High-Order Harmonic Generation from MoleculesbyNicholas L. Wagner
B.S., Case Western Reserve University, 2001

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulﬁllmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2008This thesis entitled:
High-Order Harmonic Generation from Molecules
written by Nicholas L. Wagner
has been approved for the Department of Physics

Prof. Henry Kapteyn

Prof. Margaret Murnane

Date

The ﬁnal copy of this thesis has been examined by the signatories, and we ﬁnd thatboth the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.

iii
Wagner, Nicholas L. (Ph.D., Physics)
High-Order Harmonic Generation from Molecules
Thesis directed by Prof. Henry Kapteyn and Prof. Margaret Murnane

High harmonic generation in atoms is well understood in terms of the three stepmodel (ionization, propagation and recombination), and much attention has been payed
to the dynamics of the continuum electron in the propagation step. However, a molecular
medium can be used to further understand the ionization and recombination steps. Highharmonic generation can also be used to probe the dynamics of rotational and vibrational
wave packets in molecular media.
This thesis reports on two experiments using high harmonic generation frommolecular media. In the ﬁrst experiment, an impulsive Raman pump is used to excite a rotational wave packet in the molecular system, and a second time-delayed pulse
generates high harmonics from the rotationally excited medium. Both the intensity andthe phase of the high harmonic emission are measured. The general features of theorientational dipole are extracted from the phase and intensity modulations.
A second experiment uses a vibrationally excited molecule as the high harmonic
generation medium. The coherently excited molecular vibrations modulate the intensityof the high harmonic generation. Using high harmonic generation as a probe makes it
possible to see all excited modes include two modes which were not present using acomparable visible probe. The rotational dephasing of the vibrational wave packet isalso observed.

v

Acknowledgements

Many people have contributed to this work directly and indirectly. When I started
graduate school, Randy Bartels and Tom Weinacht involved me in their work using
molecular phase modulation, which I continued with Tenio Popmintchev. I learned
about high harmonic generation experiments from Emily Gibson, with whom I alsostudied ionization pulse compression. I began studying high harmonic generation inmolecular media with Andrea Wüest, with whom I did the majority of work presented
here. He and I completed the vibrational study and laid the ground work for the studyof rotational wave packets. Robynne Lock, Xibin Zhou and Wen Li have joined the
project and continued studies on rotational and vibrational wave packets. In addition to
those who have worked with me directly, Zach Walters has been very helpful with many
theoretical discussions and determining what could be done with the analysis. Membersof our research group helped me greatly, teaching me basic experimental competence,
listening to my problems and helping develop my ideas. Emily Gibson, Etienne Gagnon,
Xiaoshi Zhang and Amy Lytle have all worked and helped maintain the laser system.
They frequently put aside their own work to allow me time to complete my studies. The
support staﬀ in JILA have made sure I have gotten paid, the machine shop has built large
parts of my experimental apparatus and the supply oﬃce has purchased much of myexperimental apparatus. Finally, my advisers Margaret Murnane and Henry Kapteyn
have spend untold hours advising me and committed undue resources to allowing me tocomplete my studies.

viContents

Chapter
1 Introduction1

1.1

Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Lewenstein Model of High Harmonic Generation . . . . . . . . . . . . .

4

1.3

Modiﬁcations of the Lewenstein Model for Molecular Media . . . . . . .

72 Experimental Apparatus

8

2.1

Ultrafast Ampliﬁed Ti:sapphire Laser System . . . . . . . . . . . . . . .

10

2.2

Mach-Zehnder Interferometer . . . . . . . . . . . . . . . . . . . . . . . .

11

2.3

Generation Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.4

Detection Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.5

Typical Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

2.6

Removing Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.7

Setting Focal Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.8

Background Subtraction and Integration . . . . . . . . . . . . . . . . . .

20

3 Theory of Impulsively-Excited Molecular Rotational Wave Packets

23

3.1

Alignment After an Impulsive Torque . . . . . . . . . . . . . . . . . . .

24

3.2

Quantization of Rotational Energy Levels and Revivals . . . . . . . . . .

28

3.3

Quantum Calculation of Rotational Wave Packet . . . . . . . . . . . . .

29

3.4

Wave Packet Simulations . . . . . . . . . . . . . . . . . . . . . . . . . .

31

vii
3.5

Sweeping of the Angular Position During Alignment . . . . . . . . . . .

33

3.6

Molecular Alignment and Nuclear Spin Statistics . . . . . . . . . . . . .

36

3.7

Pump Intensity, Pump Duration, and Rotational Temperature . . . . .

38

4 High Harmonic Generation from Rotational Wave Packets

43

4.1

Averaging HHG Over Rotational States . . . . . . . . . . . . . . . . . .

43

4.2

Intensity Measurements from Rotationally Excited CO2 . . . . . . . . .

44

4.3

Origin of the Bump at Maximal Alignment . . . . . . . . . . . . . . . .

46

4.4

Origin of the Anti-Aligned Peak . . . . . . . . . . . . . . . . . . . . . . .

48

4.5

Origin of the Baseline Reduction . . . . . . . . . . . . . . . . . . . . . .

48

4.6

Measuring the Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.7

Using Gas Mixtures to Measure the Phase of Molecular HHG . . . . . .

50

4.8

Direct Phase Measurement . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.9

Reconciling the Diﬀerences in Phase Measurements . . . . . . . . . . . .

65

4.10 A Two Center Model of HHG from CO2 . . . . . . . . . . . . . . . . . .

66

4.11 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

5 High Harmonic Generation from Vibrationally Excited Molecules

72

5.1

Exciting Vibrational Wave Packets . . . . . . . . . . . . . . . . . . . . .

73

5.2

Classical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

5.3

Observation of Vibrational Motion in SF6 using HHG . . . . . . . . . .

76

5.4

Sensitivity to Vibrational Motion . . . . . . . . . . . . . . . . . . . . . .

81

5.5

Comparison with Other Experiments . . . . . . . . . . . . . . . . . . . .

81

5.6

Decay of Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

5.7

Calculation of Modulation Amplitudes . . . . . . . . . . . . . . . . . . .

84

5.8

Vibrations from Other Molecules . . . . . . . . . . . . . . . . . . . . . .

84

5.9

Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

viii
6 Summary

91

Bibliography

92

Appendix
A Lewenstein Model of High Harmonic Generation

98

B Self-compression of Ultrashort Pulses in an ionizing medium

105

ix

TablesTable
3.1

The rotational constant and anisotropic polarizability for several common
molecules[28]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

31

The ratio of even and odd rotational state populations for several common
molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

B.1 Table of possible pulse shaping mechanisms that involve only 1-D propagation eﬀects in Ar at a pressure of 4 torr (i.e. not spatio-temporal pulse
evolution eﬀects) and the length of propagation over which they become
signiﬁcant[14]. T = pulse duration; GVD = group velocity dispersion;n2 = nonlinear index of Ar; n = linear index of Ar; I = laser intensity; ω= laser frequency; c = speed of light; unm = waveguide mode index; a= waveguide radius. The calculated gas terms are lower limits calculated
for neutral Ar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

xFiguresFigure1.1

An illustration of the three step model of HHG . . . . . . . . . . . . . .

2.1

The experimental apparatus consists of three main parts: a Ti:sapphire2

ampliﬁer, a Mach-Zehnder interferometer, and a glancing-incidence EUV
spectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92.2

Diagram of a free jet expansion.

14

2.3

Properties of the gas jet for several gases and distances away from the. . . . . . . . . . . . . . . . . . . . . .

nozzle. These values were calculated using methods shown in Atomic andMolecular Beam Methods [52].
2.4. . . . . . . . . . . . . . . . . . . . . . .

16

The transmission as a function of photon energy for EUV light throughthe 200 nm aluminum thin ﬁlm ﬁlter, which is used to reject the fundamental light. Most of our experiments are limited to observing HHG
emission in the range of ∼20-72 eV. . . . . . . . . . . . . . . . . . . . .

2.5

17

A typical high harmonic spectrum generated by a 28 fs pulse in argon gas
with a density of 1018 cm-1 . The peak intensity of the pulse was ∼1014
W/cm2 . Lower energy harmonics are on the right side of the graph andhigher on the left side. The harmonic orders are labeled in red for theﬁrst order diﬀraction from the grating and in blue for the second orderdiﬀraction.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

xi
2.6

High harmonic intensity in SF6 is plotted against the position of the lensfocusing the HHG beam. The focus is aligned to the outlet of the gas jet
at 6.5 mm; however, HHG is most intense when the focus is 2 mm before
the jet. This position also selects the short trajectory harmonics[51]. . .

3.1

An impulsive pump pulse applies a torque to a linear molecule. The
molecule is kicked in the direction of the polarization of the pump pulse.

3.2

21

25

Classical calculation of impulsive alignment of CO2 . The molecular axis
position for several time delays after the initial impulse as a function of
initial angular position is shown. The initial alignment is seen a t=0.6. .

3.3

27

a) The alignment parameter < cos2 θ > shown as a function of time forCO2 . b) The weighted angular density at a time of anti-alignment. b)
The weighted angular density when aligned. . . . . . . . . . . . . . . . .

3.4

32

The angular density shown as a function of time for CO2 . The verticalaxis is the polar angle with respect to the pump polarization and thehorizontal axis is the time evolution of the wave packet. One full periodof the wave packet is shown. The initial alignment, 1/4, 1/2, 3/4, andfull revival can be seen at 11, 22, 33, 44 ps respectively. There are also a
number of small features between the revivals . . . . . . . . . . . . . . .

3.5

34

An illustration emphasizing that one need to consider the weighted alignment distribution with the azimuthal angle integration completed, as opposed to the full angular density. The weighted alignment distribution
shows that most of the molecules are not pointed along the polarization
axis, even at maximum alignment. . . . . . . . . . . . . . . . . . . . . .

3.6

35

The angular density near the 1/2 revival in CO2 . The red line showswhere the wave packet is localized and sweeps through angular positions.

37

xii
3.7

The eﬀect of nuclear spin statistics is illustrated in the alignment parameter < cos2 θ > for N2 O, O2 , CO2 and N2 . The 1/4 revival can be
reduced, inverted, or ﬂipped depending on the ratio at which even andodd rotational states are populated. . . . . . . . . . . . . . . . . . . . .

3.8

39

Alignment parameter for CO2 around the 1/2 revival as a function ofpump intensity. Increasing the pump intensity decreases the duration of
the revival structure and increases the peak alignment.

3.9. . . . . . . . .

40

Alignment parameter for CO2 around the 1/2 revival for several initialrotational temperatures. Decreasing the temperature increases the peakalignment.

4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42a) The measured intensity of the 39th harmonic as function of time delay
for CO2 . b) The calculated alignment parameter for this rotational wave
packet. The harmonic intensity is anti-correlated with the alignment. . .

4.2

45

The intensity modulation of the 39th harmonic is shown around the 1/2
revival in CO2 . As the degree of alignment of the rotational wave packet
is increased several new features appear in the intensity modulation: 1)
A bump appears at the point of maximum alignment. Showing that with
suﬃciently strong alignment the HHG emission again begins to increase.
2) The intensity at anti-alignment gets signiﬁcantly stronger (∼4×). 3)
The baseline intensity between revivals drops below the isotropic intensity. 47

4.3

A cartoon graphic illustrating the origin of the “bump” during maximal alignment. The angular density sweeps through the minimum orphase shift causing a minimum in the high harmonic intensity. A second
minimum is created as the angular density sweeps back in the opposite
direction. There is a bump between the two minima. . . . . . . . . . . .

49

xiii
4.4

The angular density for the isotropic case (no excitation) (Black) and fortime-delays between revivals (Red). The distribution is shifted to smaller
angles between revivals. . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

51

a) The upper plot shows the calculated alignment parameter for the rotational wave packet. The lower plot shows the HHG intensity for the
31st harmonic from Kr:CO2 mixtures plotted against the time delay ofthe pump and probe pulses for several mixture ratios as stated in thelegend. b) Same as above, however, expanded near the half revival. . . .

4.6

53

a) The intensity of the 31st harmonic plotted against the fraction of CO2
in the mixture at two time delays: 21.75 ps and 21.25 ps corresponding to
the maximum anti-alignment and alignment during the revival. b) The
3rd term of eqn. 4.3 plotted against the fraction of CO2 for the sametwo time delays.

4.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55a) The magnitude (blue) and phase (red) of the dipole of CO2 retrieved
by ﬁtting the intensity from several mixtures to Eqn. 4.3. b) Same asabove, however, magniﬁed near the half revival. The alignment parameteris also plotted for reference. . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

57

a) The magnitude of the orientational dipole retrieved by ﬁtting to Eqn.
4.4 plotted against the molecular orientation. b) The magnitude andphase of the orientational dipole retrieved from the 2nd ﬁtting procedurec) The magnitude of the time dependent dipole and the ﬁt. d) The phaseof the time-dependent dipole and the subsequent ﬁt. . . . . . . . . . . .

