Dedicated to the millions of molecules I exploded.Dedicated to the millions of molecules I exploded. DOCTOR OF PHILOSOPHY (2006) McMaster University (Physics & Astronomy) Hamilton,

Dedicated to the millions of molecules I exploded.

DOCTOR OF PHILOSOPHY (2006) McMaster University

(Physics & Astronomy) Hamilton, Ontario

TITLE: Controlling Molecular Alignment

AUTHOR: Kevin Lee, BSc(Hons), MSc (University of Toronto)

SUPERVISOR: P.B. Corkum

NUMBER OF PAGES: vi, 43

ii

Abstract

The molecules in a normal gas point in random directions. With a short laser pulse, a rotational wavepacket canbe created, leading to field-free alignment of the molecules after the laser pulse has passed. Due to quantization, therotational wavepacket can revive, repeatedly aligning at regular intervals. This thesis reports work on controlling,measuring, and using molecular alignment in experiments.

The alignment of a molecule can be measured by Coulomb explosion imaging. A strong laser pulse can multiplyionize a molecule, leading to rapid dissociation. By measuring the velocities of the fragments, the alignment ofthe original molecule can be determined. Experiments using Coulomb explosion imaging to image time-dependentmolecular alignment and structure are described.

By applying multiple laser pulses, the evolution of rotational wavepackets can be controlled. An extra laser pulsecan achieve field-free three-dimensional alignment in asymmetric top molecules. Rotational wavepackets can beenhanced or annihilated by extra laser pulses by making use of the rotational revival structure. Weak laser pulses attimes of fractional revival can cause dramatic changes to the wavepacket evolution by changing the relative phaseswithin the wavepacket.

We performed a laser-induced electron diffraction experiment on nitrogen molecules, using alignment to effectivelycrystallize the gas. Early results suggest that molecular structure can be measured using this technique. We developed anew technique for measuring the angular dependence of strong-field ionization in molecules. Applying this techniqueto nitrogen, we found that it is more likely to ionize when the molecular axis is aligned with, rather than againstthe electric field of a linearly polarized ionizing pulse. These and future experiments using molecular alignment arediscussed.

Acknowledgements

While only one name can go on the cover of a thesis, thiswork would never have succeeded without the contributions ofmany other people. I begin by thanking Paul Corkum and DavidVilleneuve for accepting an unproven undergraduate into theFemtosecond science group, and patiently guiding me along thepath of science.

Soon after joining Femto, I began to work with Patrick W.Dooley and Igor V. Litvinyuk, who taught me the ways of ionimaging. This thesis would have been completely different with-out them, and all the effort they put into perfecting the CEIchamber. They did much of the analysis of our early rotationalwavepacket experiments in Section 3.3. Igor Litvinyuk did themain analysis of the nitrogen ionization experiment in Section5.2.

Thanks to Francois Legare for everything in this thesis in-volving few-cycle pulses. It was a pleasure to help him in hisquest for the molecular movie. He generated the few-cyclepulses, performed much of the experiment and analysis in Chap-ter 2, using some of the methods and computer codes I had de-veloped for earlier experiments.

Thanks to my friends who like theory, Evgeny A. Shapiroand Michael Spanner. They were a key part of developing theideas behind the wavepacket control work, and did the numericalsimulations of the control processes. Special thanks to Evgenyfor sending me over 200 emails, and putting up with over 300emails and 48 drafts of two papers from me.

Thanks to Daniel Comtois, and Domagoj Pavicic, two in-trepid postdocs whom I have hopefully helped along the road tolaser-induced-recoil-electron-self-auto-diffraction.

Thanks to Jonathan Underwood and Albert Stolow for theirwork on 3D alignment, and getting the chamber back into onepiece.

Thanks to Bert Avery, David Joines, and John Parsons fortheir expert help on all the strange things that happen in a physicslaboratory.

And to everyone else who played a part in the making of thisthesis: all the members of Femto, past and present; friends andfamily; the helpful staff at McMaster and the National ResearchCouncil; and my supervisory committee, Cecile Fradin, BrianKing, and Graeme Luke; I would like to say thank you.

iii

Publications

Chapter Publication

2 “Laser Coulomb-explosion imaging of small molecules.”F. Legare, Kevin F. Lee, I. V. Litvinyuk, P. W. Dooley, S. S. Wesolowski, P. R. Bunker, P.Dombi, F. Krausz, A. D. Bandrauk, D. M. Villeneuve, and P. B. Corkum.2005, Physical Review A, 71, 013415.

“Imaging the time-dependent structure of a molecule as it undergoes dynamics.”F. Legare, Kevin F. Lee, I. V. Litvinyuk, P. W. Dooley, A. D. Bandrauk, D. M. Villeneuve,and P. B. Corkum.2005, Physical Review A, 72, 052717.

“Laser Coulomb explosion imaging for probing ultra-fast molecular dynamics.”F. Legare, Kevin F. Lee, P. W. Dooley, A. D. Bandrauk, D. M. Villeneuve, and P. B. Corkum.2006, Journal of Physics B: Atomic, Molecular and Optical Physics, 39, S503.

3 “Direct imaging of rotational wave packet dynamics of diatomicP. W. Dooley, I. V. Litvinyuk, Kevin F. Lee, D. M. Rayner, M. Spanner, D. M. Villeneuve,and P. B. Corkum.2003, Physical Review A, 68, 023406.

“Measured field-free alignment of deuterium by few-cycle pulses.”Kevin F. Lee, F. Legare, D. M. Villeneuve, and P. B. Corkum.2006, Journal of Physics B: Atomic, Molecular and Optical Physics, 39, 4081.

“Field-Free Three Dimensional Alignment of Polyatomic Molecules.”Kevin F. Lee, D. M. Villeneuve, P. B. Corkum, Albert Stolow, and Jonathan G. Underwood.2006, Physical Review Letters, 97, 173001.

4 “Two-pulse alignment of molecules.”Kevin F. Lee, I. V. Litvinyuk, P. W. Dooley, Michael Spanner, D. M. Villeneuve, and P. B.Corkum.2004, Journal of Physics B: Atomic, Molecular and Optical Physics, 37, L43.

“Phase Control of Rotational Wave Packets and Quantum Information.”Kevin F. Lee, D. M. Villeneuve, P. B. Corkum, and E. A. Shapiro.2004, Physical Review Letters, 93, 233601.

“Coherent Creation and Annihilation of Rotational Wave Packets in Incoherent Ensembles.”Kevin F. Lee, E. A. Shapiro, D. M. Villeneuve, and P. B. Corkum.2006, Physical Review A, 73, 033403.

5 “Alignment-Dependent Strong Field Ionization of Molecules”I. V. Litvinyuk, Kevin F. Lee, P. W. Dooley, D. M. Rayner, D. M. Villeneuve, and P. B.Corkum.2003, Physical Review Letters, 90, 233003.

iv

Table of Contents

1 Introduction 11.1 Air, balloons, and crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 See, do . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Measuring molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Rotating molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Rotational wavepackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Experiments with aligned molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Coulomb Explosion Imaging 42.1 Types of Coulomb explosion imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Coulomb explosion imaging experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Detection apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.2 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Coulomb explosion imaging of triatomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.1 Assigning properties to a detected fragment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.2 Calculating structures from velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4.3 Measured structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.4 Molecular movies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Future of Coulomb explosion imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Rotational Wavepackets 133.1 Wavepackets and revivals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Making rotational wavepackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Observing rotational wavepackets in diatomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 Measuring alignment by CEI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Revival structures of diatomics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4.1 Fourier transform of revivals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4.2 Alignment of deuterium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 Field-free 3D alignment of asymmetric tops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5.1 Method of 3D alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5.2 Experimental 3D alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.6 Rotational wavepacket overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Controlling Rotational Wavepackets 244.1 Multiple-pulse alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 Strong pulse control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.1 Enhancing alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2.2 Rotational wavepacket annihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Weak pulse phase control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Other control techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Experiments Using Aligned Molecules 325.1 Ways to use aligned molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2 Alignment-dependent strong-field ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3 Laser-induced electron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Conclusion 376.1 Molecular alignment and the world. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Bibliography 38

v

List of Figures

1.1 Air, balloon, crystals . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Photograph of the moon . . . . . . . . . . . . . . . . . . . . . . 21.3 Torque on molecule in a laser field . . . . . . . . . . . . 21.4 Shape of aligned and antialigned wavepackets . 31.5 Illustration of isotropic and aligned nitrogen. . . 3

2.1 Collision Coulomb explosion imaging . . . . . . . . 42.2 Coulomb explosion imaging detection cham-

ber photograph . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Illustration of a delay line anode . . . . . . . . . . . . . . 62.4 Illustration of doughnut and timbit optics and

angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Sum momentum distribution of SO2 fragments 82.6 Measured D2O structures. . . . . . . . . . . . . . . . . . . . . 102.7 Measured SO2 structures . . . . . . . . . . . . . . . . . . . . . 102.8 Evolution of correlated kinetic energy of oxy-

gen from SO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.9 Evolution of correlated velocity and total ki-

netic energy from SO2 . . . . . . . . . . . . . . . . . . . . 112.10 Measured structures of SO2 by dissociation

channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Illustration of rigid rotor and harmonic oscil-lator energy levels . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Interferometer for making aligning and ex-ploding pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Illustration of angle ϑ . . . . . . . . . . . . . . . . . . . . . . . . 163.4 Angular distribution of aligned oxygen. . . . . . . . 163.5 Measured rotational revivals in nitrogen . . . . . . . 163.6 Measured rotational revivals in oxygen . . . . . . . . 173.7 Sixth full revival of nitrogen . . . . . . . . . . . . . . . . . . 183.8 Fourier transform of revivals in N2 and O2 . . . . 193.9 Measured rotational revivals in deuterium . . . . . 203.10 Illustration of isotropic, 1D, and 3D alignment 203.11 Molecular frame axes for SO2 . . . . . . . . . . . . . . . . 213.12 Pulse scheme for field-free 3D alignment. . . . . . 213.13 Angles for measuring 3D alignment . . . . . . . . . . 213.14 Measured 3D alignment in cold SO2 . . . . . . . . . . 223.15 Calculated alignment of full SO2 ensemble . . . . 223.16 Measured 3D alignment in hot SO2 . . . . . . . . . . . 23

4.1 Four-pulse interferometer for rotationalwavepacket control . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Swing analogy for strong pulse control . . . . . . . . 244.3 Pulse timing for enhanced alignment experi-

ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4 Surface map of enhanced alignment . . . . . . . . . . 254.5 Selected revivals from enhanced alignment

experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.6 Enhanced alignment in oxygen . . . . . . . . . . . . . . . 264.7 Calculated projections of an annulled wavepacket

in nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.8 Calculated revivals of wavepacket annihila-

tion in nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.9 Measured wavepackets showing rotational

wavepacket annulment . . . . . . . . . . . . . . . . . . . 294.10 Measured cross-shaped fractional revival in

oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.11 Polar plots of cos2 ϑ and cos2 2ϑ . . . . . . . . . . . . . . 304.12 Timing of pulses in phase control experiment. . 304.13 Measurement of coherent switching off and

on of alignment . . . . . . . . . . . . . . . . . . . . . . . . . . 314.14 Measurement of doubled alignment revival

frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1 Angular distributions of aligned and an-tialigned nitrogen . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2 Angular dependence of ionization probabilityin nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.3 Three-step model of laser-induced electrondiffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4 Geometry of two-slit diffraction . . . . . . . . . . . . . . 355.5 Illustration of interferometer for cross-

polarized pulses . . . . . . . . . . . . . . . . . . . . . . . . . . 355.6 Measured recoil momentum of singly ionized

nitrogen and argon . . . . . . . . . . . . . . . . . . . . . . . 365.7 Ratio of recoil momentum spectra of argon

and nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.1 Photograph of 3D aligned cars . . . . . . . . . . . . . . . . 37

vi

PhD Thesis –––––––––––– Kevin F. Lee –––––––––––– McMaster University - Physics & Astronomy –––––––––––– 2006

1

Introduction

1.1 Air, balloons, and crystals

The reader of this thesis is surrounded by the samekind of molecules on which I have been experimentingover the past several years, namely, nitrogen, oxygen, wa-ter, and sulphur dioxide [1]. People commonly interactwith matter in the three forms: solid, liquid, and gas. Inmany ways, a liquid is like a very dense gas, both haveparticles moving around and bouncing into each other. Asolid is quite different, with component particles that stayin place.

If you spin a

FIGURE 1.1: Air, balloon, and crys-tals.

solid around bya quarter turn,you know thatyou have rotatedthe bonds in thesolid by a quar-ter turn. If youtake an inflatedballoon though,and rotate it by aquarter turn, noth-ing much happensto the gas inside,except perhaps alittle bit of swirling. You did not know which way themolecules inside were pointing before, and even if youhad known, you would not be rotating them by rotatingthe balloon.

What if a balloonier presented you with an inflatedballoon, and asked you to make all the molecules insidepoint in the same direction? An initial reaction might beto try to freeze the balloon’s contents into a single crystal,resulting in a solid piece of aligned molecules. But thiswould not satisfy the demanding balloonier, who wantseach gas molecule left as it is, except for the directionof the molecular axes. Can molecules be manipulated aseasily and precisely as you can manipulate this book?

The simple answer is no, if only because moleculesbehave quantum mechanically, and the average personwould not consider quantum mechanical behavior tobe simple. For the specific problem of aligning gasmolecules in a balloon though, the molecules can befairly well aligned, assuming that you had an immenselypowerful laser and a transparent balloon.

1.1.1 See, doWhat are the main actions when you take a physical

copy of this thesis and rotate it in your hands? First youmight look at the thesis, and measure its current angle.Then you start the rotation by applying a torque. While itrotates, you periodically measure the angle until you aresatisfied, and you apply torque in the opposite direction tostop the rotation.

To handle individual molecules as well as you canhandle this thesis, we would like techniques that are anal-ogous to the actions described above. We need the abil-ity to measure the direction of molecules, and the abil-ity to change the rotation of molecules. The developmentof techniques for rotating and measuring molecules, andtheir application to experiment drive the work presentedhere.

1.2 Measuring moleculesThere are many ways to collect information about in-

dividual molecules. For the most part, this involves throw-ing particles at the molecule in question, and then seeingwhat happens. The intended interaction might be veryspecific, such as using light of a particular wavelength,leading to the ejection of a detectable particle such as anelectron [2]. The interaction can also be more general,such as the diffraction of x-rays [3] or electrons [4].

For the balloon problem, the simplest way to measuremolecular alignment is probably to observe the rotationof polarization of light passing through the balloon, orgas cell, as the case may be. An individual molecule isbirefringent, but a randomly aligned gas is not, due toaveraging over many molecules. With increasing align-ment of the gas, there will be an increase in the birefrin-gence of the gas. This effect has been used as a diagnos-tic in several alignment experiments [5, 6, 7, 8]. A moresophisticated version of a birefringence measurement isfour-wave mixing, which has also been used to measurealignment [9, 10].

Since many molecular interactions depend on align-ment, changing alignment has been observed in manyother ways as well. Alignment was observed by electrondiffraction [11], photodissociation [12], Raman spec-troscopy [13, 14], photoelectron imaging [15], ionizationrate [16], and high harmonic generation [17, 18].

The alignment measurements in this work were alldone using laser-induced Coulomb explosion imaging(CEI). Coulomb explosion refers to the rapid separationof charged fragments due to Coulomb repulsion. Likea microscope that expands photons from a microscopicregion to a macroscopic region, Coulomb explosion takesthe particles themselves from being Angstroms apart, to

1


being centimetres apart. To measure alignment, a shortand intense laser pulse multiply ionizes the molecule,which then Coulomb explodes. The velocities of the es-caping fragments include information about the positionsof the fragments at the time of explosion, revealing thealignment of the molecule [19, 20, 21].

Coulomb explosion does have some limitations: mea-surement can be difficult for large molecules that havecomplicated explosions; a relatively sophisticated appa-ratus is needed to measure fragment velocities; and themolecule is destroyed by the measurement. Coulombexplosion also has significant advantages. It is the moststraightforward technique in terms of understanding theresults. Other methods rely on information about amolecular property such as birefringence relative to themolecule’s orientation. With the help of some educatedguesses, a full Coulomb explosion imaging experimentshould be able to measure the orientation of a moleculewithout any prior knowledge of even the species ofmolecule. Coulomb explosion also has the advantageof giving the orientation of each molecule, rather than asingle average value that is measured by birefringence.The actual angular distribution of the molecules can bemeasured, from which average values can be calculated.

1.3 Rotating moleculesTo apply a torque to

FIGURE 1.2: Tidal forcesfrom the moon have very lit-tle effect on molecular align-ment.

a molecule, first a forcemust be chosen. Forcessuch as the gravitationaltidal force would be animpractical choice, dueto its weakness, and ourpoor ability to manipu-late it. The most effectiveways to rotate moleculesuse electric fields, whichinteract strongly withthe charged particles inmolecules. Moleculeshave been aligned usingelectric fields from electrodes [22, 23, 24], and lasers[12], two well-developed technologies. Lasers are ca-pable of generating much stronger electric fields thanelectrodes. Lasers can also modulate the electric field onthe same femtosecond timescale as molecular rotations.In this thesis, alignment experiments are done using laserpulses.

When illuminated by a laser, a molecule gains an in-duced dipole from the oscillating electrons. An infraredlaser pulse is red-detuned from the electronic transitions

in the molecule. The electrons can move quickly enoughto stay in phase with the laser field. In this case, themolecule has the lowest energy when the induced dipoleis strongest. Thus a laser pulse results in a torque pullingthe most-polarizable axis, generally the long axis, of themolecule towards the electric field of the laser, as illus-trated in Figure 1.3 [25].

