally adjacent, as pixels 1 and 2
shown in Fig. 4.8g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
C.5 A sample of the results of the cluster procedure. The upper left
image and its magnified part (upper right) is one of the 500,000 single
frames of H2 Coulomb explosions taken with linearly polarized pulses
with laser intensity of 2 ? 1015 W/cm2. The lower images are the
same image after being clustered and show only the centroids of the
clusters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
C.6 Frequency (Fx) as a function of the number of pixels in each cluster.
Laser pulses are linearly polarized with an intensity of 2?1015 W/cm2.
Upper: 500,000 frames of H2 Coulomb explosions with measured ? =
1.66, ? = 1.03, and Ntot = 2.46?106; middle: 500,000 frames of CO2
Coulomb explosions with measured ? = 4.26, ? = 3.01, and Ntot =
1.45?107; lower: 500,000 frames of NO2 Coulomb explosions with
measured ? = 2.86, ? = 1.74, and Ntot = 1.67?106. . . . . . . . . . . 161
xxi
C.7 Frequency as a function of the number of pixels in each cluster. The
data are the same as in Fig. C.6. In order to be compared to the
fitting curves, the data were plotted as stars instead of histograms.
The solid curves are Poisson distributions, g(x) = g0?xx! e??, where x
is the number of pixels in a cluster. The fit parameters are, upper
(H2): ? = 0.86 and g0 = 4.0?106; middle (CO2): ? = 3.0 and g0 =
1.2?107; lower (NO2): ? = 2.3 and g0 = 1.7?106. . . . . . . . . . . 163
C.8 Angular distributions for CO2 Coulomb explosions (CO6+2 ? O2+ +
C2+ + O2+) measured by coincidence imaging (Sec. 4.3) with different
M at the bond angle of 180?. The values of M are 7, 9, 11 fromthe top
to the bottom, respectively. The solid curves are the fitting functions
cosn ?g with the corresponding ??g (full width at half maximum)
labeled in the plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
C.9 Frequency as a function of the number of clusters in each frame. Laser
pulses are linearly polarized with an intensity of 2?1015 W/cm2. The
histograms are the measured frequencies as a function of the number
of clusters in each frame. The measured ? and ? in 500,000 frames
are, upper (H2): ? = 4.92 and ? = 2.45; middle (CO2): ? = 28.94
and ? = 6.56; lower (NO2): ? = 3.34 and ? = 2.44. . . . . . . . . . . 166
C.10 Frequency as a function of the number of clusters in each frame. The
data are the same as in Fig. C.9. Again, in order to be compared to
the fitting curves, the data were plotted as stars. The solid curves
are Poisson distributions, g(x) = g0?xx! e??, where x is the number of
clusters in a frame. The fit parameters are, upper (H2): ? = 5.81 and
g0 = 4.99?105; middle (CO2): ? = 29.72 and g0 = 4.47?105; lower
(NO2): ? = 3.84 and g0 = 4.62?105. . . . . . . . . . . . . . . . . . . 167
D.1 Three-body Coulomb explosion. The center of mass of the system is
located at the origin of the coordinate system. The x-axis is horizon-
tal and parallel to the polarization axis. Without external force, the
center of mass will not move. At t = 0, assuming symmetric geome-
try, |vectorr1 ?vectorr3| = |vectorr1 ?vectorr3| = R is the bond length. The angle between
vectorr1 ?vectorr3 and vectorr2 ?vectorr3 is the bond angle ?b. The center-of-mass angle
?CM is defined as the asymptotic value of the angle between vectorp1 and
vectorp2 as t ??. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
D.2 Simulated functions ?CM(?b) and ?b(?CM) for CO6+2 symmetric explo-
sions with R = 0.22 nm and qO = qC = 2, assuming pure Coulomb
interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
xxii
D.3 Simulated functions ?CM(?b) for symmetric explosions of four 6-fold
charged triatomic molecular ions (XY6+2 ) with R = 0.22 nm and qO =
qC = 2, as well as H2O3+ with qO = qC = 1, assuming pure Coulomb
interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
D.4 Bond lengths and bond angles where CO6+2 symmetric explosions ini-
tiate with qO = qC = 2 measured by image labelling (Sec. 4.2) at
laser intensity of 2 ? 1015 W/cm2. Bond angles ?b are determined
from measured ?CM using Fig. D.2. The error bars are the standard
deviation of four experiments under the same conditions. . . . . . . 173
D.5 Calculated bond angle as a function of bond length as ?CM is fixed
at 170?, 145? and 130?. The calculation is carried out with CO6+2
symmetric explosions and qO = qC = 2. . . . . . . . . . . . . . . . . 174
D.6 Calculated variation ??b ? ?b(R = 0.1,?CM)??b(R = 0.3,?CM) as a
function of ?CM for CO6+2 ? O2+ + C2+ + O2+ explosion with ?CM
between 125? and 180?. The maximum ??b, 0.08? or 0.05%, appears
near ?CM = 145?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
E.1 Calculated histogram of the appearance frequency of the electron at
a position (x) between ?2.7 nm as the electron is driven by the laser
field according to Eq. E.2 between t1 and t2. The initial time t1
relative to the peak of the pulse and the intensity at t1 are (a) ?130
fs and 2.2?1013 W/cm2; (b) ?110 fs and 4.8?1013 W/cm2; (c) ?90
fs and 1.0?1014 W/cm2; and (d) ?70 fs and 2.2?1014 W/cm2. The
final time t2 = 180 fs, which is roughly the end of the pulse. The
total probability in each plot is normalized to unity. . . . . . . . . . 178
E.2 Calculated histogram of the appearance frequency of the electron at
a position (x) between ?2.7 nm as the electron is driven by the laser
field according to Eq. E.2 between t1 and t2. The initial time t1
relative to the peak of the pulse and the intensity at t1 are (a) ?130
fs and 2.2 ? 1013 W/cm2; (b) ?110 fs and 4.8 ? 1013 W/cm2; (c)
?90 fs and 1.0?1014 W/cm2; and (d) ?70 fs and 2.2?1014 W/cm2.
The final time t2 = t1 + 180 fs. The total probability in each plot is
normalized to unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
E.3 Calculated final kinetic energy (Blue diamonds, left axis) by the
quasi-dynamic simulation as a function of the initial time t1 when
the electrons are born, and the laser intensity (Red stars, right axis)
at these times. The initial time is relative to the peak of the pulse. . 181
xxiii
Chapter 1
INTRODUCTION
1.1 Motivation and Goal
For decades, physicists, chemists and laser engineers have been fascinated by
the possibility of controlling molecular dynamics and chemical reactions with lasers.
This interest is driven in part by the desire to produce unusual products or to en-
hance reaction rates that are not easily obtained by statistical means. Schemes for
laser-based control have been the subject of investigation both theoretically and
experimentally (see, for example, Refs. [1] ? [8]). Control approaches have been
applied to a variety of different molecular processes ? laser-induced dissociation [3],
bond rearrangement [5], and dissociative ionization [6] are a few examples. Many
control experiments involve intense and/or ?shaped?1 laser pulses. Most often these
shaped pulses are very complicated and the affect they have on the molecular system
is not clear. Consequently, there is a need to develop ways to probe the molecular
system during excitation with these intense and exotic pulses. The research pre-
sented in this thesis is part of an effort to develop ways to probe molecular systems
during and after excitation that will reveal the physics and dynamics induced by
1The temporal and/or spectral profile (or ?shape?) of a laser pulse can be adjusted or changed
by certain devices such as a liquid-crystal or acousto-optic modulator, which can delay the phase
and/or reduce the amplitude of certain frequency components of the pulse, or eliminate some
components completely. The adjusted pulse is usually referred to as a ?shaped? pulse and the
process as pulse shaping.
1
such control pulses.
The contribution this thesis makes to this effort is two-fold: (1) the develop-
ment of image-based correlation techniques to capture snap-shots of the molecular
structure from which bond angles and bond lengths at the time the system falls
apart can be extracted and (2) the application of the developed techniques to in-
vestigate bond-angle dependence, anisotropy and alignment issues associated with
intense-field dissociative-ionization of 2- and 3-atom systems. The detection tools
developed and the knowledge gained through the experiments presented here will
aid future experiments probing more complicated aspects of the dynamics. Ulti-
mately, this approach can be applied to investigate dynamics in larger systems with
transform-limited2 and/or shaped laser pulses.
In an attempt to make direct measurements of the molecular structure, sci-
entists launched the field of Coulomb explosion imaging with fast molecular beams
(for example, see Refs. [9, 10, 11]). In these experiments, a beam of molecular
ions was accelerated through thin foils. When the molecular ions pass through the
foil, several electrons are stripped away on a sub-femtosecond time scale before the
constituent atoms can move very far. The resulting highly charged molecular ions
explode into atomic ions due to their mutual Coulomb repulsion. Since the time
scale of the beam-foil interaction is on the order of femtoseconds or even shorter,
the internal motion of the molecular ion is essentially frozen when the explosion
is initiated. Thus, the trajectories of the fragments contain the information about
2The minimum value of the time-bandwidth product of a pulse is about ?f ?? greaterorsimilar 0.5, where
?f is the frequency spread of the optical spectrum and ? is the pulse width [12]. The transform
limit is the minimum pulse width that is possible for a given spectrum. A pulse at this limit is
called transform-limited.
2
the state of the parent molecular ion prior to the explosion. In beam-foil experi-
ments, two-dimensional area detectors were used to capture the fragments providing
graphical views of the structure. However, since the molecular ions are moving in the
laboratory frame, the image must be transformed to retrieve views of the structure
in the molecular frame.
Inspired by the beam-foil experiments, Coulomb explosion experiments were
begun with intense sub-picosecond lasers [13, 14]. In the laser experiments, time-
of-flight (TOF) detectors were used initially to capture the fragments. In contrast
to the beam-foil studies, which is no longer an active field, investigations based on
laser-induced Coulomb explosions continue to be a thriving area of research both
experimentally [15] ? [69] and theoretically [70] ? [97]. This is fostered by the
flexibility laser-based experiments afford. In particular, the laser can participate
in the dynamics by exciting the precursor molecule to specific states prior to the
explosion. The disadvantage of laser-based experiments is the pulse duration, 10 ?
100 fs. At the longer times, the molecules have time to evolve during the irradiation.
While the laser-molecule interaction leads to many complications, it provides a knob
for controlling the dynamics. However, in order for control to be fully realized, a
better understanding of the laser-molecule interaction is critical.
In light of this brief overview, the two primary goals of this study are (1)
to develop quantitative detection techniques to probe laser-induced dynamics of
small molecules associated with Coulomb explosions and (2) to investigate bond-
angle-specific explosion, ejection anisotropy and dynamic alignment. With these
techniques, the experiments were performed with transform-limited pulses that had
3
widths of 100 fs (full width at half maximum, FWHM) and focused intensities of
0.1 ? 5.0?1015W/cm2. The results provide a foundation upon which future studies
involving more exotic pulses can be built.
1.2 Research Background and Contributions of This Study
Dissociative ionization of molecules, induced by short and intense laser pulses
(pulse widths ? = 50 fs ? tens of picoseconds and intensities I ? 1014 W/cm2), has
been under intensive investigation [14] ? [97] for more than fifteen years. Both ex-
perimental and theoretical investigations started with diatomics, and many different
molecules have been studied. Examples include H2 [15] ? [22], D2 [15, 19, 23], HI
[14], N2 [13, 24] ? [39], O2 [28, 32, 40] ? [42], CO [27, 28, 31, 43] ? [45], NO [46, 47],
Cl2 [37, 41, 42, 48], Br2 [41, 42], I2 [37, 41, 42, 49] ? [55]. Triatomic molecules have
also been investigated, such as CO2 [32, 36, 56] ? [63], N2O [32, 64], CS2 [65], OCS
[66], H2O [32, 67], NO2 [63, 68] and SO2 [33, 69]. Early experiments used TOF ion
mass spectroscopy to collect and analyze the atomic ions following Coulomb explo-
sions, as we have mentioned. Two-dimensional position-sensitive detectors, which
were first introduced in this lab [22], have become the primary workhorse for con-
temporary experiments because they are capable of detecting fragments ejected into
different angles simultaneously, allowing the angular distributions to be extracted
[23, 54, 62, 63, 65].
When this research started, nearly all previous analyses were based on TOF
spectra or pseudo-images composed of TOF spectra obtained by rotating the po-
4
larization axis of the laser relative to the TOF axis. The distinguishing mark of
a Coulomb explosion, however, is the molecular decay into atomic ions. Since the
ions ejected from a system with three or more atoms are not along a straight line,
in general, the explosion partners could not be collected simultaneously in previous
studies, requiring numerical simulations and momentum (and energy) conservation
to be used to identify explosion partners. Unfortunately, this identification is not
necessarily unique, even for diatomic explosions. Covariance mapping was intro-
duced by Frasinski et al. [98] to identify partners in TOF waveforms uniquely using
a covariance between different charges obtained from a large number of laser shots.
However, when used on TOF waveforms, it is difficult to extract quantitative infor-
mation for bent explosion channels. At the same time, TOF imaging and covariance
mapping are incompatible and have to be done in separate experiments. Thus, the
images acquired from TOF waveforms cannot be used to determine specific bond
angles. Consequently, there was a need for a new approach to study larger systems.
To that end, laser-induced Coulomb explosion of 3-atom systems was re-examined
with the techniques developed in this work, which are more sensitive to molecular
geometry and orientation with the laser polarization axis.
The work in this thesis employed momentum imaging based on the image spec-
trometer capable of collecting all explosion partners ejected into 4? sr introduced
by Zhu and Hill [22], which was the first study where a two-dimensional area de-
tector was employed to investigate laser-induced Coulomb explosions as well as the
first study to extend the correlation technique to two-dimensional images (a two-
dimensional analogue of covariance mapping). The 4? image spectrometer coupled
5
to a high frame-rate digital camera affords high event rates in the detection [99, 100];
several hundred ions per laser pulse can be handled in principle, significantly higher
than other imaging techniques, such as COLTRIMS (cold-target recoil-ion momen-
tum spectroscopy, developed at nearly the same time as our approach) with delay-
line anodes [23, 65, 101, 102]. The high event-rate of our approach ensures that
a large number of explosion events can be collected in a reasonable time so that
the statistical-based correlation approach can be applied to isolate ions arriving at
specific times or having specific momenta. The following techniques were developed
in this work, image labeling [99], coincidence imaging [63], and joint variance [103].
The first important observation [13] in laser-induced dissociative-ionization
(Coulomb explosion) experiments is with relatively long pulses that the total final
kinetic energy (also referred to as kinetic energy release) of the atomic fragment
ions is much less than the Coulomb potential energy determined at the equilibrium
bond length (Re) of the molecule assuming pure Coulomb repulsion. The energy
deficit is about 50% for molecules composed of light atoms (H, C, N, O, etc.),
and decreases to about 30% for molecules with heavier atoms (Cl, I, for example)
[70]. The universality of the deficit, which is a slow function of charge states,
suggests residual bonding is probably not the primary cause. The deficit is consistent
with two possibilities: enhanced ionization [14, 53, 70] ? [78] and electron screening
[79] ? [82]. The enhanced-ionization model assumes that the positively charged
molecular ion explodes at a critical internuclear separation (RC) where the molecular
ionization rate peaks. Consequently, it is far easier to reach an ionization stage
that supports the Coulomb explosion at RC. The electron-screening model assumes
6
ionization starts at Re but the positive charges are screened by the photo-electrons
that have not left the vicinity of their parent molecule. The energy deficit results
from the positive ions moving through the cloud of the photo-electrons causing them
to decelerate, so that the final kinetic energy is smaller than the original Coulomb
potential energy of the ions at Re in the absence of the electrons. Details of these
models will be discussed in the next chapter.
A second observation is that strong-field dissociative ionization induces large-
amplitude bending motion and structural deformation in triatomic systems. Large-
amplitude bending about the equilibrium bond angle has been observed in triatomic
systems, such as CO2 [56, 58, 60, 61], SO2 [33, 69] and OCS [66]. Bent molecules,
NO2 [68] and H2O [67], were found to be deformed toward the linear configuration
while linear CS2 [65] was reported to be distorted toward a bent configuration.
Furthermore, the evidence also suggested that the explosion energy can depend
on the bond angle [65]. The dynamics associated with triatomics in strong fields
has also been studied theoretically with the time-dependent Schr?odinger equation,
field-induced adiabatic states and the potential surfaces [83, 84, 85]. For CO2, these
studies showed that the laser-deformed CO2+2 potential surface favors the symmetric
bond stretching that triggers the large-amplitude bending motion of CO2+2 .
To exam laser-induced affects in 3-atom systems with 2D imaging, we focused
our studies on two systems: CO2 [62] and NO2. The purpose was to compare linear
and bent systems and their responses to light. We used the correlation techniques
developed to isolate specific precursor molecular geometries (bond angles prior to ex-
plosion). The experimental results contain several interesting features. For example,
7
the explosion energy for NO2 decreases monotonically by ?25% from the smallest to
the largest bond angle. By contrast, the CO2 explosion energy is nearly independent
of bond angle. To simulate the explosion, we extended the one-dimensional 2-atom
enhanced ionization model [70, 73, 74] into two-dimension for three charge centers
and developed a single-parameter static-screening model based on the concept of
electron-screening effect [79] ? [82]. In the enhanced-ionization model, the bond-
angle dependence of the explosion energy is primarily due to the variation of the
energy levels of the electronic states at three charge centers. In the static-screening
model, the variation of the explosion energy as a function of the bond angle is related
to the density and shape of the photo-electron distribution around the exploding
molecular ion. The predictions of both the enhanced-ionization and static-screening
models are consistent with the measured explosion energies for CO2 and NO2. At
the same time, we observed large-amplitude vibrations in explosion signals of both
CO2 and NO2.
The third important observation in laser-induced dissociative-ionization ex-
periments is that the angular distribution of the ejected atomic ions is anisotropic.
For diatomics, the ions are ejected along the polarization axis of the laser primarily
[13]. Atomic ions ejected from triatomics have a large propensity for coupling to the
polarization axis as well [58]. Specifically, the outer ions are ejected primarily along
the polarization axis while the central ion tends to be ejected in a direction that is
perpendicular to the polarization axis.
The ejection anisotropy has been the subject of constant debate for nearly
two decades [13, 36, 37, 39, 41, 42, 61, 63, 69, 86]. Two effects contribute to the
8
anisotropy, the so-called geometric [13, 86] and dynamic alignment [45, 87]. Geomet-
ric alignment relies on the induced dipole moments being larger along the molecule
than opposite. Thus, it is easier to ionize the molecule when its molecular axis is
parallel to the laser polarization axis. The molecule that explodes are aligned with
the polarization axis producing the anisotropy. Dynamic alignment refers to the
field exerting a torque on the molecular axis, inducing the molecule to rotate itself
into alignment with the field prior to the Coulomb explosion. When active, more
molecules participate in dissociative ionization than would in its absence.
Geometric and dynamic alignment are commingled in typical Coulomb explo-
sion experiments performed with linearly polarized light, making them difficult to
be distinguished. Several authors have pointed out that enhanced ionization will
ensue without dynamic alignment in circularly polarized fields since the polariza-
tion axis rotates too quickly (a period of T = ?/c = 2.67 fs, where c is the speed
of light and ? = 800 nm is the laser wavelength) for the molecule to respond to
the instantaneous torque [36, 39, 41, 42, 61]. Thus far, previous studies have pro-
vided tests for determining when dynamic alignment is active. With only a few
exceptions, however, the bulk of the work has focused on diatomics. The correla-
tion detection techniques we developed, with an ability to isolate orientations of the
molecular axis (the line connecting the two outer atoms in the system) relative to
polarization axis in triatomics, provides a unique opportunity to study alignment
issues for more complex explosions. Comparing explosion images induced by linearly
and circularly polarized fields not only allows the two effects to be distinguished, it
provides a quantitative measure of the contribution that dynamic alignment makes
9
to the anisotropy. We studied ejection anisotropy in the images of triatomic (CO2
and NO2) [63] and diatomic (H2, N2, and O2) systems. Of these systems, the ions
ejected from CO2 and NO2 were observed to have narrower distributions than N2,
O2 and H2. The narrower distributions can be explained in terms of the ionization
stage the precursor molecule reaches before the explosion, the induced dipole mo-
ment and moment of inertia of the system. The degree of dynamic alignment of H2
is the highest among the samples, due to its small moment of inertia. Surprisingly,
however, the degree of dynamic alignment for CO2 was found to be larger than those
for N2 and O2, which have smaller moments of inertia. We explained this in terms
of the length of time the precursor molecular ion spends in the field prior to the
Coulomb explosion.
This manuscript is organized as the following. Chapter 2 reviews the existing
theories and models of strong-field molecular dynamics. The apparatus including
the laser system, the image spectrometer, and the digital camera are described
in Chapter 3. The correlation detection techniques are introduced in Chapter 4.
Chapter 5 presents the experimental results and the analysis with the theoretical
models. Chapter 6 gives the summary and possible future work related to this study.
All the formulas in this thesis are in MKS units, except where explicitly specified
otherwise.
10
Chapter 2
BASIC CONCEPTS IN STRONG-FIELD MOLECULAR
DYNAMICS
This chapter provides a review of two models that have been suggested to
explain the energy deficit observed during Coulomb explosions with laser pulses
longer than 50 fs (Sec. 2.1). The mechanism behind the anisotropy of the ejected ions
observed in laser-induced Coulomb explosions of diatomic and triatomic molecules
(Sec. 2.2) is also summarized.
2.1 Kinetic Energy Release
At laser intensities higher than 1014 W/cm2, the strength of the average electric
field is more than 2.7?108 V/cm, which is a significant fraction of the internal field
of the valence electrons in atoms and molecules (on the order of 109 V/cm) leading
to ionization. Non-resonant ionization of atoms and molecules in intense laser fields
can be divided into two regimes by the Keldysh parameter ? [104],
? =
radicalBigg
Ip
2Up, Up =
e2E20
4me?2, E0 =
radicalbigg2I
0
?0c, (2.1)
where Ip is the atomic or molecular ionization potential; Up is the ponderomotive
potential, a measure of the quiver energy of a free electron in an oscillating field; e,
me, E0, ?, I0, ?0 and c are the elementary charge, the electron mass, the peak field
strength, the angular frequency, the peak intensity of the laser, the permittivity of
11
free space, and the speed of light, respectively. The Keldysh parameter represents
the ratio between the laser and electron-tunneling frequencies. For ? ? 1, the
laser frequency is much higher than the tunneling frequency and the ionization is
a multiphoton process. The atom or molecule is ionized as several photons are
absorbed simultaneously. This occurs for short-wavelength laser fields. For ? ? 1,
the high-intensity and long-wavelength regime, the electron can gain enough energy
(Up > Ip) to tunnel out of the potential barrier within one laser cycle so that
tunneling ionization dominates and the ionization rates are less dependent on the
laser frequency and the internal structure of the atom or molecule. Dissociative
ionization due to intense laser fields in the visible and infrared range has a Keldysh
parameter ? ? 1, and the multiphoton mechanism no longer applies. At the same
time, the nuclear motion makes molecular ionization more complicated than the
atomic case. The ionization rate is affected by the nuclear separation in a molecule
and dissociation and ionization are interlaced, making the theoretical predictions
more difficult. Thus, models of molecular ionization where ? ? 1 must consider
multiple charge centers and explain the dynamics of both electrons and nuclei in
laser-induced dissociative ionization. Under such circumstances, two main streams
of theoretical work have been suggested to explain the energy deficit of the Coulomb
explosion in the experiments.
The first model (Sec. 2.1.1) is based on enhanced ionization at a critical inter-
nuclear separation (RC) [14, 53, 70] ? [78]. It assumes that the positively charged
molecular ion explodes at RC. The value that RC assumes is dictated by the fact
that laser-induced ionization of the molecule to the critical ionization stage (the
12
stage where enough electrons have been removed so that the system explodes) is far
easier at RC than at Re. The final kinetic energy is determined by the relatively
small Coulomb potential energy of the fragment ions at RC. The second model (Sec.
2.1.2) assumes ionization starts at Re but the positive charges are screened by the
photo-electrons that have not left the vicinity. This electron-screening effect [79] ?
[82] has been calculated in the Thomas-Fermi description of the electrons [88, 105],
which treats the electrons as a continuous distribution. The energy deficit results
from the positive ions moving through the cloud of the photo-electrons causing them
to decelerate, so that the final kinetic energy is smaller than the original Coulomb
potential energy of the ions at Re in the absence of the electrons.
2.1.1 Enhanced Ionization and Charge Resonance
A classical over-the-barrier ionization model was first applied to calculate the
appearance intensities1 of different charge states of atoms in intense laser fields [106].
As this calculation was extended to diatomic molecules, enhanced ionization at RC
was discovered [53, 70, 73, 77]. This phenomenon was also found in the quantum
treatment of molecular dissociative-ionization [73] ? [76, 78], and interpreted by the
dynamics of the charge-resonance (CR) states, which are the two lowest eigenstates
of the electron in a symmetric double-well Coulomb potential [107, 108]. The RC
predicted by classical and quantum calculations is comparable and believed to be
the origin of the energy deficit observed in the Coulomb explosion experiments.
1Appearance intensity of a charge state is usually understood as the minimum laser intensity
at which this charge state is produced.
13
In atomic cases [106], a single valence electron is placed in the Coulomb po-
tential of the ion core and a static field consistent with the peak laser intensity. The
electron energy level is determined by the ionization potential of the atom or ion.
As the strength of the laser field increases, the Coulomb potential barrier is lowered.
Ionization is assumed to occur when the height of the barrier is equal to the electron
energy level (over-the-barrier ionization, Fig. 2.1a). Assuming the laser polarization
is along the x-axis and the origin is at the nucleus, the combined potential in one
dimension and the electron energy level can be expressed as,
Vatom = ?Kqe|x| ?eE0x, ?atom = ?Ip, (2.2)
where qe is the charge of the ion, E0 is the peak field strength of the laser, Ip is the
field-free ionization potential, and K = 1/(4??0) is the constant of proportionality in
Coulomb?s law in MKS units ? Coulomb?s constant. The potential barrier is found at
x0 where ?Vatom/?x = 0. The intensity required for over-the-barrier ionization can
be determined by equating the height of the potential barrier Vatom(x0) to the elec-
tron energy level, ?Ip. The appearance intensities predicted in such a simple model
agreed very well with the experimental results of noble gases [106] (Fig. 2.2). Since
tunneling ionization begins before the electron energy level reaches the potential
barrier, the appearance intensity and over-the-barrier ionization described in this
simplified model are actually the saturation point of tunneling ionization. However,
in a classical point of view, the terms ?appearance intensity? and ?over-the-barrier?
are justified. Thus, they will be used in the following discussion of classical field
ionization.
14
-0.75 -0.5 -0.25 0 0.25 0.5 0.75
x HnmL
-60
-40
-20
0
20
yg
re
nE
HVe
L
HaL
-0.75 -0.5 -0.25 0 0.25 0.5 0.75
x HnmL
-60
-40
-20
0
20
yg
re
nE
HVe
L
HbL
-0.75 -0.5 -0.25 0 0.25 0.5 0.75
x HnmL
-60
-40
-20
0
20
yg
re
nE
HVe
L
HcL
Figure 2.1: Classical over-the-barrier ionization model of an electron in the com-
bined atomic or molecular field and laser field for (a) atomic (Xq+), (b) diatomic
(Xq+, Xq+), and (c) linear triatomic (Xq+, Xq+, Xq+) systems. The black, blue and
red curves represent, respectively, the electron potential energy as a function of the
electron position, the electron energy level, and the laser-electron coupling eEx in
eV. The laser electric field is pointing to the left.
15
Figure 2.2: Comparison of experimentally determined and theoretically predicted
appearance intensities using a simple classical theory. The solid line corresponds
to exact agreement. (Fig. 3 of Ref. [106]) Reprinted figure with permission from S.
Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, Phys. Rev.
Lett. 63, 2212 (1989). Copyright (1989) by the American Physical Society.
For diatomic molecules, a similar field-ionization model was introduced by
Codling et al. [14, 71, 72]. The valence electron of a molecule or molecular ion is
placed in the classical Coulomb potential well of two point-like ion cores (Fig. 2.1b).
The electron energy level is determined by the ionization potential of the atom
or atomic ion shifted by the Coulomb force of the other ion. When the static
field of a linearly polarized laser is applied along the molecular axis, the potential
well is distorted. In addition to the same potential barrier of the combined field
as in the atomic case, there is also an inner barrier between the two ions in a
diatomic molecule. The heights of these two barriers depend not only on the field
strength (laser intensity), but also on the separation between the two nuclei. Later,
16
the combined potential field in diatomic over-the-barrier ionization (also in one
dimension) was derived as [53, 70, 73, 74, 77]
Vmol = ? Kq1e|x?R/2| ? Kq2e|x+R/2| ?eE0x, (2.3)
and the electron energy levels in the double-well potential (Fig. 2.1b) were given by
?+ = ?q1Ip1 ?Kq2e/R +eE0R/2,
?? = ?q2Ip2 ?Kq1e/R?eE0R/2, (2.4)
where the subscripts, + and ?, refer to the energy levels in the up-hill and down-hill
potential wells, respectively; q1e and q2e are the charges of the two ions; E0 is the
peak field strength; R is the internuclear separation (bond length); K = 1/(4??0);
Ip is the ionization potential of the atom and ionization potential of the ion with
charge qe is assumed proportional to the atomic ionization potential, I(q)p = qIp
[74]. The eE0R/2 term accounts for the Stark-shift. The laser polarization and the
molecular axis are along the x-axis, and the origin is at the center of the molecule.
At short bond lengths, near Re, the electron can move freely between the
nuclei. Ionization is difficult because the electron is confined by the potential well.
As the bond length increases and the nuclei move apart, the inner barrier increases
to block the movement of the electron. When the electron is trapped (electron
localization) in the up-hill potential well, the right well in Fig. 2.1b, the electron
energy level is Stark-shifted upward by eE0R/2 (Eq. 2.4). When the electron energy
level is high enough to overcome both the inner and outer barriers, the system
ionizes. If the bond length increases further, the inner barrier also increases so
17
that ionization requires a higher intensity. For a given molecule or ion, there is an
optimum bond length ? RC ? where the electron energy level in the up-hill well is at
the same height as both the inner and outer barriers (Fig. 2.1b), and the ionization
occurs at a minimum intensity [14, 53, 70, 71, 72, 77].
The appearance intensities of the diatomic molecular ions can be estimated in
a similar manner as in the atomic case. The inner and outer potential barriers can
be found at xinner and xouter where ?Vmol/?x = 0. The ionization field amplitude
at a specified internuclear separation can be determined by equating the height
of the inner potential barrier Vmol(xinner) to the electron energy level in the up-hill
potential well ?+. It is also required that the outer potential barrier be lower than or
equal to the inner barrier, Vmol(xouter) ? Vmol(xinner), so that the electron can pass
over the down-hill potential well and enter the continuum after it passes the inner
barrier. The lowest appearance intensity is found at RC ? 4/Ip (in atomic unit) [70],
where Vmol(xouter) = Vmol(xinner) = ?+. The kinetic energy release of the fragment
ions due to the Coulomb explosion can be predicted from Ek = Kq1q2e2/RC. The
energy deficit is defined as the difference between the kinetic energy release and the
Coulomb potential energy at Re,
ED ?EC ?Ek = Kq1q2e
2
Re ?K
q1q2e2
RC . (2.5)
The predictions were in agreement with the experimental measurements [70]. This
model can be extended to triatomic systems (Fig. 2.1c) [32, 66].