4.9

58

a) and b) The magnitude and phase of the orientational dipole retrieved
from the 2nd ﬁtting procedure for harmonic orders 21-35. c) and d) theorientational dipole magnitude and phase predicted by the two-centeremission model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

xiv
4.10 Setup for directly measuring the intensity and phase of high harmonic
emission from molecules. The two foci are 240 μm and 110 μm awayfrom the gas jet oriﬁce respectively. The diameter of the gas jet is 150
μm and the backing pressure is 700 torr. HHG from aligned and randomly
oriented molecules from two diﬀerent regions of a gas jet interfere in thefar ﬁeld.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.11 (a), (c) Interference pattern as a function of time within the 3/4 revival for
harmonic orders 27 (a) and 33 (c). (b) Intensity-scaled integrated fringes
for the 27th harmonic in the -100 fs to 100 fs interval (red squares),along with least square ﬁt (red solid line). Integrated fringes outside this
temporal window are also shown (black circles) as well as a least squareﬁt (black solid line). (d) Same as (b), but for the 33rd harmonic. The
time delay in (a) and (b) is relative to the center of the 3/4 revival. . . .

64

4.12 High harmonic spectra from aligned CO2 compared with HHG from anisotropic distribution. The black is the isotropic spectrum and the blue isthe aligned spectrum. Due to the butterﬂy shape of the angular distribution, the angular distribution can be approximated as a classical moleculealigned at 30◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.13 Anti-aligned harmonic intensity relative to the isotropic intensity for several pump powers stated in the legend. The graph shows harmonic orders
higher and lower than usual range of 19-47. The lower order were measured using a diﬀraction grate of lower line density. The high orders were
measured using a zirconium ﬁlter instead of aluminum. Peak enhancement > 1 indicates the modulation is anti-correlated with the alignment
parameter.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

xv
5.1

Normal modes of vibrations for SF6 . The wavenumber, period, degeneracy and activity of each mode is stated[28]. SF6 has three Raman-activemodes, two infrared active modes and one forbidden mode (which are notexcited in our experiment). . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

74

Relative excitation eﬃciency as a function of driving pulse duration relative to vibrational period. A duration less than half the vibrational
period is needed for eﬃcient excitation. . . . . . . . . . . . . . . . . . .

5.3

77

Red curve: Intensity of 39th harmonic generated from vibrationally excited SF6 as a function of time delay between the pump pulse and the
EUV generating probe pulse. Black curve: high harmonic emission without the pump pulse present.

5.4. . . . . . . . . . . . . . . . . . . . . . . .

77a) Discrete Fourier Transform of the 39th harmonic emission from Figure
5.3, showing the three Raman-active modes of SF6 that are excited by
our impulsive stimulated Raman scattering (ISRS) pump pulse. b) Stimulated anti-Stokes Raman scattering of a 400 fs probe pulse centered at
400nm from SF6 , after excitation by the same ISRS pump pulse used toexcite vibrations in Fig. 5.3. c) Discrete Fourier Transform of the harmonic emission data from Figure 5.3, for 0.3 ps time intervals centeredat diﬀerent times after the pump pulse (0.45 ps, 0.75 ps and 1.05 ps). .

5.5

78

Peak to peak amplitude of the observed modulation for the three vibrational peaks shown in Figure 5.4a, as a function of harmonic order; ν5(red) , ν2 (green) , and ν1 (blue).

. . . . . . . . . . . . . . . . . . . . .

80

xvi
5.6

(Upper) High harmonic signal for harmonic orders 23 - 47 as a function oftime delay. The high harmonic signal without the pump pulse present has
been subtracted in order to show the modulations in the high harmonic
signal for all harmonic orders. (Lower) Amplitude of the high harmonic
modulation by the excited Raman-active modes as a function of harmonicorder. The amplitude of all modes signiﬁcantly increases with harmonicorder. The modulation of the high harmonic signal due to the ν2 mode
is above the noise level only for harmonic orders greater than 37.

5.7

. . .

82

A cartoon illustrating the steps involved in generating high harmonics
from a system with two vibrational states. Every step can step can shift
population between the vibrational states.

5.8. . . . . . . . . . . . . . . .

A comparison of calculated vibrational amplitudes with experimentallymeasured amplitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.9

85

86

The intensity modulation and DFT of the 41st harmonic from vibrationally excited CClF3 . Two vibrational modes are visible as well as theisotopic splitting of the ν3 mode. . . . . . . . . . . . . . . . . . . . . . .

87

5.10 The intensity modulation of the high harmonic signal in vibrationallyexcited N2 O4 .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

B.1 Spectra of pulses emerging from a waveguide ﬁlled with argon at pressures
starting at 4 torr decreasing to 0 torr, front to back. Ionization induces
a spectral blueshift and broadening of the pulse. . . . . . . . . . . . . . 108

xvii
B.2 Input and output pulse characteristics at the 4 torr optimum pressure
for compression of pulse with an input laser intensity of 1015 Wcm-2 .
(a) Temporal proﬁle and phase of the input pulse (29 fs) and the ﬁnal
compressed pulse (13.3 fs), (b) Measured spectrum of the compressed
pulse, compared with the spectrum of the pulse reconstructed from thetime-domain second-harmonic FROG data. These data demonstrate the
ﬁdelity of the compressed pulse measurement.

. . . . . . . . . . . . . . 110

B.3 Comparison between the experimentally observed (reconstructed FROG)
and theoretically predicted pulse envelopes as a function of pressure for0, 2, 4, 6, 8 and 9 torr. (a) Experimentally measured pulse shapes (b)Theoretically predicted pulse shapes. The pulse compression and pulse
splitting behavior are seen both experimentally and theoretically, with excellent qualitative and good quantitative agreement between experimentand model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Chapter 1

Introduction

High harmonic generation (HHG) is the nonlinear production of harmonics of an
intense optical ﬁeld by ionization and recollision of the an electron. It was ﬁrst observedby McPherson, et al. [40] in 1987, where they observed the 17th harmonic of 248 nm inneon gas. Typically, HHG is seen by focusing an ultrafast, high-intensity, near-IR pulse
into a noble gas at intensities of ∼ 1014 W/cm2 , and observing the odd harmonics of thedriving ﬁeld. Initially, the surprising aspect of HHG is the plateau region of the highharmonic spectrum. In the perturbative region (3rd - 7th harmonic) the intensity of theeach harmonic order drops rapidly. However, in the plateau region the harmonics areroughly the same intensity over many orders. Eventually the plateau ends in a cut-oﬀ
region where the harmonic intensity drops rapidly. This plateau region was not whatis expected from perturbative nonlinear optics, which would predict that each higher
order would drop in intensity.
In 1993 Kulander[35] and Corkum[18] provided a theoretical explanation for highharmonic generation and the similar phenomena of above threshold ionization and nonsequential ionization. They explained high harmonic generation in terms of a three stepmodel, as illustrated in 1.1. First, the atoms of the atomic gas are ionized at roughly
the peak of the optical cycle. Second, the ionized electron, now in the continuum withno kinetic energy, is accelerated by the optical ﬁeld. When the optical ﬁeld changes
direction, the free electron is accelerated back toward the parent ion. Third, when the

2

Figure 1.1: An illustration of the three step model of HHG

electron returns to the parent ion it can do three things: a radiative transition to theground-state, inelastic scattering, or elastic scattering. The radiative transition resultsin HHG. The inelastic scattering is called non-sequential ionization, where the returning
electron knocks an additional electron from the atom. Finally, elastic scattering resultsin very high energy electrons in the above threshold ionization spectrum. Based on
simple classical electron trajectories in optical ﬁelds, the returning electron has a plateau
of kinetic energies which is then manifest in the high harmonic spectrum.
Most research on high harmonic generation has concentrated on the second stepof the free electron propagation: Speciﬁcally, some work has observed the long and shortelectron trajectories which both contribute to HHG[13], used the optical ﬁeld to control
and measure the electron propagation[50][33], increased the intensity to produce shorter
wavelength EUV light[24] , shaped the ﬁeld to selectively increase a single harmonicorder[8]. One notable accomplishment of this line of research is the creation of isolated
attosecond pulses[27].
The other heavily researched aspect of high harmonic generation has been phasematching the process to produce more useful (higher ﬂux) source of the EUV light. Highharmonic generation is an attractive, tabletop source of EUV light when compared withlarge, expensive synchrotron sources. However, phasematching is needed to increase
the conversion eﬃciency and improve the spatial coherence[10]. Waveguides[20], quasi-

3phasematching[74], and pulse shaping[8] have all been used to develop high harmonic
generation as a tabletop source of EUV light.
Both of these lines of research have used noble gases as the medium for HHG andhave not focused on the ﬁrst and third steps in the HHG process: strong-ﬁeld ionization
and recombination. One reason is that strong ﬁeld ionization and recombination are notparticularly interesting in isotropic atoms with no internal structure. However, whenmolecules are used as the medium for HHG, ionization and recombination become much
more interesting and complex. Both of these processes can depend on the rotationaland vibrational state of the molecule, which can potentially serve as a knob to control
HHG. Additionally, high harmonic generation can be used as a probe to understand therotational and vibrational dynamics of the molecular system.
Many aspects of of high harmonic generation in molecular media are not wellunderstood, such as how the amplitude and phase of the ionized electron varies withmolecular orientation. The initial structure of the electronic wave packet in the continuum could aﬀect the recombination amplitude. The recombination amplitude could
also be strongly aﬀected by the molecular potential distorting the incoming plane wave.
Additionally, a molecular system could break the isotropic symmetry and oﬀer the possibility of generating high harmonics polarized perpendicular to the fundamental light.
This thesis reports on two experiments using high harmonic generation frommolecules. In the ﬁrst experiment, an impulsive Raman pump is used to excite a rotational wave packet in the molecular system, and a second time-delayed pulse generates
high harmonics from the rotationally excited medium. Both the intensity and the phaseof the high harmonic emission is measured. The general features of the orientational
dipole are extracted from the phase and intensity modulations.
A second experiment uses a vibrationally excited molecule as the high harmonic
generation medium. The coherent molecular vibrations modulate the intensity of thehigh harmonic generation. The high harmonic probe could see two modes which were4
not visible using a comparable visible probe. The rotational dephasing of the vibrational
wave packet was also observed.

1.1

Outline of the Thesis
The rest of this chapter presents a brief introduction to the Lewenstein model[38]of HHG and the approach taken to extend it to molecular systems. Chapter 2 presents
the experimental apparatus used to measure the high harmonic intensity from excitedmolecules, and explains how the data is processed. The third chapter discusses thetheory of rotational wave packets excited by an ultrashort laser pulse and the detailsof the time evolution that are important for understanding the high harmonic data.
Then chapter 4 presents the data on harmonic emission from rotationally excited CO2 .
Intensity and phase measurement are discussed from the perspective of extracting theorientational dipole. Chapter 5 presents data on high harmonic generation from vibrationally excited molecules. Chapter 6 summaries the work and provides a perspective
for the future. Appendix A contains a derivation of the Lewenstein model. Appendix B
presents work on pulse compression in ionizing medium, which is another project which
I worked on during my graduate career.

1.2

Lewenstein Model of High Harmonic Generation
The theory of high harmonic generation can be divided into two parts: First, thesingle atom response is the dipole of a single atom in a strong optical ﬁeld. Second,
propagation deals with the phasematching and absorption in the medium and eﬀectively
adds up the contributions for all of the atoms generating harmonics.
The Lewenstein model[38] is a semi-classical model of high harmonic generation
which calculates the single atom response. It is derived in appendix A. The goal of thismodel is to calculate the dipole responsible for the emission of high harmonics for an
arbitrary optical pulse. The nonlinear dipole is calculated as a function of time in the5fundamental ﬁeld. This dipole is then twice diﬀerentiated to get the electric ﬁeld of theemitted harmonics. A Fourier transform then gives the high harmonic spectrum.
The main requirement for high harmonic generation is that the ionized electron
must return to the parent ion. When the electron returns to the parent ion it contributes
to a dipole response with a broad spectrum, the magnitude of the contribution depends
on the degree/rate of ionization and the energy and direction the electron is returningwith. So, to determine the dipole at a time t we ﬁrst need to determine which electron
trajectories return at that time and when they were ionized.