Returning to the

FIGURE 1.3: A laser field in-duces a dipole and torques amolecule towards the electricfield axis.

balloon problem, wecan imagine illumi-nating the balloonwith a laser that islinearly polarizedalong the directionin which we want themolecules to align.The laser gives themolecules a push to-wards alignment. Fora laser pulse much

longer than the rotational timescale, the slow increase ofthe electric field adiabatically brings molecules from theirfree-rotating states to well-aligned pendular states. At thepeak of the laser field, the molecules are well-aligned, andoscillate like pendula around the laser polarization [26].This is known as adiabatic alignment. In some cases,this is satisfactory, but in others, the presence of the laserfield is disruptive. If the laser pulse has been shaped toquickly turn off after the peak [8, 27], then the moleculeswill be left in an aligned state without the presence of thealigning field. This is known as field-free alignment.

A simpler method for achieving field-free alignment,and the one used in this thesis, is to use a pulse that isshorter than the rotational timescale. The laser kicks themolecules towards alignment, but the pulse passes beforethe molecules have had time to align. Shortly afterwards,the molecules reach their maximum alignment. As in theshaped pulse case, there is a time when the molecules arealigned and field-free. The result can be described as acoherent superposition of rotational states, known as a ro-tational wavepacket.

1.3.1 Rotational wavepacketsTo rotate a molecule, we use a laser pulse to create a

rotational wavepacket. After this, the problem of control-ling molecular rotation becomes a problem of controllingthe rotational wavepacket.

Simple linear molecules can be approximated as rigidrotors, with rotational states described by the quantumnumber J, and the projection m. The field-free behaviourof a rotational wavepacket of rigid rotor states is rela-

2


tively simple. The rotational frequencies ωJ are given byhωJ = hcBJ(J + 1), where B is the rotational constant ofthe molecule in wavenumbers. Since J can only have in-teger values, all frequencies will be multiples of a funda-mental frequency. Thus, every fundamental period Trev=1/(2Bc), a rotational wavepacket will rephase to whateverstate it was in Trev before.

FIGURE 1.4: Illustration of an aligned (middle) and an an-tialigned (right) wavepacket, for the linearly polarized align-ing pulse shown on the left.

For a rigid rotor, once there is a field-free, aligned,rotational wavepacket, the wavepacket will repeatedly re-vive as the same aligned state every Trev later. This leadsto a regular revival structure with the molecule spinningfrom aligned to isotropic and antialigned distributions, an-tialignment being the case when the distribution is per-pendicular to the aligning field, as illustrated in Figure 1.4[21]. It will be seen that significant changes can be madeto the evolution of a rotational wavepacket by using ad-ditional laser pulses at important times during the revivalstructure.

1.4 Experiments with aligned molecules

It may not be clear from the balloon problem that con-trolling molecular alignment is important. Beyond the sat-isfaction of being able to manipulate matter on a molec-ular level, it should be remembered that most molecularproperties depend on the alignment of the molecule. Thisis easily forgotten for gases, since angular effects are av-eraged away by the isotropic distribution. Many exper-iments on isotropic gas molecules would become new,deeper experiments if done with aligned molecules.

Until recently, the main experimental application ofalignment, and rotational wavepackets, was rotational co-herence spectroscopy (RCS) [28]. In this technique, therevival structure of a rotational wavepacket is measured.From the frequencies that form the revival structure,molecular properties such as the moments of inertia, andthe nuclear spin statistics can be extracted.

A more intuitive way to use alignment is simply todo an experiment on aligned molecules. For example,there have been recent experiments showing that the ef-ficiency of high-harmonic generation by molecules de-pends strongly on the alignment of the molecule relativeto the polarization of the driving field [17]. From this sim-ple observation developed a technique to tomographicallyimage electron wavefunctions of molecules [29].

Two experiments using aligned molecules are in-cluded in this thesis. One experiment measures thedependence of laser-induced ionization on alignmentrelative to the ionizing field. The other aims to measuremolecular structure by electron diffraction. These exper-iments represent the first in a new class of experimentsthat will access and exploit a type of matter that had notbeen accessible before: aligned gas molecules.

FIGURE 1.5: An illustration of randomly aligned nitrogenon the top, and aligned nitrogen on the bottom.

3


2Coulomb Explosion Imaging

2.1 Types of Coulomb explosion imaging

Coulomb explosion refers to the rapid separation oflike charges by Coulomb repulsion. In a molecule, posi-tively charged nuclei are separated by about an Angstrom,and held together by negatively charged electrons. In anideal Coulomb explosion, one would instantly remove allof the electrons, or all of the nuclei, leaving a set of like-charged particles that would then Coulomb explode. Ina real experiment, one is currently limited to removing asubset of electrons during a non-zero period of time.

The two main ionization methods are fast collisions,and strong-field laser pulse ionization. The older methodis collision-induced Coulomb explosion, where a molecu-lar ion is accelerated by an electromagnetic field towards astationary target, as illustrated in Figure 2.1. To study neu-tral molecules, accelerated negative ions can be ionizedprior to collision [30]. In modern collision experiments,the target is usually an amorphous foil of thickness 100A or less. Within about 100 attoseconds, the outer shellelectrons are stripped by the foil, and the remaining frag-ments pass through the foil within about a femtosecond[31, 32]. With no more bonding electrons, the moleculeCoulomb explodes.

Behind the foil target is the particle detector. The de-tector is generally a plane perpendicular to the travel di-rection that reports the position x,y and time t at whicheach fragment hits the detector. These can be converted tothe velocity vectors of the fragments, and then the molec-ular structure at the time of explosion.

Laser-induced Coulomb explosion might be consid-ered a less precise, but simpler and more flexible versionof collision-induced Coulomb explosion. The key differ-ence between lasers and collisions is the speed of the ion-

FIGURE 2.1: Illustration of a triatomic colliding with a foiland being detected after the explosion spatially separatesthe fragments.

ization process. If ionization takes too long, the moleculewill change shape as it adapts to the loss of each electron.For lighter constituents such as hydrogen, such rearrange-ment, essentially vibrational motion, is significant on the1 fs timescale [33].

A molecule will generally expand as electrons are re-moved and the bond is weakened. As the molecule ex-pands, laser ionization becomes easier, an effect known asenhanced, or charge-resonance-enhanced ionization. Thisis a characteristic of non-resonant, strong-field ionizationin which the laser field causes a large tilting of the nuclearCoulomb potential holding the electron. In a molecule,the potential well of a neighbouring atom will additionallydeform the Coulomb potential, especially when the inter-nuclear axis is along the laser field, leading to strongly en-hanced ionization rates for certain large internuclear dis-tances [34, 35].

Avoiding enhanced ionization is essential for trueCoulomb explosion imaging. During a slow ionization,the molecule will expand as it ionizes. As the moleculeapproaches a critical internuclear distance, the large in-crease in ionization rate means that most ionization willoccur at large internuclear separation. Regardless of theinitial molecular structure, the exploding fragments willpredominantly come from large structures aligned alongthe laser field [36].

Except for heavy molecules with slow atoms [37],laser-induced Coulomb explosion imaging had been lim-ited to the enhanced ionization regime. Techniques havesince been developed to compress energetic 50 fs, 800nm pulses from titanium:sapphire based lasers below 10fs, using self-phase modulation in a hollow core fibreto generate bandwidth, and chirped mirrors or prisms torecompress the pulse [38]. These are sometimes calledfew-cycle pulses, since the period of 800 nm light is 2.67fs.

Using few-cycle pulses on deuterium, 2H2 or D2,one of the lightest molecules, a new explosion regimewas found having a kinetic energy release greater thanmolecules in the enhanced ionization regime, but stilla little less than for a perfect Coulomb explosion [39].Regardless of how short the pulse is, the molecule willstill expand during ionization. Unless the probabilityof ionization is very high, the nuclear wavepacket willdeform at each ionization step due to the dependence ofionization rate on internuclear separation.

Few-cycle laser pulses are currently the best way to dolaser-induced Coulomb explosion. Having overcome en-hanced ionization, structure measurements can be madethat, while not as precise as collision techniques, can re-veal relative changes to molecular structure. The abilityto see changes in the molecule is the main drive behind

4


laser-induced Coulomb explosion imaging. By startinga molecular motion with one laser pulse, and measuringthe result with a later pulse at different times, a techniquecalled pump-probe, Coulomb explosion imaging can beused to watch molecular structure change in time.

2.2 AssumptionsThe simplest type of molecule is the diatomic. The

structure of a diatomic is given by the single value ofbond length. In the deuterium experiment above, onlythe deuteron kinetic energy was mentioned. To completethe imaging aspect, the following assumptions might bemade: the fragments are at rest at the time of explosion;the separation is classical [40] and governed completelyby the Coulomb potential between the two atomic ions intheir final charge state; and the other deuteron is also ion-ized.

The initial vibrational motion will violate the first as-sumption of starting from rest. In terms of measuring astructure, a Coulomb explosion has much more energythan the initial vibrational motion, so that low vibrationalenergy does not badly distort the measurement. The en-ergy released in a Coulomb explosion is usually well over10 eV. Adding a thermal contribution of 25 meV at roomtemperature would result in a less than 1% change in themeasured bond length. The motion at the time of explo-sion is more likely dominated by the energy gained dur-ing the ionization process. This is related to the secondassumption, since the ionization usually happens in steps.

For a linearly polarized pulse, as many as four elec-trons have been ionized within a laser cycle [41]. For mostexplosion events though, one might expect about one elec-tron to be removed near the peak of each half-cycle of theelectric field, possibly a result of Coulomb blockade fromnearby electrons, but this is still a current topic of research[39]. The faster the ionization, the better the second as-sumption is met. While deuterium only has two electrons,the presence of many electrons in other molecules can alsoviolate the Coulomb approximation, since the electronicstates can create a complicated potential. Since the poten-tial will always approach the Coulomb potential for largeseparations, the faster the fragments separate, the betterthe Coulomb approximation is satisfied.

The third assumption about the other charge states ismore of a technical detail. For deuterium, the assumptionis very reasonable, since the molecule will not explodeif the other deuterium is not ionized, resulting in a veryunenergetic dissociation. For any other molecule, it canbe difficult to make accurate assumptions about the chargestates of the other fragments. The safest solution is not tomake the approximation at all, and instead measure all the

fragments. There are also technical difficulties in doingthis, but in many cases, it can, or must, be done.

When detecting all fragments in an explosion, themain check is to verify that all fragments are from thesame molecule. One approach is to guarantee that thereare never two molecules in the laser focus at the sametime. With current technology, this can only be doneby reducing the gas density to keep the fragmentationrate very low. For example, if you fragment a moleculeabout once every 10 laser shots, there is about a 1 in 102

chance of having two molecules fragment in the samelaser shot. This is a very slow approach though, sincemost laser pulses do nothing. A faster solution is to calcu-late the momentum of each fragment, and verify that themomentum sum for all fragments matches the expectedmomentum of the original molecule, which is about zerorelative to the large explosion momenta.

Having made the above assumptions, the deuteron ki-netic energy can easily be converted to the bond length.The total energy of the explosion is simply taken to betwice the deuteron kinetic energy, and this energy is setequal to the Coulomb potential U between two particlesof charge q1 and q2, and bond length r: U = q1q2/4πε0r.In this case, q1 and q2 are both +e and ε0 is the vacuumpermittivity.

2.3 Coulomb explosion imaging experi-ments

2.3.1 Detection apparatusThe experiments in this thesis were all performed us-

ing the Position- and Time-Resolved Ion Correlation Kit(PATRICK) in the Femtosecond Science laboratories ofthe Steacie Institute for Molecular Sciences of the Na-tional Research Council Canada. A brief description ofthis apparatus follows; a more extensive description canbe found in the thesis of the primary chamber designer,Patrick W. Dooley [42].

The PATRICK includes a position and time sensitivedetector at the end of a constant electric-field spectrome-ter housed in a vacuum chamber, and supported by variouselectronics and vacuum components. The Coulomb ex-plosion occurs at the focus of a 5 cm focal length parabolicmirror inside the detection chamber. The focus is inside atubular stack of electrodes which creates a uniform elec-tric field that drives positive charges up towards the de-tector. The detector, a Roentdek DLD80, is an 86 mmdiameter microchannel plate (MCP) stack, backed by twohelical delay-line anodes arranged in a square grid.

As a positive ion approaches the detector, it willcontinue from the top of the spectrometer, which is at

5


FIGURE 2.2: Photograph of the PATRICK detection cham-ber with the intermediate and source chambers removed,from the view of the molecular beam, which enters by theskimmer in the centre of the photograph, mounted on thegate valve. The laser beam enters from the right side, andis focused by a parabolic mirror on the end of the manipu-lator mounted on the left side of the chamber.

ground potential, to the front of the MCP, which is at−2300 V. The acceleration in this small space gives theion enough kinetic energy to trigger an electron cascadewhen it reaches the MCP. The sudden change in voltageacross the MCP is used to determine the time of impact.The electron avalanche is picked up by the delay-linesbehind the MCP.

Each delay line is a signal and reference pair of wireslooped many times to make a detection surface behindthe MCP. A single helical delay-line is illustrated in Fig-ure 2.3. The electron pulse propagates in both directionsalong the delay line, and the time of arrival at each end ismeasured. The difference in arrival times gives the posi-tion along the wire, which corresponds to the position ofthe hit along the direction perpendicular to the wires. Itis possible to retrieve the ion time-of-flight by taking thesum of the two arrival times and subtracting the travel timeof a pulse through the entire delay-line, but the PATRICKuses the more direct measurement from the MCP. A sec-ond delay line is wrapped around the first with a relativerotation of 90◦ around the time-of-flight axis to measurethe other space dimension in the detection plane.

The signals from the MCP and delay lines are passedto the Roentdek DLATR6 unit. The DLATR6 amplifiesthe AC signal, and uses a constant fraction discriminatorto filter the signals by voltage threshold and pulsewidth,and output a well-defined ECL signal. The ECL pulsesare sent to a time-to-digital converter (TDC, Lecroy 3377)that digitizes the time signals with a 0.5 ns resolution. Forthe delay-lines, this provides a space resolution of 255 µmand 267 µm in the x and y dimensions, respectively. Theresolution of the time dimension is much better than thespace dimension, since the time-of-flight signal is also

read at 0.5 ns resolution, but with a total range of 32 µs.From the resulting (x,y, t) data, the initial velocity vec-tor of the ion can be calculated, using knowledge of thetime-of-flight electric field and the ion mass.

A molecu-

FIGURE 2.3: Illustration showing thegeometry of a single helical delay-line anode. The arrow indicatesthe direction of the space dimensionmeasured by this anode.

lar beam pro-vides cold gasmolecules for ex-plosion. It travelsperpendicularlyto both the spec-trometer axis, andthe laser propa-gation direction.The molecularbeam originatesfrom a pinhole(from MellesGriot) having adiameter from 30

to 100 µm, in the source chamber. After the pinhole,the jet passes through a 1 mm diameter skimmer be-tween the source and the intermediate chambers, another1 mm skimmer between the intermediate and detectorchambers, and a variable width piezoelectric slit whichopens horizontally. The skimmers (Model 1, nickel skim-mers from Molecular Beam Dynamics) are designed tosmoothly deflect gas that does not pass through the aper-ture away from the opening to reduce collisions withinthe molecular beam. At the laser focus, the jet is a thinribbon of gas, about a millimetre high, and about 50 µmthick.

The three chamber structure is meant to allow a highbackground pressure in the source chamber, over 10−4

Torr, while maintaining a low background pressure in thedetection chamber, below 10−6 Torr, as is needed for highvoltage operation. This kind of differential pumping sys-tem allows higher backing pressures to be used behind thenozzle, an important parameter for cooling of the molec-ular beam. The rotational temperature, and translationaltemperature in the propagation direction can be reducedby using a buffer gas, such as He, seeded with a smallpercentage of the molecule of interest [43]. By geome-try, the translational temperature perpendicular to the jetpropagation is always very low.

2.3.2 LaserA titanium:sapphire based, home-built regenerative

amplifier system was the light source for the experimentsin this thesis. The laser pulse starts as a relatively weakpulse from a mode-locked Ti:sapph oscillator. A grating

6


stretcher adds dispersion, chirping the pulse from about25 fs full-width at half-maximum (FWHM) to about 50ps. One of these stretched pulses is switched by a Pockelscell into the cavity of the regenerative amplifier while it isbeing pumped by a nanosecond pulse from a Q-switched,Nd:YLF laser (Merlin, Positive Light) [42, 44, 45, 46].

On its own, the regenerative amplifier can be madeto produce nanosecond laser pulses at around 800 nm.When it is seeded by the chirped pulse from the oscillator,the relatively intense pulse overwhelms the natural modecompetition, and is amplified during several round-trips inthe cavity. When the pulse has been maximally amplified,it is dumped by a Pockels cell.

The amplified pulse is then collimated, and thestretching is reversed by a grating compressor. Typi-cal output is an 850 µJ, 815 nm, 45 fs FWHM pulse, ata 600 Hz repetition rate. By changing the timing of thePockels cells and the pump intensity, pulse energy mightbe increased to about 1 mJ, or the repetition rate might beincreased to 1 kHz at the expense of pulse energy, to suitthe needs of the experiment.

These 45 fs pulses, when focused in the chamber,can reach well over 1016 W/cm2, and are thus capable ofremoving many electrons from molecules. This type ofpulse, with adjusted pulse energies, was used in most ofthe alignment experiments in this thesis. As discussedabove though, structural imaging requires few-cyclepulses.

To generate few-cycle pulses, about 500 µJ of the 45fs pulses are focused into two 50 cm lengths of 250 µmdiameter hollow core fibre, the first of which is filled witha little over one atmosphere of argon. Being constrainedin the fibre ensures that the pulse stays intense over thelong interaction region with argon. When emerging fromthe fibre, self-phase modulation will have stretched thepulse spectrum to span about 200 nm [38].