The electron localization and over-the-barrier ionization were also investigated
with quantum calculations [73] ? [76, 78]. The ionization rates at different inter-
18
nuclear separations of diatomic molecules were calculated from the time-dependent
Schr?odinger equation of the valence electron. A smoothing parameter (s ? 0.1
nm) was employed in one-dimensional calculations to remove the singularities of the
Coulomb potential and can be adjusted to give similar ionization potentials as the
exact three-dimensional systems [73, 76]. The maximum of the ionization rate ?
enhanced ionization ? was found at an RC similar to that predicted by the classical
calculation.
In the quantum treatment, the electron energy levels in the combined field of
the Coulomb potential and the laser radiation are assumed to be related to charge-
resonance states (Eq. 2.4) [107, 108]. There are two important factors related to
the charge-resonance states [73] ? [76]. First, their dipole transition moment grows
linearly as R/2 when the internuclear distance increases. A sufficient population
can be pumped into the upper state or upper potential well by the oscillating field.
If the internuclear separation is large enough to raise the inner potential barrier, the
electron will be confined in the upper well and the transition from the upper state
to the lower state will be suppressed ? electron localization. Second, the electron in
the upper charge-resonance state is freed when the laser field reduces the potential
barriers. Thus, the population in theupper level is removed by thefield in an efficient
and fast manner ? enhanced ionization. At smaller internuclear separations, the
population in the upper state is not sufficient to enhance the ionization. At larger
separations, although the upper state is well populated, the higher potential barriers
reduce the ionization rate. In both cases, ionization will have a lower rate or require
a higher laser intensity. In short, dynamic localization of the charge-resonance states
19
and static barrier suppression by the laser field lead to an enhanced ionization rate
at RC.
Numerical calculations using the time-dependent Schr?odinger equation have
been extended to two-electron molecules (H2) [89, 90, 91], and triatomic cases (H2+3
and H+3 ) [92] ? [95]. The critical internuclear separation in triatomic molecules,
again assuming charge-resonance transitions, leads to RC ? 5/Ip (in atomic unit)
where RC is the distance between two end atoms in the molecule [94].
Although the energy deficit in the Coulomb explosion predicted by RC (Eq.
2.5) agrees with the experimental data, the enhanced ionization model has several
weak points. First, the model does not explain how the molecule stretches to RC.
Recent quantum mechanical calculations for CO2 using time-dependent adiabatic
states and their potential surfaces [83, 84, 85] have shown that the neutral molecule
does not stretch to RC but the ionic states (CO2+2 , for example) do stretch on a time
scale comparable to the laser pulse width (? 100 fs). Second, the photo-electrons
are assumed to leave the vicinity of the molecular ion instantly after ionization.
However, a combination of the facts that the photo-electrons will be attracted back
to the positive molecular ion and that a significant part of them will be trapped by
the laser field [109, 110] means that there will be a cloud of electrons remaining in
the vicinity of the molecular ion. Thus, the electrons may need to be taken into
account in order to understand laser-induced molecular dissociative ionization.
20
2.1.2 Electron Screening Effect
The model that includes the effects of the electrons is based on a reduced
repulsive Coulomb force between the atomic ions. The observed energy deficit is
explained by the photo-electrons decelerating the exploding molecular ion, which
was first mentioned by Cornaggia et al. [29, 56].
In the electron screening model [79] ? [82], the electrons start to be removed
from the molecule at Re when the laser field is applied. The positively charged
atomic cores start to repel each other under the mutual Coulomb repulsion. As
more electrons are being removed, they do not leave rapidly, but form a cloud near
the atomic cores. As the atomic cores move through the cloud of electrons, a larger
volume of the electron cloud is enclosed in the increasing space between nuclei. The
resultant electron-screening effect decelerates the motion of the fragments signifi-
cantly such that the observed kinetic energy release is smaller.
The electron-screening effect was calculated in theThomas-Fermi model, which
treats the electrons as a continuous distribution of charged particles. A density func-
tion ?e(vectorr), instead of wave functions for individual particles, is employed to represent
the electrons in the system. The electrons, in an atom or molecule, fill all the energy
levels up to the Fermi energy in the mean field ?(vectorr), which is an average electrostatic
potential of the electron distribution and the nuclear charges. The number density
of the electrons with energies lower than the Fermi energy (?F) is given by [105]
n(vectorr) = ?e(vectorr)/me = [2me(?F ?e?(vectorr))]
3/2
3?2planckover2pi13 , (2.6)
where me is the mass of electron. In an external field, ?(vectorr) is determined self-
21
consistently by the external field and the combination of the electron distribution
and the nuclear charges. The electron density depends on the mean field. If the
external field is large enough, the electrostatic potential is bent at some vectorr where
the electron density drops below zero, which means some electrons spill over the
potential barrier and ionization occurs.
To study a molecule subject to a short laser pulse of optical frequency in this
model, a time-dependent approach is used [79] ? [82, 111] where the hydrodynamic
equations describe an electronic fluid shaken by an external oscillating field. Besides
a time-dependent mass density ?e(vectorr,t) similar to ?e(vectorr) in Eq. 2.6, a velocity field
vectorv(vectorr,t) is introduced to characterize the dynamics of such an electron fluid in a
many-electron system. The time-dependent mean field ?(vectorr,t) is determined by the
oscillating field and the strong internal interaction of the fluid self-consistently and
governs the motion of the fluid. With appropriate initial conditions ? the electron
density ?e(vectorr,t = 0) and the electrostatic field ?(vectorr,t = 0) of the molecular system
without the external field ? the density and velocity fields are found numerically on
a space grid at each temporal step. The kinetic energies of the fragment ions are
also calculated at the same time.
The Thomas-Fermi description of the electrons was first applied to calculate
the appearance intensities for atomic ionization and the results were in agreement
with measurements [88, 111]. For molecules, interlaced ionization and dissociation
were observed in the time-dependent numerical simulation. The result of Brewcyzk
et al. [79, 80, 81] clearly showed the early acceleration and later deceleration of the
nuclei in the process of the Coulomb explosion of Cl2 induced by an intense laser
22
Figure 2.3: Simulated evolution of the features of the Cl2 molecule for a linearly
polarized laser pulse with a peak intensity of 1.25 ?1016 W/cm2 and ? = 610 nm
ramped to maximum intensity in ten optical periods. Main figure shows the kinetic
energy of the atomic ion fragments vs time. Note that Coulomb explosion begins
near the peak intensity; deceleration of the fragments sets in around 20 optical
periods. Inset: Distribution of net electrical charge including both nuclear and
electronic components. (Fig. 1 of Ref. [81]) Reprinted figure with permission from
M. Brewczyk and K. Rz?a?zewski, Phys. Rev. A 61, 023412 (2000). Copyright (2000)
by the American Physical Society.
field (Fig. 2.3). The variations of ion charge (net charge in a small area around
individual atomic cores) and molecule charge (ion charge plus the negative charges
enclosed between nuclei) with time show that the number of electrons enclosed in
the space between atomic cores increases as the nuclei move apart. There are up
to two units of electron charge between atomic cores during the process when the
Coulomb explosion is simulated in two-dimensional calculations. The kinetic energy
deficit, with respect to the Coulomb explosion at Re as a result of photo-electron
screening, was confirmed at the end of the simulation. Hering et al. [82] extended
this model to Coulomb explosions of linear triatomic molecules and calculated the
23
Figure 2.4: Calculated and experimental multifragmentation channels kinetic en-
ergy releases vs total number of removed electrons in the Coulomb explosion of CO2
induced by a linearly polarized laser pulse. (Fig. 1 of Ref. [82]) Reprinted figure with
permission from Ph. Hering, M. Brewczyk, and C. Cornaggia, Phys. Rev. Lett.
85, 2288 (2000). Copyright (2000) by the American Physical Society.
kinetic energies of the ions at the end of the simulations for different charge states.
The results were consistent with the experimental data (Fig. 2.4).
Three points in this model need to be examined more carefully. First, the
electrostatic potential (mean field) neglects the instantaneous interaction between
electrons. For example, the electron-electron scattering is not taken into account.
Second, the model treats the electrons as a continuous distribution. When the
number of electrons freed through ionization is low, this statistical treatment may
not be valid. Third, the calculated kinetic energy of the ions in the simulation (Fig.
2.3) shows a large-amplitude oscillation in time. Whether the final or asymptotic
kinetic energy at the end of the simulation is sensitive to the initial or final times and
24
the stability of the simulation have not been addressed. Meanwhile, the numerical
calculation of the dynamics involved in this model is very time-consuming.
2.2 Ejection Anisotropy
As the energy deficit in molecular Coulomb explosion experiments has been
widely recognized, it is also well established that the angular distribution of the
atomic fragments ejected subsequent to strong-field induced dissociative-ionization
is anisotropic. Two effects contribute to the anisotropy, the so-called geometric
[13, 86] and dynamic [45, 87] alignment.
Geometric alignment relies on the argument that molecules are preferentially
selected along the polarization axis [13, 86] during the ionization process. The laser-
induced transition of electrons depends on ?vector? ? vectorE?, where vector? is the induced dipole
moment and vectorE is the laser electric field. Since vector? is due to either moving charge from
one side of the molecule to the other (charge resonance, dipole transitions [73]) or
induced molecular polarizability, it is largest when parallel to the molecular axis.
Consequently, ?vector?? vectorE? is largest when vectorE is parallel to vector? that is along the molecular
axis and is a function of the geometric orientation angle, ?g, defined as the angle
between the polarization and the molecular axes. Thus, molecules with their axes
more aligned with the field are easier to ionize leading to an anisotropic angular
distribution for the explosion components subsequent to dissociative ionization.
Dynamic alignment refers to thefact that moremolecules aretorqued topartic-
ipatein enhanced ionization than would in puregeometric alignment. Forsufficiently
25
long pulses, compared with the molecular rotation period, an induced polarizability
can cause dynamic alignment [45, 87] due to the development of pendular states
? superpositions of field-free rotational states of the molecules [112, 113]. In short
pulses (? 100 fs), dynamic alignment is caused by the field exerting a torque on the
molecular axis, inducing the molecule to align itself with vectorE prior to the Coulomb
explosion [36, 42, 96, 97].
Both alignment effects coexist and contribute to the ejection anisotropy in typ-
ical Coulomb-explosion experiments performed with linearly polarized light, making
them difficult to distinguish. Geometric alignment, or angular dependence of en-
hanced ionization, has been calculated by the classical field-ionization model [86]
and time-dependent Schr?odinger equation [96], while simulations by Schmidt et al.
[37] provide a convincing argument that pulses as short as 130 fs can induce signifi-
cant dynamic alignment in diatomic systems prior to the Coulomb explosion. Thus
far, previous experimental studies have provided qualitative tests for determining
when dynamic alignment is active. These tests include varying the polarization
[36, 39, 41, 42, 61], wavelength and pulse width of the field [37], laser intensity
[37, 39], the moment of inertia of the molecules [41, 42], and the charge state of
the constituent atoms [37, 39, 41, 42, 61]. The anisotropy measurement has also
been compared with a two-dimensional over-the-barrier ionization calculation [86].
In general, dynamic alignment increases as the moment of inertia decreases and the
laser intensity increases, and it is also a function of the ionization charge states
[37, 39, 41, 42, 61]. It appears to be clear that dynamic alignment is inactive in I2
[41, 42, 86] and active in H2 [86], N2 [36, 37, 39, 86], O2 [41, 42] and CO2 [36, 61].
26
In a static field, dynamic alignment is characterized by the pendular states
[112, 113]. The static field implies two conditions. The first is that the laser pulse
width needs to be sufficiently long compared with the molecular rotation period so
that the molecules have enough time to fall into pendular states. The second is
that the laser frequency needs to be much greater than the reciprocal of the pulse
width (? ? ??1) and it takes many laser cycles for the molecule to be aligned so
that the field can be averaged over the pulse duration. Then, the time-independent
Hamiltonian for the pendular states in a static field is given by
H = BJ2 ? 12E20 parenleftbig(?bardbl ???)cos2 ?g +??parenrightbig, (2.7)
where J2 is the squared angular momentum operator, B = planckover2pi1/(4?cIm) the rotational
constant with Im as the moment of inertia, E0 the peak field strength, and ?bardbl (??)
the polarizabilty component parallel (perpendicular) to the molecular axis. The
interaction potential [113]
Vp = ?12E20 parenleftbig(?bardbl ???)cos2 ?g +??parenrightbig = ?V? ??V cos2 ?g (2.8)
is attractive to the point of ?g = 0 if ? ? ?bardbl ??? > 0. The depth of the potential
well is
?V = Vbardbl ?V? = E
2
0
4B(?bardbl ???) =
?E20
4B , (2.9)
where Vbardbl,? = E204B?bardbl,? are the potential energies at parallel and perpendicular direc-
tions to the field. The torque that a molecule feels in such a potential well is
T = ??Vp??
g
= ??V sin2?g = ??E
2
0
4B sin2?g, (2.10)
27
Table 2.1: Molecular properties for H2, N2, O2, CO2, NO2 and I2. The equilibrium
bond length (Re) and the ground-state rotational constant (B0) are extracted from
experimental measurements [114]. The moment of inertia (Im) is calculated from the
ground-state Re (The value of NO2 is calculated at its ground-state bond angle ?be =
134?). The ground-state rotational time is ?e = 1/(2cB0) because the rotational
energy at the ground state is E0 = h? = hcB0J(J + 1) with ? = 1/?e and J = 1.
H2 N2 O2 CO2 NO2 I2
Re (nm) 0.074 0.110 0.121 0.116 0.119 0.267
Im (?10?40 gm cm2) 0.455 14.0 19.4 71.5 67.2 751
B0 (cm?1) 60.853 1.998 1.438 0.390 0.434 0.0374
?e (ps) 0.27 8.35 11.60 42.76 38.43 446
which increases as the laser field or molecular polarizability increases and the rota-
tional constant decreases.
However, the rotation time of small molecules listed in Table 2.1 is on the order
of tens of picoseconds or even more except for H2. The analysis of pendular states
may not be valid for the ultrafast dynamics induced by 100 fs laser pulses, at least
for the ground states of the molecules. Instead, the pendular potential provides a
confinement to the molecules, which will inevitably be ionized once trapped in the
potential well. As the depth of the well (?V ? vector?? vectorE/B) increases, the molecules
are trapped more tightly near the ploarization axis.
To understand the torque on the molecular axis exerted by the field of a short
pulse, a classical pendulum equation of motion has been solved for the dynamic
alignment of a polarized molecule [36, 42, 96, 97]. The torque (vectorT ) of a dipole
moment (vector?) in an external electric field (vectorE) has the form of
vectorT = vector?? vectorE, T = ?Esin?g, (2.11)
28
where ?g is the angle between vector? and vectorE. Thus the equation of motion is
Im ??g = ?T , ??g = ??EI
m
sin?g, (2.12)
where Im is the moment of inertia of the molecule. Assuming the induced dipole
moment is ? = ?Ecos?g and the field takes the time-average value E = E0/?2,
Eq. 2.12 becomes
d2(2?g)
dt2 = ?
E20?
2Im sin2?g. (2.13)
At the small-angle limit (?g ? 0), the pendular frequency is
?p =
radicalBigg
E20?
2Im . (2.14)
For general values of ?g, Eq. 2.13 can be solved with the elliptic integral of the first
kind ? K(sin2 ?) [115] (App. A). The rotation time for the molecule to move from
?g to parallel to the polarization direction can be expressed as
?r = K(sin
2 ?g)
?p . (2.15)
For ?g < ?/2, K(sin2 ?g) is on the order of unity.
At this point, Bandrauk and Ruel [96] assumed ? ? R3, and estimated the
rotation time as
?r ? 1E
0
radicalbigg?
m
R , (2.16)
where, ?m is the reduced mass with Im = ?mR2 and R is the molecular bond
length. However, this relationship between the polarizability and molecular size
(? ? R3) was derived for a hydrogen atom using perturbation theory, and valid
mostly for atoms or spherical molecules [105]. For a linear molecule in an intense
field, especially one elongated to RC ? 2?3?Re, this may not be valid.
29
Nevertheless, as shown in the equation of motion (Eq. 2.12), the degree of
dynamic alignment should increase as the dipole moment and electric field increase
and moment of inertia decreases. Besides, by integrating the equation of motion,
one can see that the molecules would be aligned more by spending a longer time in
the field.
30
Chapter 3
APPARATUS
This chapter describes the apparatus used in this study, including the fem-
tosecond laser system, the image spectrometer and vacuum system, digital camera
and computer system for data acquisition. The overall experimental setup is shown
in Fig. 3.1. Section 3.1 describes the lasers, their operation and beam diagnostics.
The characteristics and operation of the image spectrometer and vacuum chamber
are described in Sec. 3.2. The digital camera and computer system, which are the
foundation of the correlation detection techniques that will be described in the next
chapter, are discussed in Sec. 3.3.
3.1 Laser System
The kilohertz femtosecond laser system employed in this study was manufac-
tured by Spectra Physics/Positive Light. The gain medium in both the oscillator
and amplifier is Ti:Sapphire (Ti3+:Al2O3) crystal, where the Ti3+ titanium ion is
responsible for the laser radiation. Absorption transitions occur at wavelengths be-
tween 400 and 600 nm, while fluorescence transitions exist between 670 and about
1100 nm. Stable and high-output laser emission is available from wavelengths 690
to 1080 nm in the oscillator depending on the optics set [116]. A longitudinal pump
is utilized in both the oscillator and amplifier to ensure the pump and cavity modes
31
Ar+ Pump
Nd
:Y
LF
 P
um
p
Mode Locked
Ti:Sapphire
Ti:Sapphire
Regen
Beam
Diagnostics
Timing
Electronics
Camera,
Frame Grabber
& Computer
Digital
Scope
4pi Image & 
Time-of-Flight 
Spectrometer
Figure 3.1: The overall experimental setup includes three parts. (1) The 100 fs,
800 nm Ti:sapphire laser system (Spectra Physics/Positive Light) includes a mode-
locked Ti:Sapphire oscillator (Tsunami) pumped by an argon ion laser (BeamLok), a
Ti:Sapphire regenerative amplifier (Spitfire) with stretcher and compressor pumped
by a Nd:YLF laser (Merlin), and beam diagnostics; (2) 4? image and TOF spec-
trometer housed in a vacuum chamber; (3) digital camera and oscilloscope with
computer and timing electronics.
are collinear and overlapped within a long distance inside the Ti:Sapphire rod.
The entire system is composed of four lasers. The mode-locked Ti:Sapphire
oscillator (Tsunami, tunable over 735? 840 nm) is pumped by an All-line continuous-
wave (CW) argon ion laser (BeamLok 2060) at 5.5 W. The two most intense emission
lines from the argon laser are at 488.0 and 514.5 nm [117]. The average power output
of Tsunami is 400 mW with a 76 MHz repetition rate so the energy per pulse is
about 5 nJ. The measured pulse width in daily operation is 60 ? 80 fs with a 12
32
nm bandwidth at 800 nm central wavelength. The output pulses from the oscillator
are seeded into a chirped-pulse regenerative amplifier (Spitfire including stretcher
and compressor) pumped by a Q-switch intracavity doubled Nd:YLF laser (Merlin,
527 nm) at 8 ? 10 W. The final output power from the amplifier is as high as 1
W average power at 1 kHz repetition rate and 90 ? 120 fs pulse width with 10 nm
bandwidth at 800 nm central wavelength.
The mode-locked pulses in Tsunami are sustained by a mechanism called self
mode locking, which can be explained by the optical Kerr effect in the laser medium
[118]. As a high-intensity Gaussian beam passes through a medium, index refraction
of the material is changed by the nonlinear effect according to n(r) = n0 + n2I(r),
where r is the radial distance on the wavefront, I is the intensity, n0 and n2 are the
linear and 2nd order nonlinear index refraction. In a Gaussian beam, I drops as
r increases so that n(r) also decreases and the medium will act as a positive lens
to make the beam size smaller. This is known as self-focusing. The mode-locked
beam has a higher intensity than the CW mode with the same average power in
the cavity so that it has a tighter waist in the laser medium. As the oscillator is
longitudinally pumped by another laser beam, the spatial gain profile created by the
Gaussian shape of the pump beam acts like a soft aperture which gives the tighter
mode-locked cavity mode higher gain, or more loss for the CW mode [119]. Thus,
the mode-locked beam is favored over the CW mode by the optical Kerr effect and a
stable pulse train is sustained. At the same time, the width of the mode-locked pulse
is broadened every time it travels through the laser medium. The positive group-
velocity dispersion (GVD) or chirp is a combination of the second derivative of the
33
linear index refraction with respect to the wavelength (d2n0/d?2) and self phase
modulation by the nonlinear effect in the laser medium ? n(t) = n0 +n2I(t), where
t is time. In order to generate short pulses close to the transform-limit, negative
GVD is introduced by a four-prism sequence to compensate the positive GVD in
the laser rod [12, 116, 120, 121].
In order to obtain high peak power, the oscillator pulses need to be amplified.
For short pulse amplification, three issues have to be addressed in the design of the
amplifier, a broad bandwidth to support the full spectrum of the short pulse, a near-
saturation fluence in the medium to extract the energy asmuch aspossible, and a low
intensity to prevent material damage and nonlinear effect distorting the spacial and
temporal profiles of the pulses. While the first requirement is fulfilled by the large
gain-bandwidth of the Ti:Sapphire crystal itself, the other two are accomplished
by the chirped pulse amplification (CPA) [122]. The short pulses (80 fs) from the
oscillator are first stretched to 100 ? 200 ps by a folded diffraction grating stretcher,
which disperses different frequencies and causes the higher frequency components
to travel longer distances through the stretcher than lower frequency components.
The dispersion is positive and linear. The low peak power pulses are then amplified
in a regenerative amplifier cavity by a factor of about 106 in about 20 round trips.
After the amplified pulse exits the cavity, a grating compressor reverses the process
in the stretcher to remove the positive chirp introduced by the stretcher as well as
the extra linear dispersion introduced by the amplifier cavity. The pulse width is
compressed back to about 100 fs and the final power can be as high as 10 GW (1
mJ/100 fs) [123].
34
QWPM1 M2
M3 M4
PC1
PC2
LR
From Stretcher
OP
To CompressorPump
Beam
Figure 3.2: Cavity of the regenerative amplifier. M1 ? M4 are broadband dielectric
mirrors, LR is the Ti:Sapphire laser rod, PC1 and PC2 are the input and output
Pockels cells, QWP is the quarter wave plate, and OP is the thin film polarizer.
The schematic of the regenerative amplifier cavity in Spitfire is shown in Fig.
3.2. The pulse from the stretcher is reflected into the cavity on the right surface
of the laser rod. If both Pockels cells are off, the pulse makes a round trip in the
cavity and is reflected out by the thin film polarizer (OP). When a pulse is selected
to be amplified, the input Pockels cell (PC1) is activated by a quarter-wave voltage
(typically 3500 volts) after the pulse passes it the first time. As the incident angle
on the surface of the laser rod is close to the Brewster angle, the reflected pulse has a
vertical polarization. After it is reflected by M1 and goes through PC1 (not activated
yet) and the quarter-wave plate twice, the polarization is rotated to be horizontal.
When the pulse travels through the rest of the cavity and comes back, PC1 has
been activated and acts like a quarter-wave plate, which cancels the polarization
rotation invoked by the real quarter-wave plate. The polarization of the pulse stays
horizontal and the pulse is trapped in the cavity and amplified. After about 20 round
35
trips, the output Pockels cell (PC2) is activated also by a quarter-wave voltage. As
the now amplified pulse double-passes PC2, the polarization is rotated to be vertical
and the pulse is reflected out of the cavity by the output polarizer (OP) [123]. The
polarization of the pulse and voltages on the Pockels cells at each optical element
in the amplifier cavity are listed in Table 3.1. The activation of the Pockels cells in
middle and lower tables is labeled right before the pulse arrives. In practice, they
can be activated any time before the pulse arrives and after the pulse leaves the
last time. The electronic controller of the Pockels cells is synchronized with the
oscillator. The Pockels cells are activated and deactivated at the right times so that
one and only one pulse is allowed to circulate in the cavity at a time and only up
to 1000 pulses are selected out of the 76 million from the oscillator in each second.
Several important characteristics of the amplified pulse have been determined
in the experiments. They are pulse width, power stability, and beam divergence.
The pulse width of Tsunami and Spitfire can be measured by a home-built
single-shot autocorrelater. The schematic is shown in Fig. 3.3. The pulse from the
laser is split into two identical copies by the beamsplitter. One is reflected by the
mirrors M3, M5 and M6, and the other by M1, M2, M4 and M6. The two pulses
cross each other at the BBO crystal with a small angle (about 7?). The optical
paths in the two arms are set to be the same so that the two pulses arrive at the
BBO crystal at the same time to generate second harmonic signal, which bisects
the angle between the two original beams and detected by a CCD (Charge-Coupled
Device) camera. Mirrors M1 and M2 are mounted on a translation stage so that the
relative delay between the two pulses can be adjusted.
36
Table 3.1: Polarization (POL) of the pulse and voltages on the Pockels cells (PC1 and PC2) at each optical element in the
amplifier cavity. The labels of the elements are the same as in Fig. 3.2. The polarization states and the voltages (0 or ?/4
voltage) are the ones AFTER the pulse passes through the element. The pulse is running from left to right. The upper table
is the trip with both Pockels cells off. It starts as the pulse is reflected into the cavity by the right surface of the laser rod
(Fig. 3.2). At the end, the vertically polarized pulse is reflected out by the thin film polarizer (OP). The middle table is the
trip when the pulse is trapped. It also starts as the pulse is reflected into the cavity by the laser rod. After this trip, the pulse
stays in the cavity as PC1 stays activated. The lower table is the last round trip a pulse makes in the cavity. It starts as the
pulse goes through the laser rod toward the quarter-wave plate. At the end, the amplified pulse is ejected out by the thin film
polarizer. Two mirrors (M2 and M3) are omitted since they do not affect the polarization of the pulse.
start QWP PC1 M1 PC1 QWP LR OP PC2 M4 PC2 OP LR QWP PC1 M1 PC1 QWP
POL arrowdblbothv anticlockwise anticlockwise clockwise clockwise ? ? ? ? ? ? ? ? clockwise clockwise anticlockwise anticlockwise arrowdblbothv
PC1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
PC2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
LR OP PC2 M4 PC2 OP
POL arrowdblbothv arrowdblbothv arrowdblbothv arrowdblbothv arrowdblbothv arrowdblbothv
PC1 0 0 0 0 0 0
PC2 0 0 0 0 0 0
start QWP PC1 M1 PC1 QWP LR OP PC2 M4 PC2 OP LR QWP PC1 M1 PC1 QWP
POL arrowdblbothv anticlockwise anticlockwise clockwise clockwise ? ? ? ? ? ? ? ? clockwise arrowdblbothv arrowdblbothv clockwise ?
PC1 0 0 0 0 0 0 0 0 0 0 0 0 0 ?/4 ?/4 ?/4 ?/4 ?/4
PC2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
start QWP PC1 M1 PC1 QWP LR OP PC2 M4 PC2 OP LR QWP PC1 M1 PC1 QWP
POL ? clockwise arrowdblbothv arrowdblbothv clockwise ? ? ? clockwise anticlockwise arrowdblbothv arrowdblbothv
PC1 ?/4 ?/4 ?/4 ?/4 ?/4 ?/4 ?/4 ?/4 ?/4 ?/4 ?/4 ?/4
PC2 0 0 0 0 0 0 0 ?/4 ?/4 ?/4 ?/4 ?/4
37
CCD
BS
Translation
Stage
M1
M2
M3
M4
M5
M6
BBO
From Laser
Infrared
Filter
Figure 3.3: Schematic of a single-shot second-harmonic autocorrelater. Mirrors M1
and M2 are gold-coated broadband, while mirrors M3 ? M6 are aluminum-coated
broadband. BS is 50-50 beamsplitter. The distance that the translation stage
moves is measured by a micrometer with an accuracy of 20 ?m. BBO (beta barium
borate) is the doubling-frequency crystal. All the reflected beams are kept in the
same plane, and both pulses arrive at the crystal with the same incident angle. The
second harmonic signal bisects the angle between the two fundamental beams and
detected by a CCD camera. The transmitted fundamental beams are filtered out
by the infrared filter (Schott BG series, more than 70% transmission at 400 nm and
less than 1% at 800 nm). The crystal is mounted on a rotary holder in order to be
rotated to get phase-match when the polarization of the incident beam changes.
The principle of a single-shot autocorrelater is to transform the temporal pro-
file of the pulse into a spatial signal, which can be detected easily with a CCD array
or camera [124]. This transformation is realized by the second harmonic (SH) gen-
eration in the BBO crystal (Fig. 3.4). The SH signal is only present when the two
pulses overlap both spatially and temporally; its spatial profile can be expressed as
SH(x) ?integraltext???I(t+?t)I(t??t)dt, where 2?t is the time delay between the two pulses
as a function of position x (Fig. 3.4). The FWHM (D) of the SH(x) signal can
38
be measured with the CCD camera. It is proportional to the autocorrelation pulse
width ?auto. In order to obtain the coefficient between D and ?auto, the autocorre-
later needs to be calibrated by changing the delay between the two pulses. This can
be done by moving the translation stage. When the stage is moved by a distance ?,
the optical paths in the two arms differ by 2?. Thus the delay time is 2?/c, where
c is the speed of light. The SH signal also moves along the x direction accordingly,
and the shift d corresponds to 2?/c. Then the autocorrelation pulse width can be
calculated as ?auto = 2? ? D/(c ? d). It is not possible to retrieve the pulse shape
I(t) from SH(x) or pulse width ? from ?auto directly because SH(x) is an integral
convolution of I(t). However, since the function of I(t) is normally well defined
by the characteristics of the laser system, the ratio between ? and ?auto is easily
obtained. For Ti:Sapphire oscillator and amplifier, the pulse shape is a hyperbolic
secant squared function [116] so that
? = ?auto1.55 = 2??D1.55c?d. (3.1)
The pulse width of the oscillator is measured to be 60 to 80 fs, while that for the
amplified pulse is 90 to 120 fs.