∇p S(p, t, ti ) = ttidt (p − A(t )) = 0(1.1)

Here S is the semi-classical action and represents the phase of the electron trajectory. This equation is the result of the stationary phase approximation which tells
us which trajectories will contribute the most. p is the momentum that the electron
has when it recollides. A is the vector potential of the fundamental ﬁeld and ti is thetime at which the electron was ionized. When the fundamental ﬁeld is sinusoidal, there
are two solutions for every optical cycle that have a given recollision energy. These twosolutions are called the long and short trajectories (corresponding to the time spent in
the continuum). Solutions from the previous optical cycles can be ignored because thelateral spreading of the electronic wave function is proportional to the time spend in
the ﬁeld. So the shortest times contribute the most and trajectories in the continuum
longer than one cycle can be ignored.
After the returning trajectories are known the dipole can be calculated.

d(t) =

ai (ti )ap (ti , t)ar (t)

(1.2)

i

The dipole is the product of the complex amplitude from each step and is commonlysummed over the two contributing trajectories (the long and short trajectories). ai is

6
the ionization amplitude and is usually computed using static electric ﬁeld ionization
rates[1]. The phase of the ionization amplitude is usually neglected.
The propagation amplitude is a complex number with a constant amplitude;
however, the phase is the phase the electron accumulates during propagation in thecontinuum. Any eﬀect from the atomic potential is neglected. Because the electron accumulates phase faster than the fundamental ﬁeld, the propagation amplitude oscillates
many times faster than the fundamental optical ﬁeld, leading to high harmonics.

  t

ap (ti , t, p) = exp(iS(ti , t, p)) = exp i

ti







2

dt (p + A(t) − A(t )) /2 + Ip



(1.3)

The recombination amplitude is the expectation value of the dipole operator using
the total electronic wave function: the part in the ground state plus the part returningfrom the continuum. Only the terms mixing the ground state and returning wave
contribute to HHG. We assume the fundamental electric ﬁeld does not signiﬁcantly
aﬀect the recombination. The returning electronic wave is approximated as a planewave.
ar (t, p) =ψground μx exp(ip · x)

(1.4)

The dipole calculated here is the single atom response; however, the observed
HHG signal is emission from a macroscopic sample of atoms. The response of all of
the atoms must be added coherently. With macroscopic sample of dipoles oscillating inphase, the resulting optical ﬁeld does not have a dipole shape, but is emitted as highly
directed beam propagating in the same direction as the fundamental ﬁeld. The intensityof the emitted beam is strongly aﬀected by the degree to which the atomic dipoles are inphase. This phase is in turn aﬀected by the spatial variation of the optical ﬁeld and thetemporal slip between the harmonic ﬁeld and fundamental ﬁeld due to the medium andionization. Several phasematching technique exist to minimize this phase variation and7produce more intensity harmonics: waveguides, quasi-phasematching, and pre-ionized
media.

1.3

Modiﬁcations of the Lewenstein Model for Molecular Media
There are two immediate modiﬁcations necessary for molecular media. First theionization rate can be calculated using a method appropriate for molecules such as MOADK theory[59]. Second, the recombination amplitude is calculated by approximating
the electronic ground state as the highest occupied molecular orbital (HOMO).
There are several additional approximations which may be important in molecularsystems. For the ionization step, there are several methods of calculating the strong
ﬁeld ionization rate[59][34], and the initial phase of the electron could also be important.
For instance, is there a π shift in the electron phase when the electron comes from a
positive or negative lobe of the HOMO? Additionally, nodes in the molecular electron
density could change the shape of the outgoing electronic wave packet.
During the propagation step, the electron moves far from the parent ion and is
not signiﬁcantly aﬀected by the potential. The maximum excursion of a typical electronis ∼10 nm for an 800 nm driving ﬁeld, and the lateral spread of the electron is ∼1 nm.
During recombination it is also possible the single active electron approximation
is not valid, meaning the bound electrons as well as the returning electron need to beconsidered[46][25]. The fundamental optical ﬁeld could distort the molecular orbital
and inﬂuence the recombination amplitude, and ﬁnally the molecular potential could
distort the incoming plane wave[60]. There is no current consensus regarding whether
these eﬀects are important; and if they are, in which systems and regimes these eﬀects
matter. The experiments in this thesis show directly the orientational dependence which
a molecular medium gives to the ionization and recombination amplitudes, as well as
indirect evidence of the eﬀect of the molecular potential on the returning plane wave.

Chapter 2

Experimental Apparatus

Rotational and vibrational wave packets evolve very fast (fs - ps), and a pumpprobe technique is needed to observe the time evolution. A pump-probe technique
is used when the detection device (such as a camera) does not have a fast enoughresponse time to detect the time evolution. Instead of using the speed of the detector tocapture the motion (as might be done with a high speed video camera), the speed of theillumination is used to detect the motion. A very short pump pulse creates the wave
packet through an impulse excitation. Then, a time delayed probe pulse illuminates
the wave packet for a very short time at some point after the pump. This is repeated
for all time delays, and a complete picture of the motion is acquired. The pump-probe
technique requires that the wave packet can be excited repeatedly and that pump andprobe pulses are shorter than the time scale of the wave packet’s time evolution. Our
pump and probe optical pulses are created by a Ti:sapphire ultrafast laser ampliﬁer.
The output of the laser system is then split into a pump and probe pulse by a MachZehnder interferometer. The pump pulse excites the wave packet in a gas jet. The
probe pulse is then focused onto the same gas jet and generates high harmonics. The
EUV harmonics are then detected using a spectrometer and a EUV CCD camera. The
experimental apparatus is illustrated in ﬁgure 2.1.

Al filter 200 nm

10-5 torr

Beam imaging

Gas nozzle
0.17 mm diameter
1500 torr backingpressure

10-3 torr

Interaction Chamber

diversion forbeam diagnostics

FROG

2 deg
g angleg

50 cm lens

30 cm lens

Beamsplitter
R30% : T70%
Pump beam ffor
molecular alignment

Polarizer and half-wave plate
For varying the Intensity

Computer-controlled translation stage

Retro-reflector

Mach - Zehnder Interferometer

Figure 2.1: The experimental apparatus consists of three main parts: a Ti:sapphire ampliﬁer, a Mach-Zehnder interferometer, and aglancing-incidence EUV spectrometer.

EUV diffraction
grating

Al filter 200 nm

EUV
spectrometer10-5 torr

Detection
ChamberAndor EUV CCD camera

Interaction and Detection Chamber

Ti:sapphire laser system
800 nm
2.5 mJ/pulse, 1 kHz
25 fs FWHM
~15 mm diameter

Probe beam forf HHG

Ultrafast Laser Amplifier

9

10

2.1

Ultrafast Ampliﬁed Ti:sapphire Laser System
The laser system used in this experiment is an ampliﬁed Ti:sapphire system[7].

This laser is ideal for HHG because of the very broad bandwidth of Ti:sapphire, whichmakes it possible to generate very short pulses. This is useful in the case of HHG becausewith a short pulse, the intensity can ramp up to a very high value before ionizing the
medium. HHG typically requires ultrashort pulses < 30 fs and a per-pulse energy of∼ 1 mJ. Chirped pulse ampliﬁcation[56] is used to produce pulses of this duration andintensity.
The laser system begins with the oscillator[2]. This is a Ti:sapphire laser that iscontinuously pumped and is mode-locked through a Kerr-lens. The repetition rate isdetermined by the length of the cavity and is 86 MHz in the system used here. The
oscillator also contains a prism pair to compensate for the dispersion of the Ti:sapphirematerial, output coupler and focusing mirrors. The oscillator typically produces ∼ 13
fs pulses with an average power of 450 mW.
Next, a Pockels cell is used to select isolated pulses from the 86 MHz pulse train
coming from the oscillator. This is done primary because the 532 nm lasers used topump the ampliﬁer are limited in their repetition rate. So, the Pockels cell is used tolower the repetition rate of the seed pulse train to 1 kHz. This signiﬁcantly reduces theaverage power of the pulse train (by a factor of 86,000).
The 13 fs pulse cannot be ampliﬁed directly because it would easily become intense
enough to damage the Ti:sapphire ampliﬁcation material. So,a chirped pulse scheme isused for ampliﬁcation[56]. A grating-based stretcher makes the path length for ’blue’
colors longer than ’red’ colors. This stretches the 13 fs seed pulse to ∼ 2 ps, without
losing any spectral components.
The 1 kHz pulse train then goes through two ring ampliﬁcation stages. The
ﬁrst stage can be thought of as an high gain ampliﬁer and the second stage can be11thought of as a low gain, power ampliﬁer. The ampliﬁcation narrows the spectrum
and lengthens the minimum pulse duration. The Ti:sapphire ampliﬁcation crystals are
pumped with 532 nm pulses approximately 100 ns in duration. These pump pulses
deposit a signiﬁcant thermal energy in the crystal causing a thermal lens. This thermallens is minimized by cooling the crystals with liquid nitrogen[42]. At liquid nitrogentemperatures the thermal conductivity of the Ti:sapphire crystal is much larger which
reduces the thermal lens.
After ampliﬁcation the pulse train is compressed to its minimum duration, usinga grating compressor. The compressor performs the opposite function of the stretcher.
The ’blue’ colors travel a shorter path than the ’red’ colors. The pulses are compressed
to a duration < 30 fs. The laser system used in theses experiments typically produces
3 mJ per pulse and has an average power of 3 W. The compressor can be adjusted tocompensate for material further down the line in the experiment.

2.2

Mach-Zehnder Interferometer
The second part of this apparatus is a Mach-Zehnder interferometer that is used

to separate the original pulse into two time-delayed pulses and adjust the power, polarization and focusing of each arm of the interferometer. The laser beam is ﬁrst split
in two by a multilayer dielectric beamsplitter, usually 50% of the power in each beam.
The beam used to pump the molecular coherence is then time-delayed using a computer
controlled translation stage and a retro-reﬂector.
The retro-reﬂector greatly simpliﬁes alignment of the interferometer and insuresthat the pointing of the time-delayed beam does not change as the time-delay is adjusted.
The motor used to adjust the time-delay is also important. The coherence of molecularrotations can last up to a few 100 ps, which requires the range of the translation tobe ∼ 2 cm. However, the features of the wave packet are typically 100 fs in duration.
So, a resolution of ∼ 10 fs is required for rotational wave packets. Vibrational wave

12
packets require resolution shorter than the vibrational period. In these experiments,
it is often necessary to average several datasets to obtain a good signal-to-noise ratio.
This requires a motor that can return to its original position after moving ∼ 2 cm away.
These requirements are best met using a closed-loop motor and a position sensor.
The duration of the pump pulse can also be adjusted by putting material in thepump arm. This material disperses the pulse and increases its duration. This is usually
done when pumping rotational wave packets. A pulse of 100-200 fs is usually suﬃciently
short to impulsively excite a rotational wave packet. So by increasing the pulse length
the peak intensity can be lowered, which minimizes other processes such as ionizationand multi-photon dissociation which are undesirable. When pumping a vibrational
wave packet, the material is removed and a short sub-30 fs pump pulse is used, since itnecessary to use a pump pulse with a duration less than half of the vibrational period.
Both arms of the interferometer include a 1/2 waveplate and a polarizer, used toadjust the power on each beam. For some experiments an additional 1/2 waveplate isinserted after the polarizer to adjust the polarization of the pump pulse. Finally bothbeams are focused using lenses. The probe/HHG beam is focused with a 30 cm lens,and the pump beam is focused with a 50 cm lens. This insures that the probe spot issmaller than the pump spot in the interaction region. Because the probe spot size issmaller than the pump, it sees a more uniform excitation and does not see the lower
intensity edges of the pump pulse. Both lenses are mounted on translation stages along
the axis of propagation to adjust their positions relative to the gas jet in the generationchamber. The lenses are also mounted to allow translation perpendicular to the beamfor ﬁne adjustment of the overlap of the two beams at the gas jet.
Optical interference is used to align and ﬁnd time-zero for the interferometer.
First, both arms of the interferometer are set to approximately the same length using
a tape measure. The time delay is scanned by hand until an interference pattern isobserved. Once the interference pattern is found, the pointing of each beam can be13adjusted until the interference fringes become a bullseye, which indicates both beams
are propagating parallel to each other. The timing is then adjusted until the fringe
contrast is maximal, this allows the time-zero of the interferometer to be found within±10 fs. In the non-collinear case discussed below the bullseye pattern cannot be used.
After both beams pass through their respective lens, they are recombined. Thisis done by one of two methods: a dielectric beam splitter or a non-collinear reﬂection.
Using a dielectric beamsplitter allows both the pump and probe to propagate collinearly
and is much easier to align; however, a signiﬁcant amount of power (∼ 1/2) is lost at thebeamsplitter and the polarization cannot be rotated without changing the reﬂectance ofthe beamsplitter. A second technique is to reﬂect the pump with a mirror immediately
next to the probe beam. The angle of the mirror is set such that the beams cross at thefocus/gas jet at a very small angle < 3 degrees. This method allows for more eﬃcientuse of the laser power and for adjustment of the polarization of the pump. We did not
ﬁnd that the HHG signal depended on which geometry we used, as long as the incident
pump and probe pulses emerging were the same.