The dispersion of this broadband pulse is then over-compensated by several bounces on four dielectricchirped mirrors. The extra dispersion compensationis to correct for the dispersion that will be added bypropagation through air, and through the window of thevacuum chamber. Fine-tuning of the dispersion can bedone by adding glass windows of different thickness.A simple diagnostic of pulse duration is to maximizethe brightness from the plasma produced by the focusedbeam. A more precise way is to maximize the kineticenergy of exploded hydrogen fragments in the detectionchamber.

The few-cycle pulses we produce this way can havea few hundred microJoules of energy, and be as short as7 fs, as measured by spectral interferometry for directelectric field reconstruction (SPIDER) [47]. The pulse

can be linearly polarized by grazing-incidence reflectionson germanium. The pulse can be circularly polarizedby a thin, broadband λ/4 waveplate (Halle 500-900 nm,340µm MgF2). When focused in the chamber, the peakintensity can exceed 1015 W/cm2.

To measure time evolution, we need two pulses witha controllable time delay. This is known as a pump-probeexperiment, where a pump pulse initiates a process, andanother pulse probes the system at different times relativeto the pump pulse. Pulses are often time delayed by split-ting and recombining a pulse in an interferometer havingan arm with an adjustable path length. This is difficult toapply to few-cycle pulses since the beamsplitters wouldneed a very large bandwidth. Instead, we make use ofthe time delay imposed by the index of refraction of glass[48].

beam

�

�

�

d

nair

nglass

FIGURE 2.4: Left: Illustration of doughnut and timbit op-tics for generating two delayed few-cycle pulses. Right: topview of doughnut, showing angles used to calculate timedelay for a pulse travelling through a tilted glass plate.

The delay apparatus is made of two pieces of glass,which were originally a single window. A 5 mm diameterdisc was cut out of the centre of a 3 mm thick, 25 mmdiameter window, leaving an annulus and a disc. Usinga deep-fried pastry analogy, Canadians will often refer tothese optics as a doughnut and a timbit. When both opticsare centred on, and perpendicular to the beam path, theyact again as a single window. Besides the extra dispersionof the glass, which can be corrected by extra dispersioncompensation after the fibre, the pulse is otherwise unaf-fected.

When the annulus, or disc, is rotated so that the opticis no longer perpendicular to the beam path, the part of thepulse moving through the rotated optic will travel throughmore glass, leading to a time delay. Using Snell’s law, thetime delay, t, for a tilt angle of θ is

t =dc

[(1

cos(φ)−1

)nglass +

(1− cos(θ−φ)

cos(φ)

)nair

],(2.1)

where nglass and nair are respectively the refractive indicesof the rotated piece and the surrounding medium (approx-

7


imating group and phase velocity to be equal) and φ is theangle of deviation of the light in the doughnut relative tothe original propagation direction:

φ = arcsin(sin(θ)nair/nglass). (2.2)

The result is two separate pulses with delay controlled bythe angle of rotation. There is an additional benefit that thepulse that travels through the annulus will have a smallerfocus and higher intensity than the pulse from the smallerdisc.

In our pump-probe Coulomb explosion experiments,we rotate the annulus, and use the more intense and tightlyfocused pulse from the annulus as the exploding probe,and the more loosely focused pulse from the disc as thepump. The rotating optic will cause small changes to thepulse duration from changes in dispersion, so some ex-periments may benefit from rotating the annulus insteadof the disc.

2.4 Coulomb explosion imaging of tri-atomics

2.4.1 Assigning properties to a detected frag-ment

In our experiments with triatomics, we used few-cyclelaser pulses to Coulomb explode sulphur dioxide (SO2),heavy water (D2O), and carbon dioxide (CO2). The datareturned from an experiment is a number of events, eachof which is a set of detector hits for a given laser pulse.

For each fragment, the detector returns position andtime-of-flight. The atomic number and charge of a singlehit is not directly known. The time-of-flight of an ini-tially stationary fragment is directly related to the ratio ofthe mass and the charge, but kinetic energy from the ex-plosion will change the time-of-flight to possibly overlapwith other mass/charge ratios. In some cases, differentpeaks in the time-of-flight spectrum are well-separated,making assignment of atomic number and charge simple.In other cases, the peaks will overlap significantly, suchas light, highly charged fragments, or atomic masses thatare multiples of each other, as in the case of 32S and 16Ofrom SO2.

For imaging purposes, one must also assign fragmentsto groups of hits originating from the same molecule.Since we need to collect all ions from an explosion forCoulomb explosion imaging, we ultimately rely on themomentum sum to verify the assignments given to eachhit. It is possible, though inefficient, to simply iteratethrough all sets of three hits within an event, testingthe momentum sum while assuming different values foratomic number and charge. Usually, selection of different

FIGURE 2.5: Distribution of the summed momentum ofS3+, O3+ and O2+ fragments from exploded SO2. Thesharp peak at low momenta are fragments from the samemolecule, the broad peak is from random coincidencesfrom different molecules.

charge states allows at least one peak to be distinguishedby time-of-flight windows, so the iteration only needs tobe done to with the remaining pair of fragments. Separatetime-of-flight windows will also reduce biasing from thelower detection efficiency for a roughly 20 ns deadtimefrom the MCP and delay-line detector [49].

Having a small momentum sum does not guaranteethat all the properties assigned to a hit are correct. As inFigure 2.5, the momentum sum from correctly assignedfragments from the same molecule result in a narrow peakin the momentum sum distribution from 0 to 5×10−23 kgm/s or more, depending on the initial velocity from themolecular beam. The momentum sum from incorrectlyassigned fragments forms a broad peak with greater mo-mentum sum. The extent to which the tail of this peakoverlaps with the momentum range of the correctly as-signed fragments gives an estimate of the false coinci-dence rate of about 5% in these experiments. These falsecoincidences can be neglected since we are primarily in-terested in relative changes to average structures.

2.4.2 Calculating structures from velocities

After filtering the data by momentum sum, we are leftwith many sets of three fragments of known mass, chargeand velocity grouped by originating molecule. To com-plete the Coulomb explosion imaging process, the frag-ment velocities must be converted to molecular structure.

The structure of a triatomic molecule is determined bythree values: two bond lengths, and a bond angle. Extract-ing a structure from the fragment velocities of a triatomicis more complicated than in the diatomic case, in whichonly the bond length varies. An analytic approach to thesimpler linear triatomic case was only recently published[50] with results for nonlinear triatomics promised at alater date.

8


Using a numeric approach, I wrote an algorithm tocalculate molecular structure from the fragment data. Thecore calculation takes a given molecular structure andthe final charge states of the fragments, and returns theasymptotic velocity vectors for each fragment. The calcu-lation starts with the ions in the final charge states, at rest,in the given structure. The system then evolves forwardin time under the influence of the Coulomb potential, orsometimes an ab initio potential. The integration followsthe Runge-Kutta technique [51].

The core calculation is called by a downhill simplexoptimization algorithm [51] which compares the calcu-lated velocities for different structures to the experimentalvelocities for a particular molecule. The result is a molec-ular structure consistent with the experimental result towithin the given approximations.

The fitness function that the algorithm optimizes isusually a reduction in the sum of the magnitudes of thedifference of the three pairs of experimental and calcu-lated velocity vectors. The experimental velocities used inthe comparison are actually the measured velocities withthe centre-of-mass motion subtracted.

In some cases, it may be desirable to bias the fitnessfunction. For a linear molecule such as CO2, the carbonvelocity may be very low, making the structure sensitiveto small errors in the carbon velocity. Only matching thevelocities of the oxygen can remove this sensitivity, with-out much effect on the resulting structure.

For speed, the optimization is run twice. The firsttime, the initial structure is the known equilibrium posi-tion of the molecule, and the time step is very long, re-sulting in a fast calculation with poor accuracy, as mea-sured by energy conservation. The result of the first cal-culation is used as the starting structure for the secondrun, which is done with a small time step that conservesenergy well. The time step used depends mostly on themass of the atoms involved, as well as the charge states.Depending on the time step, the structure calculation forone molecule will take a few minutes on a personal com-puter circa 2002.

2.4.3 Measured structuresUsing few-cycle pulses, the PATRICK, and the numer-

ical structure algorithm, we measured the molecular struc-tures of D2O and SO2 [52]. Imaged structures are shownin Figures 2.6 and 2.7. The plots are clouds of an entireset of individual structures drawn on the same graph. Allmolecules are rotated into the plane of the page. The cen-tral atom is placed at the origin, and the rotation in theplane is such that the bond angle is split in half by the y-axis with the outer atoms pointing down. The molecule is

Molecule Bond Length Bond Angle ReferenceH2O 0.9587 A 103.9◦ [53]SO2 1.4307 A 119.32◦ [54]

TABLE 2.1: Equilibrium geometries of molecules imagedby Coulomb explosion. Note that equilibrium geometriesdepend on charge, and are isotope independent.

then translated so that its centre-of-mass is at the origin.The result is a set of three clouds that are approximatelythe square of the nuclear wavefunctions.

The equilibrium geometry of D2O and SO2, listed inTable 2.1, are also plotted on the graphs, as the three dot-ted circles joined by lines. Note that the average geometrywill be slightly different than the positions of the poten-tial minima due to the anharmonicity of the potential. Forexample, the average bond length in D2O is 0.01 A longerand the bond angle is half a degree wider than the equilib-rium points [53].

The measured structures are quite close to, but stilllarger than the equilibrium structures. As in the D2 exper-iment discussed earlier, these explosions are closer to theactual structure than those from long explosion pulses, butthey also do not match the actual structure. The reasonsfor the discrepancy are the same as for the D2 experiment:wavefunction deformation during ionization, and motionduring ionization.

For comparison, an image of SO2 taken using 45 fsexploding pulses is also shown in Figure 2.7, reflectedin the x-axis to separate the images. With 45 fs pulses,the bond lengths are noticeably longer. The difference inbond length indicates that the few-cycle pulse has at leastsurpassed the worst effect of enhanced ionization, whichis to make all molecules explode from the same large, eas-ily ionized geometry. Since the imaged structure is wellwithin an Angstrom of the equilibrium geometry, we canalso be confident that we can image structures to aboutAngstrom accuracy, within the approximation of zero ini-tial velocity.

To verify that the Coulomb approximation was reli-able, we also ran the numerical structure algorithm forheavy water using an ab initio potential energy surface,as calculated by S. S. Wesolowski using an augmentedcorrelation-consistent polarized-valence quadruple-zeta(aug-cc-pVQZ) basis set [55, 56]. The resulting struc-tures were very close to those calculated using a purelyCoulombic potential, with the average structures differingby less than 0.1 A. This is likely because the ab initiopotential only differs significantly from the Coulombpotential for very small separations. During a Coulombexplosion, the fragments quickly separate, spending most

9


FIGURE 2.6: Overlay of many structures of D2O as mea-sured by Coulomb explosion imaging. The dotted circlesjoined by lines indicate the positions of the equilibriumstructure of water. Explosions were from the D+ + D+ +O2+ charge state.

of their time at large separations, where the potential isCoulombic.

The main result of these measurements is that wecan roughly measure the structure of small moleculeswith Coulomb explosion induced by few-cycle laserpulses. For example, it is clear from the measured ve-locities, even without the structural computation, thatthe exploded molecules were not linear, but stronglybent. While this level of accuracy is crude comparedto spectroscopic measurements, it is precise enough toobserve major structural changes that are inaccessible byspectroscopy.

2.4.4 Molecular moviesKnowing that we can quickly ionize and explode

molecules with few-cycle laser pulses, in experimentsled by Francois Legare, we used Coulomb explosion toobserve the evolution of molecules immediately after theionization of a few electrons [33, 57]. Two few-cyclepulses with a controlled time delay illuminated SO2molecules in the gas jet.

The pump pulse initiates dissociation in SO2 by ion-ization of up to three electrons. Immediately after ion-ization, the molecule will still be close to the equilibriumstructure of the neutral molecule, but since it is no longerneutral, it will move towards a low-energy geometry ofthe new charge state.

Even before using the reconstruction algorithm, muchcan be learned directly from the measured velocities. Cor-relation maps are simple and useful tools for analyzingCoulomb explosions. Figure 2.8 shows the correlated ki-

FIGURE 2.7: Two sets of SO2 structures as measured byCoulomb explosion imaging with different exploding pulsedurations. The dotted circles joined by lines indicate thepositions of the equilibrium structure for the 8 fs data.Structures measured by 45 fs pulses are plotted with theoxygens at positive Y. Explosions were from the S3+ + O2+

+ O2+ charge state.

netic energies of the two oxygen fragments at differentpump-probe delays.

The 0 fs panel is equivalent to Coulomb explosionwithout a pump pulse. In this case, we see one spot onthe diagonal, corresponding to the explosion of a roughlysymmetric molecule, as you would expect of SO2 at equi-librium.

After ionization by the pump pulse, the spot on thecorrelation maps in Figure 2.8 begins to change shape.For clarity, the nonionized contribution has been removedfrom the later delays, using the 0 fs map as a reference.The spot breaks up into three parts: a symmetric contribu-tion lying along the diagonal, and two asymmetric contri-butions moving away from the diagonal.

The symmetric events correspond to a simultaneousexpansion of both SO bonds as the molecule dissociates.The increasing bond length means there is less storedCoulomb potential, resulting in lower explosion fragmentkinetic energies for longer delays.

For the asymmetric component, one oxygen fragmentremains energetic, while the other becomes slower. Forthis dissociation channel, one of the oxygens has sepa-rated from the molecule. The further this oxygen driftsaway, the less kinetic energy it receives from the Coulombexplosion. The SO fragment stays bound. After the ex-plosion pulse, the sulphur can strongly repel the nearbyoxygen, giving this fragment a kinetic energy comparableto that from the unpumped explosion.

10


FIGURE 2.8: A series of correlation maps taken at different pump-probe delays. Shown is the density of events havinga given pairing of oxygen kinetic energies. Explosions were from the S3+ + O2+ + O2+ charge state. The greyscale islogarithmic.

The same separation of symmetric and asymmetricchannels is seen in the correlation of the angle betweenoxygen velocities, and the total kinetic energy of all thefragments, plotted in Figure 2.9. At early times, the ki-netic energy starts decreasing, while the velocity angledoes not change much, indicating an increase in bondlength only.

As in Figure 2.8, from 60 fs, the asymmetric channelbegins to separate from the symmetric dissociation. Whilethe symmetric component continues towards lower totalkinetic energy as the atoms separate, in the asymmetricchannel, the total kinetic energy stops decreasing, sincethe bonded SO pair maintains bond length. The asym-metric component also becomes much broader in velocityangle, as the bonded SO pair rotates relative to the sepa-rated oxygen fragment.

Just by looking at correlations of fragment data, wehave identified the two major dissociation channels oflaser-ionized SO2. To see the details of the bond lengthsand angles, we can use our numerical algorithm to extractmolecular structure.

Figure 2.10 shows molecular images for different dis-sociation channels. A different plotting style has beenused to better show the asymmetric structure. The top leftpanel shows the unpumped structure. The orientation of

each structure is fixed on the same plane by placing oneoxygen, and the centre of mass of an oxygen and sulphurpair, on the x axis, with sulphur given a positive y value.The density of oxygen atoms along the x axis is plottedalong the bottom of each graph.

The lower-left panel is made with events selected byoxygen kinetic energy to be in the asymmetric dissocia-tion channel as seen in Figure 2.8. These events are thenimaged using the algorithm described earlier, and thesestructures are all overlain. The asymmetry is clearly seenin these structures. By looking at the structures of indi-vidual molecules, the rotation of the bound pair of atomsis seen. As the unbound oxygen leaves, it imparts angularmomentum to the molecule. The faster the oxygen leaves,the more the molecule has rotated and the further the oxy-gen is from the molecule. This dissociation channel re-sults in an SO rotational wavepacket. The two solid linesshow the average structure of the third of the moleculeswith the largest, and the third with the smallest bond an-gle.

The right panel was made by selecting the region ofsymmetric events from the oxygen kinetic energy corre-lation map. The molecules in this channel are essentiallythe same shape as the original molecule, but significantlyexpanded from their original size.

FIGURE 2.9: A series of correlation maps taken at different pump-probe delays. Shown is the density of events havinga given pairing of angle between the two O velocities, and the total kinetic energy. Explosions were from the S3+ + O2+

+ O2+ charge state. The greyscale is logarithmic.

11


FIGURE 2.10: Images of SO2 separated by dissociation channel. The 0 fs figure is from unpumped, neutral SO2. Theconnected squares show the average positions of each cloud. The middle figure includes events from the asymmetricdissociation channel. The lower panel is made for events following the symmetric dissociation channel. Explosionswere from the S3+ + O2+ + O2+ charge state.

Care is needed for image reconstruction in pump-probe experiments. Since the purpose of the experimentis to observe moving molecules, the zero initial velocityassumption will not be true. This is only a problem if themotion involved is slow relative to the asymptotic explo-sion speed. In this particular example of SO2 dissociation,dissociation is a fairly energetic event, and the atoms spa-tially separate, reducing the energy in the Coulomb ex-plosion. For example, S3+ + O2++ O3+ has 186 eV inthe SO2 equilibrium geometry. For the same charges witha 4 A bond length, like the symmetric dissociation struc-tures in Figure 2.10, the Coulomb potential drops to 67 eV,making the initial velocity more important. While the im-ages become quantitatively less reliable for longer delays,there is still much that can be learned from the relativechanges with time delay.

2.5 Future of Coulomb explosion imag-ing

The Coulomb explosion results in this chapter markthe end of the preliminary phase of Coulomb explosionimaging. While there is still room for improvement, laserpulses can now be made fast and intense enough to imagemolecules with enough accuracy to observe large changesto molecular structures.