The long-term stability of the average power from Spitfire was monitored by
a Scientech power meter. In daily operation, the variation of the average power
within 4 ? 8 hours after the whole system is warmed up, which takes up to 3 hours,
is less than 3%, and normally no more than 5%. The short-term energy variation is
determined by a photodiode. After the pulse is focused by a concave mirror inside
the vacuum chamber, it diverges very quickly and part of the beam goes out of the
39
1k
r
2k
r
I1(t) I2(t)
SH(x)
x
Figure 3.4: Second harmonic (SH) generation in the BBO crystal. This plot is
in the same plane as in Fig. 3.3. The SH signal is only generated where the two
pulses overlap both spatially and temporally. The fundamental pulses propagate
in the directions of vectork1 and vectork2. Their temporal profiles are I1(t) and I2(t) with
I1(t) = I2(t) = I(t). The SH signal from the overlap of the pulses is SH(x), where
x direction is parallel to the face of the crystal.
chamber through the viewport. A photodiode is located outside of the viewport
to detect it. Since the beam is diverging, the input intensity on the photodiode
decreases as it is moved away from the viewport. The location of the phototdiode
is set so that the intensity is low enough for the photodiode to operate in linear
range. The signal from the photodiode is collected by a digital oscilloscope (LeCroy
9350AM) and the waveforms of roughly every 15,000 pulses (15 seconds), which fit
onto a floppy disk, are transfered to a computer to be analyzed. The peak amplitude
of the signal on the photodiode is assumed to represent the peak intensity of the
laser pulse. The standard deviation of the signals for 15,000 pulses is about 10 ?
20% (Fig. 3.5).
The divergence of the amplified pulse was determined by measuring the small-
est spot size after the beam was focused by a concave mirror and comparing the
40
Figure 3.5: Measured variation of the peak amplitude of the signal on the pho-
todiode, which is assumed to represent the peak intensity of the laser pulse. The
photodiode is connected to the digital oscilloscope (LeCroy 9350AM) with 50 ?
impedance. In this particular run, 13,000 laser pulses were recorded and the stan-
dard deviation of the signal is 18%.
result to theory. For a focused Gaussian beam at the diffraction limit, assuming the
incident beam diameter is ?wm that contains 99% of the total energy, the waist at
the focal point is [12]
wDLf = f??w
m
, (3.2)
where f is the focal length, ? the wavelength, and wm the beam waist at the lens
or concave mirror. After being reflected by three uncoated windows (roughly 4%
reflectivity each, to reduce the beam intensity) and two aluminum mirrors, the pulse
from Spitfire was focused by a 3-meter concave spherical aluminum mirror (f = 1.5
m). A CCD camera was set at the focal point with a few neutral density filters
to reduce the beam intensity further so as not to damage or saturate the camera.
The minimum size of the focused beam was found at 1.6 m away from the mirror
41
by moving the camera along the propagation direction. The radius from the center
of the beam to where the intensity drops to 1/e of the peak value was 18 camera
pixels in the horizontal direction. The pixel size of the camera in this direction is
8.4 ?m, so r = 151 ?m. Since the intensity of a Gaussian beam drops to 1/e at
r = w/?2 (i.e. w defined as 1/e2 point of the intensity), the waist of the focused
beam is wf = ?2r ? 210 ?m. As the camera was placed in front of the concave
mirror, the unfocused beam size was bigger than the CCD and only a portion of the
beam was detected by the camera. Fortunately, this portion was larger than a half
of the beam, so the radius at the 1/e point could still be measured as roughly 450
pixels. Thus, the unfocused beam size at the mirror was wm = ?2?450? 8.4 ?m
? 5.3 mm. According to Eq. 3.2, the beam size at diffraction limit should be wDLf
= 72 ?m with f = 1.5 m, ? = 800 nm, and wm = 5.3 mm. Thus, the divergence
of the beam was about three times (R = wf/wDLf ? 3) the diffraction limit. Since
the relative divergence (R) is proportional to 1/f (Eq. 3.2), the uncertainty (?RR )
is also proportional to the uncertainty of the focal length. A 10% uncertainty of f
would cause a 10% uncertainty of R.
3.2 Image Spectrometer and Vacuum System
The spectrometer (Fig. 3.6) used to study strong-field dissociative ionization
of molecular gases consists of an image and a TOF detector [22, 100], allowing
images and TOF waveforms to be obtained from the same focal point by reversing
the electric field used to sweep the ions to the detectors. The image detector, which
42
is able to collect all ions ejected into 4? sr simultaneously, consists of a region of
space with a uniform static electric field and an image quality microchannel plate
(MCP) with a phosphor screen at its back end. The potential and spacing between
the electrode rings are set as shown in Fig. 3.6 in order to generate the uniform
static field between the image MCP and the flight tube. The typical static field
(? 260 volts/cm) leads to a maximum detectable energy of doubly charged ions as
about 95 eV. The MCP has a diameter of 50 mm with a 40-mm active area to detect
charges. Laser pulses are brought to focus at a distance l from the center of the
MCP. Charges (?qe) are swept toward the MCP by the static electric field after
they are ejected from the focal point. Upon striking the surface of the MCP, the
ions generate electrons that are amplified in the MCP and then converted to light by
the phosphor screen. The light is digitized by a digital CCD camera and recorded
to disk in real time at up to 735 Hz. The camera frame rate is synchronized with
the laser repetition rate so that each captured frame corresponds to one laser pulse.
The camera exposure time and the MCP/phosphor gain are adjusted to produce
a zero background while allowing near single-charge detection. The center of mass
of the ejection dynamics coincides with the lab frame with its origin at the center
of the image. The TOF detector (Fig. 3.6) consists of an acceleration region (the
same static electric field as in the image detector between ER3 and ER4), a field-
free drift region (inside the flight tube), a second acceleration region just prior to an
MCP, while a digital oscilloscope (LeCroy 9350AM), also synchronized with laser
repetition rate, is used to capture and analyze the waveforms.
The spectrometer is housed in an ultra-high vacuum chamber constructed
43
CC
D
CA
M
ER
A
MCP
MCP
DI
GI
TA
L
SC
OP
E
FLIHGT
TUBE
ELECTRODE
RINGSPHOSPHOR SCREENWITH FIBER
OPTIC BUNDLE
ER1 ER2 ER3 ER4
Vphos
VMCP
V0 V1V2 V
3 V4 Vfront V
back
FOCAL POINT
(INTERACTION ZONE)
Figure 3.6: Structure of the image-TOF spectrometer. There are six electrode rings
in the spectrometer. The two close to the TOF detector are not being used. The
other four (ER1 to ER4) are used to generate a static electric field between the image
MCP and the flight tube. Their inner diameters are 40 mm to match the size of the
active area of the image MCP. The voltages on the electrodes are V4 = 2V3 = 4V2
= 8V1. Positive or negative voltages (normally 2000 volts) are applied to take ion
images or TOF waveforms, separately. The front side of the image MCP is grounded
(V0 = 0). The distances between the electrodes and MCP are D4 = 2D3 = 4D2 =
8D1 = 76.2 mm (3 inches) to ensure a uniform field. The focal point of the pulse is
between ER3 and ER4 so that the distance between the interaction zone and MCP
is l = 3D4/4 = 57.15 mm and the potential Vint = 3V4/4. The potential across MCP
VMCP is up to 2000 volts. The front side of the phosphor screen is connected to the
back of MCP, while the potential between the back side and the ground is Vphos =
4800 volts. The voltages on the front and back sides of the TOF MCP are Vfront
= ?2300 volts and Vback = ?300 volts. The 140-mm-long flight tube is connected
with ER4 on one end and the other end is 30 mm away from the front side of the
TOF MCP. The diameter of the tube is 40 mm and there is no field in the tube.
from non-magnetic stainless steel with a base pressure of about 4 ? 10?10 Torr
and pumped by a turbo molecular pump. The turbo pump is a Pfeifer TMH 260
with a TCP 380 pump controller and its pumping speed is 260 liters/s at 1000
Hz. The turbo pump is backed by a Sargent-Welch 1398 mechanical vacuum pump
to keep the pressure in the fore line below 50 mTorr. To perform experiments,
gas is introduced into the chambers through a high-precision leak valve (Granville-
44
Phillips 216 with pressure control ranging from 10?11 to 1300 Torr and conductance
to vacuum of 10?12 liters/s1). Typically, experiments are run at pressures between
10?9 and 6?10?8 Torr to minimize space charge effects.
Laser pulses are introduced into the chamber through a fused-silica viewport
at normal incidence. The beam is focused with a spherical mirror (38 mm diameter,
75 mm focal length) mounted inside the chamber and its intensity was determined
as the following. In principle, the intensity of a focused Gaussian beam at the
diffraction limit is given by
I = P2(f#?)2, (3.3)
where f#? = f?/(?wm) = wDLf (Eq. 3.2) and P is the laser power [12]. For a pulsed
laser with an average power Pavg at a repetition rate frep and a pulse width ?, the
power of the pulse is P = Pavg?T /(?frep), where T is the power transmission from
the laser to the focal point. Thus, the intensity is
I = PavgT2w2
f?frep
, (3.4)
where the real beam size wf is used instead of wDLf . In our system, the unfocused
beam size before the concave mirror in the spectrometer is estimated as wm = 8 mm
so that the focused beam size at the diffraction limit is wDLf = 2.4 ?m with f =
75 mm and ? = 800 nm (Eq. 3.2). Since the divergence of the beam is about three
times the diffraction limit (Page 42), the real beam size at the focal point is wf =
3?wDLf = 7.2 ?m. For Spitfire, with Pavg = 0.5 W, frep = 1000 Hz, ? = 100 fs and
1Conductance (C) is the flux or throughput (Q) of the gas divided by the pressure difference
between the inlet (P1) and outlet (P2) of the segment, C = Q/(P1?P2), and measures how well the
gas can go through a component of the vacuum system (valve, orifice, etc.). The unit dimensions
of C and Q are volume/time and pressure?volume/time.
45
T = 74% (80% from the output of the compressor in Spitfire to the input viewport
of the vacuum chamber and estimated 8% loss at the viewport), the intensity at the
interaction zone of the spectrometer is 3.7?1015 W/cm2.
This calculation was checked with the measurement of the ion appearance
intensities of noble gases, for example, xenon. The TOF spectra of xenon ions
generated by the laser radiation with linear polarization were taken under a series
of average laser powers. Higher charged xenon ions appear one by one as the laser
power increases. The appearance intensity of a certain charge state was calculated
by Eq. 3.4 from the lowest average power where xenon ions with this charge state
were found in the TOF spectrum. The values of the appearance intensities of Xe3+,
Xe4+ and Xe5+ were measured as 4.4, 7.1 and 12?1014 W/cm2 (Fig. 3.7). These
values were found to be consistent with the appearance intensities determined in a
previous study (4, 9 and 14?1014 W/cm2) [106] with a difference of only about 15%.
Typically, an experiment is composed of capturing 100,000 to 500,000 camera
frames or 1,000 to 10,000 TOF waveforms generated by 100 fs laser pulses focused
to intensities between 1014 and 1016 W/cm2. Figs. 3.8 and 3.9 show samples of TOF
and image spectra, respectively. Each laser pulse produces tens of ions in a variety of
charge and energy states. For molecular systems, a significant number of near-zero
energy ions are generated with each pulse. These ions, molecular and atomic, collect
at the center of the image. Since an ion?s trajectory depends only on its charge and
initial kinetic energy (see App. B.1), ions with different masses but with the same
kinetic energy and angular distribution will collect at the same place on the image.
To study simple spectra such as single-component photo-electron distributions or
46
Figure 3.7: TOF spectra of Xe at laser intensities of 4.4 (top), 7.1 (middle) and
12?1014 W/cm2 (bottom) calculated by Eq. 3.4. The ions are Xe+ (8.6?s), Xe2+
(6.3?s), Xe3+ (5.2?s), Xe4+ (4.6?s) and Xe5+ (4.1?s). The sharp single peaks at 4.2
and 4.5?s are N+2 and O+2 from the background. The signal of Xe3+ (Xe4+, Xe5+)
can be barely seen in the top (middle, bottom) plot.
47
the two-component proton distribution of H2 dissociative-ionization, this is not a
major issue. However, for systems with multiple explosion channels (such as N2, O2,
NO2 or CO2), there can be great confusion from overlapping spectra of the various
atomic and molecular ions. To reduce this confusion, the fact that ions with different
masses and at different charge states arrive at the image detector at different times is
exploited to gate the potential across the MCP [100]. The potential (VMCP) is only
brought up to 2000 volts when the interested ions arrive at the MCP, and is kept
at 1000 volts otherwise so that electron amplification does not occur in the MCP.
We note that, since only VMCP is changed and V0 stays fixed (Fig. 3.6) when the
MCP is gated, the static field in the interaction region is not affected. At the same
time, although the potential across the phosphor screen (Vphos ? VMCP) is higher
when VMCP is kept low than when VMCP is brought up, no ion images appear on
the screen since electron amplification does not occur in the MCP when VMCP is
kept low.
Using the structure of the spectrometer and potentials on the electrodes (Fig.
3.6), it is possible to calculate when and where an ejected ion will strike either the
image or TOF MCP. Arrival time for an ion with zero initial momentum or ejected
parallel to the MCP to fly from the interaction zone to the image detector is given
by
t =
radicalBigg
2ml
qeE = l
radicalbigg 2m
qeVint, (3.5)
where m and qe are the mass and charge of the ion, l and Vint are the distance
and potential between the focal point and the detector (Fig. 3.6), and E = Vint/l
48
Figure 3.8: TOF waveforms resulting from CO2 Coulomb explosions obtained with
100 fs pulses focused to 2?1015 W/cm2 when the polarization axis is parallel and
perpendicular to the TOF axis. The upper traces are rescaled by a factor of two.
Note, the C structure is different in the parallel and perpendicular traces while some
O components are missing altogether from the perpendicular trace. Each trace is
an average of 1000 laser shots.
49
Figure 3.9: Surface plots of ion images resulting from NO2 Coulomb explosions
obtained with linearly polarized 100 fs pulses focused to 2?1015 W/cm2 when the
polarization axis is parallel to the image MCP. The upper plot was taken with no
gate on the MCP, all the molecular and atomic ions were collected; the large peak at
the center was formed by near zero-energy ions, the top of which had to be truncated
in order to show other structures on the image. The lower plot was taken while the
MCP was gated; only doubly- and triply-charged atomic ions were collected. Each
plot is an average of 500,000 laser shots.
50
Table 3.2: Arrival time in the unit of ns for atomic and molecular ions which have
zero initial momentum or are ejected parallel to the image MCP calculated by Eq.
3.5 with Vint = 1500 volts. The charge states marked as N/A either does not exist
or is never detected. The arrival time for an electron is 5 ns.
+1 +2 +3
H 212 N/A N/A
H2 300 N/A N/A
C 736 520 425
N 795 562 459
O 850 601 491
OH 876 N/A N/A
H2O 901 N/A N/A
N2/CO 1124 795 N/A
NO 1163 823 N/A
O2 1202 850 N/A
CO2 1409 996 N/A
NO2 1441 1019 N/A
is the static field strength. Table 3.2 lists the arrival times for relevant atomic and
molecular ions. The ions with nonzero momentum will arrive earlier or later and this
table only serves as a guideline for setting up the MCP gate to detect a particular
ion. Detailed calculations for nonzero-momentum ions as well as the flight time
to TOF MCP are given in Appendix B. Typically, gate widths of a few hundred
nanoseconds are sufficient to isolate the first few atomic charge states. Figure 3.10
displays a series of gated images for the dissociative ionization of NO2 at 2?1015
W/cm2, and shows clearly that ions with different masses and charges arrive at the
MCP at different times. Details of the image spectrum will be discussed in the next
chapter.
51
(11) O+
770 ? 870 ns
(12) O+
820 ? 920 ns
(13) OH+
870 ? 970 ns
(14) N2+,3+ ,O2+,3+
370 ? 670 ns
(15) All Ions
No Gate
(6) N2+ , O2+
520 ? 620 ns
(7) O2+
570 ? 670 ns
(8)
620 ? 720 ns
(9) N+
670 ? 770 ns
(10) N+
720 ? 820 ns
(1) H+
160 ? 260 ns
(2) H2+
260 ? 360 ns
(3) N3+
370 ? 470 ns
(4) N3+ , O3+
420 ? 520 ns
(5) N2+
470 ? 570 ns
10
Figure 3.10: Gated images of the dissociative-ionization spectrum of NO2 generated
with 100 fs, 2?1015 W/cm2 pulses. The times under each image indicate when the
MCP gate opens and closes and are relative to the arrival time of the laser pulse
to the focal point. Images (1) ? (13) were taken under the same conditions (gas
pressure, laser intensity and gate width), but have been re-scaled relative to (6) by
the following factors: (1) 1.7?, (2) 10?, (3) 4.3?, (4) 4.3?, (5) 3?, (6) 1?, (7) 6?,
(8) 95?, (9) 10?, (10) 4.3?, (11) 1.5?, (12) 1.5? and (13) 19?. Few ions were
detected in image (8). Image (14) is a composite image of all the doubly and triply
charged atomic ions (i.e., N2+, O2+, N3+ and O3+, same as the lower plot in Fig.
3.9) and image (15) is the composite image taken with no gate (same as the upper
plot in Fig. 3.9).
52
3.3 Digital Camera for Imaging
The system for image acquisition is composed of a high frame-rate digital
CCD camera, a frame grabber, a computer and a digital delay generator (DDG).
The digital camera is a Dalsa model CA-D1 with 128?128 pixels, up to 735 frame-
per-second acquisition rate, and a standard C-mount lens adapter. The pixels of the
CCD sensor are squares with a 16-?m2 size. The intensity output is 8-bit digital.
The value runs from 0 to 255 (28-1), and the noise (random noise and fixed pattern
noise) is no greater than 3 [125]. In the test runs of the camera without the laser,
there are 1% of the pixels with a value of 0, 78% with a value of 1, 21% with a
value of 2, and only 5 pixels with a value of 3, out of 3.3?109 pixels (total 200,000
frames). Therefore, the threshold for the frames is set at 3. Any pixel with a value
lower than 3 is treated as background and set to be zero. The Bitflow Road Runner
frame grabber transfers the captured frames from the camera to a computer with
an 800-Hz Pentium III CPU. The frames are stored directly onto an external SCSI
drive attached to the computer to keep up with the high frame rate of the camera.
Two 160-GB (gigabytes) SCSI drives can store up to 10 million frames each. The
Video Savant software is run under the Windows NT operating system to control
the frame grabber and the digital camera. Some sample frames taken by the camera
are shown in Fig. 4.1.
A trigger signal at the laser repetition rate from the Spitfire controller is fed
to the digital delay generator (DG535, Stanford Research Systems, Inc.). Two of
its output channels are used to form a square wave to trigger a GRX high-voltage
53
pulse generator (Directed Energy, Inc.) to gate the MCP. The high-voltage pulse
generator takes two high-voltage inputs (V+ and V?) and forms a square wave
following the trigger signal from the DDG. In our case, V+ = 2000 volts and V?
= 1000 volts. The timing relative to the laser pulse and the width of the gate are
controlled by varying the delays on the DDG channels to detect ions with different
charge and mass. A third channel with a 50-?s fixed delay synchronizes the digital
camera. Since the maximum frame rate of the camera is 735 Hz, the laser is set to
run at 500 Hz instead of 1000 Hz for the camera to keep up. Even at this rate, a
half million frames (a typical experiment) can be taken within 20 minutes.
54
Chapter 4
CORRELATION DETECTION
Correlation detection employed in this study is a type of statistical approach
to identify the ions ejected from the same explosion event so that the pre-explosion
geometry of the parent molecule or molecular ion can be determined. In order to
apply correlation detection, three requirements have to be met by the spectrometer
and data acquisition device. The first requirement is simultaneous collection of all
explosion partners ejected into 4? sr, which is fulfilled by the image spectrometer
described in Sec. 3.2. The second and third are that each single frame contains
ejected ions from one and only one laser pulse and a large number of frames need
to be collected to reduce statistical errors. The high frame-rate digital camera syn-
chronized to the laser repetition rate described in Sec. 3.3 meets these requirements
and is able to achieve this kind of data acquisition in a reasonable time (half million
frames in less than 20 minutes). Although correlation detection is able to handle
multiple events in each frame, there are two constraints on the number of events.
The first is concerned with ions from different channels landing at the same place
on the MCP, which can be minimized by keeping the sample pressure low and by
gating the MCP so that only a subset of all the ejected ions are detected (Sec. 3.2).
The second involves saturation of a specific explosion channel which exists in too
many frames. False correlation introduced by this channel needs to be eliminated
55
or minimized before real results can be extracted. The techniques of correlation
detection discussed in this chapter are image labeling [99] (Sec. 4.2), coincidence
imaging [63] (Sec. 4.3), and joint variance [103] (Sec. 4.4).
4.1 Image Spectrum
We begin our discussion by describing the characteristics of the image spec-
trum. As each laser pulse is focused at the interaction zone, a number of molecules
are ionized and subsequently explode. Depending on the timing of the MCP gate
(Page 51), tens of ions with specified charge and mass will generate bright spots on
the phosphor screen. As these bright spots are captured by the digital camera in a
single frame, up to about a hundred pixels will have values beyond the background.
The camera frames streamed to the SCSI drive from the digital camera are
stored by recording the value of each pixel. Since each frame has 128?128 pixels
and each pixel occupies one byte (8 bits), each frame takes 16 kilobytes (kB) of disk
space and a typical experiment of 500,000 frames occupies about 8 gigabytes (GB).
This huge size makes the data file extremely hard to be transfered and analyzed.
Fortunately, since only a few percent of the pixels (100 out of 16 thousand) have
values higher than the background (Page 53), these frames can be compressed dra-
matically into much smaller files [126]. The procedure of image compression and
recovery is given in Appendix C.1. Only the pixels with values higher than the
background (nonzero pixels) are recorded in the compressed image file. A typical
compressed image file containing 100,000 to 500,000 frames only occupies several to
56
a few hundred megabytes (MB) of disk space and can be stored or transfered easily.
In general, each ion striking the MCP illuminates a cluster of adjacent pixels on the
camera. The number of pixels within a cluster is typically between 1 and 5. The
procedure to group the adjacent pixels into a cluster and to determine the centroid
of each cluster is given in Appendix C.2. The information of the clusters was stored
in a separate file.
Since there are fewer than 100 nonzero pixels on each camera frame that
have more than 16 thousand pixels, the single frame looks rather like a collection
of random bright spots (upper three and lower left images in Fig. 4.1). However,
if thousands of single frames are added together to form a composite or average
image, structures corresponding to the molecular dynamics induced by the intense
laser field emerge. The lower right image in Fig. 4.1 is a composite of 500,000 frames
(synchronized to 500,000 laser pulses) and displays the momentum distribution of
doubly and triply charged N and O ions originating from the Coulomb explosion of
NO2. As the center of the image coincides with the center of mass of the explosion,
the radius R, where a charge appears on the image, is proportional to the momentum
of this charge,
p =
radicalbigg
qeEm
2l R, E =
qeE
4l R
2, (4.1)
where E = p2/(2m) is the kinetic energy, m and qe are the mass and charge of the
ion, l is the distance from the laser focal point to the image MCP (Fig. 3.6), and
E is the static field strength. (Detailed discussion about Eq. 4.1 is given in Ap-
pendix B.1.) Thus, the image is called the momentum distribution or momentum
57
+ + +
+ ? (total 500,000 frames) ?
Figure 4.1: The composite image of NO2 Coulomb explosions (lower right image)
showing the momentum distribution of Nq+ and Oq+ ions (75% q = 2 and 25% q =
3). This distribution contains 500,000 camera frames, i.e., 500,000 laser shots, each
centered at 800 nm, with a pulse width of 100 fs and linearly polarized (horizontal
in the images) with a peak intensity of 2?1015 W/cm2. The vertical peak near the
center is composed of Nq+ ions while the distribution parallel to the polarization
axis is a mixture of both Nq+ and Oq+ ions. Four samples of single frames (upper
three and lower left) are displayed.
spectrum. However, this is only true for the ions ejected approximately parallel to
the MCP with no perpendicular momentum component (p ? pbardbl and p? ? 0). In
general, the momentum distribution on the image is distorted by the ion trajecto-
ries with p? negationslash= 0. Although the ions with p? negationslash= 0 can be eliminated and the correct
momentum and angular distributions can be extracted by image deconvolution tech-
niques [100, 127] ? [130], the deconvolved images cannot be utilized to identify decay
partners.
It is important to understand that deconvolution is not required for correlation
images generated by the techniques described later in this chapter. This is because
the peak correlation signals are due to the highest concentration of ions per unit
area on the image [100, 103]. The concentration depends on the initial direction of
58
0 20 40 60 80 100
0
20
40
60
80
100
20 40 60 80 100
20
40
60
80
100
(a) (b)
(c) (d)
Figure 4.2: Locations where charges ejected isotropically into 4? sr with the same
energy and a Gaussian energy spread will accumulate on a 2D image: (a) an isotropic
distribution of charges in the absence of the static electric field form a sphere; (b)
a simulated 2D image when all trajectories are collected on the MCP; (c) a surface
plot of a quarter of the upper right image, the charge signal peaks at the edge of this
bowl-shaped distribution where land the trajectories initially ejected parallel to the
MCP (p? ? 0); (d) a corral-shaped distribution formed when only the trajectories
with p? ? 0 are collected.
the explosion trajectories. As shown in Fig. 4.2, the highest concentration of ions for
a given momentum corresponds to the trajectories with p? ? 0. As a small region
around the peak is isolated in correlation detection, the explosion events with p? ? 0
are extracted. In general, ions ejected from these events (p? ? 0) will land on the
detector at the largest distance from the center of the image, which is proportional
to the momenta of these charges (Eq. 4.1).
59
4.2 Image Labeling
Image labeling [99] is the 2-dimensional analog of covariance mapping [98] that
has been employed successfully to analyze 2-atom Coulomb explosions. The tech-
nique allows all the correlated partners to be visualized with a family of momentum
images similar to the total momentum distribution image but with only the corre-
lated partners displayed. Each of these images is called a correlation image.
Covariance mapping, introduced by Frasinski et al. [98], is capable of reducing
the ambiguity for correlating explosion partners in TOF waveforms. The covariance
matrix, which identifies the explosion partners in the waveform at times t? and t?,
is defined as
??? = ?S(t?)S(t?)???S(t?)??S(t?)?, (4.2)
where ?? means the average over N laser shots,
?S(t?)? = 1N
Nsummationdisplay
k=1
Sk(t?),
?S(t?)S(t?)? = 1N
Nsummationdisplay
k=1
Sk(t?)Sk(t?). (4.3)
The correlation coefficient is given by,
C?? ? ????
???
, (4.4)
where
?? =
radicaltpradicalvertex
radicalvertexradicalbt 1
N
Nsummationdisplay
k=1
[Sk(t?)??S(t?)?]2. (4.5)
Zhu and Hill [22] were the first to extend this technique to two-dimensional
images. To produce a correlation image, the composite momentum distribution
60
image in Fig. 4.1 is divided into sectors, S(R,?). The size of each sector is chosen
such that ?E(? R?R, see Eq. 4.1) and ?? are the same for all sectors. This is
shown in Fig. 4.3. The correlation coefficient, Cij, between sectors i and j that are
actually two-dimensional indices representing (Ri,?i) and (Rj,?j), is given by [99]
Cij = ?ij?
Si?Sj
,
?ij = ?(Si ??Si?)(Sj ??Sj?)? = ?SiSj???Si??Sj?, (4.6)
?Si =
radicalbig
?(Si ??Si?)2?,
where ?ij is the covariance between sectors i and j, ?Si is the square root of the
variance in the counts for sector i with laser shots, and ?? means the average over
N laser shots,
?Si? = 1N
Nsummationdisplay
k=1
Si(k),
?SiSj? = 1N
Nsummationdisplay
k=1
Si(k)Sj(k). (4.7)
Typically, about 500 sectors are employed and there are about 10 ? 25 pixels in
each sector. For Cij to be meaningful, it is critical that the average number of ions
generated per sector per laser shot be less than one and that the image for each laser
shot be stored individually. This matrix identifies explosion partners at locations
i and j. With two-dimensional images, i and j are two-dimensional indices and
Cij, or ?ij, is four-dimensional, which is impossible to display. It is not necessary,
however, to display or even calculate all the elements because useful correlations
can be obtained from a judicious subset of two-dimensional projections [22]. In
particular, the ith correlation image is assembled by replacing the ion counts at
61
AVERAGE IMAGE CORRELATION IMAGE
POLARIZATION AXIS
0 1
Figure 4.3: Correlation images. The concentric circles and spokes in the left panel
divide the momentum distribution image (the same as that displayed in Fig. 4.1) into
sectors. The gray arrow indicates labeled ions (i.e., a subset of ions with a narrow
momentum distribution moving downward, 6 o?clock); the white arrows indicate the
correlated sectors (ions moving toward 2 and 10 o?clock). The right panel shows the
correlation image for the Coulomb explosion where all three atomic ions are ejected
simultaneously. The gray (white) arrows indicate the final momenta of the labeled
(correlated) charges.
the pixels of the jth sector in the composite image (i.e., the left panel in Fig. 4.3)
with the correlation coefficient Cij. This is actually a two-dimensional projection
from the four-dimensional correlation matrix, which is labeled or selected by the
ith sector. The right panel in Fig. 4.3 shows one correlation image of a 3-body
Coulomb explosion of NO2. There are as many correlation images (two-dimensional
projections) as there are sectors.
The advantage provided by the correlation images is that they not only tell
us which ions are ejected simultaneously as covariance mapping of TOF spectrum
does, they offer clear pictures of both linear and bent Coulomb explosions which
can be viewed easily. Figure 4.4 shows how linear and bent three-body channels can
be isolated for Coulomb explosions of CO2 as the ions ejected together are clearly
62
0 1
POLARIZATION AXIS
LINEAR EXPLOSION BENT EXPLOSION
Figure 4.4: Two correlation images for the Coulomb explosion of CO2 taken under
the same conditions as in Fig. 4.1. Ions moving toward 3 o?clock in the left image
and those moving toward 6 o?clock in the right image are labeled. As the correlated
ions show up on the same correlation images, linear explosion events on the left and
bent events on the right are displayed and isolated.
displayed, while linear two-body and bent three-body explosion channels of NO2 are
distinguished in Fig. 4.5.
Although the Cij values for the sectors can be used to determine the degree
of correlation, it is more flexible and visual to work directly with the pixels, as
shown in Fig. 4.6. The image on the right was generated by taking the difference
between the average distribution shown on the left (all 500,000 frames) and an
average distribution composed only of frames that have nonzero counts in the ith
sector ? a selective average [99] (Fig. 4.7). The selective average and correlation
image projected from the four-dimensional Cij matrix give about the same result
but the selective average approach provides higher resolution, better control over the
size and location of the labeled sector, and an easier visualization without projection
or dimensionality reduction. As the momentum (location) of the labeled sector is
63
0 1
POLARIZATION AXIS
TWO-BODY CHANNEL THREE-BODY CHANNEL
Figure 4.5: Two correlation images for NO2 taken from the same data set as in
Fig. 4.1 showing a two-body dissociation channel (left) and a three-body channel
(right). Ions moving to the left along the polarization axis in the left image and
those moving in the direction perpendicular to the polarization axis in the right
image are labeled so that two-body and three-body events are isolated.
being varied, the ejection dynamics as a function of the momentum of a particular
partner can be studied. While the momentum of the selected charge (gray arrow in
Fig. 4.6) is predefined, the momenta of the correlated ions (white arrows in Fig. 4.6)
are found at the peaks of the selective average image. With the momenta of all
three charges ejected from the same explosion event relative to the center of mass
of the dynamics (the center of the image), the pre-explosion molecular structure
(bond length and bond angle) of the triatomics in this study (CO2 and NO2) can
be determined. Assuming a pure Coulomb interaction between the correlated ions
and energy conservation, Ek = EC, we have
EC =
3summationdisplay
i>j
Kqiqje2
rij , Ek =
1
2
3summationdisplay
i=1
p2i
mi, (4.8)
where i, j = 1, 2, 3. The subscripts 1 and 2 represent the two outer ions (oxygen)
while 3 represents the center ion (carbon or nitrogen). The mass, charge, initial
64
COMPOSITE AVERAGE SELECTIVE AVERAGE
POLARIZATION AXIS
0 1
?CM
Figure 4.6: Selective average. The image in the left panel and the gray and white
arrows are the same as in Fig. 4.3. The correlation image (selective average) in the
right panel is the difference between averaging only those frames that have nonzero
counts in the labeled sector and all 500,000 frames.
position and final momentum of each ion are given by mi, qie, vectorri and vectorpi respectively.