2.3

Generation Chamber
After the beams are focused and combined, they enter the generation chamber

which contains a gas jet and a 550 l/s turbo pump. The beams enter the chamber
through a thin (250 μm) sapphire window to minimize dispersion and nonlinear eﬀects.
When generating high harmonics it is necessary to let the gas interact with thelaser in only a small region near the focus. If there is gas present while the laser isfocusing, nonlinear eﬀects will modify the pulse temporally and spatially, which changes
the position, shape and intensity of the focus. After EUV light is generated by HHG,
it must propagate in a vacuum because it is strongly absorbed by all materials andgases. Because of these two requirements, windows can not be used to conﬁne thegas. There are, however, two common windowless methods of conﬁning gases for HHG:

14
Background Pressure PB
M<1

NozzleM<1Zone of Silence
M >> 1

Stagnation Pressure
P0 , T0, M << 1

Barrel Shock

Mach Disk ShockM<1

Jet Boundary

Figure 2.2: Diagram of a free jet expansion.

hollow-waveguides with diﬀerential pumping[20], and gas jets[52]. When using HHG asa molecular probe it is desirable to minimize propagation eﬀects, so a gas jet is preferable
because it has a very short interaction length, < 500 μm. In addition to conﬁning thegas and minimizing the interaction length, the supersonic expansion in the gas jet also
cools the temperature of the gas signiﬁcantly from room temperature to ∼ 75K.
The gas jet used in this experiment is a rather simple design. It consists of ashort piece of fused silica capillary tubing with an inner diameter of 150 μm, an outer
diameter of 1.2 mm and a length of 1 cm. One side of the capillary tube is the molecular
gas at a stagnation pressure of 700-2000 torr. The gas moves through the capillary andsupersonically expands into the vacuum chamber. The chamber has a pressure < 10−5
torr when the gas jet is turned oﬀ, and a pressure of ∼ 10−3 torr when gas is ﬂowing.
Ideally a larger turbo pump would reduce the background pressure in the chamber andallow a ’cleaner’ supersonic expansion. However, the density needed for eﬃcient HHG is> 1017 W/cm2 , and the laser is aligned to intersect the gas jet very near the end of thecapillary, usually 100-300 μm. Hence the background does not have a signiﬁcant eﬀect
on the expansion in this region. The background pressure is also low enough to avoid

15
absorption of EUV light produced through HHG. The gas jet parameters for several
gases and distances away from the nozzle are shown in ﬁgure 2.3.

2.4

Detection Chamber
The detection chamber consists of a Hettrick glancing-incidence EUV spectrome-ter and an Andor EUV camera. It is pumped by a smaller turbo pump and is typicallyat a pressure of < 10−6 torr. After the EUV light is produced at the gas jet, it leaves
the generation chamber and enters the detection chamber. First, it passes through asmall diﬀerential pumping aperture, which separates the detection chamber from thegeneration chamber. Next the EUV light passes through a 200 nm aluminum ﬁlter. The
purpose of the aluminum ﬁlter is to reject the fundamental light and prevent it fromreaching the EUV camera. The EUV light then passes through a glancing incidence
EUV spectrometer and the harmonic orders are spatially separated. Then the EUV
passed through a second 200 nm aluminum ﬁlter to further reject the fundamental lightand then is detected on the Andor EUV camera.
The Andor EUV camera is similar to a common CCD camera except for three
aspects. First, the CCD is mounted in a vacuum without any coating or glass cover, because any air, coating, or cover would absorb the EUV light before it could be detected.
Second, the CCD chip is cleaved oﬀ of the wafer and the back is lapped to be verythin (∼ 10 nm). The EUV light hits the CCD chip from the back and is transmitted
through the back-thinned chip and creates an electron-hole pair which is then detected.
The camera is also cooled to -60◦ C to minimize the thermal noise.
However, because the CCD can also detects the fundamental 800 nm light, it must
be separated from the EUV harmonics. When the fundamental light is allowed to hit
the camera it easily saturates the dynamic range of the CCD. Aluminum ﬁlters are commonly used to separate the high harmonic and the fundamental light. The fundamental
is reﬂected by the surface of the aluminum and the harmonics are transmitted through16

Characterization of gas jet properties
p0 = 1500 torr

150

N2, O2170

CO2, N2O
150
170

~12

~14

~12

~14

Quitting surface location
xq / mm
~8

~10

~18

~20

Translational temperature on axis in K
x/ m
100
~150
~180
200
~100
~110
300
~70
~80
400
~60
~60

~190
~110
~80
~70

~200
~140
~100
~80

Rotational temperature (in mol. flow region) in K
~0.1 * T0 = 30

~0.1 * T0 = 30d/ m
Mach disk locationx / cmDensity on axis in cm-3
x/ m
1 x 1019
100
3 x 1018
200
2 x 1018
300
9 x 1017
400

2 x 1019
5 x 1018
2 x 1018
1 x 1018

Pressure on axis in torr
x/ m
100
~200
200
~40
300
~10
400
~5

~300
~50
~15
~8

~190
~40
~12
~6

~250
~70
~18
~7

Off-axis half-widths in m
x/ m
100
~100
200
~150
300
~200
400
~250

~100
~150
~200
~250

~80
~140
~200
~300

~80
~140
~200
~300

1.3 x 1019 1.2 x 1019
3 x 1018
5 x 1018
1.2 x 1018 1.5 x 1018
0.9 x 1018 1 x 1018

Figure 2.3: Properties of the gas jet for several gases and distances away from thenozzle. These values were calculated using methods shown in Atomic and Molecular
Beam Methods [52].

17Figure 2.4: The transmission as a function of photon energy for EUV light through the
200 nm aluminum thin ﬁlm ﬁlter, which is used to reject the fundamental light. Mostof our experiments are limited to observing HHG emission in the range of ∼20-72 eV.

18
the thin foil of aluminum. The fraction of the light transmitted is shown in ﬁgure 2.4.
The aluminum ﬁlter passes photons with energy between 17 and 72 eV. In the systems
we studied, these energies are also those most abundantly produced by HHG. Higher
harmonic orders can be observed by using other metallic ﬁlters such as zirconium or
silver, which have transmission windows at high energies. It is necessary to use verythin 200 nm aluminum ﬁlters so a signiﬁcant fraction of the EUV light is transmitted;
however when ﬁlters are very thin, they often have small pinholes. Because of this, itis necessary to use two separate ﬁlters: one before and one after the spectrometer. Any
fundamental light which leaks through pinholes on the ﬁrst ﬁlter is allowed to diﬀract,and only a very small portion of this can then leak through any pinholes on the second
ﬁlter.
It is also important to mount the aluminum ﬁlters carefully. Being a thin 200 nm
foil up to 1 inch in diameter, the ﬁlters will break if they have too much diﬀerential airpressure on them when the chamber is pumped down. So the mount must allow air toeasily move around the ﬁlter but not allow light the leak around it. Ariel Paul[47] hasdesigned a baﬄe for this purpose, which conveniently ﬁts into a NW40 ﬂange.
No materials reﬂect EUV light at normal incidence with any signiﬁcant eﬃciency,
however at glancing incidence the fraction of reﬂected light becomes greater. Thus
glancing incidence reﬂections are typical used in EUV systems. The spectrometer used inthese experiments contains a gold mirror and a diﬀraction grating at glancing incidence.
The gold mirror is spherical; however at glancing incidence the eﬀective focal length in
the plane of incidence become very long and can be ignored[3]. At glancing incidence,the spherical mirror is eﬀectively a cylindrical mirror, which focuses the light in the
spectrally-dispersed dimension. The other dimension is allowed to diﬀract naturally.

19

2.5

Typical Raw Data
The CCD camera is 1024 by 256 pixels with 12 bit resolution; however, usually

the smaller dimension is integrated on the chip in ’full vertical binning mode’. A typical
spectrum from argon is shown in ﬁgure 2.5, where orders 19-45 can be seen. The
maximum energy/harmonic order is limited by the aluminum ﬁlter. In ﬁgure 2.5, thepeaks from the ﬁrst-order diﬀraction from the grating are labeled in red, and second
order diﬀraction is labeled in blue.
The harmonic order can be determined using two methods. First, there is a sharp
cutoﬀ in the aluminum transmission at 72.7 eV. The central wavelength of the laser is
790 nm so the last harmonic order visible below the absorption edge should be the 45th .
The harmonic orders can be counted down from the 45th . When a longer wavelength
fundamental ﬁeld of 805 nm is used, the 47th drops below the aluminum edge, however
the laser used in these experiments has a central wavelength closer to 790 nm.
The second method compares the 1st and 2nd order diﬀraction peaks. The same
spectrum must be represented in both diﬀraction orders and spaced according to thegrating equation d sin θ = nλ. This means the 46th harmonic observed in the secondorder of the grating diﬀraction emerges from the grating at the same angle as the 23rd
harmonic from the ﬁrst-order diﬀraction. We can use this property to determine theorder of the each harmonic in the spectrum. First, Label the highest harmonic observed
peak order p and the 12th peak in, order q. The 12th peak is used because that is where
the second order from the grating begins. Because the highest peak from the second
order is just below the qth harmonic,

p = 2q − 1
Additionally, the pth harmonic is the 11th harmonic above the qth so,

(2.1)

20

p = q + 2 × 11

(2.2)

These two equations can be solved to get p = 45 and q = 23. Then, the rest of
the harmonic orders can easily be labeled. Both methods give the same result.

2.6

Removing Cosmic Rays
Cosmic rays sometimes hit the CCD and cause counts on the pixels of the camera.

This can be a problem if these cosmic rays which obscure the data. The problem is acutewhen the CCD integration time is longer than 10 seconds in ’full vertical binning mode’.
However, the eﬀect of the cosmic rays can be eliminated by taking two identical spectra
one after another. Any diﬀerences between them are assumed to come from cosmic rays
hitting the CCD, and the diﬀerences are replaced by neighboring values.

2.7

Setting Focal Positions
The pump beam is focused such that the center of the focus is aligned with thecenter of the gas jet, and the most intense part of the focus is used to pump the molecularcoherence. However, the high harmonic beam is aligned such that it is focus slightly
before the gas jet, because this is where the most intense harmonics are created andthe short trajectories are selected through spatial phase matching[51]. Figure 2.6 shows
the high harmonic intensity from SF6 as a function of the high harmonic focal position.
The high harmonic intensity is most intense when the focus is ∼ 2 mm before the gas
jet.

2.8

Background Subtraction and Integration
In the analysis of the data, we would like to retrieve a number from the highharmonic spectrum which is proportional to the intensity of each harmonic order. As

21

Figure 2.5: A typical high harmonic spectrum generated by a 28 fs pulse in argon gas
with a density of 1018 cm-3 . The peak intensity of the pulse was ∼1014 W/cm2 . Lower
energy harmonics are on the right side of the graph and higher on the left side. The
harmonic orders are labeled in red for the ﬁrst order diﬀraction from the grating and inblue for the second order diﬀraction.

Figure 2.6: High harmonic intensity in SF6 is plotted against the position of the lensfocusing the HHG beam. The focus is aligned to the outlet of the gas jet at 6.5 mm;
however, HHG is most intense when the focus is 2 mm before the jet. This position also
selects the short trajectory harmonics[51].

22
can be seen in the argon spectrum in ﬁgure 2.5, the number of counts on the camera is notzero between the harmonic orders. This non-zero background must be subtracted before
each harmonic peak can be integrated. The background is due mainly to two factors:
thermal noise in the CCD and scattered EUV light. Because part of the backgroundcomes from the EUV light, the background level will change with overall EUV intensity.
So each spectrum has a unique background. The background is determined by drawing
a cubic spline through points midway between the harmonic peaks, and this backgroundis then subtracted from the spectrum. Once the background has been removed the peaks
are integrated to get the relative intensity of the harmonic order.

Chapter 3

Theory of Impulsively-Excited Molecular Rotational Wave Packets

The use of an optical ﬁeld to align a molecule is a powerful technique making it
possible to study how molecular orientation aﬀects many physical processes, including
HHG, as well as processes such as chemical reactions, photoelectron emission, and thetemporal variation of the index of refraction[55]. Optical ﬁelds can be used to alignmolecules in two regimes: static and dynamic alignment. In static alignment, theduration of the optical pulse is much longer than the rotational time constant and themolecules align adiabatically to the ﬁeld. In the dynamic case, the duration of the
optical pulse is much shorter than the rotational period and excites many rotational
levels, forming a rotational wave packet. As the rotational wave packet evolves in timethe molecules periodically align along the direction of the polarization and anti-alignperpendicular to the polarization. Typically using either method, an intensity around1013 W/cm2 is required to signiﬁcantly align a molecular sample with a rotationaltemperature of 100’s of Kelvin.
This chapter explains how molecules are dynamically aligned, speciﬁcally how therotational wave packet is excited, and the characteristics of the time-evolution of theinduced alignment.

24

3.1

Alignment After an Impulsive Torque
The impulsive pump excites rotational levels through a two-photon Raman pro-cess. The rotational levels are excited such that immediately after the pump pulse has
passed there is an initial alignment. This initial alignment can be understood in termsof a classical rigid rotor. The subsequent revivals are a result of the quantization of therotational energy levels and require a quantum mechanical description.
Starting with a classical rigid rotor, a torque can be applied to the rotor whichcauses the rotor to swing toward alignment. Consider a rotor with an initial angle θwith respect to the electric ﬁeld as shown in ﬁgure 3.1. We can ignore the azimuthal
angle φ, because the geometry is symmetric in that angle. Additionally, assume therotor is initially stationary, or more speciﬁcally assume that the impulse given by theelectric ﬁeld (torque multiplied by time) is much larger than the average thermal angularmomentum (moment of inertia multiplied by thermal angular velocity).
The electric ﬁeld induces a dipole in the molecule. The induced dipole then feelsa torque from the electric ﬁeld. The torque exerted by an electric ﬁeld is τ = μ × E
where μ is the induced dipole which is determined by the polarizability tensor, μ = α·E.
For a linear molecule the polarizability can be described by a polarizability parallel andperpendicular to the molecular axis.
⎛

⎞

⎜ α⊥
⎜
⎜
α=⎜
⎜ 0
⎜
⎝

0

0
α⊥
0

0 ⎟

⎟
⎟
0 ⎟
⎟
⎟
⎠

(3.1)

α

This polarizability tensor assumes the molecule lies along the z-axis. However,
we can rotate it through the polar angle θ.(Clockwise rotation around the y-axis.)