Using few-cycle pulses, we have performed femtosec-ond time-resolved Coulomb explosion imaging of disso-ciating SO2. Laser Coulomb explosion imaging can nowenter a development phase, where these basic techniquescan be extended to different processes and more complexsystems.

12


3Rotational Wavepackets

3.1 Wavepackets and revivalsWhile physicists like to

FIGURE 3.1: Illustrationof energy level spac-ings for a rigid rotor (J)and a harmonic oscilla-tor (n).

think in terms of eigenstates,eigenstates can be boring,since they only change phase.To describe a more generaland interesting wavefunction,you need to make a coherentsuperposition of eigenstates,known as a wavepacket. Awavepacket can act more like aclassical particle, with the pos-sibility of spatially localizedwavepackets moving around ininteresting ways [58].

In a harmonic potential,a localized wavepacket canoscillate back and forth in-definitely. This is becausethe energy levels of a sim-ple harmonic oscillator are given by En = nhω + hω/2,where n is a whole number, and ω is the fundamentalfrequency of the potential. As the components of thewavepacket evolve, they first dephase since they havedifferent energies in different eigenstates. A period of theoscillator frequency later though, all the n componentswill have gathered n2π of phase relative to the groundstate. This is equivalent to being in the original state, andthe wavepacket revives in its original position.

In the simple harmonic oscillator case, the revival usu-ally isn’t called a revival, since the wavepacket does notdephase enough to lose its shape. Anharmonic potentialsare quite different though. If you have a system with es-sentially random energy spacings, a wavepacket wouldquickly dephase, and remain dispersed for a long time. Ifthere is a continuum of eigenstates, you will likely neversee your wavepacket again. For discrete eigenstates, sincethere are only a finite number of frequencies involved, awavepacket will revive eventually [59]. Revivals on sucha long timescale are sometimes called superrevivals [60].

The energy levels of a rigid rotor are given by

EJ = hcBJ(J +1), (3.1)

where J is a whole number; h is Planck’s constant; B =h/(8π2cI) is the rotational constant, where I is the mo-ment of inertia of the rotor; and c is the speed of light,which is here because B is usually given in wavenumbers[61].

Defining ω1 = 4πBc, the frequencies of the energylevels can be written as ωJ = J(J + 1)ω1/2. Noting thatJ(J +1) is always an even integer, we see that the energylevels are all integer multiples (0, 2, 6, . . . ) of the fun-damental frequency ω1. These energy levels are like aharmonic oscillator with most of the levels missing. Thus,a wavepacket in a rigid rotor will also revive periodicallyevery Trev = 2π/ω1 = 1/(2Bc). If the wavepacket isaligned and evolving freely, there will be periodic re-vivals of this aligned state every fundamental period Trev[21].

3.2 Making rotational wavepacketsCreating a rotational wavepacket involves applying a

torque to a molecule. An unusual method for making anSO rotational wavepacket was shown in the pump-probemeasurement on SO2, where the departing oxygen ionpushes against the sulphur side of the remaining SO frag-ment, which begins rotating.

The simplest way to make a rotational wavepacket iswith a short laser pulse. At a laser wavelength of 800nm, the field oscillates slowly enough for the electrons tofollow in phase. When oscillating in phase, the energyis lower the more the molecule is polarized. Since thepolarizability of the molecule depends on its orientation,the laser field causes a torque on the most polarizable axis,usually the longest axis of the molecule, towards the laserfield.

The dependence of the energy on the polarizabilityleads to an angular dependent potential well. For a linear(one moment of inertia) or symmetric top (two differentmoments of inertia) molecule [62] in a linearly polarizedfield, the angular dependent potential, averaged over theoscillating field, is given by H =−(I/2)(α‖−α⊥)cos2 θ.Here, I is the laser intensity; α‖ is the polarizability alongthe molecular axis, α⊥ is the polarizability along all theaxes perpendicular to the α‖ axis, and θ is the angle be-tween the molecular axis and the electric field axis [63,26, 64].

Thinking in terms of this potential well, several pointscan be made. The well depth depends on the laser inten-sity, so stronger illumination will result in a deeper well.In the adiabatic alignment case, the laser pulse is long, andmolecules with less rotational energy than the well depthcan be trapped, oscillating around the bottom of the wellwhile the field is on, in what are known as pendular states[26]. If the original rotational energy is high, as is the casefor hot molecules, then the molecules can rotate in and outof the well, and will not stay in a pendular state.

In the impulsive alignment case, the laser is only ap-plied for a short time. If the rotational energy is very

13


high, a shallow well will not be able to overcome the orig-inal rotation and push the molecule towards alignment.Energetic rotations are also faster, making a short pulsefor slow rotations appear to be a long pulse for fast ro-tations. This will be more like an adiabatic alignment,which leaves the molecule unaligned after the pulse haspassed.

To achieve good impulsive alignment then, thealigning pulse should consider the original state of themolecules. The two parameters of an unshaped laser pulseare the pulse duration and peak intensity. The maximumintensity is given by experimental requirements such asavoiding ionization. The longer the pulse duration, themore energy there can be in the pulse while staying belowthe maximum intensity. The total interaction can thus beincreased by lengthening the pulse duration. The pulsemust be weak by the time the molecules reach their peakalignment though, in order to obtain field-free alignmenteither at that time, or at a later revival. The time for themolecules to reach the peak alignment is determined bythe rotational energy, which is normally greater than theoriginal rotational energy. At higher initial temperature,the molecules will rotate more quickly, and the pulse mustbe shorter.

Thinking in terms of rotational wavepackets, we needto consider the rotational states and their transitions. Thecos2 θ part of the potential H is the term of interest. Theexpansion in spherical harmonics Y J

M is [65]:

cosθ Y JM(θ,φ) =

√(J +M +1)(J−M +1)

(2J +1)(2J +3)Y J+1

M

+

√(J +M)(J−M)(2J +1)(2J−1)

Y J−1M . (3.2)

Squaring gives the matrix elementsˆcos2

θJ1,J2 = 〈J1,M| ˆcos2θ|J2,M〉 for rotational states

with quantum number J and projection M:

ˆcos2θJ,J =

(J +1)2−M2

(2J +1)(2J +3)+

J2−M2

(2J +1)(2J−1),

ˆcos2θJ,J+2 =

√(J +1)2−M2

√(J +2)2−M2

(2J +3)√

(2J +1)(2J +5),

ˆcos2θJ,J−2 =

√(J +1)2−M2

√J2−M2

(2J−1)√

(2J +1)(2J−3), (3.3)

and all other terms are zero. From these matrix elements,we can see that the potential allows for transitions be-tween J states separated by 0 or ±2, and does not coupledifferent M.

Starting from a single J,M state, many Raman transi-tions will spread the wavefunction to many other J states

in steps of 2, resulting in a rotational wavepacket. Appliedto an ensemble of molecules, the result is many wavepack-ets starting from different rotational states. The overall be-haviour is the incoherent sum over the initial distribution.For higher temperatures, there will be more initial statesinvolved. Since rotational wavepackets from different ini-tial states behave differently, increasing the temperaturetends to wash out the overall alignment signal, reducingthe peak alignment of the ensemble. Alignment is stillquite temperature robust though. Rotational revivals werefirst observed in room temperature gas cells [6].

3.3 Observing rotational wavepackets indiatomics

Our first rotational wavepacket experiments, withmany contributions from P. W. Dooley and I. V. Litvinyuk,use a linearly polarized laser pulse to create a rotationalwavepacket in the diatomics nitrogen or oxygen. We thenmeasure the evolution of the alignment using Coulombexplosion by a time delayed pulse [21]. The samePATRICK detection system and regenerative amplifier isused as in the Coulomb explosion imaging experiments.These experiments form the basis for our later, morecomplex, alignment experiments.

The gas molecules are from the molecular beam, andare thus rotationally cooled. Typical conditions for theseexperiments were 33 Torr of gas behind a 100 µm pinhole.This gives a rotational temperature roughly estimated tobe 100 K [43]. These experiments have also been donewith room temperature background nitrogen in the detec-tion chamber, which gives similar results, but with a re-duced degree of alignment.

The laser pulse from the laser is about 850 µJ of 800nm light with a full-width at half-maximum (FWHM) ofabout 45 fs . The pulse is directed into a Mach-Zehnderstyle of interferometer, illustrated in Figure 3.2. The in-terferometer outputs an aligning pump pulse and an ex-ploding probe pulse. The beam splitter and combiner arepartial reflectors, not polarizers. There is thus a secondoutput from the beam combiner, not illustrated, with dif-ferent pulse energies. Either output can be used dependingon the pulse energies needed.

The probe pulse is circularly polarized by a broadbandλ/4 waveplate (Halle 500-900 nm), and focuses to about1016 W/cm2, enough to multiply ionize and explode thetarget molecules. The beam combination occurs at closeto normal incidence to minimize the ellipticity of the cir-cularly polarized probe pulse. The rotation of the λ/4waveplate is optimized by measuring the polarization im-mediately in front of the chamber window, after reflec-tions that add ellipticity.

14


The alignment pulse is linearly polarized, and is at-tenuated by a neutral density filter to focus to less than1014 W/cm2, so that nitrogen and oxygen do not signifi-cantly ionize. A telescope reduces the pump beam size byhalf so that it focuses to a spot twice as large as the probepulse. Exploded molecules are thus exposed to a smallervariation in pump intensity, which reduces variations inalignment.

The two pulses are overlapped in time and space byobserving the interference fringes between the two beamsafter recombination. At these high intensities, the infraredpulses are easily visible as bright spots on conventionalyellow fluorescent ink as is found in highlighters, orcoloured paper. Intense 800 nm pulses are also visible tothe naked eye, but the conversion to yellow fluorescenceallows the pulses to be seen more safely through lasergoggles.

The pulses must first be roughly overlapped in space.This is easily done by visually overlapping the position ofthe beams on the reflective surface of the recombiner, thenoverlapping the beams far from the recombiner, by adjust-ing the angle of the recombiner. The first step controls therelative position of the beams at one point in space, andthe second step controls the direction of the beams.

The pulses must then be overlapped in time so thatthe relative time delay can be controlled by the movingdelay stage. If the spatial overlap was done well, zerotime delay can be quickly found by sliding the delay stageback and forth by hand, around the zero-delay position

FIGURE 3.2: Schematic of the optics for measuring rota-tional wavepackets. An interferometer provides a time de-layed pump and probe pulse to measure the time evolutionof alignment. Waveplates and neutral density filters can beadded to the arms to control polarization and intensity. Theunusual geometry is meant to reduce the ellipticity of thecircularly polarized probe pulse by minimizing large anglereflections.

as estimated by measuring the lengths of the two arms.When overlapped, the combined spots will flash due torandom phase changes between the arms. If the spatialoverlap is poor, another way to find temporal overlap is toobserve the spectrum of the two pulses. When the pulsesare close in time, the resulting modulation in the spectrummay be easier to observe than the visible flashing of thespots.

When interference fringes can be seen at zero time de-lay, the fringes can be used to accurately align the direc-tion and divergence of the two beams. When the direc-tion of the two beams differ, there will be parallel linesof fringes perpendicular to the plane defined by the twobeams. When the divergence does not match, the fringeswill curl into rings. This can be used to correct the lensspacing of the telescope.

When properly aligned, there will be no visible inter-ference fringes, only even flashing across the spot fromrandom vibrations and air currents. The separation ofthe spots at the two foci depends on the amount of thebeam over which fringes are visible. If there is one visiblefringe, this is about a π phase difference across a diameterkw, where k is a scaling factor, and w is the conventionalbeam diameter at that point. The focused spots will beseparated by πw0/(2k), where w0 is the diffraction lim-ited beam waist diameter. In our experiments, the fringesare often visible quite far to the edge of the beam, that is,k > 1. Along with the doubled focused size of the pumprelative to the probe, one fringe would mean some, butpoor, overlap. Alignment can be made much better thanone fringe, resulting in good overlap at the focus, whichcan be verified by focusing into a pinhole, and comparingthe energy passing through for the two beams.

3.3.1 Measuring alignment by CEIThe alignment is measured from the angular distribu-

tion of a few thousand exploded fragments for each timedelay. Since only the alignment of the molecule is beingmeasured, the requirements are not as strict as in the fullCoulomb explosion imaging case. The 45 fs pulses fromthe laser result in explosion at enhanced ionization bondlengths. This affects the kinetic energy of the fragments,but does not have much effect on the direction of the frag-ment velocities.

The fragments of certain charge state and atomicspecies are selected by filtering by time-of-flight. Acharge state can be further divided into different chan-nels, such as the N2 → N2++ N2+, or the N2 → N2++N3+. Both these channels produce N2+ fragments, butthe kinetic energy will be higher for the second channel,due to greater repulsion by the higher charges. Filtering

15


by channel can be used to simplify the analysis. Kineticenergy is also useful for removing parent ions that havethe same charge-to-mass ratio as some atomic fragments,and appear as very low energy fragments when treated asatomic fragments.

It is generally preferable to use higher charge statefragments. Higher charge states are more likely to beformed near the high intensity focus, reducing variationin the aligning field seen by the detected molecules.Higher charge states are also more likely to have sep-arated quickly, reducing the time spent near the otherfragment in an excited state.

As with any measurement, it is important to considerpotential biases. When the intense probe pulse illumi-nates the molecule, the molecule may rotate from thesame force used by the aligning pulse. As in the exper-iment in Section 5.2 [16], the ionization probability candepend on the alignment of the molecule relative to thelaser field. For higher charge states, enhanced ionizationwill dominate, preferentially exploding molecules withinternuclear axes lying along the laser field.

In the limit of heavy molecules or very short laserpulses, the induced rotation from the probe pulse is neg-ligible. In the limit of round molecules, or extremely in-tense laser pulses that can explode all molecules in the fo-cus, there will be no angular dependence of the explosionprobability. Our experiments are not within these limitsthough, so we use symmetry properties of the laser fieldsto avoid bias.

For these diatomic experiments, we use a circularlypolarized probe pulse, which is cylindrically symmetricaround its propagation direction. The experiment returnsan angle θ of the fragment velocity to the alignment axis(the polarization of the aligning pulse) as often appears incalculations and some experiments. The circularly polar-ized pulse may cause rotation into the polarization planethough, so we only use the angle ϑ (sometimes called“curly theta”) which is the component of the fragment ve-locity projected onto the polarization plane, as illustrated

FIGURE 3.3: Illustration of the angle ϑ. The illustratedprobe pulse is polarized in the XZ plane, and the aligningpulse would be polarized along X.

FIGURE 3.4: Measured angular distribution of aligned ro-tational wavepacket in oxygen with 〈cos2 ϑ〉= 0.70.

in Figure 3.3. The angle ϑ will not change from inducedrotation or preferential ionization because of the symme-try of the circularly polarized probe pulse.

After collecting many fragments, we have an angulardistribution for the wavepacket at each time delay. Theangular distribution can be plotted in a polar histogram,which is roughly the shape of the squared wavefunctionviewed along the laser propagation direction. A polar his-togram for aligned oxygen is shown in Figure 3.4.

To show the time evolution, the angular distributionfor each point can be reduced to the single value 〈cos2 ϑ〉,and plotted along a time axis. Such time evolution plotsare shown in Figures 3.6 and 3.5. For an isotropic dis-tribution, 〈cos2 ϑ〉 = 1/2; perfect alignment has 〈cos2 ϑ〉= 1; and perfect antialignment, with all molecules per-pendicular to the alignment axis would have 〈cos2 ϑ〉 = 0.Note that, for an angle to an axis, θ, 〈cos2 θ〉= 1/3 for anisotropic distribution.

FIGURE 3.5: Measured rotational wavepacket evolution innitrogen.

16


FIGURE 3.6: Measured rotational wavepacket evolution in oxygen.

3.3.1.1 Baseline ReferenceSeveral angular distributions are implicit in a mea-

surement of alignment. What we want to extract is the ac-tual angular distribution of the molecular ensemble, A(φ),for some type of angle φ, at the time of probing. The rawdistribution returned by the experiment, M(φ), is not nec-essarily the same as A(φ).

When illuminated by the probe pulse, the intensepulse can rotate the molecule giving a new distribution, A′.If the type of angle φ was chosen well, A′(φ) = A(φ). As afunction of other angles where induced rotation is impor-tant, the distributions will not be equal. From A′, thereis an angular dependent probability that the moleculewill explode and be detected, leading to the measurementM(φ).

In the case of nitrogen and oxygen, the circular probepulse produces fragments with velocities lying within afew degrees of the polarization plane. The very intenselaser field, and comparison to numerical simulations ofrotational wavepackets by Michael Spanner [21], suggestthat essentially all molecules will be pulled into the polar-ization plane where they can explode. We then only needto consider the angle in the polarization plane ϑ.

In the ideal case, probing an isotropic initial distribu-tion, such as one with no aligning pulse, would lead toa measurement M that is constant as a function of angle.We find that experimental distributions are not isotropic,likely a result of ellipticity in the probe pulse, spatiallyuneven detection efficiency across the microchannel platedetector, and issues of multiple fragments hitting the de-tector at the particular times and locations for differentangles and charge states. If the anisotropy is significant,we can correct the angular distributions by referring to themeasured distribution B from an isotropic distribution.

This baselining procedure is equivalent to assumingthat there is a detection efficiency α(φ) that is only a func-tion of the angle of interest. Ignoring normalization, wetake α(φ) = B(φ), and M(φ) = α(φ)A(φ). If we try to cor-

rect the baseline measurement, we get B(φ)/α(φ) = 1, anisotropic distribution.

The presence of a constant background will also ben-efit from a baseline measurement. For example, an unre-lated molecular ion that happens to be in the time-of-flightwindow can appear as a sharp peak in the distribution fora small range of angles. With no baseline correction, alarge ion peak would skew a calculation such as an aver-age squared cosine.