The initial Coulomb potential energy and final kinetic energy are EC and Ek. The
bond lengths are r13 = |vectorr1? vectorr3| and r23 = |vectorr2? vectorr3|, while the angle between vectorr13 and
vectorr23 is the bond angle. Detailed calculation is given in Appendix D.
4.3 Coincidence Imaging
While image labeling (Sec. 4.2) can isolate specific bond angles, extracting the
relative strength of the explosion signal requires a triple-coincidence ? coincidence
imaging [63]. This is because the correlation image (selective average) contains
events with a spread in p? about p? = 0, as indicated by the width of the correlated
spots (Figs. 4.6 and 4.8b). We can reduce this spread by reducing the size of
the correlated spots. Using the fact from Fig. 4.2 that the highest concentration
65
Nonzero Count
Same Labeled SectorZero Count
? =
Normalized Hits Average Selective Average
Figure 4.7: Selective average. The images in the upper row shows that a selec-
tive average is generated by taking the difference between an average distribution
composed only of frames that have nonzero counts in the labeled sector (lower right
image) and the average distribution of all 500,000 frames. The lower row shows two
single frames with zero and nonzero counts in the same labeled sector.
of charges for a given momentum allows us to extract events with p? = 0, if we
focus on a narrow region about the peak point. Figure 4.8 shows an example of
coincidence imaging for the 6-electron explosion channel of CO2. Figure 4.8a is
the momentum distribution of the ejected doubly-charged ions, while a selective
average containing only the ions ejected simultaneously is shown in Fig. 4.8b. After
the momenta of all three explosion partners (vectorpOleft, vectorpC, vectorpOright) were determined from
Fig. 4.8b, a narrow region around each momentum vector is chosen and an image
(Fig. 4.8e) can be obtained by integrating only the frames containing ions in all three
regions simultaneously. Since the ions are constrained in a very narrow region, they
are from the events with p? = 0. The strength of the signal or probability for
a particular explosion is proportional to the number of frames containing all the
66
explosion partners.
We determine if a single frame contains all the explosion partners in coinci-
dence imaging in the following way. The conditions under which the experiment
was run allowed only a few explosion events per laser shot. As the clustered images
(Page 57 and App. C.2) are employed to avoid the signal of one charge spreading
over several pixels, there are two criteria a cluster must meet to be considered to
be part of a real triple-coincidence. The first signature is that the location of the
centroid of a valid cluster must fall within a circle of radius r0 around the final
momentum of each ion (vectorpOleft, vectorpC, and vectorpOright, Fig. 4.8). The size of r0 is related to
the uncertainty in determining the location of the centroid, which typically is 1 to 3
pixels because the peak number of pixels in a cluster is roughly 1 ? 5. It is set that
r0 = 1.5 and 2.5 for triple-coincidence measurements for CO2 and NO2 respectively;
r0 = 1.5 for double-coincidence measurements for H2. The second requirement for
inclusion is that there can only be one centroid (cluster) within each circle. To
avoid invalid clusters, each of which is illuminated by two or more ions landing near
each other on the MCP, valid clusters are restricted to those having a maximum
number of pixels (M). The value of M is typically between four and ten where the
occurrence frequency of clusters as a function of number of pixels in a cluster falls
to about 10% of its peak value (App. C.2). As shown in Fig. 4.8g, a centroid may
be partially enclosed by the circle. Centroids falling entirely within the circle (block
1 in Fig. 4.8g) contribute unity to the count while those that are partially enclosed
(block 2 in Fig. 4.8g) contribute fractional counts. The contribution to the signal
strength for a specific frame is given by the product of the count from each ion circle
67
?g = 0? ?
g = 10?
(c) (d)
(e) (f)
(a) (b)
(g)
(1) (2)
Er
Figure 4.8: Center-of-mass momentum distributions of ejected C2+ (center spots)
and O2+ (outer spots) ions associated with the symmetric CO6+2 Coulomb explosion
channel, induced by linearly polarized (horizontal), 100 fs pulses at 800 nm focused
to 2?1015 W/cm2. (a) Average distribution composed of 500,000 frames. (b) Selec-
tive average showing only the momenta of the three charges ejected simultaneously
from a CO6+2 precursor with a bond angle of 163?. (c) A ball-and-stick diagram
of CO2 defining the molecular axis (dashed line) and the geometric angle, ?g = 0?,
which is the angle between the laser polarization and the molecular axis (the line
connecting two outer ions). (d) The same as (c) for ?g = 10?. (e) Triple-coincidence
image (see text) corresponding to (c). (f) The same as (e) but corresponding to (d).
The arrows in (e) and (f) represent the final momenta of the ejected ions (vectorpOleft, vectorpC,
vectorpOright). The angle between vectorpOleft and vectorpOright defines the center-of-mass angle ?CM.
(g) Grid illustrating how contributions to the signal strength are determined. Block
1 makes a unit contribution while block 2 contributes only 0.6, corresponding to the
area enclosed by the circle.
68
(1) (2)
(4)(3)
Figure 4.9: Images showing the anisotropies in the ejection angular distribution of
Coulomb explosions for selected bond angles: (1) CO6+2 at 180?, (2) CO6+2 at 163?,
(3) NO6+2 at 132? and (4) H+2 . These images were constructed by adding a series of
coincidence images similar to those shown in Figs. 4.8e and 4.8f. The dashed lines
represent the axes, parallel and perpendicular to the laser polarization, which cross
at the center-of-mass. In (1) ? (3), the largest arcs correspond to O2+ ions.
? if each centroid were only 50% enclosed, this frame would contribute 0.125 to the
total count.
By keeping the magnitudes of the momenta along with the relative angles
between them constant (compare the stick figures in Figs. 4.8c and 4.8d), the prob-
ability for strong-field induced Coulomb explosion as a function of the geometric
orientation angle (?g, defined as the angle between the molecular axis and the polar-
ization axis) for specific bond angles can be determined from the relative strength
of the explosion signal on triple-coincidence images like Figs. 4.8e and 4.8f, and an
image can be generated to reveal the anisotropic angular distribution graphically.
Figure 4.9 shows the probability for Coulomb explosion as a function of ?g for several
bond angles as well as for H2.
69
4.4 Joint Variance
When selective averages and coincidence images are contaminated with false
correlation because of saturation of one or more very strong channels, covariance
matrix (?ij in Eq. 4.6) does a better job of determining true ejection partners.
Correlating three or more components, however, requires higher order variance cal-
culations. Frasinski, et al. [64] have explored this approach and suggested
???? = ?S(t?)S(t?)S(t?)???S(t?)S(t?)??S(t?)?
??S(t?)S(t?)??S(t?)???S(t?)??S(t?)S(t?)?
+2?S(t?)??S(t?)??S(t?)?, (4.9)
as the appropriate triple variance of TOF spectrum to be calculated. However,
the image analogue to ???? is found to be quite noisy, making it difficult to extract
useful information. As an alternative, a product of pairwise-determined covariances,
or joint variance [103], has been employed successfully to reduce the effects of the
contaminating channel.
Following the definition of the regular covariance (?ij) in Eq. 4.6, joint variance
?J of B with A and C, which is actually a measure of triple-correlation of all three
charges, is given by
?JA ?BC = ?AB ??BC,
= (?SASB???SA??SB?)?(?SBSC???SB??SC?), (4.10)
where the bar over B indicates the common entity in the calculation. Let us assume
an explosion channel ABC ? A+B+C, which is the target of the correlated-partner
70
search, is contaminated by avery strong channel AC ? A+C. Even though?AC may
be significant, it does not reflect the real 3-body correlation appropriately. However,
it plays a primary, or at least, the same role as ?AB and ?BC in the triple coincidence
approach, skewing the results to higher values. By contrast, in joint variance, ?AC
is forced to play only a secondary role at best to reduce the contamination as we
will show in the next chapter.
71
Chapter 5
EXPERIMENTAL RESULTS AND ANALYSIS
Exploiting the ability of the image spectrometer [22, 100] to collect all the
ejected charges over 4? sr (Chap. 3) and correlation techniques [63, 99, 103] to iden-
tify ejections at different bond and orientation angles (Chap. 4), Coulomb explosions
of small molecules were investigated with 800 nm, 100 fs laser pulses at intensities
of 0.1 ? 5?1015 W/cm2. The explosion dependence on the bond angle of triatomic
molecules and orientation angle of diatomic and triatomic molecules was studied.
In the bond-angle studies (Sec. 5.1), CO2 [62] and NO2 (with equilibrium bond
angles of 180? and 134?, respectively) were the targets. This allowed us to compare
and contrast the dynamics in nominally linear and bent systems. Energies, momenta
and signal strengths of the fragment ions from Coulomb explosions were measured
as functions of bond angle with correlation detection techniques (Secs. 5.1.1.1 and
5.1.1.2). Large-amplitude vibrations were observed in explosion signals of both CO2
and NO2 and analyzed with the bending vibrational amplitude of the molecules (Sec.
5.1.1.3). The measured explosion energies were compared with predictions from the
enhanced-ionization (RC) and electron-screening models (Sec. 5.1.2).
In the orientation-angle studies (Sec. 5.2), the ejection anisotropy of triatomic
(CO2 and NO2) [63] and diatomic (H2, N2, and O2) Coulomb explosions were in-
vestigated with correlation detection techniques and by comparing image spectra
73
obtained with linearly and circularly polarized fields. The results are discussed in
Sec. 5.2.1. The different behavior (ejection angular spreads and degrees of dynamic
alignment) of the tested molecules was explained in terms of their ionization stages,
induced dipole moments, rotational constants as well as the time that the precursor
molecular ions spend in the field prior to the Coulomb explosions (Sec. 5.2.2).
5.1 Explosion Energy Measurement
5.1.1 Experimental Results
The focus of this study was the symmetric 6-electron explosion channel where
CO2 or NO2 was ionized six times prior to the Coulomb explosion into energetic
O2+ + C2+ + O2+ or O2+ + N2+ + O2+ ions. The experiments were taken at
peak laser intensities (I0) ? 2?1015 W/cm2, pulse widths (?) ? 100 fs, and center
wavelength (?) ? 800 nm. Each experiment consisted of 500,000 camera frames
corresponding to the same number of laser pulses. The sample gas pressures were 5
and 30?10?9 Torr for CO2 and NO2, respectively. The NO2 signal was roughly six
times weaker than the CO2 signal. The potential between the laser focal point and
the MCP in the image spectrometer was Vint = 1500 volts, so that the static field
pushing the ions to the MCP was 262 V/cm (see Fig. 3.6). The image calibration
was ? = 0.293 mm/pixel (Page 146 in App. B.1) ? one pixel on the image (camera
frame) corresponded to a distance of 0.293 mm on the image MCP. Therefore, the
momentum and energy of the ion at radius r (in the unit of pixel) on the image
spectrum were p = 1.15?qmr and E = 0.00986qr2, where q and m are the charge
74
AVERAGE
IMAGE
LINEAR
CHANNEL
POLARIZATION AXIS
BENT
CHANNEL
Figure 5.1: Momentum distributions of the C2+ and O2+ ions associated with the
symmetric CO6+2 Coulomb explosions: average image composed of 500,000 laser
pulses (left); correlation images (selective averages, Sec. 4.2) showing only the mo-
menta of the three charges ejected simultaneously from a CO6+2 parent that is linear
and parallel to the polarization axis (center) and bent at a bond angle of ? 165?
with the plane of the molecule parallel to the MCP face (right). The geometric cen-
ter of each image is the center of mass of the explosion and the arrow indicates the
polarization axis. The width of the MCP gate (Sec. 3.2) was 320 ns, which covers
all C2+ and O2+ ions and a small amount of C3+ and O3+ ions (less than 20% of
the total signal) due to the overlap of signals in time.
and mass of the ion in the units of elementary charge and atomic mass, respectively.
The momentum p and energy E are in atomic units (a.u.) and electron volts (eV),
respectively.
5.1.1.1 CO2 Results
The momentum distribution of the doubly charged atomic ions from CO2
explosions is shown in the left image in Fig. 5.1. The distribution is composed of
O2+ ions with momenta of 160 to 350 a.u. (10% of peak) and directed parallel (within
a small angular spread) to the polarization axis (pX) and C2+ ions with momenta
of 0 to 200 a.u. and directed perpendicular (within a small angular spread, too) to
the polarization axis (pY ). The ratio of the signals of the middle peak (C2+) and
75
peaks on both sides (O2+) is about 1:2.5; the small deviation from the expected 1:2
(one carbon and two oxygen ions from CO2) is due primarily to the gain variation
on the MCP. The center and right images in Fig. 5.1 are two examples where image
labeling (Sec. 4.2) was employed to isolate explosion events that occurred at two
different bond angles (?b). When C2+ ions (the white spots in the middle of center
and right images in Fig. 5.1) with specific pY (pX ? 0 in this case) are selected,
the correlated O2+ ions exist in these images as well, which allows the momenta
of all the ions ejected simultaneously to be determined. While the momentum of
the selected C2+ ions is predefined, the momenta of the O2+ ions are taken to be
at the peaks of the correlated ions (the left and right spots in the center and right
images in Fig. 5.1). At the same time, the center-of-mass angle (?CM, the angle
between the momenta of two oxygen ions) is calculated from pX and pY of the
oxygen ions ? ?CM = 2tan?1(pX/pY ). The pre-explosion bond angle (?b) and bond
length can be determined from the momenta and ?CM, as described in the next
paragraph. Simultaneous ejection means that the correlated ions must obey energy
and momentum conservation. Thus, pX for the two groups of O2+ ions in the center
and right images are equal and opposite while pY for C2+ are equal and opposite
to the sum of pY for the two groups of O2+ ions. We note that this measurement
does not assume momentum conservation. In stead, the result ? data points of
|pY| for C2+ and 2|pY| for O2+ are consistent with each other within their standard
deviations ? confirms that the momenta are actually conserved in Fig. 5.2, which
shows how |pX| and |pY| for the correlated ions depend on the bond angle of the
parent molecule at the time of the explosion. At the same time, Fig. 5.2 also shows
76
140 150 160 170 180
qb H?L
0
50
100
150
200
250
P Y
Ha.u
.L
0
50
100
150
200
250
P X
Ha.u
.L
Figure 5.2: Momenta of the three charges ejected simultaneously as a function of
the bond angle (?b) in the symmetric CO6+2 Coulomb explosion channel. Each data
point represents the momentum measured from the center-of-mass of the charge
distribution in a correlation image similar to the two right images in Fig. 5.1; the
error bars represent the standard deviation in this determination over four runs.
The upper series of points (green diamonds) correspond to |pX| for O2+; |pX| is
virtually the same for the two groups of ions so only one point is plotted for each
angle. The lower trace is |pY| for C2+ (blue stars) and 2|pY| for O2+ (red triangles);
pY is also the same for the two groups of O2+ ions. The solid curves represent the
calculated momenta assuming RC equals 0.215 nm (see App. D for details of this
calculation).
that the data points are consistent with the calculated momenta assuming a constant
bond length, indicating that RC of CO6+2 explosions is nearly independent of ?b over
the observed range [62].
The explosion energy (Ek), which is assumed to be equal to the initial Coulomb
77
140 150 160 170 180
qb H?L
0.19
0.2
0.21
0.22
0.23
R C
H
mnL
76
72
68
64
E k
HVe
L
Figure 5.3: Bond length (left axis) and explosion energy (right axis) as functions
of bond angle for CO6+2 symmetric explosions with qO = qC = 2 measured by image
labeling (Sec. 4.2). The error bars are the standard deviation of four experiments
under the same conditions.
potential energy (EC), can be calculated from the momenta of all three charges by,
Ek = 12
3summationdisplay
i=1
p2i
mi, (5.1)
where i = 1, 2, 3. The mass and final momentum of each ion are given by mi and
vectorpi respectively (see Eq. D.8). The initial molecular structure prior to the explosion,
the CO bond length R and bond angle ?b, was then extracted as described in Ap-
pendix D. The bond length (R = RC, assuming the explosion occurs at RC of the
enhanced ionization, see Sec. 2.1.1) and explosion energy (Ek) as functions of bond
angle are plotted in Fig. 5.3. The measured RC for CO6+2 explosions is constant
(0.215 nm) and varies by less than 0.1% for the observed ?b between 145? and 180?.
The explosion signal strength was measured at each bond angle by coincidence
imaging (Sec. 4.3), where a narrow region around each of the three momenta deter-
mined by image labeling, as mentioned above, was chosen and the signal strength
78
145 150 155 160 165 170 175 180
qb H?L
0.2
0.4
0.6
0.8
1
evi
tal
eR
la
ngi
S
Figure 5.4: Explosion signal strength as a function of bond angle for CO6+2 symmet-
ric explosions initiate with qO = qC = 2 measured by coincidence imaging (Sec. 4.3)
at the momenta of the three charges determined by selective averages. The error
bars are the standard deviation of four experiments under the same conditions.
was determined by the number of frames containing ions in all three regions simul-
taneously. The measured distribution is shown in Fig. 5.4 and peaks near ?b = 173?.
This observation will be discussed later in Sec. 5.1.1.3.
5.1.1.2 NO2 Results
While the measured momenta and energies of CO2 explosions are consistent
with a constant RC, that for NO2 is totally different. The momentum distribution
for the doubly charged atomic ions of NO2 is shown in the left image in Fig. 5.5. The
distribution is composed of O2+ and N2+ ions with momenta of 90 to 300 a.u. (10% of
peak) and directed parallel (within a small angular spread) to the polarization axis
(pX) and N2+ ions with momenta of 0 to 220 a.u. and directed perpendicular (within
a small angular spread) to the polarization axis (pY ). The ratio of the signals of the
79
middle peak (N2+) and peaks on both sides (O2+ and N2+) is about 1:4.5. This large
deviation from the expected 1:2 (one nitrogen and two oxygen ions from NO2) is due
primarily to a two-body explosion channel, as will be discussed later. Similar to the
CO2 case, Fig. 5.6 shows how |pX| and |pY| for the correlated ions depend on the
pre-explosion bond angle of the parent molecule. Although the consistency between
data points of |pY| for N2+ and 2|pY| for O2+ confirms momentum conservation,
data points of both |pX| and |pY| deviate from the constant-RC curves. Especially,
|pX| points deviate further as the bond angle increases. It is clear that RC for NO2
explosions is not constant. The bond length (R = RC) and explosion energy (Ek)
as functions of bond angle, derived from the measured momenta with the same
method as in the CO2 case, are plotted in Fig. 5.7, which confirms that RC for
NO2 explosions is not constant. The curve of RC (Ek) shows a monotonic increase
(decrease) as ?b approaches 180?.
Structurally, the momentum distribution of NO2 looks similar to that of CO2,
but the dynamics are quite different. As shown in the center plot of Fig. 5.5, a
very strong two-body correlation along the polarization axis was revealed by image
labeling, which was not observed in the CO2 spectrum. The fact that the ratio of
signals in the middle and side peaks in the momentum distribution (about 1:4.5,
left image in Fig. 5.5) deviates from the expected 1:2 confirms that there are ions
from channels other than three-body explosions in the side peaks. This channel
probably involves an N?O explosion because there is a strong 2-photon dissociation
for NO2 ? O + NO with little kinetic energy. As one oxygen leaves the molecule,
NO ? N2+ + O2+ explodes as a diatomic. The second possibility is an O2+?O2+
80
AVERAGE
IMAGE
TWO-BODY
CHANNEL
POLARIZATION AXIS
THREE-BODY
CHANNEL
Figure 5.5: Two- and three-body explosion channels of NO2. Left: average of
N2+ and O2+ ions over 500,000 laser shots; center: selective average of two-body
correlation channel where the bright spot to the right is the selected sector; right:
selective average of three-body correlation channel where the bright spot in the
middle is the selected sector. Polarization axis is horizontal. The width of the MCP
gate (Sec. 3.2) was 330 ns, which covers all N2+ and O2+ ions and a small amount
of N3+ and O3+ ions (less than 20% of the total signal) due to the overlap of signals
in time.
explosion with the nitrogen atom dropped as a neutral. The impurity of NO and
O2 in the NO2 gas cylinder (less than 0.5%) is not enough to explain the observed
two-body explosion signal. However, due to the fact that the ejection energy (about
10 eV) and mass of N and O are very close to each other, correlation imaging (Sec.
4.2) could not discriminate N?O and O?O explosions on the image spectrum. At
the same time, due to the overlap of signals from two- and three-body channels,
covariance mapping of TOF spectrum (Sec. 4.2) could not produce definite results
to reveal which explosion, N?O or O?O, was more dominate, either. Discriminating
N?O from O?O explosion could have interesting consequences for laser control and
should be further pursued. This two-body channel was treated as noise for the
current study.
As verified by coincidence imaging measurements (Sec. 4.3), the strength of the
81
100 110 120 130 140 150 160 170 180
qb H?L
0
50
100
150
200
250
300
350
P Y
Ha.u
.L
0
50
100
150
200
250
P X
Ha.u
.L
Figure 5.6: Momenta of the three charges ejected simultaneously as a function of the
bond angle (?b) in the symmetric NO6+2 Coulomb explosion channel determined by
image labeling. The error bars represent the standard deviation in this determination
over seven runs. The upper series of points (green diamonds) correspond to |pX|
for O2+; |pX| is virtually the same for the two groups of ions so only one point is
plotted for each angle. The lower trace is |pY| for N2+ (blue stars) and 2|pY| for O2+
(red triangles); pY is also the same for the two groups of O2+ ions. The solid curves
represent the calculated momenta assuming RC equals 0.26 nm (see App. D).
two-body channel (double-coincidence) is more than 10 times stronger than linear
three-body explosions (triple-coincidence). The momenta of the ejected ions for the
three-body Coulomb explosion obtained by image labeling and the signal strength
obtained by coincidence imaging could be distorted, especially near ?CM (as well as
?b) close to 180?, where the spectra of these two channels overlap (Fig. 5.8).
As shown in the top plot of Fig. 5.8, the O2+ momentum increases smoothly
as C2+ momentum decreases from the edge of the selected group to the image
82
110 120 130 140 150 160 170 180
qb H?L
0.22
0.26
0.3
0.34
0.38
0.42
R C
H
mnL
7065
60
55
50
45
40
35
E k
HVe
L
Figure 5.7: Bond length (left axis) and explosion energy (right axis) as functions
of bond angle for NO6+2 symmetric explosions initiate with qO = qC = 2 measured
by image labeling (Sec. 4.2). The error bars are the standard deviation of seven
experiments under the same conditions.
center (center of mass) and peaks at linear configuration (?CM = 180?) for CO2 3-
body explosions. By contrast, the O2+ momentum from NO2 explosions determined
by selective average (center plot of Fig. 5.8) has an abrupt change near ?CM ?
180?. As will be shown, this distortion is caused by the strong two-body channel.
When an N2+ ion near the image center (zero momentum) is selected, the correlated
O2+ will only appear near the polarization (horizontal) axis as the momentum is
constrained by momentum conservation. At the same time, because of its larger
signal strength and higher occurrence frequency, O2+ ejected from the two-body
channel also appears in the selective average along the polarization axis, causing
accidental correlations in the three-body image-labeling measurement. This tends to
shift the positions of the correlated O2+ ions to smaller momenta because O2+ from
the 2-body channel has a smaller momentum than that from the 3-body channel.
83
As the selected N2+ moves away from the center of the image (larger momentum
perpendicular to the polarization axis), the correlated O2+ picks up perpendicular
momentum as well and the contamination from the two-body channel becomes less
significant.
Since the image labeling (selective average) measurement was strongly dis-
torted by the 2-body channel, joint variance [103], as described in Sec. 4.4, was
employed to analyze the signal of the 3-body explosion. Joint variance is able to
emphasize the existence of the N?O correlation in the triple-correlation and force
the correlation between two outer ions to play only a secondary role to reduce the
contamination from the two-body channel. As SA, SB and SC in Eq. 4.10 are taken
to be SOleft (signal of O on the left side of the image), SN (signal of N in the middle)
and SOright (signal of O to the right), joint variance (?J) should be calculated as
SN samples along the axis perpendicular to the polarization axis (similar to the
selected ions in image labeling) and SOright (SOleft) samples through the right (left)
half image to search for the peak position of ?J, where the real correlated O2+ from
the 3-body channel is located, in principle. However, since the selective average
already provides the approximate range for the momenta or locations of O2+ in the
real 3-body correlation and the accidental correlation from the 2-body channel, ?J
only needs to be calculated as SOright (SOleft) samples in the neighborhood around
the O2+ momenta determined by the selective average. As the correlated peaks
on selective averages are no larger than 20?20 pixels, ?J is calculated as SOright
(SOleft) samples in a 25?25-pixel box around these peaks, which is 50% larger than
the largest peak found in selective averages and only occupies less than 8% of the
84
Figure 5.8: Contour plots of average images with both selected (middle group)
and correlated (groups on both side) momentum vectors in three-body correlation
marked. Top and center: CO2 and NO2 images with the locations (momenta)
of correlated ions determined by selective average; bottom: NO2 images with the
locations (momenta) of correlated ions determined by jointvariance. The green (red)
circles mark the selected C2+ (N2+) ions while the blue ones mark the correlated
O2+ ions. The green and yellow arrows in the center plot indicate that the selected
N2+ at the bottom of the middle group is correlated with O2+ at the top of both
side groups. As the selected N2+ or C2+ is moved up along the center group, the
corresponding O2+ on each side moves down along its arc.
85
half image, to make sure no 3-body correlation would be missed as well as to save
computational time. The O2+ momenta of the 3-body channel are determined at
the weighted centers of the ?J distributions in the 25?25-pixel boxes, and the ex-
plosion signal strengths are calculated by integrating ?J in a 3?3-pixel area at the
weighted centers. These centers of the joint variance distribution are not necessarily
at the same locations as the peaks in the selective averages. In fact, the centers of
joint variance and peaks of selective average separate more as the bond angle gets
closer to 180?, as shown in Fig. 5.8. The momenta of the real 3-body channel are
thus taken to be at the weighted centers of joint variance. In the bottom plot of
Fig. 5.8, it is obvious that the abrupt change of the momentum near ?CM ? 180?
is reduced except at one point right at ?CM = 180?, where the signal strength of
the 3-body explosion drops to almost zero, as shown in Fig. 5.9. For bond angles
smaller than 150?, correlated O2+ momenta determined by both selective average
and joint variance measurements actually overlap with each other (Fig. 5.8).
Explosion signal strengths measured by joint variance and coincidence imaging
(triple-coincidence) at the momenta determined by joint variance along with double-
coincidence for the two O2+ are shown in Fig. 5.9. While the double-coincidence
points increase monotonically and triple-coincidence points stay roughly constant as
?b approaches 180? from 140?, the joint variance points peak near ?b = 140?, which is
fairly close to the equilibrium bond angle of NO2 in its ground state (?be = 134?). At
the same time, data points from all three measurements follow similar trends for ?b <
140?. Since double- and triple-coincidence signals follow the same trend at ?b < 140?
(Fig. 5.9), this is consistent with that the signals come from the same 3-body channel
86
110 120 130 140 150 160 170 180
qb H?L
0.5
1
1.5
2
2.5
evi
tal
eR
la
ngi
S
Figure 5.9: Explosion signal strength as a function of bond angle for NO6+2 symmet-
ric explosions initiate with qO = qN = 2 measured by joint variance (blue square),
triple-coincidence (red triangle), at the selected N2+ momentum and correlated O2+
momenta determined by joint variance, along with double-coincidence (green star)
at the O2+ momenta. The error bars are the standard deviation of seven experiments
under the same conditions.
and there is no excess correlation from the 2-body channel to contaminate the triple-
coincidence measurement. Thus, it is believable that the signal strength measured by
triple-coincidence (Fig. 5.9) as well as the momenta determined by selective average
(Fig. 5.8) for ?b < 140? belong to the real 3-body explosion channel. Furthermore, as
the momenta and relative signal strength determined by joint variance are essentially
the same as those determined by selective average and triple-coincidence for ?b <
140? (Figs. 5.8 and 5.9), the joint variance measurement is also believable. As
the bond angle approaches 180?, the departure of the signal strength measured
by double-coincidence from that measured by triple-coincidence (Fig. 5.9) confirms
that there are additional O?O correlations (evidently from the 2-bodychannel) other
than that due to the 3-body channel. The signal measured by triple-coincidence is
87
enhanced by the accidental coincidence from the 2-body channel as ?b approaches
180?. The fact that joint variance emphasizes the existence of the N?O correlation in
the triple-correlation and reduces the contribution from the false 2-body correlation,
along with that the relative probabilities determined by joint variance and triple-
coincidence as well as the momenta determined by joint variance and image labeling
scale together at smaller bond angles (< 140?), gives credence to our believe that
the probabilities and momenta determined by joint variance at larger angles are also
valid.
Figure 5.10 shows how |pX| and |pY| determined by joint variance for the
correlated ions depend on the pre-explosion bond angle of the parent molecule.
Although the data points still do not follow constant-RC curves, the deviation near
180? is smaller than that in the result of image labeling (Fig. 5.6). In the same
fashion as in the CO2 case, the explosion energy (Ek) and the initial molecular
structure (RC and ?b) can be determined from the momenta of the ions and ?CM.
The bond length (RC) and explosion energy (Ek) as functions of bond angle are
plotted in Fig. 5.11. The average RC is 0.26 nm while RC at the peak of explosion
signal (?b = 140?, Fig. 5.9) is 0.254 nm. As ?b increases from about 110? to 180?, RC
(Ek) increases (decreases) from 0.211 to 0.315 nm (71.3 to 46.6 eV) monotonically
beyond the standard deviation of the measurement (Fig. 5.11), indicating that NO2
appears to behave differently from CO2 in the intense laser field even for the same
3-body explosion channel.
88
100 110 120 130 140 150 160 170 180
qb H?L
0
50
100
150
200
250
300
350
P Y
Ha.u
.L
0
50
100
150
200
250
P X
Ha.u
.L
Figure 5.10: Momenta of the three charges ejected simultaneously as a function of
the bond angle (?b) in the symmetric NO6+2 Coulomb explosion channel determined
by joint variance. The error bars represent the standard deviation in this determi-
nation over seven runs. The upper series of points (green diamonds) correspond to
|pX| for O2+; |pX| is virtually the same for the two groups of ions so only one point
is plotted for each angle. The lower trace is |pY| for N2+ (blue stars) and 2|pY| for
O2+ (red triangles); pY is also the same for the two groups of O2+ ions. The solid
curves represent the calculated momenta assuming RC equals 0.26 nm (see App. D).