25

Figure 3.1: An impulsive pump pulse applies a torque to a linear molecule. The molecule
is kicked in the direction of the polarization of the pump pulse.

26
⎛
⎜ α
⎜
⎜
T
α(θ) = R αR = ⎜
⎜
⎜
⎝

⎞

− Δα cos2 θ0Δα sin 2θ

0

α⊥

0

Δα sin 2θ0(α⊥ + Δα cos2 θ)

⎟
⎟
⎟
⎟
⎟
⎟
⎠

(3.2)

where Δα = α − α⊥ . Now, we can use the rotated polarizability tensor todetermine the torque on the molecule. At this point, we’ll set the electric ﬁeld to bepolarized along the z-axis.

τ = μ × E = (α · E) × E = −Δα sin 2θEz2 ŷ(3.3)

The torque is zero when the molecule lies parallel or perpendicular to the electric
ﬁeld, and the torque is maximum at 45◦ . This formula for the torque can be inserted
into the angular version of Newton’s second law to get a diﬀerential equation for theclassical time evolution. This equation will tell us how a rotor with initial position θ0
will evolve with time, θ(t).

I

d2 θ(t)= −E 2 (t)Δα sin 2θ
dt2

(3.4)

Here I is the moment of inertia of the rotor. Dynamic alignment requires animpulsive torque. The time scale of the rotational motion isI
E 2 Δα so the electric ﬁeld

pulse must be shorter than this, so a delta function E 2 (t) = βδ(t) impulse is used. If
the rotor is initially not moving and is at an angle θ0 to the electric ﬁeld, then theexpression for the position after the pulse isθ(t) = θ0 −

βΔαsin 2θ0 t
I(3.5)

This equation relates the angular position of the rotor to its initial position. The
result is plotted in ﬁgure 3.2 for several time delays. At t = 0.6 (Green curve), there isan alignment because a large number of initial angles correspond to positions near zero.

27

βΔα

Figure 3.2: Classical calculation of impulsive alignment of CO2 . The molecular axis
position for several time delays after the initial impulse as a function of initial angular
position is shown. The initial alignment is seen a t=0.6.

28This shows there is an initial alignment shortly after an impulse torque. It is not useful
to extend this classical model further because is does not exhibit the revivals, which are
due to the discrete rotational energy levels.

3.2

Quantization of Rotational Energy Levels and Revivals
The rotational wave packet is periodic in its time evolution. This periodic natureis due to the wave packet’s composition from rotational eigenstates with discrete energies. Rotational eigenstate, |JM , has an energy BJ(J + 1). Here B is the rotational
constant of the molecule and deﬁnes the time scale of the rotational dynamics. The
2

rotational constant is inversely related to the moment of inertia: B = h̄2I . A rotational wave packet can be expressed in terms of the rotational eigenstates with complex
coeﬃcients, cJM , and energy-dependent phase accumulation.

ψ(t) =J,M

cJM e−iEJ t/h̄ |JM  =

cJM e−iBJ(J+1)t/h̄ |JM 

(3.6)

J,M

When no electric ﬁelds are present(H = BJ 2 ), the coeﬃcients are independentof time, and the dynamics comes from interference between rotation levels of diﬀerent
energies. If we consider a time t = πh̄
B , then the phase of each state is πJ(J +1). Because
J is restricted to positive integer values, the phase at this time is a multiple of 2π forall possible values of J. This means the wave packet at this time(and multiples of thistime) is the same as the wave packet at t = 0; the wave packet periodically repeats
itself. The time period between the repetition is called the revival time. Because there
is an initial alignment after an impulsive torque(as described in the previous section),
this initial alignment will repeat itself at the revival time. This is called the full revival.
At fractions of the revival time (1/2, 1/4, 1/8, . . .), there are fractional revivals during
which only a portion of the states are in phase.

29

3.3

Quantum Calculation of Rotational Wave Packet
Rotational wave packets are usually characterized by the alignment parameter< cos2 θ >; however, this is not a full description of the time evolution. In fact, somefractional revivals are not seen in < cos2 θ >, but are seen in full angular density. The
goal of calculating the rotational wave packet is to calculate the time evolution of thealignment parameter and the angular density. The calculation presented here follows
Ortigoso[45].
We start by deriving the Hamiltonian for a linear molecule in an optical ﬁeld. A
molecule interacts with light through the dipole operator.

1
H = H0 − μ · E
2

(3.7)

The dipole operator can be expanded as a Taylor series in the electric ﬁeld.

μ = μ0 (θ) + α(θ) · E + . . .

(3.8)

We have chosen the laboratory frame where the direction of the electric ﬁeld is constant,so the permanent dipole μ0 (θ) and polarizability α(θ) are functions of the molecular orientation. We only expand the dipole to the term which is linear in the ﬁeld because therotational alignment is caused by the induced dipole (the second term). Dropping highorder terms ignores higher-order non-linear eﬀects. Next, we put the Taylor expanded
dipole into the Hamiltonian.

11
H = H0 − μ0 · E + (α · E) · E
2
2

(3.9)

Now, we will restrict the electric ﬁeld to the optical frequencies, or frequencies
much larger than the rotational time scale, ω >> B/πh̄. The typical timescale formolecular rotations is picoseconds and an optical cycle is 1-2 femtoseconds. During an
optical cycle the position of the molecule is frozen. The permanent dipole term averages

30
to zero over the optical cycle because it is linear in the ﬁeld. However, the polarizability
term does not average to zero because it depends on the square of the electric ﬁeld andremains positive from one half cycle to the next.
We have already determined the polarizability for the linear molecule as a functionof the polar angle θ.
⎛
⎜ α
⎜
⎜
T
α(θ) = R αR = ⎜
⎜
⎜
⎝

⎞

− Δα cos2 θ0Δα sin 2θ

0

α⊥

0

Δα sin 2θ0(α⊥ + Δα cos2 θ)

⎟
⎟
⎟
⎟
⎟
⎟
⎠

(3.10)

Now, we will set the electric ﬁeld to be polarized in the z-direction and insert thepolarizability into the Hamiltonian. Additionally, the ﬁeld-free Hamiltonian is just therotational energy operator, BJ2 .
1
H = BJ2 + E2 (t) Δα cos2 θ + α⊥4

(3.11)

This Hamiltonian is most easily dealt with in the |JM  basis. In the matrix
form, JM |H| J  M   is non-zero only for J = J  or J = J  ± 2 and M = M  . The timedependent Schrödinger equation can be expressed as an inﬁnite number of diﬀerential
equations where each diﬀerential equation corresponds to a |JM  state. Because stateswith J > 40 are not typical populated thermally or excited, the equations for J > 40
can be neglected. The system of diﬀerential equations can be solved numerically.
The initial condition for this problem is a thermally populated ensemble of rotational states, which are populated in a Boltzmann manner and have incoherent phaserelationships. This can be modeled by calculating the time evolution many times andadding the results incoherently. The time evolution is calculated using each thermally
excited |JM  state as an initial condition. The initial condition is a pure |JM  state.
The observable such as < cos2 θ > or angular density ψ|ψ is calculated for each initialstate. The observable is then added incoherently weighted by the thermal populations.

31
The result is the thermally averaged alignment parameter or angular density.
There are several parameters speciﬁc to individual molecules that go into thesecalculations. The rotational constant deﬁnes the energy level spacing. The anisotropic
polarizability sets the strength of the interaction. Table 3.1 lists these parameters forsome small molecules.
Molecule
N2
O2
CO2
N2 O

Rotational Constant, B
1.99 cm−1
1.43 cm−1
0.39 cm−1
0.43 cm−1

anisotropic polarizability, Δα
0.7 Å3
1.1 Å3
2.1 Å3
2.8 Å3

Table 3.1: The rotational constant and anisotropic polarizability for several common
molecules[28].

3.4

Wave Packet Simulations
The alignment parameter < cos2 θ > is 1/3 for an isotropic, unexcited ensemble.

For an ensemble perfectly aligned along the z-axis, < cos2 θ >= 1. For an ensemble lying
in a plane perpendicular to the z-axis < cos2 θ >= 0. When the alignment parameter isgreater than 1/3 the sample is ’aligned’ and when the alignment parameter is less than
1/3 the sample is ’anti-aligned’, meaning that distribution lies in a plane perpendicular
to the polarization and has azimuthal symmetry.
The calculated alignment parameter for CO2 is shown in ﬁgure 3.3 a). The wave
packet is excited with a 140 fs pulse with a peak intensity of 5 × 1013 W/cm2 . The
pump pulse comes at t = 0, and after the pump pulse has passed there is an initialalignment. The initial alignment is repeated after one full rotational period (42.6 ps forCO2 ). There are also half and quarter revivals in CO2 . In between the revivals, thewave packet is not static. There is still a lot going on, but it is not visible in < cos2 θ >.
These smaller fractional revivals, eighth etc., can be seen in the time evolution of theangular density shown in ﬁgure 3.4. The angular density is generally a function of the

32

a)

0.7
0.6

<cos2θ>Full Revival

InitialAlignment

Baseline
Alignment

0.5
0.4
0.3
0.2
0.1
0

10

20

30

40

Time (ps)

b)

c)

0

0

30

330

60

300

120

210

150

60

300

90

270

240

30

330

90

270

240

120

210

150

180

180

Anti-Aligned

AlignedFigure 3.3: a) The alignment parameter < cos2 θ > shown as a function of time forCO2 . b) The weighted angular density at a time of anti-alignment. b) The weighted
angular density when aligned.

33
polar and azimuthal angles; however, a linear polarized optical ﬁeld can not change themagnetic quantum number m. Because of this, the wave packet retains the cylindricalsymmetry of the initial thermal ensemble. The angular density shown in ﬁgures 3.3 and3.4 is averaged over the azimuthal angle and weighted by the sine function from theangular diﬀerential, sin θdφdθ.
 2π

P (θ) =

0

Pf ull (θ, φ) sin θdφ(3.12)

Here Pf ull (θ, φ) is the full angular density and is has a functional dependence on boththe polar and azimuthal angles. However, in most cases the azimuthal dependence isunimportant due to the cylindrical symmetry of the wave packet. P (θ) the angulardensity with azimuthal integration already done. This somewhat more useful form ofthew angular density for visualizing the eﬀect of the molecular alignment.

3.5

Sweeping of the Angular Position During Alignment
When the molecule is aligned it cannot be thought of as having the molecular axis

parallel to the pump pulse polarization, as illustrated in the 3.5. The angular density
P (θ, φ) is maximized parallel to the polarization. However, the important quantity is thedensity at each polar angle, P (θ, φ) sin θdφ. The sine term in the azimuthal averaging
forces the weighted angular density to zero at θ = 0, and the angular distribution is abutterﬂy pattern, which peaks at some angle near zero but not at zero. As the alignment
pulse intensity is increased or the temperature is decreased, a higher degree of alignmentis achieved and the butterﬂy pattern peaks closer to zero.
In summary, during alignment the wave packet is localized around an angle nearzero. At times near the maximum alignment the wave packet is still roughly localized
and sweeps through angles until the maximum alignment is reached. Then the wave
packet sweeps through the angular positions in the opposite direction. This can be seen

180

160

140

120

100

80

60

40

20

0

5

10

15

25

Time (ps)

20

30

35

40

45

Figure 3.4: The angular density shown as a function of time for CO2 . The vertical axis is the polar angle with respect to the pumppolarization and the horizontal axis is the time evolution of the wave packet. One full period of the wave packet is shown. The initialalignment, 1/4, 1/2, 3/4, and full revival can be seen at 0, 11, 22, 33, 44 ps respectively. There are also a number of smaller features
between the revivals

Angle of Molecular Axis (degrees)

0

34

35

Figure 3.5: An illustration emphasizing that one need to consider the weighted alignment distribution with the azimuthal angle integration completed, as opposed to the full
angular density. The weighted alignment distribution shows that most of the molecules
are not pointed along the polarization axis, even at maximum alignment.

36
by looking at the angular density near the 1/2 revival in CO2 , shown in ﬁgure 3.6.

3.6

Molecular Alignment and Nuclear Spin Statistics
In molecules consisting of identical atoms, such as diatomics, the rotational wave

packet can vary signiﬁcantly from one molecule to the next. This diﬀerence is due tonuclear spin statics and how they aﬀect the populations of even and odd rotational
levels[29]. Consider N2 as an example. The nitrogen-14 nuclei have a spin of 1 and arebosons. Therefore the total molecular wave function must be symmetric upon exchangeof the nuclei. The wave function can be divided into electronic, rotational, and nuclear
spin parts.