To understand baselines with constant signals, we canwrite the measurement as M(φ) = M0(φ)+ I(φ), and thebaseline as B(φ) = B0(φ) + I(φ), where the 0 indicateswhat the distribution would be without the constant ioncontribution, and I(φ) is the ion contribution.

Ideally, we would have A(φ) = M0(φ)/B0(φ). Follow-ing the usual baseline procedure, we instead get A(φ) =M(φ)/B(φ) = (M0(φ) + I(φ))/(B0(φ) + I(φ)). Wherethere is no ion signal, I(φ) = 0, and the baseline will be-have normally. In the limit of the ion signal being muchstronger than the desired signal (I(φ) � B0(φ),M0(φ))the desired signal becomes negligible, and the distribu-tion will appear to be an isotropic distribution. For I(φ)'B0(φ), variations in the measurement will have an effect,but be partially diluted by the constant ion contribution.

In the case of a sharp ion peak over a moderatelyaligned distribution, the baseline correction will reducethe skewing effect of the sharp peak by making this regionappear isotropic, which is closer to the true distribution.Any skewing is further diminished if an average is takenover the entire distribution, since the ion peak will onlycover a small percentage of the angular space. If there is abroad, constant signal, a baseline correction will removethe constant contribution, while diluting the variation inthe actual signal.

A baseline correction was used for most of the align-ment measurements in this thesis. By referring to theknown isotropic distribution, we can correct for the im-perfect detection system to retrieve the original angulardistribution.

17


3.4 Revival structures of diatomicsLooking at the revival structures in Figures 3.6 and

3.5, there are many features and differences to discuss.Starting from the left, the curve starts with a 〈cos2 ϑ〉 of0.5 at negative time. This is the isotropic thermal distri-bution measured before the aligning pulse, which is cen-tred at time 0. At time 0, the molecules are kicked bythe aligning pulse, and the curve begins to increase as themolecules align. The 45 fs aligning pulse would be barelyvisible if plotted on the figure, and is shorter than the risetime of this prompt alignment.

After the prompt alignment, the alignment returnsto a lower value, but slightly above 0.5. We call thisincoherent alignment, because it occurs even when thewavepacket is dephased. Incoherent alignment occursbecause the laser field couples the J rotational quantumnumber, but not the projection number M. For strongaligning fields, the average J value can increase signif-icantly, while M remains less than the original J value.This gives a preferential tilt to the angular momentumvector, resulting in some net alignment.

For oxygen, Trev is 11.601 ps [66], and for nitrogen,Trev is 8.383 ps [67]. The revivals at natural numbermultiples of Trev after the aligning pulse are known asfull revivals. The centre of the revival is a replica of themolecule immediately after the laser pulse, and is thusnot strongly aligned. Like the prompt alignment that it isrepeating, it moves from an isotropic distribution towardsalignment.

Looking at the later full revivals of oxygen in Figure3.6, the shape progressively changes for longer times,rather than exactly repeating every Trev. This gradualchange of the revival structure happens because oxygenis not really a rigid rotor. In the rigid rotor approxima-tion, there are two point masses with a constant distancebetween them. The point mass approximation is onlyweakly broken by the finite size of the nuclei and theelectron distribution.

The constant bond length approximation is brokenby vibrations and centrifugal distortion from energeticrotations. There are almost no vibrational excitationsfrom temperature or the laser excitation. This is easilyverified by looking at the measured revival times. Fornitrogen, the rotational constants in the ground and firstexcited vibrational states are B0 = 1.9895924 cm−1 andB1 = 1.9722227 cm−1 [67]. This corresponds to revivaltimes of 8.383 ps and 8.457 ps respectively, a significantdifference, with our experiments showing revivals muchcloser to the ground vibrational state. Figure 3.7 showsa measured rotational revival (circles), and a numericalcalculation by Michael Spanner for our experimental

FIGURE 3.7: Circles show the measured sixth full revivalof nitrogen. The solid curve is a theoretical calculation.

conditions, using only molecules in the ground rotationalstate (solid curve).

Noticeable chirping of the revival is visible in Figure3.7 from centrifugal distortion from high rotational en-ergy. The bond length stretches for faster rotations. Thisappears as a higher order correction in the energy spec-trum, slightly shifting the levels from the rigid rotor val-ues. These shifts progressively deform the wavepacket asthe chirp increases with time.

3.4.1 Fourier transform of revivalsA Fourier transform of the revival structure reveals

the frequency components involved, from which the ro-tational states can be deduced. To understand the Fouriertransform, we will approximate the plane-projected angleϑ with the theoretically simpler angle to the laser polariza-tion axis θ. The revival structure plots the time evolutionof 〈Ψ(t)|cos2 θ|Ψ(t)〉, where

Ψ(t) = ∑J,M

aJ,Me−i(EJ/h)t |J,M〉 (3.4)

is the field-free evolving rotational wavepacket, EJ is theenergy of level J, and aJ,M = |aJ,M|e−iφJ,M is the initialamplitude and phase of component J,M.

Using this wavefunction, and noting that ˆcos2θJ,J+2 =

ˆcos2θJ+2,J (see Equation 3.3) we can expand 〈 ˆcos2

θ〉:

〈 ˆcos2θ〉 = 〈Ψ(t)| ˆcos2

θ|Ψ(t)〉= ∑

J,M|aJ,M|2 ˆcos2

θJ,J +2|aJ+2,M||aJ,M|

×cos(∆ωJ,J+2t +(φJ+2,M−φJ,M)) (3.5)

where ∆ωJ,J+2 = (EJ+2−EJ)/h = 2π(4J+6)/(2Trev), us-ing Eqn. 3.1. The time dependence of the evolution isthus determined by pairs of J states separated by 2. Thefrequency for a pair J,J + 2 is ∆ωJ,J+2, and the strengthof the frequency component is proportional to the am-plitudes in both J states. Written in normal frequency,

18


FIGURE 3.8: Fourier transforms of the revival structuresof nitrogen and oxygen. J state refers to the lower of theJ,J +2 pair.

∆νJ,J+2 = (4J +6)/(2Trev), meaning that frequency com-ponents will appear at integer multiples of 1/(2Trev) in theFourier transform.

The Fourier transforms for nitrogen and oxygen re-vivals are shown in Figure 3.8, with the frequency axisconverted to the corresponding J,J + 2 states. The aver-age frequency corresponds to the oscillation speed of thealignment signal during a revival. In terms of energiesscales, the rotational temperature was about 100 K, whichcorresponds to about J = 6, and the peak well-depth fromthe alignment field corresponds roughly to J = 26 for bothnitrogen and oxygen.

The oxygen transform is at a higher frequency reso-lution because the data was over 2Trev of delay, while thenitrogen data was over 1Trev, at a similar time resolution.

Oxygen only has odd J states, and nitrogen has a 2:1ratio of even:odd J states, which is reflected in the Fouriertransforms. The odd and even types of a molecule areknown as ortho and para, with ortho usually denoting thepredominant type. The Raman interactions with the laserfield follows the selection rule ∆J = 0,2, so molecules inodd or even J states remain in odd or even states after thealigning pulse.

The ratio of even to odd states is a consequence ofthe nuclear spins of the atoms in the molecule. Nitrogenatoms 14N have a nuclear spin I = 1 [61]. For a homonu-clear diatomic, an exchange of the identical atoms musthave the correct symmetry. Since 14N has an integer spin,it is a boson, and the total wavefunction should be sym-metric with respect to exchange.

For even T values, the nuclear spin function is sym-metric due to the way the spins pair up. For example,T = 2 has the possible projection MT = 2, which corre-

sponds to both nuclear spins having the same sign, whichis symmetric to exchange. The remaining contributions tothe wavefunction, known as the coordinate wavefunction,must then be symmetric to maintain symmetry. For oddT , which is antisymmetric, the coordinate function mustbe antisymmetric.

The coordinate wavefunction includes the electronic,vibrational, and rotational contributions. Electronicwavefunctions are sometimes symmetric or antisymmet-ric. The vibrational contribution is symmetric, since itdepends only on the internuclear separation. From the na-ture of the spherical harmonics, odd rotational J states areantisymmetric, and even rotational J states are symmetric.

For a given electronic state then, the even:odd ratioof the nuclear spins are reflected in the even:odd ratioof the J state populations. For nitrogen with I = 1, thetotal nuclear spin of 14N2 can then be T = 2, 1, or 0.For each T , there are 2T + 1 possible projection statesMT = T,(T − 1),(T − 2), . . . ,−T . From T = 2 and 0,there are 5 and 1 states, and from T = 1 there are 3 states.This results in a 2:1 symmetric to antisymmetric ratio inthe nuclear spin function, which in turn results in a 2:1even:odd ratio of J states, as seen in Figure 3.8. Oxygenatoms, 16O, have a nuclear spin I = 0 [61]. There is onlyT = 0 for O2, which results in only odd J states, as seenin the lower panel of Figure 3.8. Carbon dioxide is anexample of a molecule with only even J states [68].

In nitrogen, the imbalance between even and odd Jstates results in incomplete cancellation at the quarter re-vivals, resulting in the small revival seen at 2 ps in Figure3.5. In oxygen, there are no even states to cancel the quar-ter revivals of the odd J states, leaving them comparableto the full revival such as the alignment at 3 ps in Figure3.6.

3.4.2 Alignment of deuteriumAs the simplest neutral molecule, hydrogen is the

favourite molecule of physicists. Aligning hydrogenis technically difficult though, because it rotates veryquickly, and it is difficult to make intense, short pulsesthat can align hydrogen. Since we have access to time-delayed, few-cycle pulses, we were able to align andprobe the alignment of deuterium: 2H2 [69].

Using the doughnut-timbit stage in Figure 2.4, wealigned and probed with the same linear polarization.The measured alignment is shown in Figure 3.9. Thealigning pulse here had a peak intensity low enough toavoid strong ionization, roughly 2× 1014 W/cm2, and aFWHM of about 10 fs. The probe pulse had the sameduration as the pump pulse, and was intense enough tofully ionize molecules in the focus. The translational

19


FIGURE 3.9: Rotational wavepacket evolution in deu-terium, using few-cycle laser pulses. The points are mea-surements, and the curve is a numerical simulation of theexperiment.

temperature of the molecular beam was measured to be240 K. This temperature is consistent with the illustratednumerical simulation of a wavepacket in deuterium with240 K rotational temperature, using the same laser pulse,calculated by E. A. Shapiro.

In Figure 3.9, we see the prompt alignment during thefirst 50 fs. Trev is 557.7 fs for deuterium [70], and it hasthe same 2:1 even:odd ratio as nitrogen [61]. The smallerrevival at 150 fs is thus the partially suppressed quarterrevival, and the revival near 300 fs is the half revival.

Using a linear probe means that we are not able torule out induced rotation from the probe pulse. The mea-sured alignment is thus not necessarily quantitatively cor-rect, but does qualitatively show the changes of alignmentrelative to an isotropic distribution. A baseline measure-ment, taken with the pump pulse blocked, was set to have〈cos2 θ〉 = 1/3, where θ is the angle of the deuteron ve-locity to the aligning pulse polarization axis.

The fast rotations of hydrogen make it especially in-teresting for ultrafast index modulation. Rotationally ex-cited hydrogen has been used to generate bandwidth forcompressing ultraviolet pulses to less than 10 fs [14].

3.5 Field-free 3D alignment of asymmet-ric tops

Beyond linear and symmetric top molecules is themost general asymmetric top, which has three differentmoments of inertia. When all three axes of an asymmetrictop are simultaneously aligned, this is called 3D align-ment, after the three molecular axes that align to the threeaxes of space, as illustrated in Figure 3.10.

Earlier efforts to achieve 3D alignment were donein the adiabatic regime, where alignment occurred in thepresence of the aligning pulse [71, 72]. The aligning pulsewas elliptically polarized, causing the most-polarizableaxis to align with the major axis of the electric field,

FIGURE 3.10: An illustration of randomly aligned tri-atomics on the top, 1D alignment on the left, and 3D align-ment on the right.

the second-most-polarizable axis to align with the minoraxis, and the third axis automatically aligns with thelaser propagation direction. Experiments in the presenceof an aligning field are problematic though. The extralaser field may interfere with the process being studied,and the long aligning pulse may cause excitations of themolecules. As is in the linear rotor case, field-free 3Dalignment would be preferred over in-field alignment [73,74].

3.5.1 Method of 3D alignmentTo study 3D alignment, we chose the molecule SO2, a

bent triatomic, with sulphur in the centre. The O–O axis isthe most polarizable, and has the heaviest rotational iner-tia. The axis of symmetry in the molecular plane, betweenthe two oxygen atoms and passing through the sulphur isthe next most polarizable, and has low inertia for rotationaround the O–O axis. To be more general, I will speak ofa heavy and a light axis, assuming that the heavy axis isslow, and the most polarizable; and the light axis is fast,and the second most polarizable.

A short, linearly polarized pulse will align the heavyaxis as it would a diatomic [9], but the light axis will con-tinue to tumble around the heavy axis, as in the left panelof Figure 3.10. There can be no 3D alignment, simplybecause a linear pulse is too symmetric.

Three 3D alignment schemes were compared numer-ically by Underwood et al.. [75]. The schemes studiedwere a short elliptical pulse; a long elliptically polarized

20


FIGURE 3.11: From left to right, illustration of the heavyO–O axis, the light axis in the molecular plane and throughthe sulphur, and the axis perpendicular to the molecularplane. The alignment of the first two axes are plotted inthe data presented below.

pulse with a rapid turnoff; and one linear pulse, followedby an orthogonally polarized pulse during a revival of theheavy axis. The orthogonal pulse scheme was found togive the best 3D alignment.

To 3D align SO2, we used a modified version of theorthogonal pulse scheme [76]. Since the rotational levelsof an asymmetric top are more complex than that of therigid rotor, they do not revive strongly. Strong revivals aremore likely to occur at low rotational temperatures on thescale of 1 K, since there are fewer states involved. Evenat such low temperatures though, the revivals will still beweaker than the prompt alignment immediately after thefirst aligning pulse. To avoid this problem, we work onlywith the prompt alignment, ignoring the revivals.

To 3D align, we first use a horizontally polarizedpulse to begin the alignment of the heavy axis, as illus-

Field

Time

Inte

nsi

ty

x

y

x

y

FIGURE 3.12: Illustration of pulse scheme used to 3Dalign sulphur dioxide. The first pulse, polarized along x,1D aligns the O–O axis. The next pulse, polarized along y,rotates the molecule around the O–O axis, leading to 3Dfield-free alignment after both pulses have passed.

Axis Scaled RotationalPolarizability Constant

parallel to O–O 0.4692 2.0273543in molecular plane 0.2983 0.344174

through Sperpendicular to 0.2325 0.293527molecular plane

TABLE 3.1: Polarizability [77] and rotational constants [78]for 32S16O2. The total polarizability of SO2 is 3.90 A3, thevalues in the table are the scaled principal components ofpolarizability and sum to 1.

trated in Figure 3.12. Before the heavy axis reaches peakalignment, but when there is already significant alignmentof the heavy axis, we apply a vertically polarized pulse.In the limit of perfect heavy axis alignment, the verticalpulse will have the greatest possible effect on the align-ment of the light axis and no effect on the heavy axis. Forreasonable heavy axis alignment, the second pulse willalign the light axis and, being lighter, it can align morequickly than the heavy axis. By adjusting the pulse timingand intensity, the alignment of both axes can be made topeak near the same time, resulting in good, field-free, 3Dalignment.

FIGURE 3.13: Illustration showing the circularly and hor-izontally polarized probe pulses, and their correspondingangles ϑxz and ϑzy, which are symmetric relative to thelaser polarization.

21


3.5.2 Experimental 3D alignmentThe main aspects of our 3D alignment experiment are

the molecules, the aligning pulses, and the measurement.Diagnosing 3D alignment requires the measurement oftwo molecular axes relative to the laboratory frame. ForSO2, we use the heavy O–O axis relative to the polariza-tion axis of the first pulse, and the light axis in the molec-ular plane through the sulphur relative to the second align-ing pulse.

We diagnose alignment by Coulomb explosion withintense 45 fs laser pulses. When selecting probing po-larization and geometry, we again take care to removepossible bias from induced rotation and preferential ex-plosion in the enhanced ionization regime. We start witha circularly-polarized probe pulse, which is symmetric inthe polarization plane. We use this probe to measure thealignment of the O–O axis projected onto the polarizationplane, relative to the first aligning pulse. This angle is ϑxz,as shown on the top of Figure 3.13. The notation ϑAB in-dicates the angle between the projection of the vector onthe plane AB, and the axis A. The x axis is along the polar-ization axis of the first, horizontally polarized pulse; the zaxis is along the polarization axis of the second, verticallypolarized pulse; and the y axis is along the laser propaga-tion direction.

For the second measurement, we use a horizontallypolarized exploding pulse and look at the projected an-gle ϑzy of the light axis through S. In these two cases, themolecule is considered 3D aligned when 〈cos2 ϑxz〉 for theO–O axis and when 〈cos2 ϑyz〉 for the plane perpendicularare both above the isotropic average of 1/2. By using

FIGURE 3.14: Experimental 3D alignment of cold SO2.The upper panel shows ϑxz for the heavy axis measuredby circularly polarized light; the lower panel shows ϑzy forthe light axis measured by linearly polarized light.

symmetry, we know that any anisotropy in the measure-ment is due to alignment of the gas, and not an effect ofthe probe.