5.1.1.3 Large-Amplitude Vibration
The explosion signal strength (relative probability) of CO2 and NO2 (Figs. 5.4
and 5.9) shows large-amplitude vibration. We will compare this observation to the
bending vibrational amplitude at Re of the triatomic molecules. The vibrational
amplitude (??b) can be estimated from kR2e??2b/2 = (v + 1/2)hc?2 assuming har-
monic oscillation, where v is the quantum number of the energy level, k and ?2
89
110 120 130 140 150 160 170 180
qb H?L
0.225
0.25
0.275
0.3
0.325
0.35
0.375
R C
Hmn
L
70
65
60
55
50
45
40
E k
HVe
L
Figure 5.11: Bond length and explosion energy as functions of bond angle for
NO6+2 symmetric explosions initiate with qO = qN = 2 measured by joint variance.
The error bars are the standard deviation of seven experiments under the same
conditions.
are the force and vibrational constants for the bending motion at Re, h is Plank
constant, and c is the speed of light. The constants of CO2 (NO2) are k = 57 N/m
(152 N/m) and ?2 = 667 cm?1 (750 cm?1) at Re = 0.116 nm (0.119 nm) [131].
Therefore, for CO2 in the ground state (v = 0), ??b is 7.5?. In classical
harmonic oscillation, the molecule will spend most of its time near the end points or
turning points of its oscillation. Thus, the most probable location of the molecule
in the bending vibration should be at ?b = 172.5?. This value is in very good
agreement with Fig. 5.4, where the measured signal strength peaks near ?b ? 173?.
The maximum observed bending (minimum ?b at about 145?) corresponds to v ?
10, or Ev = (v+1/2)hc?2 ? 0.9 eV. For NO2 in the ground state (v = 0), ??b = 4.8?,
thus the peak signal should appear near ?b ? 129.2 and 138.8?, which is in very good
agreement with the joint variance measurement ? the measured signal strength peaks
near ?b ? 140? (Fig. 5.9). The reason we do not observe two peaks at both 129.2
90
and 138.8? is probably due to the angular resolution in the measurement is not high
enough. The maximum observed bending (?110?) at ?b < ?be = 134? corresponds to
v ? 12 (Ev ? 1.2 eV), and the maximum bending (? 170?) at ?b > ?be corresponds
to v ? 25 (Ev ? 2.4 eV). There are three possible reasons for this asymmetric
vibrational amplitude on the two sides of the equilibrium bond angle (?be). The first
is that the anharmonic term becomes more significant as the bond angle approaches
180? so that this simple model for vibrational energy assuming harmonic oscillation
is not valid any more. The second is that the laser field distorted the vibrational
motion, which made it easier to increase than to decrease the bond angle. A third
reason could be that the distortion on the explosion signal as a function of bond
angle caused by the 2-body channel was not corrected completely by joint variance.
The real signal at ?b > 150? might be smaller than what was measured.
The observed explosion signal as a function of bond angle for both CO2 and
NO2 shows large-amplitude vibrations (v greaterorsimilar 10 and Ev greaterorsimilar1 eV) beyond the Maxwell-
Boltzmann distribution (? e?E/(kT) with E is the energy level, k is the Boltzmann
constant and T is the temperature) at the room temperature, which is consistent
with previous observations [56, 58, 60, 61]. On the other hand, as shown in Fig. 1
of Ref. [132], the molecular vibrational energy of NO2 in the ground state shows an
asymmetric potential well, where the energies at ?b = 110? and 180? are equal and
the minimum appears at ?be = 134?. This is consistent with our observed signal
distribution of NO2, which peaks near 140? and extends to 110? and 180?. This
evidence, along with the fact that the peaks of the explosion signal distributions
for both CO2 and NO2 appear near their equilibrium bond angles, shows that the
91
molecular vibrational energy in the laser field has similar minimum and potential
well structure as in the ground state of the neutral molecule, at least for the 6-
electron explosion channels of CO2 and NO2.
At the same time, the experimental results show that NO2 explosion energies
decrease monotonically by more than 25% from the smallest to the largest bond
angles observed (Fig. 5.11), in contrast to CO2, which exhibits explosion energies
that are nearly independent of bond angle (Fig. 5.3). This observation is different
from the result of CS2 measured by a position-sensitive detector [65], where the
explosion energy increases as the bond angle increases. The explosion energy as a
function of the bond angle will be analyzed with two models in the next section.
5.1.2 Explosion Energy Analysis
In this section, the measured explosion energies of the 6-electron explosion
channels for CO2 and NO2 are compared to the calculations in the field-ionization
(RC) and electron-screening models.
5.1.2.1 Triatomic 2-D Field Ionization
To compare the experimental results to the field-ionization model, the one-
dimensional diatomic field-ionization model (Sec. 2.1.1) is extended to 2-dimension
with three charge centers. The coordinate system employed is shown in Fig. 5.12.
The center atom (q3) is set at the origin (r3 = 0), and the line between the outer
atoms q1 and q2 is parallel to the x-axis as well as the laser field (vectorE), which is pointing
92
?b
q3
q1q2
R
Er
1r
r
x
y
2r
r
Figure 5.12: Coordinate system employed in triatomic 2-D field ionization calcula-
tion.
to the?x direction. The vectors vectorr1 andvectorr2 are pointing from the origin (center atom)
to the outer atoms, and their lengths are the bond length (r1 = r2 = R), as the
symmetric configuration is considered. The angle between the vectors vectorr1 and vectorr2 is
the bond angle (?b).
Therefore, the 2-D potential energy (adapted from Eq. 2.3) is given by
V(x,y) = ? Kq1e
2
radicalbig(x?Rsin(?
b/2))2 + (y?Rcos(?b/2))2 +s2
? Kq3e
2
radicalbigx2 +y2 +s2
? Kq2e
2
radicalbig(x +Rsin(?
b/2))2 + (y?Rcos(?b/2))2 +s2
+eEx, (5.2)
where e is the elementary charge, qie with i = 1,2,3 are the charges of the three
nuclei, E is the laser field strength, and s is the smoothing parameter to remove
the singularity of the Coulomb potential and make the two-dimensional calculation
agree with measured RC and ionization potentials [73, 76]. The adapted electron
93
Table 5.1: Ionization potentials of H and the first two ionization stages of C, N,
and O atoms in eV [133], where I(q)p is the ionization potential from X(q?1)+ ? Xq+.
q H C N O
1 13.6 11.262 14.536 13.619
2 24.386 29.604 35.122
energy levels of the charge-resonance states (Eq. 2.4) are given by
?+ = ?1 = ?Ip1 ?K
parenleftbiggq
3e2
R +
q2e2
2Rsin(?b/2)
parenrightbigg
+?eERsin(?b/2),
?? = ?2 = ?Ip2 ?K
parenleftbiggq
3e2
R +
q1e2
2Rsin(?b/2)
parenrightbigg
??eERsin(?b/2), (5.3)
?c = ?3 = ?Ip3 ?K
parenleftbiggq
1e2
R +
q2e2
R
parenrightbigg
,
where the first term (Ipi) is the ionization potential for X(qi?1)+ ? Xqi+ (Table 5.1),
the second and third terms lowers the energy levels because of the presence of the
other two positive charge centers and the fourth term in ?+ and ?? accounts for
shifting of the energy level in the presence of the laser field. The sin(?b/2) in the
fourth term accounts for the classical Stark shift in an external field. However,
without a detailed quantum calculation, it is not known how this level will shift due
to the overlap between the electron wave functions at the three charge centers with
the presence of the laser field. Thus, an adjustable parameter, ?, is employed with
values between 0 (no shift) and 1 (charge resonant shift [73, 75, 107]). As a simple
and convenient starting point, ? is assumed to be independent of the bond angle.
As ?b was changed from 100? to 180?, RC for enhanced ionization was deter-
mined via two different electron pathways. The first, shown in the upper-left panel
of Fig. 5.13, is through the saddle points of the potential, which will not be along the
94
Saddle Points Potential Barrier
Figure 5.13: Two over-the-barrier ionization pathways for a triatomic ion at
?b = 134?: saddle-point ionization (upper left) and quasi-diatomic, parallel to the
polarization axis and across the O?O barrier (upper right). Arrows in the upper
plots indicate the electron pathways. The lower plots are the corresponding poten-
tial curves along and centered at the O?O axis. The contours in the upper plots
and the short line segments in the lower plots indicate the electron energy levels in
three potential wells. The right plots correspond to 5% reduction in RC relative to
the left but at a cost of a 10% increase in ionization intensity. Other parameters in
the calculation are ? = 1.0, s = 0.01, q1 = q2 = q3 = 2, and Ip = 35.1 eV which is
the ionization potential for O+.
polarization axis when ?b negationslash= 180?. As the potentials at the inner and outer saddle
points equal to the electron energy level (?+) in the upper (right) well (blue contour),
the ionization takes place and RC, as well as Eia ? the ionization field strength, are
found. At this time, ?+ is still lower than the potential barriers along the O?O
bond, as shown in the lower-left panel of Fig. 5.13. The second is across the O?O
barrier, parallel to the polarization axis regardless of the value of ?b (right panels in
95
Fig. 5.13). The ionization takes place and RC is found as the potentials at the inner
and outer barriers along the O?O axis equal to the electron energy level (?+) in the
upper (right) well, similar to the diatomic case discussed in Sec. 2.1.1. The latter
pathway, a quasi-diatomic pathway, typically requires higher ionization intensities
(Iia = c?0E2ia/2) and yields smaller RC?s relative to the saddle-point pathway.
In general, RC increases as ? increases, since larger ? increases the energy shift.
Figure 5.14 shows the RC and the ionization intensity (Iia) curves as functions of
?b calculated with the quasi-diatomic and saddle-point pathways for a series of ??s
between 0 and 1. The RC curves are almost equally spaced, and increase (decrease)
as ?b increases for the quasi-diatomic (saddle-point) pathway. The ionization in-
tensity decreases as ?b increases for both pathways. Both RC and Iia curves for
the quasi-diatomic pathway have a larger curvature than those for the saddle-point
pathway.
At the same time, RC decreases as s increases. A smaller s makes the wells
deeper. The curvature, however, is nearly independent of both ? and s. Figure 5.15
shows RC and Iia curves as functions of ?b calculated with the quasi-diatomic and
saddle-point pathways at a series of s?s between 0 and 0.05 nm. The RC curves
increase (decrease) as ?b increases for the quasi-diatomic (saddle-point) pathway.
The ionization intensity decreases as ?b increases for both pathways. As V(x,y)
(Eq. 5.2) needs to remain physical ? the wells must support the bound states, for
example ? s is restricted to certain values. At the lower end, s is bounded by 0; below
about 0.01 nm, however, s no longer influences the value of RC in both pathways.
For an upper limit, s = smax is determined when one of the three energy levels (Eq.
96
Figure 5.14: Calculated RC (upper plots) and Iia (lower plots) as functions of ?b
(Eqs. 5.2 and 5.3) with ? = 0 (violet, dashed), 0.25 (green, solid), 0.5 (orange,
short-dashed), 0.75 (blue, dot-dash), 1.0 (red, long-dashed) for the quasi-diatomic
(left panels) and the saddle-point (right panels) pathways. Other parameters in the
calculation are s = 0.01 nm, q1 = q2 = q3 = 2, and Ip = 35.1 eV which is the
ionization potential for O+.
5.3) is at the bottom of a well as shown in Fig. 5.16; shifts proportional to ?ER
(? = 1.0, with the + sign corresponding to the uphill well and the ? sign to the
downhill well) lead to s ? 0.07 nm for Ip = 35.1 eV, the ionization potential for O+.
As ? decreases, smax also decreases.
In order to make a quantitative comparison with the experiment results, the
adjustable parameters of the model need to be determined first. In the enhanced-
ionization model, the transition moment between the charge-resonance states is
proportional to R/2 for diatomics [107, 108], while this value in linear H2+3 and H+3
was calculated to be ? 0.4R where R is the distance between two outer H [93].
Thus, if the charge-resonance is assumed to account for the shift of the energy levels
97
Figure 5.15: Calculated RC (upper plots) and Iia (lower plots) as functions of ?b
(Eqs. 5.2 and 5.3) with s = 0 (violet, dashed), 0.01 (green, solid), 0.02 (orange,
short-dashed), 0.03 (blue, dot-long-dash), 0.04 (red, long-dashed), 0.05 (light blue,
dot-dash) for the quasi-diatomic (left panels) and the saddle-point (right panels)
pathways. Other parameters in the calculation are ? = 0.5, q1 = q2 = q3 = 2, and
Ip = 35.1 eV which is the ionization potential for O+.
in Eq. 5.3, ? should take values between 0.5 and 0.8. The smoothing parameter has
been employed in 1-D and 2-D quantum calculations for the enhanced ionization
[73, 74, 76, 93, 96], and the value was taken between 0.5 and 2 a.u. (0.03 ? 0.1 nm)
to match the 3-D results. Therefore, the lower limit of s is taken as 0.03 nm, and
the upper limit is the physical limit of s (Fig. 5.16), 0.06 ? 0.08 nm depending on ?
and Ip.
For CO2, in our 2-D field-ionization model, if the ionization potential is as-
sumed to be 13.614 eV for O (q = 1, Table 5.1), the lower limit of RC is calculated
to be ? 0.23 nm at ?b = 180? with ? = 0 and s = 0.06 nm for both saddle-point and
98
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
x HnmL
-70
-60
-50
-40
-30
-20
-10
0
yg
re
nE
HVe
L
Figure 5.16: Extreme value of the smoothing parameter. The potential structure
at the upper limit of s, 0.08 nm for Ip = 35.1 eV, R = 0.26 nm, and ? = 1.0 at an
intensity of 3.3?1014 W/cm2. The short line segments indicate the energy level of
the bound electron in each well. In this example, the bottom of the left potential
well coincides with the energy level of the electron.
quasi-diatomic pathways, which is still larger than the experimental value (0.215
nm). At the same time, RC?s predicted with C (Ip = 11.264 eV and q = 1) and
C+ (Ip = 24.376 eV and q = 2) are even larger than the one predicted with O.
Consequently, the best agreement between theory and experiment for RC is associ-
ated with ionizing O+ (Ip = 35.146 eV), which also agrees with the electron being
removed from the upper potential well instead of from the middle well. Evidently,
since RC for CO+2 is larger than that for CO3+2 , the first three electrons must be
removed before the appropriate RC is reached or RC is not determined by this sim-
ple one-electron model. Similarly, for NO2, if the ionization potential is assumed
to be 13.614 eV for O, the lower limit of RC is calculated to be ? 0.26 and 0.27
nm at ?b = 140? with ? = 0.5 and s = 0.08 nm for saddle-point and quasi-diatomic
pathways, respectively, which are larger than the experimental value (0.254 nm).
99
Consequently, the enhanced ionization of NO2 also appears to be associated with
ionizing O+.
The model curves for both pathways are shown in Fig. 5.17 for the specific
values of s and ? given in the figure caption. These s and ? produce values for
RC that matched the measured values at ?b = 180? for CO2 and ?b = 140? for
NO2. This choice is not unique. While different choices would produce different
curves, the new curves would have nearly the same contour as those shown ? the
slop of the model curve RC(?b) for either pathway characterized by ? =|RC(110?)?
RC(180?)|/RC(180?) varies between 0.02 and 0.1 within the allowed range (0 ? ? ? 1
and 0 ? s ? smax), regardless of the values of RC predicted. If it is constrained
that the predicted RC needs to match with the measured values, the variation of ?
is only about 0.02.
As shown in Fig. 5.17, the model curves RC(?b) with the same Ip and ? give
a relatively reasonable match within the standard deviations to both CO2 and NO2
data at bond angles larger than 140?. This may be an indication that the enhanced
ionization behaves roughly the same way in both systems for large bond angles.
However, it is also interesting to note that while the curves generated by both
ionization pathways are consistent with CO2 observations, which is consistent with
the fact that these two pathways do not differ from each other much at bond angles
close to 180?, neither pathway is consistent with all the NO2 data points ? the
curvature is wrong. The saddle-point ionization, which might seem to be the more
plausible pathway, predicts RC should increase slightly as ?b decreases, but the
measured RC actually decreases as ?b decreases (Fig. 5.17). Ionization along the
100
100 120 140 160 180
qb H?L
0.2
0.225
0.25
0.275
0.3
0.325
0.35
R C
H
mnL
Figure 5.17: Experimental values of RC(?b) for CO6+2 (blue triangles) and NO6+2
(red squares) explosions taken from Figs. 5.3 and 5.11. The RC-model curves with Ip
= 35.146 eV, q = 2, and ? = 0.8 for both saddle-point and quasi-diatomic pathways
are represented by the short-dashed and dashed curves respectively. For saddle-
point ionization s = 0.074 nm (0.055 nm) for CO2 (NO2) while for quasi-diatomic
ionization s = 0.074 (0.046) nm for CO2 (NO2).
quasi-diatomic pathway predicts the correct sign of the curvature. Nevertheless,
this curve diverges from the data points at small bond angles for NO2 by more than
one standard deviation. It needs to be emphasized that a different choice for ? and
s, or even Ip, in the acceptable range will not change the slop of RC(?b) (?) to more
than 10%, which is by no means able to account for the observed variation (greaterorsimilar 25%)
of NO2 data between 110? and 180?.
In the calculation described above, we assumed ? to be constant for all the
bond angles. However, the energy shift of the electron?s state due to the overlap
between the electrons? wave functions at the three charge centers could change as the
bond angle changes. In the charge-resonance point of view, this shift corresponds
to the energy difference between the charge-resonance states in a laser field. The
101
charge-resonance states and their energy levels in a triatomic ion could very well
depend on the bond angle. As shown in Figs. 5.11 and 5.17, RC decreases as ?b
decreases for NO2 data points. A smaller RC indicates a smaller ?, as we mentioned
before (Fig. 5.14). Thus, it appears to be that, at least for NO2, the energy shift
between the charge-resonance states (?) decreases as ?b decreases. We recall that
the value of ? for H2 from quantum calculations [107, 108] as well as other diatomics
[70] from classical calculations (see Sec. 2.1.1) is ?di = 0.5, which is smaller than
that calculated for linear H3 molecular ion (?tri = 0.8) [93] ? the only triatomic
charge-resonance calculation of which we are aware. Therefore, one might imagine
that the smaller ? (than ?tri = 0.8) at small bond angles for NO2 could be related to
a tendency for the charge-resonance states to become diatomic-like as ?b decreases
to 110? and as RC approaches 0.21 nm ? the value of NO diatomic explosions
[47]. A possible explanation of this tendency is that the triatomic charge-resonance
states (including the electron wave functions at all three charge centers) becomes
spatially restricted (only charge-resonance between adjacent nuclei ? N and O) as
the bond angle decreases. In linear cases, the electron can move from one end of
the molecule to the other directly ? triatomic charge-resonance. However, in bent
configurations, the motion of the electron may occur in two steps ? from one end
nucleus to the central nucleus to the other end ? which establishes charge resonance
between adjacent nuclei.
To quantify the variation of ?, we keep a value of 0.8 at ?b = 140?, which
we have used to match our constant-? calculation to the measurement, and assume
? = ?di = 0.5 at ?b = 110?, where RC approaches its diatomic value. Furthermore,
102
for simplicity, we assume ? varies as a linear function of sin(?b/2), same way as the
length of the molecule varies (Eq. 5.3). Therefore,
?(?b) = a+bsin(?b/2). (5.4)
With a = ?1.539 and b = 2.489, ?(110?) = 0.5 and ?(140?) = 0.8 for NO2, which
gives ?(180?) = 0.95. For CO2, the value of a is taken to be ?1.639 to match the
data points, which gives ?(180?) = 0.85 and ?(145?) = 0.73. Inserting Eq. 5.4 into
Eq. 5.3, RC ? ?b curves were generated through the saddle-point pathway (Page
94) ? the electron moving from one oxygen to the nitrogen to the other oxygen, at
small bond angles, instead of directly between two oxygen nuclei, as discussed in
the last paragraph. The curves are shown in Fig. 5.18. This method works better
(?2 = 0.0033 nm2 is smaller) than the constant-? curves (?2 = 0.0107 nm2 for the
saddle-point pathway and ?2 = 0.0055 nm2 for the quasi-diatomic pathway, see Fig.
5.17) for NO2 and implies that electron dynamics becomes more diatomic-like as the
bond angle decreases, where ?2, the sum of the squares of the deviations between
the measured and calculated RC, is defined as
?2 =
summationdisplay
?b
parenleftBig
RC, EXP(?b)?RC, CAL(?b)
parenrightBig2
. (5.5)
The model curve is consistent with all the data points except the two end points
at ?b ? 110? and 180?. At the same time, although the curvature of the model
curve for CO2 is slightly larger than the data curve, the new curve falls within the
uncertainty of all the CO2 data points.
Although this RC ??(?b) model gives smaller ?2 than the constant-? calcula-
tion, because the variation of ? is taken into account, the description of the ?(?b)
103
100 120 140 160 180
qb H?L
0.2
0.225
0.25
0.275
0.3
0.325
0.35
R C
H
mnL
Figure 5.18: Experimental values of RC(?b) for CO6+2 (blue triangles) and NO6+2 (red
squares) explosions taken from Figs. 5.3 and 5.11 and the saddle-point RC ??(?b)-
model curves with Ip = 35.146 eV, q = 2, and ? = a + bsin(?b/2) with b = 2.489
and a = -1.639 (-1.539) for CO2 (NO2). The smoothing parameter is s = 0.074 nm
(0.055 nm) for CO2 (NO2).
function (Eq. 5.4) is a bit arbitrary and without much theoretical support. First,
the values for ? for diatomic and linear triatomic charge-resonance states ? ?di =
0.5 and ?tri = 0.8 ? were calculated from H+2 and H2+3 , which almost certainly will
not reflect all of the physics of multi-electron molecular ions. Second, the expres-
sion of ?(?b) (Eq. 5.4) is simplified, where the difference between CO2 and NO2 is
only accounted for by a shift in the constant a. A detailed quantum calculation
is necessary to determine how ? depends on ?b. However, if we assume Eq. 5.4
reflects some of the physics, a fit of the RC curve to the data points with a and b as
fit parameters could provide some information about how the energy shift changes
as ?b changes. For example, we can see that the curvature of the model curve for
CO2 (NO2) is too large (small) (Fig. 5.18). The curvature of the NO2 curve can be
104
increased to get a better match to the data points by raising ?(?b = 180?) and/or
lowering ?(?b = 110?). This implies that the energy shift of the states in NO2 at
large (small) bond angles could be larger (smaller) than that in H3 (H2). Similar
analysis for CO2 would have the opposite implication.
5.1.2.2 Static Screening
Besides the enhanced-ionization (RC) model, the explosion energy can also
be predicted by the electron-screening model (Sec. 2.1.2). The downside of the
dynamic-screening (DS) model is a more involved set of calculations. As will be
shown below, however, that the DS-model gives the same explosion energy as the
single-parameter RC-model suggests that screening might be characterizable with a
single parameter as well. Specifically, a charge defect (?) [62] may be introduced
such that the potential at the critical ionization stage is set by an effective charge,
qeff ? q ? ?. Assuming ? depends only on q, it can be derived by equating the
Coulomb potential energies at RC with full charge q with that for Re with effective
charge qeff,
EC(RC,q) = 2Kq
2e2
RC +
Kq2e2
2RC sin(?b/2),
EC(Re,q ??) = 2K(q ??)
2e2
Re +
K(q ??)2e2
2Re sin(?b/2), (5.6)
EC(RC,q) = EC(Re,q ??).
Then, ? can be defined as
? ? q(1?
radicalbig
Re/RC), RC = Req
2
(q ??)2. (5.7)
105
3 4 5 6 7 8 9
Ionization Stage
25
50
75
100
125
150
noi
sol
px
E
yg
re
nE
HVe
L
Figure 5.19: A comparison between the linear explosion energies for COZ+2 (Z =
3?9) calculated by the dynamic screening model of Ref. [82] (blue triangles) and
by qeff (red squares). The experimental value for RC (= 0.215) nm measured for
the 6-electron channel along with Re = 0.116 nm were used to determine ? and qeff
(Eqs. 5.6 and 5.7).
Figure 5.19 shows that this mapping of RC onto ? leads to the same explosion
energies for symmetric ionization stages between 3 and 9 for linear CO2 as calculated
by the hydrodynamic equations of the DS-model [82] (Sec. 2.1.2). This surprisingly
simple result suggests that a quasi-static electron distribution that envelopes the
nuclei for early times during the explosion captures the essence of the dynamics,
which would explain why an RC concept works so well. What remains to be done,
however, is to look at the behavior of ? as a function of bond angle and to determine
if the measured ? is in agreement with the theory for 3-atom systems.
For simplicity, a static electron distribution is employed in the calculation. Its
shape and density do not vary in time ? Static Screening (SS). The energy deficit
in the process of dissociative ionization is assumed to be solely determined by the
Coulomb potential between electrons and the three ions at their initial bond length
106
and bond angle. The energy deficit, or the charge defect derived from the energy
deficit, can be used to compare with the experimental measurement. To determine
the angular behavior of EC in the SS-model, the energy deficit is defined as,
ED ?
3summationdisplay
i=1
integraldisplay
A
K?(x,y)qie2dxdyradicalbig
(xi ?x)2 + (yi ?y)2 +s2, (5.8)
the interaction energy between the positive ions (xi, yi) and the cloud of electrons
represented by a 2-dimensional distribution ?(x,y) confined to an area A with the
x-axis along the laser polarization direction. The smoothing parameter s is intro-
duced to remove the singularity of the Coulomb potential. Since ED is equal to the
difference between Coulomb potential energies calculated at Re with qi = 2 and with
qeffi = qi ??, we have
ED = EC(Re,q)?EC(Re,q ??), (5.9)
=
parenleftbigg2Kq2e2
Re +
Kq2e2
2Re sin(?b/2)
parenrightbigg
?
parenleftbigg2K(q ??)2e2
Re +
K(q ??)2e2
2Re sin(?b/2)
parenrightbigg
,
? can be extracted once ED is determined,
? = q?
radicalBigg
q2 ? 2ReED sin(?b/2)Ke2(1+ 4sin(?
b/2))
. (5.10)
Then, ? or EC(Re,q ??) can be compared with the measured kinetic energy Ek,
Ek = EC(Re,q ??) = EC(Re,q)?ED. (5.11)
The positions of the ions (xi, yi) in Eq. 5.8 are set so that the CO or NO bond
length in CO2 or NO2 is the ground state Re; the bond angle (?b) is allowed to vary
so that the angular dependence of ED or ? can be determined. The distribution
107
(?), which is due to electrons being liberated at different intensities throughout the
pulse, is approximated by a two-dimensional Gaussian function,
?(x,y) = ?0e?x2/x20?y2/y20. (5.12)
It is centered at the center of mass of the triatomic molecule, which is also the origin
of the coordinate system (same as the one shown in Fig. D.1). The line between the
two outer atoms is parallel to the x-axis as well as the laser polarization axis. The
strength of the distribution ?0 = Qel/(?x0y0) is determined by Qel = integraltextA ?(x,y)dxdy.
The area A is a 4x0 ?4y0 rectangle, out of which the Gaussian distribution ?(x,y)
drops to less than 2% of its peak value. The total number of electrons in the
distribution, Qel, is bounded by 6, the total number of electrons freed from the
system and constrained to be the same for CO2 and NO2, since the free electrons
should behave approximately the same in both systems in the presence of the laser
field along with three positive charge centers.
As Eq. 5.12 is inserted into Eqs. 5.8 and 5.10, it is obvious that ED as well
as ? are functions of x0, y0, Qel, and s. In general, ED(?b) or ?(?b) decreases with
increasing s. In contrast to the RC model, s has a significant effect on the contour
of the curve, bending the curve more as s decreases. At the same time, as x0 or y0
increases, ?(?b) decreases, and the curve bends more as y0 decreases (Fig. 5.20).
Again, in order to make quantitative comparison with the experiment results,
the adjustable parameters of the model need to be determined first. The size of the
electron distribution is set by kinematics ? the oscillation radius of the electrons,
?0, due to the ponderomotive force in the field along the polarization axis (x-axis),
108
Figure 5.20: Energy deficit (ED) as a function of ?b calculated by SS-model (Qel =
1). Top: x0 = 0.57, 1.27, 1.98, 2.69, 3.39 nm with y0 = 0.014 nm and s = 0.02 nm;
middle: y0 = 0.007, 0.014, 0.021, 0.042, 0.085 nm with x0 = 1.27 nm and s = 0.02
nm; bottom: s = 0.005, 0.01, 0.02, 0.03, 0.05, 0.1 nm with x0 = 1.27 nm and y0 =
0.021 nm.
109
and the bending vibration range along the y-axis. Along the polarization axis, x0
depends on ?0, which is given by [109, 134],
?0 = eE0m
e?2
, (5.13)
where e is the elementary charge, E0 is the peak field strength of the laser, me is
the electron mass, and ? is the laser frequency. Under the current condition, the
intensity I0 ? 1015 W/cm2 (E0 =
radicalBig
2I0
c?0 ? 8.7 ? 10
10 V/m) gives an ?0 ? 2.7
nm. Since electrons are freed at different times during the pulse (hence at different
intensities), x0 should be set roughly by the time-averaged ?0. If x0 = 1.27 nm is
taken, the distribution drops to 1% of its peak at x = 2.7 nm, which is roughly
the upper limit of the electron radius in the distribution. The value x0 = 1.27 nm
probably reflects the fact that ionization starts at intensities in the low 1014 W/cm2
range. Again, since the free electron dynamics in the field should be the same, x0 is
constrained to be the same in both CO2 and NO2.
As the length of the electron distribution (x0) is determined by the oscillating
laser field, the width (y0) will depend on the effective molecular width, which is
related to the amplitude of the bending vibration, since the source of electrons are
the atoms in the molecule. From Figs. 5.3 and 5.4, CO2 apparently vibrates between
145? and 215?. Assuming the molecule bends through these angles at its Re, 0.116
nm, it is straightforward to show the O moves about 0.035 nm away from either
side of the axis. If y0 is taken as ? 0.021, the distribution falls to about 6% of its
maximum value at y = 0.035 nm and the model curve agrees with the experiment.
For NO2, the vibration is roughly between ?b = 120? and 170?. With Re = 0.119
110
100 120 140 160 180
qb H?L
0.5
0.6
0.7
0.8
s
HeL
Figure 5.21: Experimental values of ?(?b) for CO6+2 (blue triangles) and NO6+2 (red
squares) explosions derived from Figs. 5.3 and 5.11 by Eq. 5.7. The SS-model ?(?b)
curves with Qel = 1.85, x0 = 1.27 nm, y0 = 0.021 nm (0.014 nm), and s = 0.03 nm
(0.01 nm) for CO2 (NO2) are represented by the solid curves.
nm, O moves about 0.05 nm. If y0 is taken as 0.014 nm, the distribution falls to less
than 5% at y = ? 0.025 nm.