ψtotal = ψe ψrot ψspin(3.13)The electronic part is symmetric because nitrogen has a σ orbital. The rotational
states are antisymmetric for odd J and symmetric for even J. Because the nuclear spin
is 1, the nuclei have 9 possible spin states: 3 are antisymmetric and 6 are symmetric.
All of these states are populated equally at experimental temperatures. Because thetotal wave function must be symmetric, symmetric spin functions are paired with even-J
rotational functions, and anti symmetric spin functions with odd-J rotational functions.
This pairing causes even-J rotational states to be populated in a 6:3 ratio compared to
odd-J states.
The rotational excitation has selection rules of J = J  + 0, ±2 and M = M  . So
the initial thermal/spin-weighted ratio of even to odd rotational states is maintainedin the wave packet. Molecules with diﬀerent nuclear spins and electronic symmetries
have diﬀerent ratios, shown in table 3.2. When calculating the rotational wave packet,
the initial thermal distribution of state is created with ratio of even and odd state
appropriate for the molecule. The ratio is maintained because the excitation does not

37

0
10
20

Angle (deg.)

30
40
50
60
70
80
90
20.5

20.75

21

21.25

21.5

21.75

22

Time (ps)Figure 3.6: The angular density near the 1/2 revival in CO2 . The red line shows where
the wave packet is localized and sweeps through angular positions.

38
mix even and odd rotational states.
Molecule
N2
O2
CO2
N2 O

even
2
0
1
1

odd
1
1
0
1

Table 3.2: The ratio of even and odd rotational state populations for several common
molecules.

The ratio of even and odd rotational states aﬀects the character of the wave
packet, as is shown in ﬁgure 3.7. The main diﬀerence is in the quarter revivals. The
N2 O molecule has no symmetry because the oxygen atom is on one side of the linear
molecule. Because of this lack of symmetry, even and odd states are populated equally.
The main eﬀect of this is that no quarter revivals appear in the time evolution of
the < cos2 θ >. Oxygen and carbon dioxide have inverted structures at the quarter
revivals, and nitrogen has quarter revivals which are reduced in magnitude. At thequarter revivals, even and odd states align separately and each alignment is 180◦ out ofphase. So, when the even and odd states are equally weighted there is no net alignment
and the < cos2 θ > remains unchanged; however, the quarter revival is still present in
the angular density.

3.7

Pump Intensity, Pump Duration, and Rotational Temperature
Typically, it is desirable to achieve the highest possible degree of alignment. The

obvious way of doing this is to increase the pump intensity, which increases the population in high rotational states which allows for a sharper alignment. However, increasing
the optical intensity too much causes other undesirable processes, such as multiphoton ionization, to turn-on. The energy content of the pump can also be increased bylengthening the duration of the pump pulse while keeping the intensity constant. The
duration can be increased so long as the process is impulsive. A rule of thumb is to set

39

Figure 3.7: The eﬀect of nuclear spin statistics is illustrated in the alignment parameter< cos2 θ > for N2 O, O2 , CO2 and N2 . The 1/4 revival can be reduced, inverted, or
ﬂipped depending on the ratio at which even and odd rotational states are populated.

40

×
×

θ

×
×
×
×

Figure 3.8: Alignment parameter for CO2 around the 1/2 revival as a function of pumpintensity. Increasing the pump intensity decreases the duration of the revival structure
and increases the peak alignment.

41
the pump pulse duration at 1% of the rotational period of the molecule. The initial rotational temperature also has an eﬀect of blurring out the alignment, so the maximum
degree of alignment can be increased by lowering the initial rotational temperature.
Commonly the intensity is high enough and the temperature low enough that the rotational states populated by the excitation are not already thermally populated. The
temperature and intensity also aﬀect the duration of the revivals as is shown in ﬁgures
3.8 and 3.9 . Higher temperatures and intensities lead to wave packets made of higher
energy rotational states, which narrow the duration of the revival structures.

42

Figure 3.9: Alignment parameter for CO2 around the 1/2 revival for several initialrotational temperatures. Decreasing the temperature increases the peak alignment.

Chapter 4

High Harmonic Generation from Rotational Wave Packets

One goal of using high harmonics to probe rotationally excited molecules is tounderstand how HHG emission varies with molecular orientation. Ideally, we would
like to be able to align the molecules perfectly in one direction and then vary thepolarization of the high harmonic generating ﬁeld. This would allow us to easily see how
the high harmonic emission varied with the molecule’s orientation. The resulting angulardependence of high harmonic emission (or orientational dipole) could then be compareddirectly with theoretical calculations, and used to extract structural information about
the molecule[30]. Unfortunately, because the molecules cannot be perfectly aligned,
the orientational dipole cannot be retrieved simply by rotating the polarization; theaveraging over the rotational distribution must be taken into account.

4.1

Averaging HHG Over Rotational States
Initially, we will consider a single harmonic order, q. The total high harmonic

emission from the rotational wave packet is weighted by the angular density of the wave
packet and integrated over all polar angles.
 π

Iq (t) = 

0

2


P (θ, t)Bq (θ)dθ (4.1)

The azimuthal integration has already been carried out using the cylindrical symmetry and is contained inside the angular density P (θ, t) along with the sinusoidal

44weight. The orientational dipole Bq (θ) describes the angular dependence of high harmonic emission for harmonic order, q. In general, the orientational dipole can be acomplex number and represents the amplitude and phase of the harmonic emission
from a molecule oriented at the corresponding angle. In the above equation, we haveignored any harmonic emission polarized perpendicular to the fundamental polarization;
this assumption is valid so long as the pump and probe polarization are parallel. Whenthe pump and probe polarizations are parallel, perpendicularly polarized harmonics are
not generated for symmetry reasons[23].

4.2

Intensity Measurements from Rotationally Excited CO2
This section will report the measured intensity from rotationally excited CO2 , andrelate the features in intensity modulation to features of the orientational dipole. CO2
is used for two main reasons. First, it is relatively easy to achieve a strong alignment
because it’s anisotropic polarizability is larger than other small molecules. Additionally,it has a ionization potential of 13.8 eV, which means it generates a broad range ofharmonic compared with lower ionization potential molecules. It is important to notethat the ability to resolve features in the orientational dipole is limited by the maximum
degree of alignment in the wave packet. Even with an alignment of 0.6 the CO2 molecule
is localized to only ±15◦ .
The measured intensity of the 39th harmonic is shown in ﬁgure 4.1a along withthe alignment parameter for the rotational wave packet in ﬁgure 4.1b. The harmonic
intensity is normalized to the harmonic intensity from isotropic CO2 . The main featureof the intensity modulation is the revival structure which occurs at the same delay timeas those in the alignment parameter. The revival structure in harmonic intensity is
anti-correlated with the alignment parameter. As the alignment parameter increases
the harmonic intensity decreases, and when the intensity is maximum the alignment issmallest. This anti-correlated property suggests that the orientational dipole for CO2

45

Figure 4.1: a) The measured intensity of the 39th harmonic as function of time delay
for CO2 . b) The calculated alignment parameter for this rotational wave packet. The
harmonic intensity is anti-correlated with the alignment.

46
can be approximated as 1− < cos2 θ > as a ﬁrst guess with a constant minus thealignment parameter. The negation will invert the revival structure in the alignment
parameter which will match the HHG intensity modulation. In this data the maximum
degree of alignment is ∼0.54.
However as we increase the intensity of the pump/alignment pulse (and consequently the degree of alignment) new features begin to appear. This can be seen in
4.2, where there are three new features: 1) A bump appears at the time of maximumalignment; 2) The intensity at anti-alignment becomes signiﬁcantly stronger (∼4×); and
3) The baseline intensity between revivals drops below the intensity corresponding to
an isotropic distribution.

4.3

Origin of the Bump at Maximal Alignment
The bump can be explained by looking at how the angular distribution changes

near the maximal alignment, as described in section 3.5 where the wave packet is localized, sweeps toward smaller angle and then returns. An important point is that theharmonic intensity goes almost to zero on both sides of the bump. There are two possible explanations for why the harmonic intensity would go to zero: 1) The wave packet iswell localized, and the orientational dipole has a minimum which is seen as the localized
wave packet sweeps through angular positions near maximal alignment. 2) There is a πphase shift in the orientational dipole, and the intensity goes to zero as the wave packet
sweeps through the phase shift. The observed minima in the modulation could be due
to either or both of these eﬀects.
This explanation is illustrated in ﬁgure 4.3. The ﬁgure shows the magnitude andphase of a possible orientational dipole for CO2 . There is a minimum in the magnitude
and a π phase jump around 30◦ . Also, shown is the angular density for a speciﬁctime delay. As the angular density evolves in time, it sweeps through 30◦ to maximumalignment and then sweeps back through 30◦ . The observed high harmonic intensity is

47

Figure 4.2: The intensity modulation of the 39th harmonic is shown around the 1/2
revival in CO2 . As the degree of alignment of the rotational wave packet is increased
several new features appear in the intensity modulation: 1) A bump appears at the
point of maximum alignment. Showing that with suﬃciently strong alignment the HHG
emission again begins to increase. 2) The intensity at anti-alignment gets signiﬁcantly
stronger (∼4×). 3) The baseline intensity between revivals drops below the isotropic
intensity.

48
the integral of orientational dipole weighted by the angular density. A sharp alignment
eﬀectively selects a small part of the orientational dipole under the angular density.
The minimum in the magnitude and the phase shift both could cause a minimum in theobserved intensity.

4.4

Origin of the Anti-Aligned PeakThe bump when the molecule is aligned tells us about the orientational dipole

at small angles (0◦ -45◦ ); similarly, the peak of the modulation when the molecule isanti-aligned gives information about larger angles (60◦ -90◦ ). As the power/intensityof the pump pulse is increased the peak near the anti-alignment becomes stronger andnarrower. The angular density also becomes more localized around 90◦ . The high harmonic intensity is increasingly enhanced with higher degrees of anti-alignment (smaller< cos2 θ > as is shown in ﬁgure 4.2. Because there are no features on the peak, this isindicative of the orientational dipole increasing towards 90◦ . If the orientational dipole
decreased near 90◦ , we would expect a small dip to appear at the top of the peak.

4.5

Origin of the Baseline Reduction
This third feature of a reduced baseline between revivals indicates a phase shift in

the orientational dipole. The angular density is not isotropic between revivals, because
the rotational excitation populates many higher J-states; however, the magnetic quantum number M is unchanged by the excitation and still reﬂects the thermal distribution
after the pump pulse. This leads to a weakly aligned state between the revivals. This is
the reason the < cos2 θ > is slightly larger than 1/3 between the revivals. Compared to
an isotropic distribution, the angular density between revivals is shifted toward smaller
angles as shown in ﬁgure 4.4. Between the revivals, the high harmonic intensity canbe dramatically decreased. For some orders, the high harmonic intensity is less thanhalf the isotropic intensity. How can a small shift toward smaller angle lead to a dra-

49

Figure 4.3: A cartoon graphic illustrating the origin of the “bump” during maximalalignment. The angular density sweeps through the minimum or phase shift causing aminimum in the high harmonic intensity. A second minimum is created as the angulardensity sweeps back in the opposite direction. There is a bump between the two minima.

50
matic drop in intensity? If the HHG where incoherent, this dramatic eﬀect would not
be possible. However, if there is a π phase jump in the orientational dipole, this dramatic decrease makes sense. The slight shift toward smaller angles weights each sideof the π phase jump more equally and contributions from each side of the phase jump
destructively interfere leading to the reduced intensity between revivals.

4.6

Measuring the Phase
The intensity modulations in CO2 are strongly suggestive of the π phase jumpin the orientational dipole. If this is the case the bump would correspond to harmonic
emission which is 180◦ out of phase compared with emission from other time-delays. It
is desirable to conﬁrm this by measuring the phase of the harmonic emission, and thiswas done using two diﬀerent methods. First, we used gas mixtures where an atomic gas
is used as a reference emission. And second, we directly measured the phase using ainterferometric double focus arrangement. One focus is used as a reference and directlyinterferes with emission from the other focus.

4.7

Using Gas Mixtures to Measure the Phase of Molecular HHG
To measure the phase we use a mixture of an atomic and a molecular gas. The

pump pulse will only rotationally excite the molecular gas. As the emission from themolecular gas evolves with the rotational wave packet, the emission from the atomicgas remains constant and can be used a reference to determine the relative phase of theemission from the molecular gas.
Consider a mixture of an atomic (Kr) and a molecular (CO2 ) gas. The emission
intensity of a single harmonic order (n) resulting from the mixture is then the coherent
sum of the emission from both species.

Iq (a) = |aDCO2 exp(iφCO2 ) + (1 − a)DKr |2

(4.2)

51

Figure 4.4: The angular density for the isotropic case (no excitation) (Black) and fortime-delays between revivals (Red). The distribution is shifted to smaller angles between
revivals.