A measurement showing 3D alignment is shown inFigure 3.14. The first aligning pulse, polarized along x, isapplied at -400 fs, and is stretched to about 180 fs FWHMby passage through 13 mm of SF6 glass. The secondpulse, vertically polarized along z, is applied at time 0,and is about 45 fs FWHM. The pulse stretching is meantto better match the aligning pulse to the rotational speedof the axis being aligned. Both pulses had peak intensi-ties of about 2× 1013 W/cm2. The SO2 molecules werefrom a continuous expansion of 500 Torr of 0.5% SO2 inargon through a 75 µm pinhole, with an estimated rota-tional temperature of 10 K.

For each data point, tens of thousands of fragmentswere collected. Of these, about two thousand correlatedpairs of O3+ and O2+ were selected by comparison withcharacteristics found from triply correlated fragments.The correlation filter for the oxygen pair required O3+

fragments to have a kinetic energy between 50 and 80 eV,O2+ fragments to have a kinetic energy between 30 and 50eV, and a pair sum momentum of magnitude 30× 10−23

and 55× 10−23 kg m/s. From the correlated pairs, theheavy O–O axis and the light axis through S were ap-proximated by the difference and sum respectively of thenormalized oxygen velocity vectors.

Looking at Figure 3.14, the first aligning pulse at -0.4 ps starts the alignment of the heavy axis in the upperpanel, while leaving the light axis isotropic in the lowerpanel. The second aligning pulse at 0 ps does damagethe alignment of the heavy axis, which would otherwise

FIGURE 3.15: Calculated 3D alignment of SO2 of thewhole ensemble, for the experimental conditions. Thesolid line is for the angle between the heavy axis and the xaxis, the polarization of the first aligning pulse. The dashedline is for the angle between the light axis, and the z axis,the polarization of the second aligning pulse.

22


FIGURE 3.16: Experimental 3D alignment of hot SO2. Thesolid line shows ϑxz for O2+ fragments measured by cir-cularly polarized light, the dashed line shows ϑzy for S3+

fragments measured by linearly polarized light.

have continued to increase in alignment, but the decreaseis slow enough that the light axis aligns while the heavyaxis is still well aligned. At 0.5 ps, there is good alignmentof both axes, indicating field-free, 3D alignment of SO2.

The curves in Figure 3.14 are classical simulations ofSO2 in the experimental conditions as calculated by J. G.Underwood. The calculation includes the probability ofexplosion as a function of the O–O axis to the laser po-larization, by using data from the explosion of unalignedmolecules, and yields generally good agreement.

Figure 3.15 shows the calculated alignment of theheavy (solid) and light (dashed) axes, of the entire ensem-ble, using the angle of the axis to the respective aligningpulse polarization without projection. In this unprojectedangle θ, the molecules start at the isotropic value of 1/3.When the heavy axis begins to align, the light axis valuedrops slightly below 1/3, but this is just an artefact of thistype of angle, rather than an indication of 3D alignment.The second pulse aligns the light axis, and there is peak3D alignment near 500 fs, as in the experiment.

We have also observed field-free 3D alignment inexperiments with minimal rotational cooling. To reducecooling, the expansion was from less than 10 Torr of SO2alone, with an estimated rotational temperature on thescale of 100 K . The first aligning pulse was at -40 fs withpeak intensity 8× 1013 W/cm2, and the second aligningpulse was at 0 fs, with peak intensity 5× 1013 W/cm2.Neither aligning pulse was stretched in this case, sincethe rotational speed of the heavy axis is too high at thistemperature for a 180 fs pulse.

The measured field-free alignment is shown in Figure3.16. For this plot, correlation was not used due to a lowernumber of measured fragments. The heavy axis is here ap-proximated by the velocity of individual O2+ fragments,

and the light axis by individual S3+ fragments. For thislevel of alignment, the angle between the fragment veloc-ity and the molecular axis will not cause much reductionin the 〈cos2 ϑ〉 value relative to a correlated measurement.At 0.22 ps, peak field-free, 3D alignment is observed. Al-though not as high as in the rotationally cooled case, 3Dalignment is still possible in rotationally hot gas.

3.6 Rotational wavepacket overviewIn this chapter, short laser pulses create rotational

wavepackets in molecules, aligning linear molecules, and3D aligning asymmetric tops. To experimentally mea-sure alignment, intense laser pulses Coulomb explodethe molecules, imparting alignment information to themeasurable fragment velocities. We measure alignmentin the linear molecules nitrogen, oxygen, and the verylight molecule deuterium. Field-free 3D alignment wasobserved in sulphur dioxide.

There are other ways to make a wavepacket. Using ashort, linearly polarized pulse, as in this chapter, alignsa linear molecule to the polarization axis, as well as adisc perpendicular to the axis during revivals, as in Fig.1.4. An aligning pulse can be circularly polarized, inwhich case linear molecules would be pulled into a plane,and later revive aligned along the laser propagation direc-tion. Calling alignment to a plane “aplanement,” a circu-lar alignment pulse can aplane a planar molecule such asbenzene into the polarization plane, and later revive an-tiaplaned with a common axis along the laser propaga-tion direction. For a linear molecule, antiaplanement bya circularly polarized pulse will result in molecules lyingalong the laser propagation direction. Careful choice ofan elliptically polarized pulse can result in alignment ofa linear molecule along the major axis of the ellipse, andthen along the laser propagation axis during a revival [79].

Shaped pulses have also been used to create rotationalwavepackets. They usually take the form of a pulse witha slow increase in intensity which adiabatically aligns themolecules, and a rapid decrease that leaves a wavepacketof aligned molecules that are free to evolve without a field[27, 8].

The ability to align and measure the alignment ofmolecules is the basis for experiments using aligned orrotating molecules. In the next two chapters, experimentswill be done on rotational wavepackets, using the creationand measurement techniques described in this chapter.

23


4Controlling Rotational

Wavepackets

4.1 Multiple-pulse alignmentA few years ago, it was observed that adding a second

laser pulse to a rotational wavepacket could change the re-sulting alignment at a later time [80]. In this chapter, wecontrol the form and evolution of rotational wavepacketswith multiple laser pulses. Control can be an interestingproblem on its own; when applied to rotational wavepack-ets, it also becomes a tool for new experiments or appli-cations.

Rotational wavepacket control is of fundamental inter-est as an experimental implementation of the rigid rotor,an important quantum system. The rigid rotor has longbeen a staple of quantum chaos research [81], and rota-tional wavepackets were recently proposed as a quantumlogic system [82, 83, 84].

In any experiment where the alignment quality is im-portant, control can help by increasing the degree of align-ment. When exploiting molecules, such as using their fastrefractive index modulation, controlling the revival struc-ture controls the temporal behaviour of properties like therefractive index.

This chapter describes some ways to control rota-tional wavepackets with additional laser pulses. Thesecontrol pulses are replicas of the aligning pulse, sharing

FIGURE 4.1: Optical schematic for the rotationalwavepacket control experiments in this chapter. The op-tics are identical to that in Figure 3.2, with the addition oftwo interferometers in the pump arm, yielding a total of fourspatially overlapped pulses with variable time delay. Inten-sity and polarization of each pulse can be controlled usingadditional waveplates and neutral density filters.

the same propagation direction, polarization, and pulseduration. The control parameters are the time delaysbetween pulses, and the pulse intensities. The controlmethods are divided into two limiting cases, where thecontrol pulses are either strong or weak. With strongcontrol pulses, the control pulses can apply forces com-parable to the aligning pulse. Weak pulse control uses asubtler interaction, adjusting the quantum phases withinthe wavepacket.

4.2 Strong pulse controlStrong pulse control refers to using control pulses

that are similar in intensity to the original aligning pulse.The aligning pulse creates a rotational wavepacket, andthe control pulse is applied at different times, resultingin improved alignment, destruction of the wavepacket, orsomething in between.

Strong pulses can be likened to the pushing of a swingas in Figure 4.2. The first push of the swing is like thealigning pulse turning the molecules from an isotropic to-wards an aligned state. While the swing is in motion, theswing can be pushed again when the swing passes throughthe low point. Depending on the direction the swing ismoving, the swing will either gain or lose energy from theextra push.

FIGURE 4.2: Swing analogy for strong pulse control. Fromleft to right, pushing the swing corresponds to aligning, en-hancing, and annihilating.

For a rotational wavepacket, the times when the swingis at the low point correspond to the centres of full and halfrevivals. At these times, the wavepacket is again isotropic,and the molecules are moving towards alignment during afull revival, or towards antialignment during a half revival,as in Figure 3.5.

If a pulse is applied when the molecules are alreadyaligning, the molecules will gain energy and have betteralignment since alignment occurs over a shorter period oftime. If a pulse is applied when the molecules are movingaway from alignment, the molecules will be slowed. If thepulse is weak, the molecules are slowed, but continue to-wards antialignment; if the pulse is strong, the moleculeswill be reversed, and pushed back towards alignment; and

24


if the pulse is just right, almost all of the energy given bythe first pulse can be removed, destroying the wavepacket.

The significance of kicks spaced by revival times wasseen in theoretical studies of periodically kicked, planarquantum rotors [81, 85], as part of quantum chaos stud-ies. The point of interest in these studies was the effectof quantization on the way a rotor takes energy from pe-riodic kicks. At quantum resonance, corresponding to en-hancing alignment, a rotor absorbs energy more quicklythan a classical system. At quantum antiresonance, cor-responding to wavepacket annihilation, the rotor does notincrease in energy, as the pulses alternately add and re-move energy. In a classical system, the rotors would pickup energy more slowly since, without quantum resonance,the kicks can remove energy as well.

4.2.1 Enhancing alignmentWhen aligning a molecule, there will be some maxi-

mum intensity above which undesirable nonlinear effectswill occur. For example, at the ionization saturation inten-sity, most molecules will ionize. Since the laser is nonres-onant and short, linear effects should be rare. Enhancingalignment with multiple pulses is a useful technique forachieving strong field-free alignment while minimizingnonlinear effects, such as ionization. Rather than usinga single, intense, aligning pulse, the revival structure canbe exploited to effectively divide the single pulse into twoor more pulses with intensities below a given threshold,while giving better alignment than is possible with onlyone of the pulses.

The use of quantum resonance for enhancing align-ment was first studied theoretically[86], and a morepractical analysis with finite kick number was presentedshortly after [87]. We experimentally show enhancedalignment with a second laser pulse at a full revival [88]:the case of pushing with the swing’s motion. The generaltiming is shown on an unmodified revival structure innitrogen in Figure 4.3. There are three laser pulses: analigning pulse at time 0, a control pulse at the first fullrevival at 8.4 ps, and a probe pulse measuring the fullrevival at 16.8 ps. The alignment and control pulseshad the same polarization, and peak intensity of roughly1.4× 1014 W/cm2. The control pulse was scanned overthe first full revival, resulting in a series of measuredrevivals at 16.8 ps as a function of the control pulsetiming.

The resulting series of revivals are shown as a surfacemap with alignment as a function of the control pulse timeand the probe time in Figure 4.4. The curves projected onthe walls are the revivals measured with only the aligningpulse (left), and only the control pulse (right), to show the

FIGURE 4.3: An unmodified rotational wavepacket in ni-trogen to show timing. The control pulse is applied in the� hatched region near 8 ps, the � hatched region at 17ps is observed for enhanced alignment.

magnitude of the alignment from the individual pulses.The peak shows that there is an optimal time where thecontrol pulse enhances the alignment, and other times leadto reduced alignment.

The same data is replotted in Figure 4.5. These plotsare like side-views of Figure 4.4 along the time axes. Inpanel (a), the full revival from only the aligning pulse isshown, marked as 1. The timing of the control pulses isgiven by the position of the vertical lines. The ends ofthe vertical lines show the amplitude of the revival mea-sured at 16.8 ps for that timing of the control pulse. Thetwo revivals for the vertical lines marked © and 4 areshown in panel (b), and a revival from the control pulseonly, marked 2, shows the original magnitude of the re-vival. When the control pulse is too early, the revival is

FIGURE 4.4: Alignment as a function of the control andprobe time. The peak alignment is higher than the align-ment from the individual aligning pulses, shown by the re-vivals on the walls as green and blue dashed lines.

25


FIGURE 4.5: (a) Amplitudes of the measured revivals at 17ps are shown by the vertical lines at the time of the controlpulse. Curve 1 is alignment with the aligning pulse only. (b)Curve 2 shows alignment from the control pulse only. Thecurves marked © and 4 show the revivals correspondingto the strongest alignment and antialignment.

deformed, favouring antialignment over alignment in the4 revival. Alignment is optimized with a control pulse inthe middle of the revival as in the case of ©.

To show general applicability, enhanced alignment foroxygen, a molecule with more centrifugal distortion, isshown in Figure 4.6. The control pulse was applied at32.08 ps, or 2.75 Trev after the first aligning pulse. Again,the alignment from two pulses is better than that from ei-ther of the pulses alone.

The timing for optimal alignment will depend on theintensities, and molecule involved. For example, centrifu-gal distortion will change the shape of the revivals, andthus the optimal timing. Good enhancement has also beenseen with the control pulse applied soon after the aligning

FIGURE 4.6: The solid curve shows enhanced alignmentin oxygen. The alignment from the first aligning pulse onlyis marked 1, and the alignment from the second aligningpulse only is marked 2.

pulse during the prompt alignment, and with the aligningpulse being weaker than the control pulse [89].

Soon after our demonstration, two-pulse alignmentwas applied to the measurement of the dependence of ion-ization rate on molecular alignment in carbon monoxide[90], and used to help understand the effects of rotationalwavepacket behaviour in spectroscopic experiments [10].

4.2.2 Rotational wavepacket annihilationRotational wavepacket annihilation is the case of

pushing against the swing of Figure 4.2, and correspondsto quantum antiresonance. To annihilate, the aligningpulse makes a wavepacket from a thermal distribution,and the control pulse, applied at Trev/2, removes theenergy given by the aligning pulse, annihilating thewavepacket and leaving isotropic molecules. More gener-ally, the two pulses are a zero-effect pulse pair that returnthe wavepacket to whatever state it would have been inwithout the pulse pair. An observation made after bothpulses should be unable to tell whether or not the pulsepair had been applied. The second pulse should annulthe effect of the first pulse. This ideal annulment can beapproximately achieved with two laser pulses.

Rotational wavepacket annihilation was seen previ-ously in nitrogen, and interpreted in a planar-rotor approx-imation, with J �M [91]. In this section, we consider themore general concept of the zero-effect pulse pair [92].

Following the analysis of E. A. Shapiro [92], we startby considering the effect of laser pulses of duration τ,which are much shorter than the average rotational pe-riod of the molecule. A pulse applies the potential V (θ, t)during τ, where θ is the angle between the rotor and theelectric field, and t is time. If the field-free motion duringthe pulse is negligible, H0τ� 1, then the propagator Ukickcan be reduced to an angular dependent phase shift:

Ukick(τ) = e−iR

τ0 (H0+V (θ,t))dt ≈ e−i

Rτ0 V (θ,t)dt (4.1)

Omitting terms that are independent of θ since they onlyadd an overall phase to the whole wavefunction, we have,as in section 3.2, the interaction V (θ, t) =−(I(t)/2)(α‖−α⊥)cos2 θ. This leaves −

Rτ

0 V (θ, t)dt = εcos2 θ, whereε =

Rτ

0 (I(t)/2)(α‖−α⊥)dt.

4.2.2.1 Annulment in 2D planar rotorAnnulment is clearest in the two dimensional, planar

rotor model, with only even J states. A planar rotor isrestricted to rotate only in a plane, and is described by asingle angle to the laser field, θ. This reduces the angularmomentum to ˆL2D = −ih∂/∂θ, which has eigenfunctions(1/√

2π)eiMθ, and energy spectrum hcBM2, with M aninteger.

26


A zero-effect pulse pair can then be written as the se-quence Uzepp2D:

Uzepp2D = eiε ˆcos2θ Trev2D/2 eiε ˆcos2θ. (4.2)

Trev2D/2 represents evolution over half of the revival timeTrev2D. Note that we use the more relevant semiclassi-cal definition Trev2D= 1/(4Bc) [93] in this two dimen-sional analysis, which is half the value of Trev used inthis thesis for rotations in three dimensions. For nitrogen,Trev= 8.383 ps [67].

For proper annulment, Uzepp2D should be equivalent toTrev2D/2, as it would be if the pulses had not been applied.Using the eigenfunctions and energies above, we can eval-uate the operators in the basis of the even M states:

ˆcos2θ =

. .. 1/2 1/4

1/4 1/2 1/41/4 1/2 .

. .

,

ˆsin2θ =

. .. 1/2 −1/4−1/4 1/2 −1/4

−1/4 1/2 .. .

= 1− ˆcos2

θ,

Trev2D/2 =

.

1−1

1.

(4.3)

with 1 being the unity operator. Multiplication shows that

Trev2D/2 ˆcos2θ = ˆsin

2θ Trev2D/2, (4.4)

and expanding the exponential in Uzepp2D gives

eiε ˆcos2θ = 1+ i ˆcos2θ− 1

2( ˆcos2

θ)2 + ..., (4.5)

Using equations (4.4) and (4.5), one can show that

Trev2D/2 eiε ˆcos2θ = eiε ˆsin2θTrev2D/2. (4.6)

Since ˆsin2θ = 1− ˆcos2

θ, [ ˆsin2θ, ˆcos2

θ] = 0. With eAeB =eA+Be

12 [A,B], the sequence Uzepp2D can then be rewritten as

Uzepp2D = eiε ˆcos2θeiε ˆsin2θTrev2D/2 = eiεTrev2D/2. (4.7)

That is, up to a global phase, applying Uzepp2D to any ini-tial state Ψ0 results in Ψ0 with Trev2D/2 time evolution, asit would have been if no laser pulses had been applied.

This two dimensional, even J case represents ideal an-nulment. If there are odd J states, the annulment will notwork because Trev2D/2 is different for odd J, although thisis not so in three dimensions.