For the model curves, once x0 and y0 are fixed, Qel and s are used as the
primary fit parameters. Since the electron dynamics should be very similar in the
two systems, it is not surprising to recognize Qel ? 2 for both systems. It is also
not surprising that all the electrons are not trapped in the field; only those born at
the right phase of the pulse will revisit the origin for several cycles [109].
The SS-model curves for both triatomics are shown in Fig. 5.21 for the specific
values of Qel, x0, y0, and s given in the figure caption. It is clear that the SS-
model gives a pretty good match to the measurements for both CO2 and NO2. The
parameters of the curves provide some insight into the physics. First, since the
length of the electron distribution (related to x0) as well as the charge contained
111
within (Qel) are the same for CO2 and NO2, the difference between the two systems
would be associated with the width of the distribution, as well as s, the smoothing
parameter. The best fit to the data is obtained when y0 for CO2 is 50% larger
than it is for NO2. This would imply a smaller charge density in the cloud for CO2
making ED smaller. A smaller ED leads to a smaller ? (see Eq. 5.10) as observed.
Second, ? decreases with ?b because the electron distribution is longer than it is
wide ? its length is ? 1 nm while its width is ? 0.01? 0.02 nm. During a linear
explosion, for example, the electrons continue to screen the nuclei as they separate.
During a bent explosion, where there is a substantial transverse component of the
momenta, screening is less efficient as the nuclei move out of the high density area
more quickly ? ? is smaller. Finally, ? changes more rapidly for NO2 than it does
for CO2 between about 170? and 145?. This is also due to the fact that the CO2
electron cloud is wider. A wider cloud means that for the same ?b, CO2 experiences
a smaller reduction in the screening than NO2.
5.1.2.3 Discussion
The experimental results show that NO2 explosion energies decrease mono-
tonically by more than 25% from the smallest to the largest bond angles observed,
in contrast to CO2, which exhibits explosion energies that are nearly independent
of bond angle. It is shown that both the enhanced-ionization model with ?(?b)
and the static electron-screening model are consistent with the dynamics of highly
symmetric Coulomb explosions for both CO6+2 and NO6+2 , as shown in Fig. 5.22.
112
100 120 140 160 180
qb H?L
40
45
50
55
60
65
70
E k
HVe
L
Figure 5.22: Experimental values ofEk(?b) for CO6+2 (blue triangles) and NO6+2 (red
squares) explosions taken from Figs. 5.3 and 5.11. The RC ? ?(?b) model curves
(dashed) for CO2 and NO2 are derived from Fig. 5.18 by Eq. D.8 with R = RC. The
SS-model curves (solid) are derived from Fig. 5.21 by Eq. 5.6. The model parameters
are the same as those in the figure captions.
The result that the RC ??(?b) model is consistent with the measurement indicates
that the electron dynamics appears to depend on the bond angle and the energy
shift between charge-resonance states has a tendency to decrease as the bond angle
decreases. This is the simplest application of RC model. More complicated calcula-
tions, such as moving two or more electrons between the charge-resonance states at
a time or allowing the nuclei to move, and a better description of the ?(?b) function,
should be able to produce a better match to the experiments and provide more in-
formation about the behavior of the charge-resonance states in triatomic molecular
ions. To understand the physics behind this phenomenon, a full quantum treatment
may be required.
At the same time, although the fact that a single parameter ? the charge defect
113
(?, Eqs. 5.6 and 5.7) ? generates the same explosion energy as the dynamic-screening
model (Fig. 5.19) suggests an average distribution may be sufficient to capture most
of the physics, it is obvious that the static-screening model is over-simplified. The
size of the electron distribution should depend on the time-dependent laser intensity,
according to Eq. 5.13. In fact, we have performed two simple simulations to check
how the initial time, when the electrons are freed by the laser field, would affect
the final kinetic energy of the Coulomb explosion under the electron-screening effect
(App. E). The first one, based on Eq. 5.13, was to simulate the electron distribution
as a function of the initial time. The distribution is directly related to the probability
for the electron to appear at different positions in the space when the electron
is driven by the oscillating field. The result showed that the calculated electron
distribution depended on the initial time when the electron was removed by the laser
field, as well as the time interval and the laser intensity during the time interval when
the electron was oscillating near the molecular ion. The second simulation was to
simulate the explosion energy as a function of the initial time with a quasi-dynamic
screening calculation. The explosion of three positive charges was simulated under
their mutual Coulomb repulsion and the influence of electron wave packets moving
according to the radius of vibration of a free electron in the laser field (Eq. 5.13).
The result showed that the final kinetic energy of the explosion was sensitive to the
initial time when the electrons were born and trapped near the nuclei by the laser
field. Both simulations showed the instability of the dynamics ? the results were
very sensitive to the initial time ? which posed a question whether electron screening
was a practical and physical description of the dynamics. In order to explore the
114
validity of the electron screening model further, a full quantum mechanic treatment
is required as well as the knowledge about the exact time, relative to the laser pulse,
at which each electron is freed may be required.
In addition, experimental results showed that the explosion channels and en-
ergies are about the same for enhanced ionizations in both linearly and circularly
polarized fields with either the same laser intensity [36] orsame electric field1 [41, 42].
This evidence questions the validity of the electron-screening model because the dy-
namic screening predicted that the explosion induced by circularly polarized laser
pulses would result in the production of ions in higher charge states and larger kinetic
energies when linearly and circularly polarized lasers have equal field strength [81].
Although the kinetic energy of the ions was predicted to be comparable for linearly
and circularly polarized laser pulses with the same intensity [81], this prediction is
conflicted with the argument that the electrons driven by a circularly polarized laser
field will not be able to come back near the parent molecular ion [135, 136] so that
the electron effect in circularly polarized fields should be significantly different from
that in linearly polarized fields. If screening were correct, the kinetic energy release
should be different in linearly and circularly polarized fields.
Step-by-step experimental evidence of the dynamics on shorter time scale is
necessary to reveal the details of the physics behind dissociative ionization. One
possible experiment is a pump-and-probe, similar to the idea proposed by Stapelfeldt
et al. [137]. A 100-fs pulse pumps the molecule, a sub-10-fs pulse serves as a probe,
1With the same intensity (electric field), the electric fields (intensities) in linearly and circularly
polarized lasers are EL = ?2EC (IL = IC/2).
115
while the delay between two pulses is varied. As the short pulse explodes the
molecule at different stages of dissociative ionization induced by the long pulse,
the kinetic energy as well as the angular spread of the ejected ions may be observed
as a function of time at femtosecond resolution. The results should provide detailed
information about how the molecular structure, or even electronic dynamics, evolves
during ionization.
5.2 Ejection Anisotropy Measurement
In this section, the explosion dependence of the orientation angle (ejection
anisotropy) in small triatomic and diatomic molecules are investigated. The exper-
imental determination of the ejection anisotropy in enhanced ionization of a series
of small molecules induced by intense laser fields includes two steps. First, the an-
gular distribution of the energetic charge ejection subsequent to Coulomb explosion
is determined by isolating a series of specific orientations (?g, geometric orientation
angle, the angle between the molecular and polarization axes) of the precursor ions
and measuring their relative signal strengths by coincidence imaging (Sec. 4.3). Sec-
ond, the degree of dynamic alignment is determined quantitatively by comparing
explosion signals induced by linearly and circularly polarized fields. The results are
discussed in terms of the ionization stages, induced dipole moments, rotational con-
stants of the molecules as well as the time that the precursor molecular ions spend
in the field prior to the Coulomb explosions. The experimental results are presented
in Sec. 5.2.1, while Sec. 5.2.2 gives the discussion.
116
5.2.1 Experiments and Results
In the determination of the ejection anisotropy in enhanced ionization of small
molecules induced by intense laser fields, the experiment conditions are the same
as described in Sec. 5.1.1, and the intensity of the circularly polarized field is twice
that of the linearly polarized field so that the electric field is the same for the
comparison. Small molecules (N2, O2, CO2, and NO2) composed of atoms with
similar mass were used as samples to compare the ejection anisotropy in diatomic
and triatomic Coulomb explosions, and the results were also compared with that for
H2 which has been found to be dynamically aligned in a short pulse (? 50 fs) [86].
5.2.1.1 Angular Spread Measurement
In the first step of the determination of ejection anisotropy, the explosion
angular distributions of two triatomic systems [63], CO2 (nominally linear) and NO2
(bent in its ground state with an equilibrium angle, ?be = 134?), were compared with
that of H2, N2, and O2. For the triatomics, the symmetric 6-electron channel, the
dominant channel under our experimental conditions, is chosen (XO6+2 ? O2+ +
X2+ + O2+, where X is C or N). In the symmetric channel, the magnitudes of the
momenta of the two O ions are equal in the center of mass frame. Coincidence
imaging (Sec. 4.3), with the ability to isolate precursor orientations and geometries,
is used to make a series of measurements. The probability (signal strength) of
strong-field induced Coulomb explosions is measured as a function of ?g for a fixed
bond angle (?b, see Figs. 4.8c and 4.8d). This series of measurements was repeated at
117
Table 5.2: Molecular constants and properties for CO2, NO2, H2, N2, and O2.
The quantities are extracted from experimental measurements: Re, the equilibrium
bond length [114]; RC, the critical separation [this study]; ?be, the equilibrium bond
angle [114]; B0, the ground state rotational constant [114]; n, the exponent of cosine
distribution [this study]; ??g, the angular spread of the distribution [this study].
CO2 NO2 H2 N2 N2 O2 O2
Explosion Channel 2,2,2 2,2,2 1,1 1,2 2,2 1,2 2,2
Re (nm) 0.116 0.119 0.074 0.110 0.110 0.121 0.121
RC (nm) 0.22 0.26 0.25 0.22 0.23 0.23 0.23
?be (Deg) 180 134 N/A N/A N/A N/A N/A
B0 (cm?1) 0.39 0.434 60.853 1.998 1.998 1.438 1.438
n 39 25 19 7 15 11 15
??g (Deg) 22 27 31 50 34 40 34
several bond angles between 110? and 180?. The probability for Coulomb explosions
as a function of ?g was also measured for diatomics under the same conditions at
their peak explosion momenta.
The probability of Coulomb explosions for CO2 and NO2 at their equilibrium
bond angles (180? and 134? respectively) are shown in Fig. 5.23 along with that of
H2. These distributions were fit to cosn ?g and the full widths at half maximum
(??g) were extracted; the results are given in Table 5.2. There is a monotonic
increase in the width of the distribution as one goes from CO2 (the smallest) to H2
(the intermediate), and to N2 and O2 (the largest). Curves representing ??g vs ?b
for CO2 and NO2 are shown in Fig. 5.24. It should be noted that ??g increases with
decreasing n. Thus, the remarkable flatness of these curves is an indication that n
is nearly independent of ?b.
Since the induced dipole moment depends on the separation of charge between
the two ends of the molecule, one might be tempted to argue that the ejection
118
-40 -20 0 20 40
qg H?L
0
0.2
0.4
0.6
0.8
1.
evi
tal
eR
ytil
ib
ab
or
P
CO26+
cos39q
-40 -20 0 20 40
qg H?L
0
0.2
0.4
0.6
0.8
1.
evi
tal
eR
ytil
ib
ab
or
P
NO26+
cos25q
-40 -20 0 20 40
qg H?L
0
0.2
0.4
0.6
0.8
1.
evi
tal
eR
ytil
ib
ab
or
P
H22+
cos19q
Figure 5.23: A comparison between the relative probabilities for Coulomb explosion
as a function of ?g: (upper) the symmetric CO6+2 explosion channel at 180?, (middle)
the symmetric NO6+2 channel at its equilibrium bond angle (134?), and (lower) H2.
The error bars correspond to the standard deviation for four different runs. The
solid curves are cosn ?g fits where n = 39, 25 and 19, respectively for CO2, NO2 and
H2. The FWHM, ??g, of the distributions are respectively 22?, 27? and 31?.
119
145 150 155 160 165 170 175 180
Bond Angle H?L
10
20
30
40
50
qDg
H?L
CO26+
110 120 130 140 150 160 170 180
Bond Angle H?L
10
20
30
40
50
qDg
H?L
NO26+
Figure 5.24: The widths (??g) of the orientation distributions vary as a function
of bond angle for the symmetric CO6+2 (upper) and NO6+2 (lower) Coulomb explo-
sion channels. The data are extracted from distributions similar to those shown in
Fig. 5.23. The error bars are the standard deviation over four different runs. The
solid lines are the equilibrium ??g values (22? for CO2 and 27? for NO2).
distribution in the Coulomb explosion of triatomics should depend on bond angle.
Such an argument would be based on the fact that when the systems bend, thelength
of the molecules will change according to 2Rsin(?b/2) (where R is the bond length
between the outer and central atoms). For CO2 (NO2), this separation changes by
2.5% (15.5%) over the range of bond angles observed for a fixed bond length. While
the scatter in the data in Fig. 5.24 is consistent with variations of these magnitudes,
120
there is no clear increase from one side to the other in either system. Since the
ionization is at least in part sequential, ionization is distributed throughout the
pulse, giving the triatomic systems time to execute bending vibration. The ground
state vibrational frequencies, 667 and 750 cm?1, correspond to periods of 50 and
44 fs respectively for CO2 and NO2. Higher vibration levels and ion states tend
to be stiffer, leading to even shorter periods. Consequently, the flat distribution in
Fig. 5.24 could be an indication of angular washout, resulting from various stages
of ionization occurring at different bond angles.
5.2.1.2 Dynamic Alignment Measurement
In the second step of the determination of ejection anisotropy, the degree of
dynamic alignment is measured by comparing the ion yield ratio in linearly and
circularly polarized fields. The dynamic alignment is characterized by a parameter
?, the degree of dynamic alignment, which ranges from 0 (no dynamic alignment)
to 1. These limits can be linked to the relative number of atomic ions, R, generated
under linearly (NLP) and circularly (NCP) polarized fields;
R = NLP/NCP. (5.14)
In the absence of dynamic alignment, NLP,G < NCP and R?RG. The subscript G
means the population ionized solely through geometric alignment. When there are
N0 molecules in the laser focus,
NLP,G = 1?N0
integraldisplay ?/2
??/2
cosn ?gd?g, (5.15)
121
where cosn is the angular distribution of the explosion, while
NCP = 1?N0
integraldisplay ?/2
??/2
d?g = N0. (5.16)
At intensities of 1015 W/cm2, all molecules parallel to vectorE (?g = 0) are ionized. The
entire population is ionized in circularly polarized fields because all polarization
directions are sampled. Therefore,
RG(n) = NLP,GN
CP
= 1?
integraldisplay ?/2
??/2
cosn ?gd?g = 1???
parenleftbiggn + 1
2
parenrightbigg
/?
parenleftbiggn+ 2
2
parenrightbigg
. (5.17)
When all molecules are torqued into alignment with vectorE (complete dynamic align-
ment), NLP ? NCP and R? 1. Thus, the degree of alignment is defined as
? ? (R?RG)/(1?RG), (5.18)
which is a measure of the population being torqued.
To measure NLP and NCP, linearly polarized radiation was generated directly
from the Ti:Sapphire regenerative amplifier (Sec. 3.1). Elliptical polarization was
generated with a quarter-wave plate. The experiments were run with an ellipticity,
?, of about 0.92, where ? = E</E>. The electric field along the major (minor) axis
is indicated by the superscript > (<); E< = 0 corresponds to linear polarization.
The direction of the major axis was controlled with a half-wave plate. Enhanced
ionization is assumed to occur at the critical radius, RC, which is determined by
the field strength [73]. Thus, to keep the field the same, the circular polarization
intensity was twice that of the linear polarization. When ? = 0.92, the fields are
nearly the same for elliptical (EP) and linear (LP) polarizations ? E>EP ?= 1.04ELP
and E<EP ?= 0.96ELP. approximate circular polarization more closely, we averaged
122
E
Detec
tor Pl
ane
kz
?
?
x
y
X
Y
M
l
Figure 5.25: Coordinate systems in the focal and detector planes, separated by a
distance l, with the x and y axes are parallel to the X and Y axes. The laser field
(vectorE) is polarized in the xz-plane at an angle ? and the molecular axis ( vectorM) is oriented
at an angle ? relative to vectorE.
four images with vectorE>EP, confined to the xz-plane (see Fig. 5.25), pointing in two
orthogonal directions by rotating the half-wave plate through 180?.
The ion yield ratioRis extracted from Fig. 5.26, which contains: (1) linear po-
larization images obtained at 2?1015 W/cm2 (top), (2) circular polarization images
obtained at 4?1015 W/cm2 (middle) and (3) an average of 18 linear polarization
images each taken with a different ? separated by 10? between 0? and 170? obtained
at 2?1015 W/cm2 (bottom). In these experiments, each image contained 200,000
laser shots. The CO2 images contain doubly and triply charged ions, N2 and O2
images contain doubly charged ions while the H2 images are exclusively protons.
Since the physics is independent of ?, the NLP should be approximately the same
in the top and bottom rows. This is clearly not the case, which is due primarily to
gain variation over the MCP. Since the charges are collected over the same area of
the plate in the two lower rows, gain variation divides out. Consequently, using the
the two lower rows gives a better estimate of R. To obtain ? from Eq. 5.18, RG(n)
123
needs to be estimated from Eq. 5.17 first. Lacking detailed information of pure ge-
ometric alignment, the exponent of the cosine function (n) representing the angular
distribution of the explosion in the absence of dynamic alignment is assumed to be
roughly the same as the n measured in linear polarization (Table 5.2). If ?, the
degree of dynamic alignment, is small, this assumption should be pretty safe. Even
if ? is large, as the molecule is torqued and ionized at a final orientation angle ?g,
the ionization rate or probability at ?g should still obey the angular distribution of
pure geometric alignment and the measured value of n should be approximately the
same with or without dynamic alignment.
Figure 5.27 displays R vs. n for the four systems along with an estimate of ?,
which graphically is related to the relative distance to the upper and lower curves.
Clearly, ? depends on the value of n. The values plotted correspond to neff, a
weighted average of the primary channels for each system as discussed in the next
paragraph. The values of n for specific channels are listed in Table 5.2. While at
a given intensity n is independent of bond angle for CO2, it is affected by several
other parameters for all the systems. First, it depends on the intensity. At high
enough intensities, ionization will saturate and all molecules within an angle around
the polarization axis will ionize. This angle grows with intensity, broadening the
distribution and reducing the apparent n. To explore this behavior, the intensity
was varied between 1 and 4?1015 W/cm2 and found n to be constant to within 7%.
Wider intensity excursions caused n to exhibit larger variations. This is due in part
to the opening and closing of different ionization channels; n increases monotonically
with ionization stage [37, 39].
124
H2 N2 O2 CO2
1.0
0.75
0.92
1.0
0.29
0.35
1.0
0.27
0.35
1.0
0.40
0.38
H+ N2+ O2+ C2+,3+,O2+,3+
Figure 5.26: Momentum distributions of Coulomb explosions obtained with a linear
polarization at I0 = 2?1015 W/cm2 and ? = 0? (top), circular polarization at 2I0
(middle) and average image composed of 18 linear polarizations at I0 with ? ranging
from 0? to 170? in steps of 10? (bottom). Beneath each image is its relative ion yield
normalized to the second row in each column. The images in the top and middle
rows are composed of 200,000 laser shots and the bottom 18 ? 200,000 shots.
Since the images in Fig. 5.26 are composed of several channels, except for
H2, n and ? will be somewhere between the values for the dominant channels. The
dominant channels for each system are listed in Table 5.3 along with the contribution
this channel makes to the images in Fig. 5.26. Also given in the table are neff
(the sum of n?s for the primary and secondary channels weighted by the channel?s
contribution) for each system and an effective full width, ??eff, associated with
neff. It is noted that neff is only approximate for CO2 because the correlation
signals between the doubly and triply charged ions were too weak to determine the
125
10 20 30 40
neff
0.2
0.4
0.6
0.8
1.
R
16% 16.5%
90%
29%
Figure 5.27: Ion yields, R vs neff for H2 (green triangle, 0.92), N2 (blue star, 0.35),
O2 (red circle, 0.35) and CO2 (magenta square, 0.38). The upper (lower) solid curve
is R = 1 (R = RG). Values for ? (%) are also given.
angular distributions for mixed charge-state channels. The fact that n increases
with the charge state means the neff given in the table should be considered a lower
limit for CO2. Since RG(n) has a negative slope, ? for CO2 is also a lower limit.
5.2.2 Data Analysis and Discussion
Regarding the observed anisotropies (n and ??g in Table 5.2), three issues
are worth discussing. First, why are triatomics, especially CO2, significantly nar-
rower than diatomics? Second, why is NO2 wider than CO2? And third, why is H2
narrower than the other two diatomics? The fact that the anisotropy is strongly
correlated with the polarization axis suggests that the responsible mechanism must
depend on the dipole moment, vector?, induced by the field. While any molecule that is
more polarizable along one of its axes will have an effective dipole given by ??vectorE,
126
Table 5.3: Dynamic alignment parameters for H2, N2, O2, and CO2 obtained in the
experiments: the primary (secondary) channel and its contribution (%), its n and
??g taken from Table 5.2, ??eff (??da) the effective (apparent) angular spread for
neff (nda), Re and RC taken from Table 5.2, ? the degree of dynamic alignment (Eq.
5.18), R the relative number of atomic ions generated under linearly and circularly
polarized fields (Eq. 5.14), ?R the number of molecules torqued in the dynamic
alignment, and Ti/Ti(O2) the interaction time relative to that for O2 calculated
at ??Rmol? = (Re + RC)/2 by the two methods discussed in the text, where Rmol,e
(Rmol,C) = 2Re (2RC) for CO2.
H2 N2 O2 CO2
Primary Channel 1,1 (100%) 2,2 (62%) 2,1 (62%) 2,2,2 (80%)
Primary n (??g) 19 (31?) 15 (34?) 11 (40?) 39 (22?)
Secondary Channel none 2,1 (38%) 2,2 (38%) 2,3,2/3,2,2 (20%)
Secondary n (??g) N/A 7 (50?) 15 (34?) N/A
neff 19 12.1 12.5 ? 39
??eff 31? 38.6? 37.8? 22?
nda 0.13 4.67 4.67 3.88
??da 179.4? 60.9? 60.9? 66.5?
Re (nm) 0.074 0.110 0.121 0.116
RC (nm) 0.25 0.23 0.23 0.22
? 90% 16% 16.5% 29%
R 0.92 0.35 0.35 0.38
?R 0.74 0.12 0.13 0.25
Ti/Ti(O2) 1.4, 17 0.8, 0.9 1, 1 2.7, 2.4
where ? is the polarizability matrix and vectorE is the applied field, intense fields also
establish a charge dichotomy between the two end atoms, promoted by charge res-
onance states. Transitions involving charge resonance states increase in strength
with R [73, 93] eventually dominating the transition moments. As a result, the
field becomes very effective at moving charge back and forth across the molecule
establishing a very strong dipole, which interacts with the applied field.
When the pulse is very short compared with the rotational period of the
molecule, the field essentially provides a coherent kick to the system, an effect that
127
has been exploited to align linear molecules (see, for example [138] and reference
therein). At the other extreme, the field has sufficient time not only to torque
the system but to restrict the rotational motion severely leading to pendular states
[112, 138]. In the latter case, the depth of the confining potential is proportional
to |vector? ? vectorE/Bo| (Page 28), the ratio between the dipole interaction energy with the
field and the rotation energy. Since B0 (Table 5.2) is about two orders of magnitude
larger for H2 and roughly four times larger for N2 and O2, the triatomics will find
themselves in a much deeper and narrower potential, for the same field strength. In
addition, the charge-resonance dipole moment is proportional to the molecular size,
so that triatomics would have vector? twice as large as diatomics, for the same charge
disparity.
At the same time, both CO2 and NO2 lose more electrons before they explode
than do O2 and N2, while H2 loses the least electrons. It is clear from both long
and short pulse experiments that the anisotropy becomes more acute as electrons
are removed. At 130 fs, for example, the width of N2 explosions are observed to
sharpen from about 58? for the 3-electron channel (N3+2 ? N2+ + N+) to about
28? for the 6-electron channel (N6+2 ? N3+ + N3+) [37]. This same study showed
sharpening in the Cl2 explosion spectra with ionization stages as well. At 40 fs,
the 2-electron channel of N2 (N2+2 ? N+ + N+) was measured to have a width
of 50? while the 5-electron channel (N5+2 ? N3+ + N2+) was observed to have a
width of 36? [39]. The same sharpening, for both N2 and O2 from (1,2) to (2,2)
channels, has been observed in our measurement (Table 5.2). This implies that the
angular selection rule of the enhanced ionization ?i|vector??vectorE|f?n, thus the angular spread
128
due to geometric alignment, should become narrower as the charge state increases.
Therefore, considering the confining potential and molecular ionization stage, the
degree of alignment (n in angular distribution cosn ?g) should follow the order of
CO2 (NO2) > O2 (N2) > H2 for the explosion channels in this study (Table 5.2),
thus the widths of the distributions should follow CO2 (NO2) < O2 (N2) < H2.
While CO2 and NO2 have comparable moments of inertia and lose the same
number of electrons in the final channel, the width of the distribution for CO2 is
considerably narrower. The reason could be that NO2 bends through smaller angles
than CO2. If a dipole moment were established between the outer and central atoms,
perpendicular to the molecular axis, the confining potential of pendular states as
well as the angular selection of the ionization could be broadened so that NO2 has a
larger angular spread. A perpendicular dipole could also make the field less efficient
at torquing NO2. Consequently, that CO2 has a narrower distribution than NO2 is
very reasonable.
While the angular distribution of H2 determined by the confining potential and
molecular ionization stage might be broader than all other molecules in this study,
it has the highest degree of dynamic alignment (Fig. 5.27). In fact, its dynamic
alignment is much higher than other molecules and almost complete, due to its
significantly smaller moment of inertia, which causes its free rotation to be faster
as well as making it easier to torque. As almost all the population is torqued into
ionization, it is very reasonable for H2 to produce a narrower angular distribution
than O2 and N2 even with a lower molecular ionization stage.
Since the dynamic alignment of H2 is almost complete, it is possible that
129
the angular spread of H2 explosion could be actually broader in pure geometric
alignment than the measurement made in the linearly polarized field where dynamic
alignment is active. Since ? depends on the angular spread (??g or n) in pure
geometric alignment (Eqs. 5.17 and 5.18), different n or ??g would result in different
?. However, even if ??g is assumed tobe aslargeas60? (double thevaluein Table5.2
and corresponding to n ? 5), ? of H2 would still be more than 80%, still much larger
than O2 or N2 (Fig. 5.27). Nonetheless, the analysis described in the last paragragh
would stand.
Following Fig. 5.27 and Tables 5.2 and 5.3, it is obvious that angular spread as
well as dynamic alignment is essentially the same for N2 and O2. This is consistent
with the fact that their RC?s, charge states, and moments of inertia are all about
the same. However, an ? for CO2 higher than N2 and O2 is surprising. The degree
of dynamic alignment, or the number of molecules torqued by the laser field, should
be governed by the classical equation of motion for ?g (App. A),
??g = ?vector?? vectorE
Im = ?
?E
Im sin?g = ?
?eERmol
Im sin?cos?, (5.19)
where vector? is the induced dipole moment, vectorE is the laser field, and Im is the moment
of inertia. While |vector?| ? ?eRmol cos? (App. A) sustained by charge-resonance states
[73], where Rmol is the total length of the molecule, increases relative to N2 and O2,
Im of CO2 is also larger (? R2mol/4), leading to an overall reduction in the torque
at both Re and RC given in Table 5.2. At the same time, a higher ionization stage
is consistent with the precursor ion spending more time in the field [37, 39]. The
ionization stages are H2+2 , N4+2 , O4+2 and CO6+2 . Evidently, a longer interaction time
130
with the field, perhaps combined with multiphoton ionization prior to enhanced
ionization, leads to a significant enhancement in the number of molecules torqued
beyond that expected from Eq. 5.19.
The exact nature of the enhancement requires more knowledge of the evolution
of the systems in the field than that we currently have. However, it is possible to see
that the enhancement for CO2 is consistent with a longer interaction time with the
field, Ti. The simplest model assumes that Ti ? the number of molecules torqued;
longer interaction times will allow more molecules to be aligned. Under the field
strengths in this study, molecules nearly aligned within the polarization axis will be
trapped in pendular states [112], so no molecules will be lost once aligned with the
field. Thus, ?R = R?RG from Fig. 5.27, which is a measure of the molecules
torqued, can be exploited to estimate the relative interaction time, Ti(CO2)/Ti(O2)
? ?R(CO2)/?R(O2).
To estimate the relative time, however, the difference in the moments of inertia
and molecular lengths of the two systems must be accounted for. The coefficient can
be determined from Eq. 5.19. Assuming the molecules start from rest, this equation
can be solved to give a rotation time from ?g to 0 (Eq. A.18),
T(?g) = K(sin2 ?g)
radicalBigg
mRmol
2e?E , (5.20)
where K is the elliptic integral of the first kind (App. A). Thus, to compensate for
the differences in the molecular size the relative interaction time can be written as
Ti(CO2)
Ti(O2) =
?R(CO2)
?R(O2)
radicalBigg
?O2Rmol(CO2)
?CO2Rmol(O2). (5.21)
As discussed in Appendix A, theoretical calculations for charge resonance showed
131
that ? ? 0.5Rmol and 0.4Rmol for H+2 [73] and H2+3 [93], respectively, resulting
radicalBig
?H2/?H3 = 1.1 ? 1. Since no theoretical calculations, of which we are aware,
have been published for N2, O2 or CO2, we assumed ? to be the same for all the
molecules in this study. Therefore, Eq. 5.21 becomes
Ti(CO2)
Ti(O2) =
?R(CO2)
?R(O2)
radicalBigg
Rmol(CO2)
Rmol(O2) . (5.22)
Values for the relative times are given as the first number in the last row in Table
5.3. Since it is unknown where the systems are torqued most efficiently, an average
?R is chosen for these calculations between Re and RC. This model shows that CO2
spends about 2.7 times longer in the field than O2. While O2 and N2 spend about
the same amount of time, this model suggests H2 spends about 40% more time in
the field than O2 and N2. It is important to know that these ratios depend on the
values selected for ?R. If, for example, H2 were torqued more efficiently near its Re,
while O2 closer to its RC, this model would suggest H2 spends 20% less time than
O2. Since H2 loses only two electrons while the other systems lose more, it is not
unreasonable to expect H2 to be torqued closer to Re than the other systems.
It is also possible to estimate Ti more directly from T(?g) (Eq. 5.20), if it is
recognized that themolecules torqued by thefield fora specific neff lieoutside ??eff,
the width of the angular distribution given in Table 5.3. In the absence of dynamic
alignment, only the molecules within this angular spread, RG (Eq. 5.17), would be
ionized. For R to exceed RG, extra molecules must be rotated into this angle prior
to enhanced ionization. Since K(sin2 ?g) increases monotonically with ?g from 0 to
90?, it gets harder and harder to rotate molecules as ?g increases. One can estimate
132
the width of the distribution of the molecules responding to dynamic alignment,
??da, from an apparent exponent, nda, which is obtained by equating RG(nda) with
the measured R. The interaction time, Ti, can be estimated by calculating how long
it takes molecules at ?da (= ??da/2) to rotate to ?eff (= ??eff/2), which is given
by T(?da)?T(?eff). The relative interaction times in this approach are given as the
second number in the last row in Table 5.3. Except for H2, the two methods are
in agreement. Again, the time for H2 is most likely too large not only because of
uncertainty in the molecular lengths but because it is assumed that the molecules
are not rotating. A combination of the facts that K(sin2 ?g) ?? as ?g ? 90? and
that ?da lessorsimilar 90?, explains why the time for H2 is so long. Since the H2 molecules are
rotating rather rapidly compared with the other systems, it is not necessary for the
field to do all the work; most of the molecules will rotate to much smaller angles.