5222+ (1 − a)2 DKr+ 2DCO2 DKr a(1 − a) cos φCO2
Iq (a) = a2 DCO
2

(4.3)

Here DCO2 and DKr are the magnitudes of the electric ﬁeld emitted from CO2
and krypton respectively, while φCO2 is the phase of the emission from CO2 relative
to krypton. The experimentally controllable parameter is the molar fraction of CO2 ,a. After exciting the rotational wave packet in CO2 , the magnitude (DCO2 ) and phase(φCO2 ) of the harmonic emission depends on the wave packet evolution and will varywith the time delay between the pump alignment pulse and probe harmonic-generation
pulse.
The total emission from CO2 is a coherent sum of dipole responses from molecules
distributed over many angular orientations. The orientational dipole response must
therefore be weighted by the rotational wave packet probability density to determine
the molecular contribution to the harmonic emission.
 π

DCO2 (t) exp(iφCO2 (t)) =0

B(θ)P (θ, t) sin θdθ

(4.4)

The rotational wave packet angular probability density is well understood and canbe calculated[45]. It is therefore possible to retrieve the orientational dipole from themeasured time-dependent harmonic emission from rotationally excited molecules andatomic gas mixtures. The orientational dipole B(θ) can be thought of as the complex
harmonic dipole from perfectly aligned CO2 . Harmonic generation from a rotationallyexcited wave packet combines emission from many orientations of the molecule. Therefore, measuring only the intensity of the harmonic signal is insuﬃcient to retrieve theorientational dipole B(θ).
Figure 4.5a shows the intensity of the 31st harmonic as a function of time delay
between the alignment and harmonic generating beams, for mixture ratios of 0%, 19.1%,
41.0%, 78.7% and 100% of CO2 with Kr. (We have switched to the 31st harmonic

θ

θ

53

Figure 4.5: a) The upper plot shows the calculated alignment parameter for the rotational wave packet. The lower plot shows the HHG intensity for the 31st harmonicfrom Kr:CO2 mixtures plotted against the time delay of the pump and probe pulses forseveral mixture ratios as stated in the legend. b) Same as above, however, expanded
near the half revival.

54
because for this dataset the 31st harmonic is less noisy than the 39th harmonic.) The
range of time delays in Fig. 4.5a corresponds to the ﬁrst three revivals of the rotational
wave packet in CO2 . The pure Kr curve (black) shows no noise because it was measured
from a single spectrum and does not vary with delay time. In the case of pure CO2 (redcurve), several additional features not seen in other experiments with CO2 are apparent,
particularly at the peak of the alignment (time delays where < cos2 θ > is maximized;
i.e. 21.25 ps delay) where the harmonic intensity increases slightly from the minima oneach side. This is due to the higher degree of alignment in this experiment (< cos2 θ >= 0.67) compared with previous experiments (< cos2 θ > = 0.6)[32][64] as explained
in section 4.3. As the fraction of CO2 is decreased, the amplitude of the modulation
decreases as expected; however, the shape of the modulation changes signiﬁcantly as
well. In pure CO2 (red curve) at the revival, the enhancement (with respect to thesignal between revivals) is three times as strong as the suppression. When the CO2
fraction is reduced to 19.1% (orange curve), both the enhancement and suppression
are of the same magnitude. This suggests that the EUV emission from CO2 changes
phase during the revival. This can be seen more clearly in Fig. 4.6a, which shows the
harmonic intensity as a function of CO2 fraction at two time-delays - corresponding tobest alignment (harmonic suppression, red curve) and best anti-alignment (harmonicenhancement, blue curve) near the half revival.
The data shown in Fig. 4.6 was ﬁt to Eqn. 4.3 using a least squares procedure.
The magnitude and the phase of the CO2 dipole were used as ﬁt parameters. Figure 4.6ashows the experimental data as squares, while the ﬁt to the data is shown as solid lines.
Because the magnitude of the dipoles of Kr and CO2 can be measured directly in a puregas, it is possible to subtract the ﬁrst and second terms in Eqn. 4.3 from the data, andinvestigate only the 3rd term that is responsible for interference. Figure 4.6b plots thesame data as in Fig. 4.6a with the ﬁrst and second terms subtracted and then normalizedwith respect to the magnitude of emission from each species. In this graph, constructive

55

Figure 4.6: a) The intensity of the 31st harmonic plotted against the fraction of CO2 in
the mixture at two time delays: 21.75 ps and 21.25 ps corresponding to the maximum
anti-alignment and alignment during the revival. b) The 3rd term of eqn. 4.3 plottedagainst the fraction of CO2 for the same two time delays.

56
interference between the Kr and CO2 harmonic emissions corresponds to positive values,
while destructive interferences correspond to negative values. The data clearly show thatwhen CO2 is aligned, the harmonic emission interferes destructively with that fromkrypton, while when anti-aligned, the interference is constructive. This least-squares
ﬁtting procedure was applied to all measured time-delays, and the magnitude and phaseof the harmonic emission is displayed in Fig. 4.7. We can see that when the CO2 isapproximately isotropically distributed (at time delays between revivals), the emission
from CO2 is 90◦ out of phase from that of Kr. However, when the CO2 is aligned, thetotal harmonic emission is suppressed, and the phase diﬀerence between the emissions
from the two species approaches 180◦ .
At any time delay, the harmonic emission from CO2 results from a superpositionof emissions from diﬀerent molecular orientations that are distributed in a rotational
wave packet. The total (rotationally averaged) emission can be calculated from Eqn.
4.4. We can apply a least squares ﬁtting procedure a second time to determine themagnitude and phase of the orientational dipole B(θ), which is the nonlinear harmonic
response from a perfectly-aligned molecule. To accomplish this, we expand B(θ) as aseries of Legendre polynomials -B(θ) = c0 + c1 cos2 θ + c2 sin2 2θ

(4.5)

where c0 , c1 , and c2 are the ﬁt parameters. Such an expansion has been used previously
in other experimental and theoretical work[21][31]. The angular distribution is neverparticularly sharp in impulsively excited rotational wave packets (the FWHM of theCO2 distribution is 15◦ at the maximum alignment and 60◦ at the maximum antialignment under our experimental conditions of a rotational temperature of 70 K, anda pump pulse peak intensity of 3.8 × 1013 W/cm2 with a duration of 140 fs FWHM.)
Therefore, we limited our expansion to three terms. The ﬁt parameters c0 , c1 , and57

θ

Figure 4.7: a) The magnitude (blue) and phase (red) of the dipole of CO2 retrieved
by ﬁtting the intensity from several mixtures to Eqn. 4.3. b) Same as above, however,magniﬁed near the half revival. The alignment parameter is also plotted for reference.

58

Figure 4.8: a) The magnitude of the orientational dipole retrieved by ﬁtting to Eqn.
4.4 plotted against the molecular orientation. b) The magnitude and phase of theorientational dipole retrieved from the 2nd ﬁtting procedure c) The magnitude of thetime dependent dipole and the ﬁt. d) The phase of the time-dependent dipole and thesubsequent ﬁt.

59
c2 are allowed to take on complex values because the orientational dipole B(θ) and thetotal time-dependent dipole from the ﬁrst ﬁt are complex, whereas the rotational angulardensity P (θ, t) is real and positive. This eﬀectively gives 6 real-valued ﬁt parameters.
The expansion in terms of Legendre polynomials requires that the orientational dipole
have mirror symmetry about the 0◦ and 90◦ axes - the same as the molecular structure.
Figure 4.9 shows the Legendre polynomial ﬁt to the data for the 31st harmonic order.
The retrieved orientational dipole B(θ) magnitude and phase are shown in Fig. 4.9 (a)and (b). The retrieved orientational dipole shows a π phase shift at 30◦ , corresponding
to a minimum in the magnitude of B(θ). The ﬁt to the magnitude and phase of therotationally averaged emission from the ﬁrst ﬁtting step is also shown in Fig. 4.8.
Our analysis up to this point has concentrated on the emission from the 31st
harmonic order. Ideally we would perform the same analysis for all observed harmonicorders. In practice, however, we are limited to harmonic orders that have suﬃcient
intensity to achieve a good signal-to-noise ratio. Figures 4.9 (a) and (b) show theresults of a ﬁt to the orientational dipole magnitude and phase for harmonic orders
21-35. In each harmonic order, the data show a minimum in the magnitude of B(θ),
and a corresponding phase shift at the position of the minimum. As the harmonic orderincreases, the angular position of the minimum in the orientational dipole B(θ) moves
from smaller angles to larger angles, while the depth of the minimum increases. The
size and sharpness of the phase shift also increase with harmonic order. The two highest
observed harmonic orders, 33 and 35, do not follow this trend as closely. The minima
seen in our data are in agreement with the results of Vozzi[64], who also observed aminimum around 30◦ in the 31st harmonic order intensity (the harmonic phase was notmeasured in that experiment).
The trends in the magnitude and phase of the CO2 orientational dipole B(θ) that
we extract from our data are similar to those predicted by the very simple and approximate two-center interference mode. This model assumes that the angular variation of

60

Figure 4.9: a) and b) The magnitude and phase of the orientational dipole retrieved
from the 2nd ﬁtting procedure for harmonic orders 21-35. c) and d) the orientational
dipole magnitude and phase predicted by the two-center emission model.

61high harmonic emission for small molecules is dominated by the recombination of theelectron with the molecule. This model was developed theoretically by Lein, et al[37]and explored experimentally by Kanai[32] and Vozzi[63]. In this model, the recombination step is modeled by approximating the returning electron as a plane wave. The
orbital into which it recombines is modeled as two centers that are symmetric or antisymmetric, depending on the symmetry of the molecule. If we assume the two centers
are Dirac delta functions, the angular variation of the orientational dipole magnitude
and phase can be calculated from Eqn. 4.6 and 4.7 (for the anti-symmetric case).





2πR

cos θ
|B(θ)| ∝ 1 − cos

λq



2πR
π
cos θ
φ(θ) = × sgn sin
2
λq

(4.6)



(4.7)

The distance between the two centers is R. In CO2 this corresponds to the distance
between the two oxygen atoms, 0.232 nm. The electron wavelength is directly related tothe harmonic order q by: λq = 2πh̄/ 2m(qh̄ω − Ip ) where m is the mass of an electron,
ω is the frequency of the fundamental, and Ip is the ionization potential of the molecule.
We have calculated the electron wavelength as described by Vozzi[64], and the results
are shown in Figs. 4.9 (c) and (d), respectively. An angular minimum and phase shift
ﬁrst appear for the 27th harmonic order, and increase in angle with harmonic order.
We note that in this analysis, we consider only the microscopic dipole, while neglecting the propagation phase of the harmonic light (e.g. phasematching eﬀects) andany harmonic emission from molecules that is polarized perpendicular to the fundamental driving laser. These are reasonable assumptions for short interaction lengths∼0.4 mm and low densities of ∼ 7 × 1017 molecules/cm3 . Moreover, we observe strong
destructive interferences in our data indicating that the harmonic emission from the Kr
and CO2 are polarized parallel to one another.

62

Figure 4.10: Setup for directly measuring the intensity and phase of high harmonic
emission from molecules. The two foci are 240 μm and 110 μm away from the gas jet
oriﬁce respectively. The diameter of the gas jet is 150 μm and the backing pressure is
700 torr. HHG from aligned and randomly oriented molecules from two diﬀerent regionsof a gas jet interfere in the far ﬁeld.

4.8

Direct Phase Measurement
We have also measured the phase of the EUV emission from CO2 . This was doneusing a double focus geometry as shown in ﬁgure 4.10. The high harmonic probe beamis split into two using two glass plates. The glass plates spatially separate the halves ofthe beam and each beam focuses to a separate spot on the gas jet. The high harmonic
emission from each spot then overlaps and interferes on the camera. The interference
pattern is along the non-spectral dimension of the spectrometer. A non-collinear pumpis used to pump only one of the high harmonic generation spots. Any phase diﬀerence
between the high harmonic emission from each spot is seen as a shift in the interference
pattern on the camera.
The interference patterns for the 27th and the 33rd harmonics are shown in ﬁgure
4.11. The pump-probe time delay corresponds to the 3/4 revival in CO2 . Figures 4.11
a and c show the interference pattern along the vertical axis and time delay along thehorizontal axis. A shift of the interference pattern is clearly seen for the 33rd harmonic
for time delays corresponding to maximal alignment. This phase shift is also exactly
π. No phase shift is seen in the 27th harmonic. Figures 4.11 b and d show lines of
the interference pattern for time delays corresponding to the phase shift, and show the63phase shift more clearly. No phase shift is seen in harmonic orders below the 29th while
those above the 29th exhibit a π phase shift.
Using this method the phase is measured relative to phase of the emission fromisotropic CO2 . In contrast using the gas mixtures, the phase was measured relative
to emission from krypton. This point is relevant only when looking at the phase as afunction of harmonic order. When we look at the phase as a function of time-delay in therotational wave packet or the phase of the orientational dipole for a speciﬁc harmonicorder, relative measurement will only add a constant term to the phase.
This π phase shift suggests that the emission from CO2 can be described as areal number instead of a complex number. A π phase shift is simply a change froma positive number to a negative number. Because the angular density is also a realpositive number, Eqn. 4.4 restricts the orientational dipole to real number as well. The
phase shift also occurs at the same time delays as the bump described in section 4.3,
which agrees with the explanation of the bump being due to a phase shift.
Additionally, if the orientational dipole is a real number and a π phase shift
is observed in the time-delayed signal, then the intensity must go to zero when thephase jumps. This is because a real number must pass through zero to change sign.
(A complex number can go around zero in the complex plane.) This is roughly inagreement with the experimental results shown in ﬁgure 4.2. There are minima at theplaces where the phase jump occurs and the minima are close to zero intensity. However
there is a measurable high harmonic intensity at these minima. This problem with themeasured intensity modulation interferes with the ﬁtting procedure used to extract theorientational dipole. We have tried these ﬁts and they don’t produce convincing results.
It seems the next step is to improve the measurement of the intensity modulations which
will lead to better ﬁt results.