4.2.2.2 Annulment in 3DA more realistic model is a rotor in three dimensions.

In three dimensions, noting that we have returned to thethree dimensional revival time, the sequence of operationsfor the zero-effect pulse pair becomes Uzepp:

Uzepp = eiε ˆcos2θ Trev/2 eiε ˆcos2θ. (4.8)

The matrices, for either of the even J basis, or the odd Jbasis, are, as in Equation 3.3, given by:

ˆcos2θJ,J =

(J +1)2−M2

(2J +1)(2J +3)+

J2−M2

(2J +1)(2J−1),

ˆcos2θJ,J+2 =

√(J +1)2−M2

√(J +2)2−M2

(2J +3)√

(2J +1)(2J +5),

ˆcos2θJ,J−2 =

√(J +1)2−M2

√J2−M2

(2J−1)√

(2J +1)(2J−3),

ˆsin2θ = 1− ˆcos2

θ

Trev/2 =

.

1−1

1.

, (4.9)

where M is the additional projection quantum number,with the field only coupling states having the same M (seealso Equation 3.2).

The quality of annulment depends on the equality ofTrev/2 ˆcos2

θ and ˆsin2θ Trev/2. For states with |M| � J, and

J � 1, the matrix elements approach the two dimensionalvalues, so annulment should be good in this regime, not-ing that Trev/2 is the same for odd and even J in three di-mensions. If J− |M| � J and J � 1, the ˆcos2

θ matrixelements become small, so the pair of pulses will havelittle effect.

Evaluating Trev/2 ˆcos2θ and ˆsin

2θ Trev/2 for general

J,M shows that the off-diagonal elements match, butthe diagonal elements are not equal since the diagonalelements of ˆcos2

θ are not 1/2, as in the two dimensionalcase. To understand the effect, one can imagine dividingthe first pulse into two subkicks. The first subkick isthe ˆcos2

θ matrix with the diagonal elements artificiallyset to 1/2. The second subkick is a diagonal matrix thatcompensates for the artificial diagonal elements of thefirst subkick.

27


FIGURE 4.7: Calculated projections of the annulled wave-function on the original wavefunction for nitrogen by rota-tional state J0,M0. A projection less than unity indicatesundesired population movement to a different rotationalstate.

In such a scenario, the first subkick changes the initialstate in a way that could be perfectly annulled. The sec-ond subkick modifies the phases of this ideal wave packet,changing the way it revives Trev/2 later. Since the diagonalterms depend on both J and M, the diagonal elements ofdifferent states will differ from 1/2 by different amounts.Thus, applying the full pulse results in a J and M depen-dent chirp relative to what one would have if all the di-agonal terms were 1/2. This chirp changes the revival atTrev/2, so all states cannot be perfectly annulled, but theannulment can still be good, as seen in our experimentsand calculations.

Figure 4.7 shows numerical simulations by E. A.Shapiro measuring the quality of annulment. The calcula-tion was for an aligning pulse of peak intensity 7.7×1013

W/cm2, and annihilating pulse of 6.7 × 1013 W/cm2,4.21 ps after the aligning pulse. Both pulses had 50 fsfull-width at half-maximum durations. The J0 and M0axes indicate the rotational state, and the height indicatesthe value of the projection of the annulled wavepacketon the original wavepacket. For perfect annulment, allpoints would be 1. A value less than 1 indicates that thepopulation of that state has changed. The correspondingrevival structure is shown in the top panel of Figure 4.8.

The projections have values from 0.57 for low J andM states, up to 1.0. The worse recovery of lower energystates may be due to the stronger influence of the lowerpotential when the pulse intensity is not strong. When av-eraged over a thermal distribution, the average projectionis 0.867 at 100 K, and 0.912 at 300 K, which has morepopulation in higher energy states.

FIGURE 4.8: Simulated revivals for annihilation in nitro-gen. The upper plot shows annihilation at the half revival,at 100 K. The lower plot shows annihilation at the tenthhalf revival at 300 K, to show annihilation still works withstrongly chirped revivals. The triangles indicate the time ofthe annihilating pulse.

To test the importance of the rigid rotor approxima-tion, the lower panel of Figure 4.8 shows calculated an-nihilation in 300 K nitrogen at 88.14 ps, the tenth halfrevival. Because of the strong chirping of the revival fromthe late time and the high rotational energy, the annulmentis not as good as in the upper panel, but there is still signif-icant annulment, with the following revival showing littlealignment.

4.2.2.3 Experimental annulment in nitrogenIn this experiment, we used two laser pulses to create,

then annihilate a rotational wavepacket in a thermal dis-tribution of nitrogen. To verify the zero-effect pulse pairinterpretation, we applied these two pulses to an existingwavepacket, and observe the return of the original revivalsafter both pulses.

There are three alignment pulses in this experiment.Pulse 1 is the first pulse of the zero-effect pulse pair. Pulse2 was optimized to annihilate the wavepacket made frompulse 1 only. Pulse 0 creates the test wavepacket.

The pulse pair 0 and 1 had peak intensities of about3.0×1013 W/cm2. Pulse 1 was placed 1.33 ps (not a sim-ple fraction of Trev) after pulse 0, where it clearly dis-rupted the revivals from pulse 0. Pulse 2 was 4.23 ps(about Trev/2) after pulse 1, with a peak intensity of about2.7×1013 W/cm2.

The measured revivals for various combinations of thethree pulses are shown in Figure 4.9. The top curve showsthe creation and annihilation of a wavepacket from an

28


FIGURE 4.9: Experimental wave packet annihilation in ni-trogen. These are the revival structures from various com-binations of the three laser pulses. The numbered trian-gles indicate which pulses were applied, and their timing.In the top curve, pulse 1 creates a rotational wave packet,which is annihilated by pulse 2. In the lowest curve, pulse0 creates a test wave packet which is distorted by pulse1 until pulse 2 cancels pulse 1, restoring the revivals frompulse 0.

initial thermal distribution by pulses 1 and 2. The re-vivals from each pulse alone are shown in the middle threecurves.

The bottom curve shows the revival resulting from allthree pulses. Pulse 0 creates a test wavepacket at time0. Pulse 1 distorts the test wavepacket enough to preventalignment at 4.2 ps, Trev/2 after pulse 0. Pulse 2 then an-nuls the effect of pulse 1. While there has been some dam-age to the wavepacket, the revivals are clearly dominatedby the original test wavepacket, seen by comparing thebottom two curves.

Annulment can be used in many other experiments.It can be combined with other multiple-pulse controltechniques to cancel the control effects when they are

no longer needed. Annihilation allows for arbitrary tim-ing of the revivals, which would otherwise be limitedto simple fractions of Trev. By annihilating the previouswavepacket, a new wavepacket can be created at any latertime. Annihilation can also rotationally separate mixturesof molecules having different moments of inertia. Byapplying the second pulse at the half revival of a particu-lar component, it will annihilate the rotational excitationof that component only, with possible application inrotational spectroscopy.

4.3 Weak pulse phase controlTo control a rotational wavepacket with relatively

weak pulses, we cannot apply large forces to the molecule;instead, we adjust the phases within the wavepacket.Since the revivals are interferences of the wavepacketcomponents, changing the phases can have dramaticeffects.

To access the quantum phases, we make use of frac-tional revivals [94]. In oxygen, fractional revivals at oddmultiples of Trev/8 are a combination of aligned and an-tialigned components. Looking at 〈cos2 ϑ〉 though, thereis no peak. Projected onto the measurement plane as inFigure 4.10, the revival has a cross shape, which has nonet alignment. The revival can be seen by plotting thehigher order function 〈cos2 2ϑ〉, which is itself shaped likea cross, as shown in Figure 4.11. Two cross revivals areshown in the lower panels of Figure 4.12.

A cross revival allows separate addressing of the twoparts of the wavepacket. A laser pulse at a cross revivalwill polarize the aligned component more than the an-tialigned component. While the pulse is on, the two com-ponents evolve in different potentials, gathering a relative

FIGURE 4.10: Measured angular distribution during across revival in oxygen at Trev/8 after the alignment pulse.

29


FIGURE 4.11: Polar plots showing the shape of functionsfor characterising alignment. Left: cos2 ϑ for alignment toan axis. Right: cos2 2ϑ for alignment to a cross shape.

phase shift. The more intense the pulse, the greater theaccumulated phase shift by the end of the pulse. Thus,a laser pulse at a cross revival provides control over therelative phases in a rotational wavepacket.

For an organized approach to the many possiblephases that can be applied, we borrow a quantum logicformalism used for vibrational wavepackets [82], andmodel our rotational wavepackets as qubits, and ourphase control as single-qubit operations.

In quantum computing, there are three basic single-qubit gates: the Hadamard gate; the π/4 phase gate; andthe π/8 phase gate, also known as a T gate. Any single-qubit operation can be approximated by different combi-nations of these three gates [95]. If we can demonstrate

FIGURE 4.12: The upper panel shows alignment in an un-modified wavepacket. The vertical, dashed lines show thetiming of the different phase control pulses. The lowerpanels show measured cross revivals at the 3/8 and 7/8revivals.

these gates, then we can approach any qubit state. Thiscorresponds to controlling the combination of alignmentand antialignment at a revival.

To describe the wavepacket as a qubit, we start withthe two states |A〉 and |AA〉, which are defined by theshape of the wavepacket when it is aligned or antialignedat a revival. From these, we write our two qubit levels:

|0〉= (|A〉+ |AA〉)/√

2, (4.10)

|1〉= (|A〉− |AA〉)/√

2. (4.11)

Like Floquet states, these two states are eigenstates of dis-crete time evolution in the 2D rotor case. That is, everyTrev/8, these states will revive, with a phase determined bytheir quasienergies ε0 = 0 and ε1 = 4π/Trev [94]. In the3D case, the revival is slightly offset from Trev/8 due to thedifferent energy spectrum.

In a normal rotational wavepacket, the wavepacket isaligned, which can be written in terms of the eigenstatesas: |A〉= (|0〉+ |1〉)/

√2. As this state evolves, the differ-

ent quasienergies cause relative phase changes. This pro-gressive oscillation of the phase leads to the usual aligned,antialigned, and cross-shaped revivals. Every Trev/8, thetwo eigenstate components gain a net phase difference ofπ/2 or, equivalently, an opposite π/4 phase shift to eachstate. This is the action of the π/4 phase gate, so timeevolution over Trev/8 is a π/4 phase gate.

A Hadamard operation takes a superposition state toan eigenstate, and an eigenstate to a superposition state.To demonstrate a Hadamard operation, we start with thenormal aligned wavepacket, which is a superposition,and take it into an eigenstate. We do by going to thefirst cross revival at Trev/8. This cross can be written as(|0〉+ i|1〉)/

√2 = eiπ/4(|A〉 − i|AA〉)/

√2. Since a laser

pulse changes the phases between the |A〉 and |AA〉 com-ponents, we use a laser pulse to add π/2, thus puttingthe wavepacket into an eigenstate. Once in the eigen-state, the wavepacket should no longer show revivals innet alignment, since the cross shape will repeat itselfindefinitely.

To do this experimentally, we aligned oxygen at about100 K, with a 50 fs pulse with about 1×1014 W/cm2 peakintensity. Using Coulomb explosion, we measured eighthrevivals as shown in Figure 4.12. We then put a controlpulse on a cross revival, and optimized its timing and in-tensity to minimize later revivals in net alignment. Asan extra test, we applied a second control pulse at a latercross, to add another π/2 of phase, and put the wavepacketback into the superposition state, recovering revivals ofalignment. The resulting revival structure is shown in Fig-ure 4.13. The molecules are aligned at time 0; a controlpulse at 4.23 ps and 2.0×1013 W/cm2 switches the align-

30


FIGURE 4.13: Coherently switching the alignment off andon in oxygen. The dashed curve is an unmodified refer-ence wavepacket at arbitrary vertical scale.

ment off; and a control pulse at 10.06 ps at 2.3× 1013

W/cm2 switches alignment back on.To demonstate a T operation, we need to bring the

wavepacket into a state with a π/4 shift between the com-ponents, such as (|0〉+e−iπ/4|1〉)/

√2. This phase leads to

uneven cross shapes every Trev/8. An uneven cross has netalignment, so the alignment revival frequency is doublethe normal frequency, since this state shows net alignmentevery Trev/8.

In the experiment, we used a control pulse with thesame timing as the switch-off pulse for the Hadamard op-eration, but with a peak intensity of 0.9× 1013 W/cm2

to apply π/4 phase. This results in the correct shape ofuneven cross Trev/8 later, but without the correct relativephase. At the uneven cross at 5.7 ps, we apply anothercontrol pulse at 1.9×1013 W/cm2 to correct the phase byadding π/2. This puts the wavepacket in the desired state.As seen in Figure 4.14, we observe new revivals wherethere were previously only crosses, in addition to the re-vivals at the usual times.

This set of experiments showed implementations ofHadamard and T operations in oxygen [83]. These pulsesequences were not gates, meaning that they would notperform the same operation on all initial states. The fullgates are similar though, requiring one extra laser pulse atthe beginning of the sequence [84].

Thus, for a rotational wavepacket single qubit, phasecontrol should be able to approximate all single-qubit op-erations. At the moment though, rotational wavepacketsare not a proper quantum computing system, since theydo not fulfill all of the criteria for quantum computing.The main criteria for true quantum computing are: initial-ization, efficient read-out, scalability, low decoherence,and a universal set of gates. The main problem for ro-tational wavepacket computing is difficulty coupling tworotors, a necessary condition for having a universal set of

FIGURE 4.14: Measurement of a doubled revival fre-quency of alignment in oxygen. The dashed curve isan unmodified reference wavepacket at arbitrary verticalscale.

gates. This is a common problem for quantum computing.Qubits must be both resistant to decoherence, while beingable to couple to each other when needed. If a method forcoupling was found though, rotational wavepackets maybecome an interesting quantum computing platform, sincethey can be formed at room temperature, and decoheremostly from collisions, which can be partly controlled.

4.4 Other control techniquesOther ways to control rotational wavepackets have

been proposed or demonstrated. Pulse sequences for gen-erating semiclassical catastrophes, including enhancedalignment, was proposed by Averbukh et al. [86] andLeibscher et al. [96]. Hoki et al. [97] have proposedoptimal control methods for optimizing aligning and ori-enting laser pulses. Horn et al. [98] used an evolutionaryalgorithm to control revival shapes. Renard et al. haveused spatial light modulators [99] to add spectral phasesteps to their aligning pulses. They have shown controlof the relative excitation of odd and even rotational statesin nitrogen [100], and enhanced alignment by alteringthe shape of the revivals [91]. Daems et al. [79] haveused elliptically polarized aligning pulses to yield whatis like alignment to a linear polarization axis, and thelaser propagation axis. The elliptical polarization breaksthe symmetry that results in the usual antialignment to aplane, leading to similar, linearly shaped, alignment andantialignment distributions.

Combined with the control techniques demonstratedin this chapter, very detailed control of molecular rota-tion and alignment is now possible. As the body of ro-tational wavepacket control techniques continues to grow,rotational wavepackets will become even more useful as away to study and exploit molecules.

31


5Experiments Using Aligned

Molecules

5.1 Ways to use aligned moleculesThe behaviour of a molecule is very dependent on its

alignment. Alignment can play a role in any experimentor application using gas molecules.

Rotational wavepackets in gas molecules were first re-ported by Heritage et al. [6]. This report noted the changein birefringence of the rotationally excited molecules,and proposed possible spectroscopic applications. Thishappened a decade later as Ahmed Zewail’s group beganto perform rotational spectroscopy in the time-domainby observing the refractive index modulation from rota-tionally excited molecules [28]. Time-domain rotationalspectroscopy, sometimes called “Raman-induced polar-ization spectroscopy” (RIPS) or “rotational coherencespectroscopy” (RCS) is still used to measure molecules[101, 9].

Many other applications of rotational wavepackets be-gan to appear at the turn of the millenium. The ultrafastindex modulation at a rotational revival was used as a wayto generate bandwidth for pulse compression [102, 103,104, 14]. At about the same time, rotational wavepacketsbegan to be recognized as a way to achieve a high, some-times called macroscopic, degree of field-free alignmentof gas molecules [19].

After birefringence, the next molecular property seento vary with alignment was strong-field ionization. De-scribed in Section 5.2, our experiment showed a strong de-pendence of the ionization probability on the angle of themolecule to the ionizing laser field [16]. Further experi-ments measuring ionization from aligned molecules weredone in P. B. Corkum’s group at the National ResearchCouncil Canada [105, 106], and R. R. Jones’s group at theUniversity of Virginia [90].

The dependence of high harmonic generation hadbeen studied in adiabatically aligned molecules [107] butthe presence of the aligning laser field made interpretationproblematic. With field-free aligned molecules, an orderof magnitude variation in the harmonic generation effi-ciency from nitrogen was seen [17]. This effect quicklybecame a probe to measure electronic wavefunctions bytomographic reconstruction from the harmonic efficiencyas a function of angle between the molecule and the laserfield [29].

Molecular alignment is now being applied to diffrac-tion measurements of molecular structure. Structuralmeasurements by diffraction of x-rays or electrons from

a gas results in the equivalent of a powder diffractionexperiment. With the construction of new x-ray sources,alignment is being investigated as a way to improvediffraction studies by making the experiment more likecrystal diffraction [108, 109]. The diffraction of anelectron taken from a molecule and driven back by thelaser field [110, 111] also benefits from alignment inour laser-induced diffraction experiments described inSection 5.3.

5.2 Alignment-dependent strong-fieldionization

For a molecule in a strong laser field, the probabilityof ionization can depend strongly on its alignment relativeto the laser field. This is an important effect in strong-fieldscience, since ionization is an important element of manystrong-field processes. For example, ionization is the firststep in generating high-harmonics, so the alignment de-pendence of ionization will also play a role in the totalefficiency of harmonic generation [17].