For example, if ?da = 59?, R(H2) = Re and R(O2) = RC, the rotation times for H2
and O2 would be about the same.
In summary, a general technique is developed for measuring the contribution
that geometric and dynamic alignment make to the ejection anisotropy induced by
strong laser fields, and is employed in the quantitative angular dependence study
of 2- and 3-atom Coulomb explosions on a time scale of 100 fs. The experimental
results show clearly that triatomic systems can have narrower distributions than
diatomic systems at comparable ionization stages. This is probably because of the
higher ionization stage, larger induced dipole and smaller rotational constant of the
triatomics. The width of the CO2 distribution is narrower than that of NO2, but
both are nearly independent of bond angle. While the difference in the widths could
133
be associated with a larger induced dipole moment perpendicular to the molecular
axis, which is caused by NO2 bending to smaller bond angles, the independence of
bond angle could be an exhibition of angular washout, resulting from various stages
of ionization occurring at different bond angles. The argument that the small mo-
ment of inertia of H2 allows it to be more easily torqued even by a short pulse is
confirmed by its degree of dynamic alignment. However, the potential restricting
rotational motion (to pendular states) established by the field is broader for a larger
rotational constant. Thus, systems with a smaller rotational constant relative to H2
(e.g., CO2, NO2, N2 etc.) will experience a narrower confinement potential when
sufficient time is allowed for them to be trapped. This is confirmed recently by align-
ment experiments with 300 ps laser pulses [139, 140], where the degree of alignment
is found in the order of CS2 > CO2 > N2 > H2, and consistent with our analysis.
The results also show that the anomalous dynamic alignment in the linear triatomic
CO2 is consistent with the removal of more electrons at the time of the Coulomb
explosion, thus the precursor ion spends more time in the field than diatomic sys-
tems such as N2 and O2. While the simple classical model sheds some light on the
underlying physics, a full quantum mechanical treatment is necessary to confirm
these ideas about the interaction times. At the same time, the effect of the valence
electron structure on ejection anisotropy in enhanced ionization, which is neglected
in this study, and the potential contribution that angular momentum introduced by
circular polarization [35] makes to the anisotropy need to be investigated.
134
Chapter 6
SUMMARY AND FUTURE WORK
6.1 Summary
Correlation detection techniques (image labeling [99], coincidence imaging [63],
and joint variance[103]) developed in this study arebased on the imagespectrometer
that is capable of collecting charges ejected over 4? sr and the digital camera that
can be synchronized with the laser repetition rate at up to 735 Hz [22, 99, 100]. The
collection capability in both spatial and temporal domain ensures that all the ejected
ions with momenta up to ? 80?qm a.u., where qe is the charge and m is the atomic
mass, can be detected and analyzed with enough statistics. The advantage provided
by these techniques is that both diatomic and triatomic molecular explosion channels
with different momenta (energies), bond angles, and/or orientation angles can be
isolated; thus, the pre-explosion molecular configurations (bond lengths and/or bond
angles) and orientations as well as their distributions can be derived. These results
draw a clear picture about the molecular dynamics. As these techniques are applied
in the study of strong-field dynamics, details of how the molecules respond to the
ultra-fast intense fields are revealed quantitatively. Furthermore, as different decay
channels can be easily isolated, these techniques should be useful tools for laser
control studies to detect transient states in molecular dynamics as shorter (broader
bandwidth) pulses are applied and as the incoming pulses are being shaped.
135
With the correlation detection techniques, molecular dynamics, specifically
Coulomb explosion as a function of bond angle in linear and bent triatomics and
ejection anisotropy of both triatomics and diatomics under the influence of intense
100-fs laser fields, were investigated in this study.
The experimental results revealed that the NO2 explosion energies decrease
monotonically by more than 25% from the smallest to the largest bond angle. This
is in sharp contrast to the behavior of CO2, where the explosion energies are nearly
independent of bond angle [62]. We developed 2-D enhanced-ionization and static-
screening models to analyze the measured explosion energies as a function of bond
angle. In the enhanced-ionization model, the energy shifts of the charge-resonance
states at the three charge centers, represented by the parameter ? (Eqs. 5.3 and
5.4), is recognized to be the primary reason for the bond-angle dependence of the
explosion energy. Smaller shifts (smaller ?) lead to larger explosion energy and vise
versa. In the static-screening model, the shape and density of the photo-electron
distribution around the exploding molecular ion, represented by the parameters x0,
y0 and ?0 (Eqs. 5.8 and 5.12), are believed to be responsible for the variation of
the explosion energy as a function of the bond angle. A broader or thinner electron
distribution causes a larger explosion energy. As we have shown, the predictions
of both the enhanced-ionization and static-screening models are consistent with
the explosion energies of highly symmetric Coulomb explosions for both CO6+2 and
NO6+2 (Fig. 5.22). However, the instability of the electron-screening calculation is an
issue that needs to be addressed further. At the same time, the observed explosion
signals as a function of bond angle for both CO2 and NO2 showed large-amplitude
136
vibrations and peaked at the equilibrium bond angles of the neutral molecules.
The contributions geometric and dynamic alignment make to the ejection
anisotropy observed following molecular Coulomb explosion induced by strong laser
fields were measured with correlation techniques [63] and by comparing the image
spectra obtained in linearly and circularly polarized fields. The magnitude of the
spread in the observed angular distributions (??g, Table 5.2) in increasing order was
found to be CO2 (NO2) < H2 < O2 (N2). The narrower angular distributions of the
triatomic systems was explained in terms of the precursor molecular parent system
reaching a higher ionization stage prior to explosion and larger induced dipole. A
narrower distribution of H2 than that of O2 (N2) was seen to be due to a dynamic
alignment being nearly complete. The measured degrees of dynamic alignment (?,
Table 5.3) in decreasing order was found to be H2 > CO2 > O2 (N2). The near-
complete dynamic alignment of H2, the highest among the samples, is due to the
fact that it has the smallest moment of inertia. The analysis, based on the classical
pendulum equation, showed that the anomalously large dynamic alignment in the
linear triatomic CO2 is consistent with the field removing more electrons prior to
and during the Coulomb explosion, thus the precursor ion spends more time in the
field than do diatomic systems such as N2 and O2.
In summary, this study provided a better understanding of the bending and
alignment motion of the molecules in an intense field and is one step in the path
to understand laser control of the molecules, which will become practical as more
details of the dynamics are revealed and better laser and detection technologies are
developed.
137
6.2 Future Work
Despite the experiments and the theoretical analyses being in agreement in
this work, important details undergirding the physics responsible for the molecular
dynamics in strong fields are still unknown. On the theoretical side, a full quantum
treatment is necessary to understand the motion of nuclei as well as electrons as a
function of time. On the experiment side, other triatomic molecules, such as N2O
(linear) and SO2 (bent), should be studied under the same conditions and the results
should be compared to the models we developed in order to check the validity and
generality of the theories. Furthermore, molecular ionization leading to Coulomb
explosion under different pulse widths and pulse shapes should be investigated to
understand the evolution of the molecules in the field. In particular, if the laser
pulse is shaped so that the 2-body and 3-body channels of NO2 are separated, details
of both channels might be revealed with less confusions. For ejection anisotropy,
experiments with different laser pulse widths can be used to measure the degree of
dynamic alignment as a function of time. If pump-probe schemes are employed to
pre-align the molecules, the ionization probability as a function of orientation angle
can be measured to obtain information about the geometric alignment effect. In this
study, thearrival time of the charges collected on the image was determined to within
a few tens of nanoseconds. If a multinode photon-multiplier could be employed
[141, 142] along with the digital camera to achieve better temporal resolution (on the
order of 200 ps as claimed in Ref. [142]) in charge detection without compromising
the high detection rate, abundant details of the dynamics could be obtained.
138
Appendix A
Classical Pendulum Equation of Motion
This appendix shows that a classical pendulum equation of motion can be
solved for general values of the angle and the rotation time can be derived with the
elliptic integral of the first kind [115].
A classical pendulum equation of motion for the dynamic alignment of a po-
larized molecule with a dipole moment (vector?) in an external field (vectorE) has the form of
[36, 96]
?? = ?vector?? vectorE
Im = ?
?E
Im sin?, (A.1)
where ? is the angle between the field and the molecular axis, Im = mR2mol/2 is
the moment of inertia of the molecule with m as the mass of the atom at both
ends of the molecule. The damping term (2 ?Rmol ??/Rmol) [42, 97], essentially the
variation of Rmol in time, is ignored because it has been shown to be small at
intensities of 1015 W/cm2 [97]. The dipole moment can be estimated with the
concept of charge-resonance states [107, 108]. In enhanced ionization, the electron
displacement or field-induced dipole moment is mostly due to the charge-resonance
states and proportional to the bond length [73], so that the dipole moment can be
taken as ?CR ? ?eRmol cos?, where ? is nearly constant and at best a slowly varying
function of Rmol and ?, depending on how ?CR and Rmol are related in diatomic or
triatomic molecules. For diatomics, the value is ? = 0.5 [107, 108], while it was
139
shown as ? 0.4 in H2+3 and H+3 [93]. As ?CR is employed, the equation of motion
(Eq. A.1) becomes
?? = ??eERmol
Im sin?cos?,
?? = ??eERmol
Im sin?, (A.2)
where ? = 2?, and the characteristic frequency of the pendulum at small ? approx-
imation can be expressed as
?CR =
radicalbigg 2?eE
mRmol. (A.3)
Although ?CR by no means reflects the complexity of the ejection anisotropy in en-
hanced ionization quantitatively, it provides qualitative analysis for how the strength
of dynamic alignment of molecules may vary. It reads from the expression of ?CR
(Eq. A.3) or
?CR = 2??
CR
= ?
radicalBigg
2mRmol
?eE , (A.4)
that dynamic alignment grows as the mass of the atom, size of the molecule, or laser
frequency decreases and as the laser field increases.
For general values of ?, the equation of motion,
?? = ??2CR sin?, (A.5)
can be integrated over ?, and yield [115, 143],
integraldisplay ?
0
d2?
dt2 d? = ?
integraldisplay ?
0
?2CR sin?d?. (A.6)
Using
integraldisplay ?
0
d2?
dt2 d? =
integraldisplay ?
0
d2?
dt2
d?
dt dt =
1
2
integraldisplay ?
0
d
parenleftbiggd?
dt
parenrightbigg2
= 12
parenleftBig?
?2 ? ??2|?=0
parenrightBig
(A.7)
140
and
?
integraldisplay ?
0
?2CR sin?d? = ?2CR(cos??1) = ?2?2CR sin2 ?2 , (A.8)
Eq. A.6 becomes
1
2
parenleftBig?
?2 ? ??2|?=0
parenrightBig
= ?2?2CR sin2 ?2 . (A.9)
If the maximum angle of the motion is ?0 = 2?0 and ??|?=?0 = 0,
??2|?=0 = 4?2CR sin2 ?0
2 . (A.10)
Equation A.9 becomes
??2 = 4?2CR(sin2 ?0
2 ?sin
2 ?
2 ). (A.11)
Define k = sin2 ?02 < 1, then k = sin2 ?0, and
d?
dt = 2?CR
radicalbigg
k?sin2 ?2 , (A.12)
or
dt = d?
2?CR
radicalBig
k ?sin2 ?2
. (A.13)
Define sin2 ?2 = ksin2 ? = sin2 ?02 sin2 ? with ??0 ? ? ? ?0 and ??/2 ? ? ? ?/2,
then radicalbigg
k?sin2 ?2 =
radicalBig
k(1?sin2 ?) =
?
kcos?, (A.14)
From the definition of ?,
?
kcos?d? = 12 cos ?2 d?,
d?
2
radicalBig
k ?sin2 ?2
= d?2?kcos? = d?cos ?
2
= d?radicalbig1?ksin2 ?. (A.15)
141
Therefore,
dt = d??
CR
radicalbig
1?ksin2 ?, (A.16)
As the molecule moves from ? = ?0 (? = ?0) and ? = ?/2 to ? = 0 (? = 0) and
? = 0, it takes a quarter of the period of the motion,
T?0 = ?/4 = 1?
CR
integraldisplay ?/2
0
d?radicalbig
1?ksin2 ? =
K(k)
?CR , (A.17)
where the integral is the elliptic integral of the first kind, K(k) = integraltext?/20 d??
1?ksin2 ?
,
with k = sin2 ?0 [115]. The rotation time (T?) can also be expressed by the small-
angle period (?CR) as
T? = K(sin
2 ?)
?CR =
K(sin2 ?)?CR
2? = K(sin
2 ?)
radicalBigg
mRmol
2?eE . (A.18)
142
Appendix B
Charge Arrival Position and Time of Flight
Given the initial velocity of the ejected ion along with its charge and mass, it
is possible to calculate where and when it will strike either the image or TOF MCP
of the spectrometer (Sec. 3.2). The structure of the spectrometer is shown in Fig.
3.6. The calculations of arrival position and time on the image MCP [100] are given
in Sec. B.1, while the flight time to the TOF MCP is given in Sec. B.2.
B.1 Charge Arrival Position and Time on Image MCP
The two coordinate systems used in the calculation for the image side are
shown in Fig. B.1. The first, xyz, has its origin located at the focal point of the
laser with its x axis parallel to k?vector of the laser beam. The second, XYZ,
is attached to the surface of the MCP a distance l from the focal point with the
XY-plane in the surface. The origin of the XYZ system locates the center of mass
of the dynamics in the XY-plane. The x and Y axes are parallel as are the line
joining the origins of the axes and the static electric field. A charge, qe with e as the
elementary charge, ejected from the focal point in the direction (?, ?) with initial
kinetic energy E (=mv20/2), has velocity components:
v0x = v0 ?sin??cos?,
v0y = v0 ?sin??sin?, (B.1)
143
v0z = v0 ?cos?,
in the xyz system and land at point (X,Y) on the MCP given by:
X = 2lcos?? (
radicalBig
sin2 ??sin2 ?+??sin??sin?),
Y = 2lcos??sin?? (
radicalBig
sin2 ??sin2 ? +??sin??sin?), (B.2)
where ? = qeEl/E is the ratio between the electrostatic energy acquired in the field
E pushing the charge toward the MCP (numerator) and the initial kinetic energy
(denominator); l is the distance between the focal point and the detector. The time
it takes the charge to reach the MCP is given by
t =
radicalBig
m2v20y + 2mqeEl?mv0y
qeE . (B.3)
In the limit of large ? (?25), which is true for most of the experiments, Eq. B.2
reduces to
X ? 2lcos??? , Y ? 2lcos??sin??? . (B.4)
For the ? ? 0 trajectories, Eq. B.2 further reduces to
X ? 2lcos??? , Y ? 2lsin??? , (B.5)
and the ions fall on a circle of radius RE given by
RE = 2
radicalBigg
lE
qeE. (B.6)
In a general case, Eq. B.6 does not hold at most of the locations on the
image because the ions are ejected into all directions. Even the image produced
by an isotropic mono-energetic distribution of charges has contribution at R < RE
144
z
o
X
Yx
y ZO
l
?
?
Er
ov
r
Detector
Interaction 
Region
Trajectory
kr
Figure B.1: Coordinate systems used for calculating the ion trajectory on the image
side. The laser is focused at ?o? and the wave vector vectork is in the x-direction. The
center of mass of the dynamics is ?O?, which is the center of the image. The initial
velocity of the ion is vectorv0, which is pushed toward the detector by the static field vectorE.
The field strength is E = Vint/l (Fig. 3.6). The detector, XY?plane, is parallel to
xz?plane.
because of the ? negationslash= 0 components for large ?. When the ions have multiple energies,
the higher energy trajectories with ? negationslash= 0 will distort the distribution of the lower
energy components. However, for a given energy and ejection angle ?, trajectories
with ? ? 0 have the largest density of points on the image. These trajectories collect
on a circle on the image as ? changes and Eq. B.6 holds for them. On the other hand,
if ? were restricted to small angles (i.e., charges ejected nearly parallel to z?axis)
and the angular distribution were cylindrically symmetric about the z?axis which
is true in a laser field with the linear polarization also parallel to z?axis, energetic
charges would appear at a distance from the center of the image given by Eq. B.6.
Thus, the energy (E) and momentum (p = mv0) can be determined from the peak
145
position along the X?axis (Xpeak = RE as Y = 0) on the image as
E ? qeE4l X2peak, p ?
radicalbigg
qeEm
2l Xpeak. (B.7)
In a multi-component ejection, Eq. B.7 holds for each component if the peaks are
spatially separated on the image. In practice, the potential Vint = E ? l or V4 =
4Vint/3 (Fig. 3.6) is obtained directly and Rpeak is used for peaks on and off the axis
instead of Xpeak. Thus, Equation B.7 is rewritten as
E ? qeVint4l2 R2peak = 3qeV416l2 R2peak,
p ?
radicalbigg
qemVint
2
Rpeak
l =
radicalbigg
3qemV4
2
Rpeak
2l . (B.8)
Since the radius R is proportional to the momentum p, the image is often called the
momentum distribution or momentum spectrum.
The radius on the image is measured by the pixel on the images. The corre-
spondence of the pixel and the image size on the phosphor screen is calibrated by
taking an image of a thin ruler placed at the screen. In order to accommodate the
full size of the screen in the camera frame, the calibration ?calib is adjusted to be
about 0.3 mm per pixel by varying the distance from the camera to the screen and
changing the focus of the camera lens.
In order forEqs. B.2 and B.3 to hold exactly, thestatic field in thespectrometer
needs to be uniform. The uniformity of the static field was checked by measuring the
energy and momentum of protons ejected from H2 under the influence of a linearly
polarized laser field while the static field was varied. In linear polarization, protons
are ejected mostly within a small angular spread around the z-axis so that their
146
momenta nearly obey Eq. B.7 or B.8. As the static field V4 increases, the peak radii
of the protons decrease. However, if the laser intensity is fixed (1.5?1015 W/cm2),
the proton momenta or energies do not change so that the product R?V4 or R2V4
(Eq. B.8) should stay constant as V4 varies. Figure B.2 shows the measured momenta
are consistent with a constant for each ejection channel within the uncertainty of
the measurement, and confirms the uniformity of the static field. Since the energy
is proportional to V4, the uncertainty of energy measurement can be treated as
an estimate of the non-uniformity of the static field. The uncertainty of energy
measurement at five different V4 (6%) was calculated as [144]
?? =
radicaltpradicalvertex
radicalvertexradicalbt 1
N(N ?1)
Nsummationdisplay
i=1
?2i, (B.9)
with N = 5. Thus, the non-uniformity of the static field was estimated as about
6%. At the same time, the inner peak (H + H+ dissociation) of protons should
have an energy of about 0.5 eV [21] under current conditions (800 nm, 100 fs, and
1.5?1015 W/cm2), which corresponds to a momentum of 8.2 a.u.. Assuming these
values (0.5 eV and 8.2 a.u.) are correct, the errors in the measurement (0.53 eV
and 8.4 a.u.) were about 6% and 3% for energy and momentum, respectively, which
gave an overall estimate about the uncertainty of the determination of energy and
momentum with Eq. B.8, including the uncertainties of l, V4, and calibration ?calib.
B.2 Time of Flight
On the TOF side of the spectrometer, as shown in Fig. B.3, the flight time of
am ejected ion can be calculated as described in this section.
147
P (
pix
el?
Vo
lt1/
2 )
P (
pix
el?
Vo
lt1/
2 )
V4 (Volt)
2R
2R
Figure B.2: Momenta of protons ejected from H2 under the influence of a linearly
polarized laser field at 1.5 ? 1015 W/cm2 as functions of the static field in the
spectrometer (V4, see Fig. 3.6). Since the image calibration, charge qe, mass m and
distance l (Fig. B.1) are the same, the momentum p is expressed by Rpeak ??V4
(Eq. B.8), where Rpeak is the radius of the proton peak in the unit of pixel. The
data points were taken with V4 varying from 500 to 2500 volts with 500 volts at
each step. The dashed lines are the average of the five points in each plot. The
error bars correspond to ?1 pixel which is the uncertainty of the determination
of the peak positions. The upper plot is for the inner peak from the low-energy
dissociation, while the lower one for the outer peak from Coulomb explosion. The
insets (blue rectangles) are the image taken at V4 = 2000 volts to show where Rpeak
was measured.
148
z
o
?
ov
r
DE
TE
CT
OR
TRAJECTORY
FLIGHT TUBE
E1
l1
E2 = 0
l2 l3
E3
FIRST
ACCELERATION
REGION
SECOND
ACCELERATION
REGION
x
Va1 Va3
Figure B.3: Structure and coordinate system of TOF spectrometer. The laser is
focused at ?o? and propagates in or out of the paper. The x?axis is parallel to the
external fields vectorE1 and vectorE3 and the flight tube, while z-axis is parallel to the detector,
which is a 50-mm-diameter MCP with 40-mm active area. The initial velocity of
the ion is vectorv0. The distances and potentials are, according to Fig. 3.6, l1 = 19.05
mm, l2 = 140 mm, l3 = 30 mm, and Va1 = V4/4 = 500 volts, Va3 = 300 volts.
The ion (qe, m) with initial velocity vectorv0 is pushed by vectorE1 in the first accelera-
tion region, drifts through the flight tube, then is accelerated again in the second
acceleration region to strike the detector (Fig. 3.6). The flight times in these three
regions are
t1 = ml1qeV
a1
(
radicalbigg
v20 sin2 ? + 2qeVa1m ?v0 sin?),
t2 = l2
slashbiggradicalbigg
v20 sin2 ? + 2qeVa1m , (B.10)
t3 = ml3qeV
a3
(
radicalbigg
v20 sin2 ? + 2qe(Va1 +Va3)m ?
radicalbigg
v20 sin2 ? + 2qeVa1m ),
where Va1 = V4/4. The total time-of-flight is tTOF = t1 + t2 + t3. The transverse
displacement on the detector is dT = v0tTOF cos? and all the ions with dT(v0,? =
0) ? RMCP, where RMCP = 20 mm is the radius of the MCP, are detected. For ions
with dT(v0,? = 0) > RMCP, only the ones with an ejection angle ? greater than a
149
certain limit ?m are detected, which can be determined by solving the equation
dT = v0tTOF cos?m = RMCP. (B.11)
There are two solutions with ?m+ > 0 and ?m? < 0 for forward and backward ions
initially ejected toward and away from the detector and their acceptance angles are
defined as ?Acc? = ?/2 ?|?m?|. The acceptance angle can be measured from the
experiment. When the linear polarization axis of the laser is rotated relative to the
TOF axis from parallel to perpendicular, the TOF signal strength would decrease
due to a finite acceptance angle since the ions generated are ejected primarily along
the polarization direction. The acceptance angle can be estimated at the rotation
angle where the signal strength drops to a half of its value at parallel direction. The
acceptance angles estimated for N2+ from N2 explosions (about 11 eV) and H+ from
H2 (about 2.5 eV) are roughly 40? and 45?, respectively.
Under the condition v0 = 0 or ? = 0, Eq. B.10 reduces to
t1 = l1
radicalbigg
2qeVa1
m ,
t2 = l2
radicalbigg m
2qeVa1, (B.12)
t3 = l3V
a3
radicalbigg2m
qe (
radicalbig
Va1 +Va3 ?
radicalbig
Va1).
If v0 negationslash= 0 and ? negationslash= 0, the time difference of the forward and backward ions is
?t = tTOF(??)?tTOF(?) = 2ml1v0 sin?qeV
a1
. (B.13)
Then the longitudinal momentum component pbardbl = mv0 sin? and the energy associ-
ated with it Ebardbl = m(v0 sin?)2/2 can be calculated from ?t as
pbardbl = Va12l
1
qe?t, Ebardbl = V
2
a1
8ml21(qe?t)
2. (B.14)
150
Appendix C
Image Processing
The raw images (camera frames) streamed to the SCSI drive from the digital
camera are stored by recording the value of each pixel. Since only a few percent of
the pixels (100 out of 16,384) have values higher than the background (Page 53),
these frames can be compressed dramatically into much smaller files [126]. The
procedure of image compression and recovery is given in Sec. C.1. At the same
time, each ion striking the MCP illuminates a cluster of adjacent pixels on the CCD
camera. The procedure to group the adjacent pixels into a cluster and to determine
the centroid of each cluster is given in Sec. C.2.
C.1 Image Compression and Recovery
The flowchart of image compression is shown in Fig. C.1. The basic idea is
to record only the pixels with values higher than the background noise (nonzero
pixels) and get rid of the rest (zero pixels), which occupy most of the storage space
of camera frames [126]. In the compressed frames, each nonzero pixel is recorded as a
pair of bytes, where the first byte stores the signal value of this pixel and the second
byte stores the position. In general, the position of a pixel in a 128?128 image could
be as high as 214?1, which cannot be stored in one byte. Thus, a relative position
smaller than 256 (28) counted from a marker instead of an absolute position counted
151
from the first pixel of the frame is used. The marker is inserted before the byte pair
of the pixel when necessary, and also recorded as a pair of bytes. The first byte of
the marker is always zero to distinguish from the pairs for the nonzero pixels; the
second byte stores the shift in the unit of 256 from the last marker or the first pixel
if this is the first marker in this frame. Therefore, the absolute position of a pixel is
the summation of the shifts in all previous markers multiplied by 256 and plus the
relative position. A pair of zeros is placed at the end of each frame as a separator
in a compressed file with multiple frames. A counter file, containing the numbers of
nonzero pixels and pairs of bytes used for each frame, and a summary text file are
generated along with the compressed image file for easy indexing. In such a manner,
the size of the compressed images is only a few percent of the size of the camera
frame file. A typical compressed image file containing 100,000 to 500,000 frames
only occupies several to a few hundred MB instead of a few GB of disk space and
can be stored or transfered to a CD or a flash drive easily. First five nonzero pixels
in a sample camera frame and their corresponding pairs of bytes in the compressed
image file including the markers are listed in Table C.1.
The flowchart of image recovery from the compressed file is shown in Fig. C.2.
As the nonzero pixels are placed to their corresponding positions in a blank frame,
and the rest of the pixels are set to be zero, this compression?recovery process also
eliminates the background of the frames. Figure C.3 shows a comparison of two
images before and after the background elimination.
152
Shift = 1,Flag = 0
Index = i*Width+j
Set initial variables, Flag = 0, Shift = 0, Counter = 0,
Threshold = 3, j = 0, i = 0, Width = 128, Height = 128
Index modulo 256 = 0 
and (i > 0 or j > 0)
Flag = 1
Flag = 0
p(j,i) < Threshold
Counter = Counter + 1
j=j+1. If j=Width, then j=0 and i=i+1
i<Height
End
False
True
True
False
True
False
True
True
False
False
Start
Shift = Shift + 1
Read value p(j,i) 
from raw image
Output (0, Shift)
Set Flag=1
Output ( p(j,i), Index mod 256)
Figure C.1: Flowchart of the image compression process. Flag and Shift are used
to compute the shift which will be placed in the marker. Counter is used to count
the number of nonzero pixels in this frame.
153
Set initial variables,
Shift = 0, Width = 128, Height = 128
Initiate a blank array, 
Image(Width, Height),
to store the recovered image
p + q = 0
p = 0
End
False
True
True
False
Start
pos = Shift ? 256 + q
Shift = 
Shift + q
y = pos/Width
x = pos ? y ? Width
Read a pair of bytes (p,q)
from compressed image
Image(x,y) = p
Figure C.2: Flowchart of the process of image recovery from the compressed image
file.
Figure C.3: Comparison of two images before (left) and after (right) the background
elimination. Both images are composed by 10,000 single frames, each of which
corresponds to one laser pulse.
154
Table C.1: First five nonzero pixels in a camera frame and their corresponding pairs
of bytes (a, b) in the compressed image file including the markers (the pairs with a
= 0). The position of each pixel (P) is computed as P = x +y ?w, where x and y
are the coordinates of the pixel on the frame and w = 128 is the width of the frame.
The position can also be computed from (a, b) as Pi = bi + 256?summationtextj<i,aj=0 bj.
i Signal Coordinates (x, y) Position (P) pair of bytes (a, b)
1 marker 0, 21
2 4 89, 43 5593 4, 217
3 4 90, 43 5594 4, 218
4 marker 0, 1
5 4 89, 44 5721 4, 89
6 4 90, 44 5722 4, 90
7 marker 0, 1
8 4 53, 47 6069 4, 181
C.2 Image Clustering
In general, each ion striking the MCP illuminates a cluster of adjacent pixels
on the CCD camera. The number of pixels within a cluster is typically between
1 and 5. The following procedure is carried out to determine the centroid of such
a cluster. First, all the adjacent nonzero pixels around the one with the largest
signal value on an image are identified as a cluster, then this step is repeated for
the nonzero pixels left on the image. Each time after a cluster is identified, it is
tested to determine whether it is a new cluster or a part of one previously identified
one, depending on the distance between the two clusters and the values of their
pixels. Then, the pixel with the largest value in a cluster is defined as the center or
centroid. If there are two or more pixels with the same peak value, the one closest
to the center of mass of the cluster is defined as the center or centroid, where the
signal of a pixel is used to calculate the center of mass. Details of the procedure are
155
shown as a flowchart in Fig. C.4.
A sample of an H2 image before and after its pixels were grouped into clusters
is shown in Fig. C.5. Five clusters were identified in the magnified part of the image.
The measured occurrence frequencies (F) as a function of the number of pixels in a
cluster on the images of five molecular Coulomb explosions are listed in Table C.2.
The molecules are H2 (H+), N2 (N2+), O2 (O2+), CO2 (C2+ and O2+) and NO2 (N2+
and O2+). The probability (P) can be calculated as the frequency divided by the
total number of clusters (Ntot, sum of all the frequencies)
Px = FxN
tot
, Ntot =
Nsummationdisplay
x=1
Fx, (C.1)
where x is the number of pixels in a cluster, N is the maximum possible number of
pixels found in a cluster (the limit of N is 128?128, where the entire image is found
to be one cluster). The mean (?) and standard deviation (?) of the frequency curve
can be calculated as
? =
Nsummationdisplay
x=1
x?Px,
? =
radicaltpradicalvertex
radicalvertexradicalbt Nsummationdisplay
x=1
(x??)2 ?Px. (C.2)
The measured parameters (?, ?, and Ntot) are listed in Table C.3. The histograms
showing how the frequency varies as a function of the number of pixels in each
cluster for H2, CO2 and NO2 images are shown in Fig. C.6.
In an event-counting experiment similar to the occurences of the clusters de-
scribed above, a Poisson distribution, which is the limit of a binomial distribution
[145], is often used to fit the probability (Px). If the probability for event A to occur
156
Find all the adjacent untagged nonzero pixels around 
Aand define Aas the center of this new cluster C
Find the pixel (A) with the maximum value among the untagged 
nonzero pixels (which have not been grouped into any clusters)
Cluster Cis a new one 
and tag all its pixels
Cluster Cis a part of C?, merge them into one 
cluster and tag all its pixels
The pixel with the max value and closest to the center of mass of this cluster is the center
This pixel is the center of this cluster
End
False
True
True
False
True
False
False
True
True
False
False
True
True
False
Start
Chas 2 or less pixels 
or it?s adjacent but not diagonally adjacent to a previously
found cluster C?
Is Aadjacent to a pixel
(B?) in C?with center A?
Are there any untagged nonzero
pixels left on the image
only ONE pixel holds the
max value in this cluster
Distance |A A?|less than or equal to 2
A > B?
A < B?
Figure C.4: Flowchart of the procedure to determine the centroid of a cluster on an
image. The adjacent pixels p1(i1,j1) and p2(i2,j2) are determined when |i1?i2|? 1
and |j1 ? j2| ? 1. In the special case of |i1 ? i2| = |j1 ? j2| = 1, they are called
diagonally adjacent, as pixels 1 and 2 shown in Fig. 4.8g.
157
Table C.2: Occurrence frequency as a function of the number of pixels (x) in a
cluster measured in H2 (H+), N2 (N2+), O2 (O2+), CO2 (C2+ and O2+) and NO2
(N2+ and O2+) images. Laser pulses are linearly polarized with an intensity of
2?1015 W/cm2. The measured ?, ? (Eq. C.2) and Ntot, as well as the number of
frames involved, are listed in Table C.3.
x H2 N2 O2 CO2 NO2
1 1469367 82847 204837 2351480 427780
2 609562 40252 101575 2694049 448808
3 226543 21661 53661 1841137 219042
4 97789 11857 26926 2336344 322707
5 32359 4098 10185 1306536 104465
6 14026 1883 4649 1135043 76893
7 5667 785 1867 834526 39714
8 2264 249 661 655483 18766
9 865 87 222 429286 6463
10 351 31 77 267040 1822
11 143 5 39 195071 769
12 37 6 10 134708 267
13 27 2 95006 113
14 3 1 65930 34
15 1 0 44568 14
16 2 1 29150 3
17 19503 2
18 12811 2
19 8279
20 5417
21 3479
22 2178
23 1320
24 842
25 501
26 330
27 178
28 85
29 60
30 36
31 22
32 9
33 2
34 3
35 1
36 1
37 1
158
Figure C.5: A sample of the results of the cluster procedure. The upper left
image and its magnified part (upper right) is one of the 500,000 single frames of
H2 Coulomb explosions taken with linearly polarized pulses with laser intensity of
2 ? 1015 W/cm2. The lower images are the same image after being clustered and
show only the centroids of the clusters.
Table C.3: The measured ?, ? (Eq. C.2) and Ntot from the occurrence frequency
as a function of the number of pixels (x) in a cluster for H2, N2, O2, CO2 and NO2
images (Table C.2), as well as the number of frames involved.
H2 N2 O2 CO2 NO2
? 1.66 1.93 1.92 4.26 2.86
? 1.03 1.25 1.24 3.01 1.74
Ntot 2459042 163761 404713 14470415 1667664
frames 500000 200000 200000 500000 500000
159
is P, the probability for the number or frequency of the occurrence of event A to be
x in N independent tests is given by a binomial distribution,
P(x|N) = N!x!(N ?x)!Px(1?P)N?x, (C.3)
where the factor N!x!(N?x)! is the number of combinations of x objects chosen from a set
of N divided by the number of permutations. As N ?? and P ? 0, the product
P ?N approaches to a finite value ?, and Eq. C.3 becomes a Poisson distribution,
Px = ?
x
x!e
??. (C.4)
It is easy to show that both the mean and variance of Poisson distribution are equal
to ? (? = ?2). In practice, the value of x observed in one set of N tests will vary
randomly ? x can be any value between 0 and N. Thus, the set of N tests would
be repeated many times (Ntot) and the frequency for event A to occur x times
(x = 1,2,???N) is counted in the Ntot sets. The frequency as a function of x should
approach a Poisson distribution as N approach infinity.
In our case, if we consider the probability to find x nonzero pixels in a cluster,
finding a nonzero pixel in a cluster can be seen as event A and the maximum possible
number of pixels in a cluster can be seen as the sample space ? N is the number of
the tests. The act of reading the value of each pixel and finding x nonzero pixels
is equivalent to running N tests and finding x occurrences of event A. However,
the tests (pixels) are not independent since a cluster requires its nonzero pixels are
all adjacent. Thus, the probability for a cluster to have x pixels does not follow
Eq. C.3 because the number of combinations of x adjacent pixels chosen from a set
of N does not follow N!x!(N?x)!. This can be seen in Fig. C.7, where the frequencies
160
2 4 6 8 10
Number of Pixels
0
0.2
0.4
0.6
0.8
1
1.2
1.4
yc
ne
uq
er
F
H?
016 L
2.5 5 7.5 10 12.5 15 17.5 20
Number of Pixels
0
0.5
1
1.5
2
2.5
yc
ne
uq
er
F
H?
016 L
2 4 6 8 10 12 14
Number of Pixels
0
0.1
0.2
0.3
0.4
yc
ne
uq
er
F
H?
016 L
Figure C.6: Frequency (Fx) as a function of the number of pixels in each cluster.
Laser pulses are linearly polarized with an intensity of 2 ? 1015 W/cm2. Upper:
500,000 frames of H2 Coulomb explosions with measured ? = 1.66, ? = 1.03, and
Ntot = 2.46?106; middle: 500,000 frames of CO2 Coulomb explosions with measured
? = 4.26, ? = 3.01, and Ntot = 1.45?107; lower: 500,000 frames of NO2 Coulomb
explosions with measured ? = 2.86, ? = 1.74, and Ntot = 1.67?106.
161
as a function of the number of pixels in each cluster for H2, CO2 and NO2 images
are plotted with fitting Poisson distributions. The fitting curves are narrower than
the experimental frequencies; the values of ? for all three molecules calculated by
Eq. C.2 are larger than the corresponding fitting parameters. The experimental
frequencies are apparently super-poissonion.
As shown in Table C.2, the maximum number of pixels in a cluster for CO2
is much higher than H2 and NO2. It is obvious that the more pixels are included
in a cluster, the more likely this cluster could be actually illuminated by more than
one charges landing closely on the detector. Thus, it needs to be tested for CO2
whether the number of pixels in a cluster would affect the results of coincidence
measurement (Sec. 4.3). As described in Sec. 4.3, valid clusters in the coincidence
measurement are restricted to those having a maximum number of pixels M to
minimize the inclusion of clusters, each of which corresponds to two or more ions
landing near each other. To test how M would affect the coincidence measurement,
angular distributions for CO2 Coulomb explosions (CO6+2 ? O2+ + C2+ + O2+)
were measured by coincidence imaging (Sec. 4.3) with different M at two bond
angles (180? and 173?). The test showed that the value of M had little effect on the
angular spread measurement (Fig. C.8). The standard deviation of angular spreads
(??g), measured with M between 6 and 15, was less than 1% for both bond angles
in the test. However, a cluster with large number of pixels could still correspond to
multiple charges landing closely on the detector. Therefore, in coincidence imaging
measurements, M was chosen so that the probability, Px|x=M (Eq. C.1), drops to
about 10% of the peak value in order to eliminate counting multiple charges as one.
162
2 4 6 8 10
Number of Pixels
0
0.2
0.4
0.6
0.8
1
1.2
1.4
yc
ne
uq
er
F
H?
016 L
2.5 5 7.5 10 12.5 15 17.5 20
Number of Pixels
0
0.5
1
1.5
2
2.5
yc
ne
uq
er
F
H?
016 L
2 4 6 8 10 12 14
Number of Pixels
0
0.1
0.2
0.3
0.4
yc
ne
uq
er
F
H?
016 L
Figure C.7: Frequency as a function of the number of pixels in each cluster. The
data are the same as in Fig. C.6. In order to be compared to the fitting curves,
the data were plotted as stars instead of histograms. The solid curves are Poisson
distributions, g(x) = g0?xx! e??, where x is the number of pixels in a cluster. The fit
parameters are, upper (H2): ? = 0.86 and g0 = 4.0?106; middle (CO2): ? = 3.0
and g0 = 1.2?107; lower (NO2): ? = 2.3 and g0 = 1.7?106.
163
For H2, CO2 and NO2, the value of M was 4, 9, and 9, respectively.
Another distribution, invoked in the clustering process, is the number of clus-
ters or charges collected in each frame. The ultimate limit of the number of clusters
in a frame is the frame size ? 128?128, which is the sample size N = 16,384, as the
entire frame serves as the sample space. Then, the act of reading the entire frame
and finding x clusters is equivalent to running N tests and finding x occurrence(s)
of an event. This act is repeated as each frame is taken ? Ntot is the total number
of frames. Although these tests are still not absolutely independent because the
ions ejected from the same parent molecule would land on the MCP with correlated
momenta or positions, they are much more random than the tests for the number
of pixels in a cluster. Thus, the probability or frequency of the number of clusters
in a frame should more or less follow a Poisson distribution. The fitting Poisson
distributions (solid curves) and the experimental frequencies for the frames in H2,
CO2, and NO2 images (same data sets as in Fig. C.7) agree with each other rather
well as shown in Fig. C.9.
164
-40 -20 0 20 40
qg H?L
0
0.2
0.4
0.6
0.8
1
evi
tal
eR
la
ngi
S
cos38.3414 qg
Dqg=21.7238
-40 -20 0 20 40
qg H?L
0
0.2
0.4
0.6
0.8
1
evi
tal
eR
la
ngi
S
cos38.7684 qg
Dqg=21.6046
-40 -20 0 20 40
qg H?L
0
0.2
0.4
0.6
0.8
1
evi
tal
eR
la
ngi
S
cos37.9767 qg
Dqg=21.8273
Figure C.8: Angular distributions for CO2 Coulomb explosions (CO6+2 ? O2+ +
C2+ + O2+) measured by coincidence imaging(Sec. 4.3) with different M at the bond
angle of 180?. The values of M are 7, 9, 11 from the top to the bottom, respectively.
The solid curves are the fitting functions cosn ?g with the corresponding ??g (full
width at half maximum) labeled in the plots.
165
4 8 12 16 20
Number of Clusters
0
2
4
6
8
yc
ne
uq
er
F
H?
014 L
10 20 30 40 50 60
Number of Clusters
0
1
2
3
yc
ne
uq
er
F
H?
014 L
4 8 12 16
Number of Clusters
0
2
4
6
8
yc
ne
uq
er
F
H?
014 L
Figure C.9: Frequency as a function of the number of clusters in each frame. Laser
pulses are linearly polarized with an intensity of 2?1015 W/cm2. The histograms
are the measured frequencies as a function of the number of clusters in each frame.
The measured ? and ? in 500,000 frames are, upper (H2): ? = 4.92 and ? = 2.45;
middle (CO2): ? = 28.94 and ? = 6.56; lower (NO2): ? = 3.34 and ? = 2.44.
166
4 8 12 16 20
Number of Clusters
0
2
4
6
8
yc
ne
uq
er
F
H?
014 L
10 20 30 40 50 60
Number of Clusters
0
1
2
3
yc
ne
uq
er
F
H?
014 L
4 8 12 16
Number of Clusters
0
2
4
6
8
yc
ne
uq
er
F
H?
014 L
Figure C.10: Frequency as a function of the number of clusters in each frame. The
data are the same as in Fig. C.9. Again, in order to be compared to the fitting
curves, the data were plotted as stars. The solid curves are Poisson distributions,
g(x) = g0?xx! e??, where x is the number of clusters in a frame. The fit parameters
are, upper (H2): ? = 5.81 and g0 = 4.99?105; middle (CO2): ? = 29.72 and g0 =
4.47?105; lower (NO2): ? = 3.84 and g0 = 4.62?105.
167
Appendix D
Three-body Coulomb Explosion Kinematics
In general, Newton?s equations of motion of a three-body system do not have
analytic solutions. This appendix describes how the three-body Coulomb explosion
encountered in this study is solved numerically so that the molecular structure where
the explosion occurs is extracted from the momenta of the ejected ions measured in
the experiments.
As a triatomic molecular ion (CO6+2 or NO6+2 , for example) explodes at t = 0
from a bond length R, the Hamiltonian in the center of mass coordinate system
(Fig. D.1) takes the following form,
H = 12
3summationdisplay
i=1
mi ?r2i +
3summationdisplay
i>j
Kqiqje2
rij , (D.1)
assuming pure Coulomb forces between the three charges. The subscripts 1 and 2
represent the two outer ions (oxygen) while 3 represents the center ion (carbon or
nitrogen). The mass, charge, position and momentum of each ion is given by mi,
qie, vectorri = xi?x +yi?y and vectorpi ? mi?vectorri(t), respectively, where e is the elementary charge.
The separation between charges is given by vectorrij = vectorri?vectorrj. The angle between vectors
vectorr13 and vectorr23 at the time of the explosion, t = 0, is the bond angle ?b. Then the
equations of motion are,
d
dtpxi = ?
?H
?xi,
mi?xi = ? ??x
i
parenleftbiggKq
iqje2
rij +
Kqiqke2
rik
parenrightbigg
, (D.2)
168
= Kqiqje
2(xi ?xj)
((xi ?xj)2 + (yi ?yj)2)3/2 +
Kqiqke2(xi ?xk)
((xi ?xk)2 + (yi ?yk)2)3/2,
where i, j, k = 1, 2, 3. For symmetric 3-body Coulomb explosion, mO = m1 = m2,
mC = m3, qO = q1 = q2, qC = q3, and r13 = r23 = 0 at t = 0. The subscripts O and
C represent outer and center ions. Since the origin of the coordinate system is the
center of mass and the third charge moves along the y-axis, xO = x1 = ?x2, xC =
0, yO = y1 = y2, yC = y3, and the equations of motion reduce to
mO?xO = Kq
2
Oe
2
4x2O +
KqOqCe2xO
(x2O + (yO ?yC)2)3/2,
mO?yO = KqOqCe
2(yO ?yC)
(x2O + (yO ?yC)2)3/2, (D.3)
mO?yC = 2KqOqCe
2(yC ?yO)
(x2O + (yO ?yC)2)3/2,
with the initial conditions,
xO(0) = Rsin(?b/2),yO(0) = ? Rcos(?b/2)1 + 2m
O/mC
,yC(0) = Rcos(?b/2)1 +m
C/mO/2
,
?xO(0) = ?yO(0) = ?yC(0) = 0. (D.4)
Without external force, the center of mass of the system does not move, m1y1 +
m2y2 + m3y3 = 0, then yC = ?2mOm
C
yO. The equations of motion further reduce to
mO?xO = Kq
2
Oe
2
4x2O +
KqOqCe2xO
(x2O +y2O(1 + 2mO/mC))3/2,
mO?yO = KqOqCe
2yO(1 + 2mO/mC)
(x2O +y2O(1+ 2mO/mC))3/2, (D.5)
with the initial conditions,
xO(0) = Rsin(?b/2),yO(0) = ? Rcos(?b/2)1 + 2m
O/mC
,
?xO(0) = ?yO(0) = 0. (D.6)
169
?b
q3, m3
q1, m1q2, m2
R
3r
r
2r
r
1r
r
Center of Mass
3p
r
2p
r
1p
r
Polarization
Figure D.1: Three-body Coulomb explosion. The center of mass of the system is
located at the origin of the coordinate system. The x-axis is horizontal and parallel
to the polarization axis. Without external force, the center of mass will not move.
At t = 0, assuming symmetric geometry, |vectorr1?vectorr3| = |vectorr1?vectorr3| = R is the bond length.
The angle between vectorr1?vectorr3 and vectorr2?vectorr3 is the bond angle ?b. The center-of-mass angle
?CM is defined as the asymptotic value of the angle between vectorp1 and vectorp2 as t ??.
Since there is no analytic solution, the differential equations were solved nu-
merically with Mathematica (Wolfram Research, Inc.) between t0 = 0 and a final
time tf. For the molecules in this study (CO2 and NO2), if q = 2 and R = 0.1?0.3
nm, ?xO and ?yO change less than 3% between t = 200 and 2000 fs. Thus the final time
tf is set at 2200 fs to ensure ?xO(tf) and ?yO(tf) approach their asymptotic values as
t ? ?. The center of mass angle ?CM between vectorp1 and vectorp2 is evaluated as ?CM =
2tan?1 ?xO(tf)?y
O(tf)
, which can be measured by image labelling.
For a given R and ?b, there is a unique solution of final momentum vectorpO =
mO(?xO(tf)?x + ?yO(tf)?y) and ?CM. The final momenta of all three charges can be
written as
vectorp1 = mO(?xO(tf)?x + ?yO(tf)?y),
170
100 120 140 160 180
qb H?L
120
130
140
150
160
170
180
q
MC
H?L
120 130 140 150 160 170 180
qCM H?L
100
120
140
160
180
qb
H?L
Figure D.2: Simulated functions ?CM(?b) and ?b(?CM) for CO6+2 symmetric explo-
sions with R = 0.22 nm and qO = qC = 2, assuming pure Coulomb interaction.
vectorp2 = mO(??xO(tf)?x+ ?yO(tf)?y), (D.7)
vectorp3 = mC ?yC(tf)?y = ?2mO ?yO(tf)?y.
If R and ?CM are given, ?b can be found by solving the equation ?CM = f(R,?b)
numerically, where f is not analytical in most cases. As the bisection root-finding
method [146] is implemented, the solution converges to a relative error smaller than
1?10?6 within 30 iterations. The functions ?CM(?b) and ?b(?CM) are plotted in Fig.
D.2 for CO6+2 explosions assuming R = 0.22 nm. The functions ?CM(?b) are plotted
in Fig. D.3 for symmetric explosions of four 6-fold charged triatomic molecular ions
(XY6+2 ) as well as H2O3+ with R = 0.22 nm.
171
100 120 140 160 180
qb H?L
120
130
140
150
160
170
180
q
MC
H?L
H2O
CO2
NO2
SO2
CS2
Figure D.3: Simulated functions ?CM(?b) for symmetric explosions of four 6-fold
charged triatomic molecular ions (XY6+2 ) with R = 0.22 nm and qO = qC = 2, as
well as H2O3+ with qO = qC = 1, assuming pure Coulomb interaction.
Therefore, the initial molecular structure (R and ?b) can be evaluated from
the final momenta (pO, pC) and ?CM measured by image labelling with energy con-
servation,
EC =
3summationdisplay
i>j
Kqiqje2
rij =
2KqCqOe2
R +
Kq2Oe2
2Rsin(?b/2),
Ek = 12
3summationdisplay
i=1
p2i
mi =
p2O
mO +
p2C
2mC , (D.8)
Ek = EC.
In the linear configuration where ?b = ?CM = 180? and pC = 0, Eq. D.8 becomes
2KqCqOe2/R +Kq2Oe2/(2R) = p2O/mO, (D.9)
so the bond length for the linear configuration is
Rlin = KqO(4qC +qO)e2mO/(2p2O). (D.10)
Assuming the same Rlin for ?CM negationslash= 180, ?b can be determined by the bisection
172
140 150 160 170 180
qb H?L
0.19
0.2
0.21
0.22
0.23
R
H
mnL
Figure D.4: Bond lengths and bond angles where CO6+2 symmetric explosions
initiate with qO = qC = 2 measured by image labelling (Sec. 4.2) at laser intensity
of 2?1015 W/cm2. Bond angles ?b are determined from measured ?CM using Fig.
D.2. The error bars are the standard deviation of four experiments under the same
conditions.
method, and R for this ?b can be determined by Eq. D.8. Figure D.4 shows the
result of CO6+2 explosions induced by linearly polarized field at 2?1015 W/cm2.
Although the bond lengths plotted in Fig. D.4 are approximately constant for
different bond angles, this may not be true in general. The bond length in a bent
configuration (?b and ?CM < 180?) may not be the same as Rlin. On the other
hand, ?b determined from ?CM may also depend on R. As shown in Fig. D.5, ?b as
a function of R, for fixed ?CM, monotonically decreases as R increases from 0.1 to
0.3 nm.
Therefore, the variation ??b ? ?b(R = 0.1,?CM) ? ?b(R = 0.3,?CM) can be
used as a measure of how accurate ?b can be determined from ?CM assuming R =
Rlin. Figure D.6 shows that the variation ??b is less than 0.08? or 0.05% for CO6+2
? O2+ + C2+ + O2+ explosions with ?CM between 120? and 180?. For the other
173
0.1 0.15 0.2 0.25 0.3
R HnmL
173.2
173.4
173.6
173.8
174
qb
H?L
qCM = 170?
0.1 0.15 0.2 0.25 0.3
R HnmL
151.2
151.4
151.6
151.8
152
q b
H?L
qCM = 145?
0.1 0.15 0.2 0.25 0.3
R HnmL
129.2
129.4
129.6
129.8
130
q b
H?L
qCM = 130?
Figure D.5: Calculated bond angle as a function of bond length as ?CM is fixed at
170?, 145? and 130?. The calculation is carried out with CO6+2 symmetric explosions
and qO = qC = 2.
174
130 140 150 160 170 180
qCM H?L
-0.075
-0.05
-0.025
0
0.025
0.05
0.075
0.1
qDb
H?L
Figure D.6: Calculated variation ??b ? ?b(R = 0.1,?CM)??b(R = 0.3,?CM) as a
function of ?CM for CO6+2 ? O2+ + C2+ + O2+ explosion with ?CM between 125?
and 180?. The maximum ??b, 0.08? or 0.05%, appears near ?CM = 145?.
3-body Coulomb explosion discussed in this study, NO6+2 ? O2+ + N2+ + O2+,
the upper limit of ??b is about the same (0.08? or 0.055%). Thus, the functions
?b(?CM) and ?CM(?b) can be considered independent of R. Consequently, ?b can be
determined from ?CM assuming any R between 0.1 and 0.3 nm.
In summary, the motion of the three ejected ions from a symmetric 3-body
Coulomb explosion is determined numerically. The relationship between the final
center-of-mass angle and initial bond angle is found to be independent on the bond
length. After the bond angle is determined from the center-of-mass angle assuming
any bond length between 0.1 and 0.3 nm, the bond length can be calculated using
Eq. D.8. Therefore, the bond length and bond angle where the Coulomb explosion
takes place can be fully determined from the measured momenta and center-of-mass
angle of the ejected ions.
175
Appendix E
Two Simulations for Electron Screening
Two simple simulations were performed to check how the initial time (t1),
when the electrons are freed by the laser field in molecular dissociative-ionization,
would affect the final kinetic energy of the Coulomb explosion assuming the electron
screening effect (Secs. 2.1.2 and 5.1.2). The first one was to simulate the electron
distribution as a function of t1. The second was to simulate the explosion energy
as a function of t1 with a quasi-dynamic screening calculation. Both simulations
showed certain degree of instability of the dynamics ? the results were sensitive to
the initial time.
In the electron distribution simulation, one electron was assumed to follow the
oscillation of the laser field since it was born at t1. The laser pulse was assumed as
a single-frequency sech2 pulse whose electric field and intensity were given by
E(t) = E0 cos(?t)sech(t/?),
I(t) = I0 cos2(?t)sech2(t/?), (E.1)
where ? = 2.35?1015Hz, which corresponds to ? = 800 nm, ? = 50 fs with which the
corresponding intensity pulse envelop has a width of about 90 fs, and E0 = 8.7?108
V/cm so that I0 = 1015 W/cm2. Thus, following the radius of vibration of a free
electron in the laser field (Eq. 5.13) [109, 134], the position of the electron as a
176
function of time can be expressed as
xe(t) = eE(t)m
e?2
,
= eE0m
e?2
cos(?t)sech(t/?), (E.2)
where e is the elementary charge, and me is the mass of electron.
Since the maximum field was E0 = 8.7? 108 V/cm (I0 = 1015 W/cm2), the
maximum radius of electron was 2.7 nm. The probability or distribution of the
electron in the space was therefore presented by the histogram of the appearance
frequency of the electron at a position (x) between ?2.7 nm as the electron was
driven by the laser field according to Eq. E.2 between the initial time t1 when the
electron was removed from the molecule and a final time t2 relative to the peak of
the pulse. For a fixed t2 = 180 fs at the end of the pulse, as t1 varied from ?130 to
?70 fs, a double-peak structure stayed fixed near the center of the plot (Fig. E.1),
which corresponded to the low intensity part of the pulse at the falling edge (as the
time approached t2). However, the distribution on both sides of the double-peak
moved toward larger radii (?x) as t1 moved closer to the peak of the pulse (Fig.
E.1). The variation of the electron distribution became more significant as a fixed
?t = t2 ? t1 was used instead of a fixed t2 (Fig. E.2). Both width and height of
the double-peak structure varied non-monotonically, which was related to the laser
intensity variation between t1 and t2. These results showed that the calculated
electron distribution depended on and was sensitive to the initial time when the
electron was removed by the laser field, as well as the time interval and the laser
intensity during that time when the electron was oscillating near the molecular ion.
177
(a) (b)
(c) (d)
Figure E.1: Calculated histogram of the appearance frequency of the electron at a
position (x) between ?2.7 nm as the electron is driven by the laser field according
to Eq. E.2 between t1 and t2. The initial time t1 relative to the peak of the pulse
and the intensity at t1 are (a) ?130 fs and 2.2 ? 1013 W/cm2; (b) ?110 fs and
4.8?1013 W/cm2; (c) ?90 fs and 1.0?1014 W/cm2; and (d) ?70 fs and 2.2?1014
W/cm2. The final time t2 = 180 fs, which is roughly the end of the pulse. The total
probability in each plot is normalized to unity.
If this electron distribution is assumed to be the source of the screening effect in
the electron screening model of the molecular dissociative ionization, the dynamics
described by the model will also be sensitive to the initial time.
In the quasi-dynamic screening calculation, two electron wave packets repre-
sented by two 2D Gaussian density functions located at ?xe were moving according
to the radius of vibration of a free electron in the laser field (Eq. E.3),
xe(t) = eE(t)m
e?2
, (E.3)
178
(a) (b)
(c) (d)
Figure E.2: Calculated histogram of the appearance frequency of the electron at a
position (x) between ?2.7 nm as the electron is driven by the laser field according to
Eq. E.2 between t1 and t2. The initial time t1 relative to the peak of the pulse and
the intensity at t1 are (a) ?130 fs and 2.2?1013 W/cm2; (b) ?110 fs and 4.8?1013
W/cm2; (c) ?90 fs and 1.0?1014 W/cm2; and (d) ?70 fs and 2.2?1014 W/cm2.
The final time t2 = t1 + 180 fs. The total probability in each plot is normalized to
unity.
where E(t) = E0?sech(t/?) was the time-dependent electric field envelop of the laser
pulse (peaking at t = 0) with ? = 50 fs and the corresponding intensity pulse envelop
I(t) = I0 ?sech2(t/?) had a width of about 90 fs. Then, the electron distribution,
which was time-dependent through xe, was given by
?(x,y) = ?0
bracketleftBig
e?(x?xe)2/x20 +e?(x+xe)2/x20
bracketrightBig
e?y2/y20, (E.4)
where ?0 = Qel/(2?x0y0) was determined by Qel = integraltext ?(x,y)dxdy, the total number
of electrons in the density distribution.
179
Three positive charges were released at t = t1 from their initial positions sepa-
rated by the equilibrium bond length (Re) for the ground-state molecule, q1(Re,0),
q2(?Re,0), and q3(0,0) where the linear configuration and symmetric charge distri-
bution (q1 = q2 = q3 = q = 2) were considered. The charges begin to move under
the Coulomb repulsion between them and the Coulomb force from the electron dis-
tribution. As both molecules in this study (CO2 and NO2) are symmetric in mass
(m1 = m2 = mO), the motion of the charges would also be symmetric, x1 = ?x2,
y1 = y2 = 0, and x3 = y3 = 0. Thus, the equations of motion reduced to
m1?x1 = Kq
2e2
4x21 +
Kq2e2
x21
?
integraldisplay Kqe2?(x,y)(x
1 ?x)dxdy
[(x1 ?x)2 +y2 +s2]3/2 , (E.5)
where s is the smoothing parameter to remove the singularity of the Coulomb po-
tential. The first two terms are the Coulomb repulsion from the other two charges,
and the third term is the force from the electron distribution ? the screening effect.
The propagation of Eq. E.5 was carried out on discrete time intervals, t = t1,
t2, t3???, where the step size ?t = ti+1?ti = 0.5 fs was fixed. The initial conditions
at t = t1 are
x1(t1) = Re, ?x1(t1) = 0, ?x1(t1) = 0. (E.6)
At each step (t = ti), the acceleration ?x1(ti) was calculated from x1(ti?1) and ?(x,y)
with xe(ti) by Eq. E.5. Then, the velocity and position,
?x1(ti) = ?x1(ti?1) + ?x1(ti)??t,
x1(ti) = x1(ti?1) + ?x1(ti)??t, (E.7)
180
-130 -120 -110 -100 -90 -80 -70
Initial Time HfsL
50
100
150
200
250
cit
eni
K
yg
re
nE
HVe
L
5
10
15
20
25
lai
tin
I
yti
sn
etn
I
H01
31 W
?mc
2 L
Figure E.3: Calculated final kinetic energy (Blue diamonds, left axis) by the quasi-
dynamic simulation as a function of the initial time t1 when the electrons are born,
and the laser intensity (Red stars, right axis) at these times. The initial time is
relative to the peak of the pulse.
can be determined. At the end of the calculation, normally 800 steps and roughly
300 fs after the peak of the laser pulse at t = 0, the kinetic energy of the oxygen ion
(q1) was given by
Ek1 = m1 ?x12(tf)/2, (E.8)
where f = 800. Since the central ion would gain no momentum in a linear and
symmetric configuration, the total kinetic energy release of this electron-screened
Coulomb explosion was
Ek = 2Ek1 = m1 ?x12(tf). (E.9)
As t1 was being varied, kinetic energy release Ek as a function of t1 can be plotted.
The values of the parameters were chosen as the following: x0 = 0.28 nm so
that the two Gaussian functions would barely overlap at the origin; y0 = 0.021 nm
and s = 0.026 nm to roughly match with the values in the static screening model
181
(Sec. 5.1.2); Qel = 4 so that the minimum on the curve of Ek vs t1 was about 60
eV, which was roughly the same as the observation in the experiments; Re = 0.119
nm for NO2, and the peak intensity was I0 = 1015 W/cm2. As t1 moved from 130
to 75 fs earlier than the peak of the laser pulse and the laser intensity at t1 changed
from 2.2?1013 to 1.8?1014 W/cm2, the final kinetic energy varied by a factor of 4
between roughly 60 and 260 eV and the curve was not monotonic, as shown in Fig.
E.3. The result showed that, in this simple quasi-dynamic screening calculation, the
final kinetic energy was sensitive to the initial time when the electrons were born
and trapped near the nuclei by the laser field, and the variation of the kinetic energy
was not a monotonic function of the initial time.
182
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