Currently, the only available method for measuring thealignment-dependence of ionization is by using field-freealigned molecules [16]. The concept of the experiment issimple. A pump pulse starts a rotational wavepacket, andthe resulting alignment is measured. An ionizing pulseionizes some of the molecules, and the number of ions arecounted and compared for different angular distributions.

In this experiment, we measured the ionization of N2.Calculations of the angular dependence of ionization inN2 and other molecules have been done using molecularADK theory [112, 113], an adaptation of the Ammosov-Delone-Krainov (ADK) model of atomic ionization [114].A simple summary of their calculated results is that theionization rate is high when the laser field lies along a di-rection of high electron density for the state being ionized.For example, O2 has a πg highest occupied molecular or-bital (HOMO), and is thus expected to ionize preferen-tially with the internuclear axis at 45◦ to the laser field[113].

Nitrogen has a σg HOMO, which lies along the inter-nuclear axis. Nitrogen is thus expected to be more easilyionized when the molecule is lying along the laser polar-ization axis than for other alignments. We measure theionization rate for two geometries, one with the moleculesaligned along the laser field, and one with the moleculesantialigned to the laser field (in a disc perpendicular to thefield). We access these two geometries by changing thetiming of the ionizing pulse to be slightly before or afterthe centre of the half revival. For more general measure-ments, one can also rotate the polarizations of the aligningand ionizing fields.

32


We aligned nitrogen with a linearly polarized, 40 fspulse with 5×1013 W/cm2 peak intensity. For the ioniza-tion measurements, we used the peak alignment at 4.15 psafter the aligning pulse, and the peak antialignment at 4.33ps. The angular distributions, measured by an explodingpulse as in Chapter 3, are shown in Figure 5.1.

Having found the right timing, we applied the ionizingpulse, with an intensity of 2×1014 W/cm2, to the alignedor antialigned distribution, and counted N2+ ions for thesame time period for each distribution. H2O+ and O+

2 ionrates from the background gas were used to verify systemstability between the two measurements. The aligned dis-tribution was found to produce 1.23 times more ions thanthe antialigned distribution, in qualitative agreement withmolecular ADK theory.

Since the angular distribution was measured, a morequantitative angular dependence can be extracted byassuming a simple functional form for the angular de-

FIGURE 5.1: Polar histograms of the measured angulardistributions for aligned (left) and antialigned (right) nitro-gen used to measure ionization. The laser fields are at 0◦,and the origin indicates zero counts.

pendence of the ionization probability Π(θ), where θ

is the angle between the molecular axis and the elec-tric field of the ionizing pulse. One possible form isΠ(θ) = (εcos2 θ + 1)/(ε + 1). The parameter ε can be fitto the experimental results by comparing the relative ion-ization probability when integrating over the two angulardistributions for different ε. The full angular distribu-tion is approximated by rotating the measured projecteddistribution around the polarization axis.

The probability distribution Π fit to the experimentalresults is shown in the solid curve in Figure 5.2, keepingin mind that the experimental curve is a fit to two datapoints. Two theoretical curves, courtesy of X. M. Tongand C. D. Lin, are also shown for intensities slightly aboveand below our experimental intensity with good agree-ment between the three curves. In the fit, ε = 3.5± 0.5,which means that, for our laser pulses, nitrogen was about4 times more likely to ionize when aligned with the laserfield, than perpendicular.

An important point in this type of experiment is theextent of rotation induced by the ionizing pulse. The in-terplay of induced rotation (dynamic alignment) and ion-ization probability for a given alignment (geometric align-ment) during Coulomb explosion was the subject of sev-eral experiments in the past [115, 116]. In this experi-ment, we want to be in the geometric alignment regime sowe can ignore rotation during the ionization pulse. Thisis likely true in our case. By looking at the alignment af-ter an aligning pulse similar in intensity to the ionizingpulse in this experiment, we see that it takes over 100 fsafter the peak of the pulse for the isotropic distribution toreach peak alignment [21], which is longer than the 40 fsionizing pulse duration in this experiment.

FIGURE 5.2: The solid line shows the ionization probabilityas a function of the angle of the molecule to the electricfield with the assumed function fit to the experimental datawith a 2×1014 W/cm2 pulse. The dashed curve shows themolecular ADK dependence at 1.6× 1014 W/cm2, and thedotted curve is for 3.2×1014 W/cm2.

33


This experiment has shown that there is a significantdependence of the ionization probability on molecu-lar alignment in nitrogen, and demonstrated a generalmethod that has already been applied to the measurementof angular dependent ionization in other molecules [90].

5.3 Laser-induced electron diffractionIn a diffraction experiment, a wave is directed towards

an object that deflects the wave in different directions de-pending on its orientation and internal structure. Thesewaves are usually x-rays [117], neutrons [118, 119], orelectrons [4]. By measuring the direction and relative in-tensities of the diffracted waves, information about thestructure of the diffracting medium can be found.

Diffraction is the most effective for single crystals,where the diffracted signals from many layers of mate-rial add together. For other materials, such as powders,the random orientations within the material make thediffracted signal more difficult to analyse. When diffract-ing from a gas, the gas can be effectively crystallized byaligning the molecules [11, 109].

We used this approach in experiments attempting tomeasure the structure of the nitrogen molecule by laser-induced electron diffraction [120, 110, 111, 121]. In laser-induced electron diffraction, the diffracting wave is one ofthe molecule’s own valence electrons being driven by thelaser field. In the three-step model [122], the laser fieldfirst tunnel ionizes an electron from the molecule near thepeak of the laser electric field. In the second step, the elec-tron is pulled away by the laser field, and then driven back.In the third step, the electron may return to the molecule.Many interesting events can happen upon the return, suchas recombination with the release of a high harmonic, at-tosecond pulse [123]; double ionization of an additionalelectron in the molecule [124]; or diffraction of the elec-tron by the charge distribution of the molecule.

An experiment for measuring laser-induced electrondiffraction from nitrogen might be to have a nitrogenmolecule aligned, say, vertically, and to have an ioniz-ing laser linearly polarized horizontally. Most electronswould simply fly past the molecule, and leave in a hor-izontal direction. Diffracted electrons would gain avertical momentum component, leaving the horizontalpolarization axis. One might then have a screen to ob-serve the deflected electrons, and by observing the angleof the deflection, determine the molecular structure, justas one would determine the spacing of two slits fromdiffracted light.

For an actual experiment in the PATRICK chamber,some modifications of the experimental concept above areneeded. Since the PATRICK measures ions, we measure

FIGURE 5.3: Cartoon of the 3-Step model for laser-induced electron diffraction. An electron is ionized from themolecule. The electron is pulled away, then driven back bythe oscillating laser field. The returning electron diffractsfrom the molecule.

the recoil momentum of the ion from the diffracted elec-tron, rather than the electron itself. The recoil is verysmall, so the initial momentum of the molecule must beknown accurately. This leads to some geometric require-ments when planning the experiment.

The molecular beam in the PATRICK chamber is verycold in the directions perpendicular to its propagation.This is because of the geometry of the beam: moleculeswith motion perpendicular to the beam would not passthrough both skimmers. There is a much larger velocitydistribution along the beam. While there is some cool-ing from the expansion of the gas, the thermal spreadin momentum is still much greater than the recoil mo-mentum. Thus, we can only measure recoil momentumperpendicular to the molecular beam. It is also preferableto measure the momentum along the time-of-flight axis.Measurements of other axes are based on the spatial res-olution of the delay-line anode system, which is coarserthan the time-of-flight measurement, and subject to biasfrom uneven response across the detector surface.

Taking these geometric requirements into account, ourexperiment uses an aligning pulse linearly polarized alongthe vertical time-of-flight axis to align the molecules ver-tically. The ionizing pulse is linearly polarized along thehorizontal molecular beam propagation axis. The recoilmomentum pz is only measured along the time-of-flightaxis.

34


FIGURE 5.4: Geometry of a simple two-slit diffraction. ∆kis independent of |k| for a given diffraction order.

For a rough guide, we use a simple model of nitro-gen as two slits separated by the internuclear separation of1.1 A [125]. The conventional way to think about diffrac-tion is to have a monochromatic wave diffracted at a cer-tain angle where the extra path length is the wavelength ofthe incoming wave. Since we do not produce monochro-matic electron waves, taking a different perspective sim-plifies the problem, by noting that the gain in momen-tum perpendicular to the original wave propagation is in-dependent of the original wavelength. Using the geome-try in Figure 5.4, |∆k| = |k|sinθ. For a diffraction maxi-mum, sinθ = nλ/d = 2πnk/d, so |∆k|= 2πn/d, where nis an integer. For d = 1.1A, the corresponding momentumtransfer along the molecular axis would be 0.60× 10−23

kg m/s or 3.0 atomic units of momentum. In atomic units,h, the electron mass, and the electron charge all have val-ues of 1 [126, 127]. One atomic unit of momentum ish/a0 = 0.1993× 10−23 kg m/s, where a0 is the Bohr ra-dius.

The laser pulses in our experiment were from an op-tical parametric amplifier (OPA), pumped by the sameregenerative amplifier used in the other experiments de-scribed above. An OPA splits an 820 nm pump photoninto a signal and an idler photon with the same total en-ergy. In this experiment, the signal was 1480 nm, and theidler was 1840 nm. The signal and idler emerge from theOPA with orthogonal linear polarization, so we split andrecombine the pulses with polarizing beamsplitters, as il-lustrated in Figure 5.5. A computerized translation stagecontrols the time delay of the idler with a retroreflectionas in the other alignment experiments.

The signal beam is vertically polarized, aligning thenitrogen molecules along the time-of-flight axis. The sig-nal intensity is attenuated by decreasing the aperture of aniris, simultaneously preventing ionization by the aligningpulse, and increasing the focal spot size of the alignmentpulse. The ionizing idler is horizontally polarized, driv-ing electrons into the molecules along the molecular beamaxis. By measuring ionization rates as a function of idlerpulse energy, we verified that the idler was over the ion-ization saturation intensity. Both beams are overlapped by

FIGURE 5.5: Illustration of the interferometer for control-ling the time delay between the signal and idler laserpulses. The blue beam is the signal which aligns themolecules. The red beam is the idler, which can be de-layed to probe the molecules at a later time.

focusing through a pinhole with a 2 m focal length con-cave mirror. The pulse overlap in time was observed bymixing the beams in a β-BaB2O4 (BBO) nonlinear crys-tal.

To align the nitrogen molecules, the signal and idlermust have an appropriate time delay. Unlike oxygen, thetimes of alignment in nitrogen only occur either slightlybefore or after the centre of a revival, as in Figure 3.5.This means that the time of best alignment will changedepending on rotational temperature, and the intensity andduration of the aligning pulse. To find the best time delay,we measure the ionization rate from the idler as a func-tion of time delay. From the ionization experiments inSection 5.2, we know that nitrogen preferentially ionizeswhen the molecule is aligned with the ionizing field. Theionization rate as a function of time delay thus traces outthe rotational revival, from which we can find the time ofbest alignment. For the experiment, we used a time delayof 4.0 ps.

The resulting momentum distribution pz for nitrogenis shown in Figure 5.6. Only the momentum distributionfor time-of-flight less than the peak value is used becausethe 15N isotope, and signal reflections in the delay-lineanodes interfere with the measurement at longer time-of-flight.

Also plotted in Figure 5.6 is the momentum distribu-tion for argon, which was measured simultaneously by us-ing a gas mixture of 5% N2, 5% Ar, and 90% He. The ion-ization potential of argon and nitrogen are similar, leadingto similar ionization rates. Argon is meant to provide asingle-slit reference, emulating the diffraction from a sin-gle nitrogen atom. It is not clear that this is a satisfactory

35


FIGURE 5.6: Measured recoil momentum distributions forsingly ionized nitrogen and argon. The upper curve is ar-gon, the curve just below is nitrogen. The lowest crossesare background counts for argon, the circles just above arethe background counts for nitrogen.

reference. Another possibility is to measure ionizationperpendicular to the molecular axis, since this directionshould contain no structural information.

The points at the bottom of Figure 5.6 show the con-tribution from the background gas. This distribution fromthe molecular beam was selected by filtering for the loca-tion on the detector where molecules from the jet landed.The background contribution was measured by applyingthe same filter to a region on the opposite side of the detec-tor, where no jet molecules would land due to their well-defined speed. The same number of molecules ionizedfrom the randomly moving background gas should landin both filtered regions. We were able to achieve a lowbackground count by using a 90% He gas mixture. He-lium is light and fast, increasing the speed of N2 and Arin the jet. This makes them easier to separate from theslower moving background gas which lands closer to thecentre of the detector than the faster jet.

The ratio of the nitrogen and argon momentum spectraare shown in Figure 5.7. Ions are only detected to about3.5 atomic units. The maximum in ion momentum comesfrom the maximum energy that the electron can pick upfrom the ionizing field. This maximum energy is about3.17 Up [122], where Up, is the ponderomotive energy,the average kinetic energy of an oscillating free electronin the laser field [128]. For a 1014 W/cm2 laser field at800 nm, Up = 6.0 eV.

The ponderomotive energy varies with the intensity,and the square of the wavelength of the laser field. Theponderomotive energy cannot be increased arbitrarily byincreasing the intensity. Except for single-cycle laserpulses, the molecules will mostly ionize near the ioniza-tion saturation intensity. Increasing the pulse intensity

FIGURE 5.7: Ratio of the simultaneously measured recoilmomentum distributions of singly ionized argon and nitro-gen.

will only move the ionization to an earlier part of thepulse. To increase the energy of the diffracting electronthus requires a longer wavelength, which is beyond theabilities of our current equipment.

The ratio in Figure 5.7 suggests that there may be apeak at 3 atomic units, although there are not many countsat this high momentum range. The decrease around 1.5atomic units may correspond to the diffraction minimum.While this spectrum is promising, it does depend stronglyon the comparison with argon. Our group is currentlybuilding an electron imaging chamber to observe thediffracted electrons directly, at much higher count rates.A similar chamber has been built to observe diffrac-tion using the Advanced Laser Light Source (ALLS) inVarennes, Canada, where intense, long wavelength pulsesare available.

If successful, laser-induced electron diffraction willprovide a way to measure structure with ultrafast time res-olution. The same techniques used to produce single at-tosecond pulses by controlling the ionized electron [123]can be used to allow only one diffraction event within alaser pulse. As the probe pulse in a pump-probe exper-iment, laser-induced electron diffraction would observemolecular structure changing on the attosecond timescale.

36


6Conclusion

6.1 Molecular alignment and the worldWhen explaining research to a random person, one of

the first questions is always “why?” There are many re-sponses on many levels. At the most immediate level, Ican say that it is fun to make molecules do what you tellthem to do. While some people customize their cars, orplant intricate gardens, I overlap four laser pulses with mi-croradian and femtosecond precision, and spin moleculesaround. At the broadest level, one can start explaining theentire concept of science, and point out how we wouldnever have computers or compact disc players withoutquantum mechanics. This is a long explanation though,and I would not want to suggest that rotational wavepack-ets are as revolutionary as quantum mechanics.

The answer that I usually give is intermediate to thelast two. Controlling molecular alignment provides a ba-sic tool that could be useful to any experiment with gas orliquid molecules [129]. After the demonstration of strong,field-free alignment [19, 20], basic impulsive alignmentwas quickly applied to a new class of strong-field experi-ments as described in Chapter 5.

Two-pulse enhanced alignment is the most immedi-ately useful alignment control technique. Most exper-iments with aligned molecules would benefit from im-proved alignment. Given sufficient laser power, it is sim-ple to split an aligning pulse into two, gaining better align-ment, and less interference from intense aligning pulses.Two-pulse enhanced alignment is already being used inexperiments [10, 90].

The other way to use rotational wavepackets is as amethod of controlling the properties of a medium on ultra-fast timescales. This has so far been applied to the refrac-tive index of gases in hollow core optical fibres [14, 103,

104]. Rotational wavepacket annulment and phase controlwould be more applicable to these types of uses. It wouldbe possible to control the evolution of the wavepacket, andthus the property of interest, with femtosecond precisionover almost arbitrarily long timescales. By combiningwith the high spatial resolution of laser pulses, it shouldalso be possible to create arrays of closely-spaced rota-tional wavepackets, creating complex structures such astime-dependent photonic crystals [130], or a quasi-phasematched high harmonic source.

The Coulomb explosion imaging experiments inChapter 2 have a clear goal of molecular imaging.Coulomb explosion of a molecule into ionic fragmentsbrings the miniscule molecular structure into the measur-able macroscopic domain, along with femtosecond timeresolution.

Molecular alignment may eventually prove to be evenmore important than CEI as an integral part of new imag-ing techniques. Alignment is an essential part of molec-ular orbital tomography, which measures the wavefunc-tion into which an electron recombines during high har-monic generation [29]. Related experiments have alsoused alignment to extract molecular structure from highharmonic spectra [131, 132, 133, 134].

In addition to the laser-induced diffraction experi-ments described in Section 5.3, aligned molecules are be-ing investigated for use at x-ray facilities such as the Ad-vanced Photon Source at Argonne, USA, and the new freeelectron lasers, where femtosecond lasers can be coupledwith intense x-ray sources [108, 109].

At the beginning of this thesis, I asked whether we canmanipulate molecules as easily as we manipulate macro-scopic objects. While not quite as easy, we can now exertsignificant control over the alignment of a molecule. Be-ing able to rotate macroscopic objects is a basic abilitythat we use many times a day. Now that we can easilyrotate molecules, alignment will become a basic tool forusing and studying molecules.

FIGURE 6.1: Photograph of 3D aligned cars.

37


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Dedicated to the millions of molecules I exploded.Dedicated to the millions of molecules I exploded. DOCTOR OF PHILOSOPHY (2006) McMaster University (Physics & Astronomy) Hamilton,