64

Figure 4.11: (a), (c) Interference pattern as a function of time within the 3/4 revival
for harmonic orders 27 (a) and 33 (c). (b) Intensity-scaled integrated fringes for the
27th harmonic in the -100 fs to 100 fs interval (red squares), along with least square ﬁt(red solid line). Integrated fringes outside this temporal window are also shown (black
circles) as well as a least square ﬁt (black solid line). (d) Same as (b), but for the 33rd
harmonic. The time delay in (a) and (b) is relative to the center of the 3/4 revival.

65

4.9

Reconciling the Diﬀerences in Phase Measurements
The phase measured using the gas mixtures and the direct methods disagree

signiﬁcantly. Using the gas mixtures we saw a phase shift for every harmonic order andthe magnitude of the phase shift varied greatly where the direct method gave a π phaseshift for orders greater than 29. The phase shifts do occur at the same time-delays,
namely when the molecules are maximally aligned; however, the mixtures method alsosees some phase variation at the 1/4 revival during anti-alignment.
The direct measurement is more solid because it does not rely on intensity varying
quadratically with mixture ratio and the direct method gives the unique result of an
exact π phase shift. The phase shift would likely not be exactly π is there was asystematic error.
The variation of the intensity with mixture ratio is likely not completely describedby Eqn. 4.3. This can be be seen in the phase variation around the 1/4 revival in ﬁg. 4.5
where the ﬁtted phase variation has a similar shape to the intensity modulation. It is
possible the ﬁtted phase variation has been contaminated with the intensity modulation.
There are several possible sources of error in Eqn. 4.3. First, the supersonic
expansion of the gas jet changes signiﬁcantly as the mixture ratio is adjusted: mainly thedensity and the rotational temperature. Second, the probe spot size may not be smallenough to sample a uniform pump intensity. Then the high harmonic emission would
be averaged over a distribution of rotational wave packets. Third, the high harmonic
emission from a gas jet may not be completely coherent spatially. This incoherent part
would not participate in the interference. Fourth, the observed discrepancy could be
due to neglecting propagation and phase matching.

66

4.10

A Two Center Model of HHG from CO2
Lein numerically investigated HHG in H2 using an ab-initio calculation[37]. He

observed minima and phase shift in the high harmonic spectrum, and explained thesefeatures in terms of a two center emission model. First, it is assumed that the recombination process dominates the orientational dependence of HHG while the ionization
rate does not change signiﬁcantly with polar angle. The two center model then tells us
how the recombination amplitude varies with orientation of the molecule. The groundstate wave function is approximated as two delta functions separated by the internuclear
distance. These delta functions are either symmetric or antisymmetric depending on
the symmetry of the molecular HOMO. The incoming electron is modeled as a plane
wave.


ar =

(δ(x − R/2) ± δ(x + R/2)) x exp (ik · x) dx(4.8)

Here R is the internuclear distance and k is the propagation constant of theincoming electron plane wave. The dipole and the intensity for both the symmetric andantisymmetric cases are shown below.

dant (k) = i sin(0.5kR cos(θ))(4.9)

dsym (k) = cos(0.5kR cos(θ))(4.10)

Iant ∝

1 − cos(kR cos(θ))
2(4.11)

Isym ∝

1 + cos(kR cos(θ))
2(4.12)

67
This model shows intensity extrema at angles such that the internuclear axis isequal to the electron wavelength projected onto the internuclear axis. These extrema
are due the interference of the high harmonic emission from each center. The conditions
for maxima and minima are shown below:( for an antisymmetric orbital the conditions
are switched.) After passing through each minima there is an accompanying π phaseshift.

R cos θ = (2m + 1)λ/2(4.13)

R cos θ = mλ/2

(4.14)

Determining the electron wavelength from the harmonic order is somewhat uncertain. Lein’s calculations show that the eﬀective wavelength of the electron is its
wavelength at the bottom of the molecular potential, λq = 2πh̄/ 2m(qh̄ω). However,experiments by Vozzi, et al. [64] showed better agreement with the wavelength of theelectron in the continuum: λq = 2πh̄/ 2m(qh̄ω − Ip ).
This agreement was determined by looking at the high harmonic spectrum ofaligned CO2 . The CO2 aligns in the butterﬂy distribution and is mostly aligned at30◦ . The minimum in the spectrum corresponds to the ﬁrst destructive interference
minimum. Figure 4.12 shows such data from our experiment. There is a clear minimumin the aligned spectrum at 31st harmonic which agrees with the experiment by Vozzi[64]using the electron wavelength in the continuum. Additionally we have measured the highharmonic phase and we see no phase shifts for the 29th harmonic and lower; however,we see π phase shifts for the 31st harmonic and above.
An additional prediction of the two center model is that we will see the intensity
modulation ﬂip as we look at the emssion from higher harmonic orders. At an interference maximum the high harmonic intensity will be correlated with alignment instead

68

Figure 4.12: High harmonic spectra from aligned CO2 compared with HHG from anisotropic distribution. The black is the isotropic spectrum and the blue is the aligned
spectrum. Due to the butterﬂy shape of the angular distribution, the angular distribution can be approximated as a classical molecule aligned at 30◦ .

69
of anti-correlated. For CO2 aligned at 30◦ , the interference maximum is at the 79th
harmonic. We would expect the modulation to ﬂip somewhere between the minimum atthe 31st harmonic and the maximum at the 79th harmonic. Vozzi[64] claimed to haveseen this ﬂip at the 49th harmonic; however, the experimental noise makes the data lessthan convincing. We have measured the intensity modulations in CO2 for harmonic orders 13-55 and the modulation is always anti-correlated with the alignment parameteras shown in ﬁg. 4.13. The intensities at an anti-aligned time-delay in this ﬁgure are relative to the isotropic intensity, so intensities greater than 1 indicate an anti-correlation
with the alignment parameter. It is possible that a ﬂip in the modulation will appear
in harmonic orders closer to the 79th .

4.11

Outlook
Using HHG in combination with rotational alignment has been suggested as amethod of ultrafast molecular imaging. The ﬁrst goal would be to extract the a static
structure as has been done with N2 [30]. Second, this imaging could then be extended
to transient molecular structures such as a dissociating molecule[26]. It is currently notclear how the complexities in the molecular potential, ionization, and multi-electron
eﬀects can be overcome to achieve this goal. However, we have demonstrated the importance of ﬁrst obtaining an accurate measurement of the orientational dipole.
One avenue of further work in this area would be improving the measurementof the orientational dipole. Creating a wave packet with a sharper alignment would
allow us to see more features in the intensity modulation and hence extract a moredetailed orientational dipole. Excluding propagation eﬀects would make the ﬁtting
method better and allow for more trust in the retrieved orientational dipole. This could
be done by using a counter propagating blue pulse to create even harmonics and limiting
the interaction length. Decreasing the interaction length will minimize phasematchingand propagation eﬀects. Additionally, a more powerful laser system would allow the70

Pump/Alignment Power (mW)

Figure 4.13: Anti-aligned harmonic intensity relative to the isotropic intensity for several
pump powers stated in the legend. The graph shows harmonic orders higher and lower
than usual range of 19-47. The lower orders were measured using a diﬀraction grate withlower groove density. The high orders were measured using a zirconium ﬁlter instead ofaluminum. Peak enhancement > 1 indicates the modulation is anti-correlated with thealignment parameter.

71
pump pulse to have a larger spot size which would lead to more uniform wave packet
over the region which is being probed.

Chapter 5High Harmonic Generation from Vibrationally Excited Molecules

In this experiment, we demonstrate that it is possible to use high harmonic generation as a probe of internal dynamics in molecules. We impulsively excite[69][71][11] a
vibrational wave packet in SF6 using a short laser pulse, and observe oscillations in theextreme-ultraviolet high-order harmonic emission generated from these molecules using
an intense probe pulse. SF6 was chosen for being a spherically symmetric molecule thatcan be excited using the impulsive Raman interaction[68] while not being rotationallyexcited.
The impulsive excitation populates the ﬁrst vibrational level of each of the 3
Raman active vibrational modes of SF6 . All three modes are observed in the intensity
modulation of the high harmonic probe, in contrast to an optical probe where only thestrong symmetric stretch is observed. Furthermore, we observe rotational dephasing of
the vibrational modulations. First, we will discuss exciting vibrational wave packets, andthe vibrational modes of SF6 . Then, we present the intensity modulation observed in
excited SF6 , and a Fourier analysis of the modulation. This analysis is then comparedwith a similar optical experiment. Finally, we will discuss vibrational wave packets
observed in other molecules using HHG.

73

5.1

Exciting Vibrational Wave Packets
Vibrational modes can be excited using methods that we already employed for ro-tational excitation: both are excited through a two photon Raman processes. The main
diﬀerence is the spacing of the energy levels. Rotational levels are typically separatedby ∼ 1 cm−1 whereas vibrational levels are separated by 100-1000 cm−1 . Additionally,
there are a limited number of bound-state vibrational levels before reaching the dissociation limit, whereas rotational states as high as J ≈ 500[62] can be excited before thecentrifugal force pulls the molecule apart.
Because the vibrational energy levels are widely spaced, usually only the ﬁrst
vibrational state is signiﬁcantly populated in our experiments. This leads to a wave
packet whose evolution is much simpler than is the case for the rotational wave packet.
The vibrational wave packet consists of the ﬁrst excited state beating against the groundstate, leading to sinusoidal modulation of the mean vibrational coordinate.
Each vibrational mode has a Raman (two-photon accessible) and an infraredcharacter (one-photon accessible). The Raman character is due to the polarizabilitychanging with the vibrational coordinate, and the infrared character comes from thepermanent dipole changing with the vibrational coordinate. In molecules that are centrosymmetric (such as diatomics, CO2 or SF6 ), each vibrational mode is exclusively
Raman or infrared in nature. Infrared modes are excited resonantly with an infrared
electromagnetic ﬁeld, and Raman modes are excited non-resonantly by a two color Raman process. In the experiments discussed here, we are limited to exciting Raman
modes due to the non-resonant nature of our pump. The vibrational modes of SF6 areshown in ﬁgure 5.1. There are 3 Raman active modes, 2 infrared modes and one mode
which is forbidden. The forbidden mode changes neither the dipole nor the polarizability
signiﬁcantly.

74

υ1

775 cm-1
43 fs
Raman-active
very strong

υ2643 cm-1
52 fs
Raman-active
Doubly Degenerate
weak

υ3

948 cm-1
35 fs
Infrared-active
Triply Degeneratevery strong

υ4615 cm-1
54 fs
Infrared-active
Triply Degeneratevery strong

υ5525 cm-1
63 fs
Raman-active
Triply Degenerate
weak

υ6

351 cm-1
94 fs
Forbidden
Triply Degeneratevery weak

Figure 5.1: Normal modes of vibrations for SF6 . The wavenumber, period, degeneracy
and activity of each mode is stated[28]. SF6 has three Raman-active modes, two infrared
active modes and one forbidden mode (which are not excited in our experiment).

75

5.2

Classical Model
Molecular vibrational modes can be modeled as simple harmonic oscillators witha fundamental frequency equal to the vibrational frequency[9]. The optical ﬁeld drives
the vibration through an induced dipole. We start with Newton’s 2nd Law.

mq̈ = −kq + Fdrive
Here the fundamental frequency of the vibrational mode is ω0 =

(5.1)


k/m and q is

the vibrational coordinate. The driving force can be found using the potential of anelectric ﬁeld acting on a dipole.

V
Fdrive11
μ · E = ᾱ(q) · E · E
22
1 ∂ ᾱ(q)
∂V
=−
·E·E
= −
∂q
2 ∂q

=

(5.2)

Here ᾱ(q) is the polarizability tensor expressed as a function of the vibrational
coordinate. We have ignored the permanent dipole because this term will average to zero
when the electric ﬁeld oscillates at optical frequencies: ωoptical >> ω0 . This condition issatisﬁed with a near IR driving ﬁeld frequency of 13, 000cm−1 and the fastest molecularvibrations < 5000cm−1 . Next, we set the electric ﬁeld to be linearly polarized, and set
1 ∂ ᾱ(q). Now, the equation of motion becomesan interaction parameter B = mc
0 ∂qq̈ + ω02 q = −BI(t)(5.3)

where I(t) is the intensity envelope of the optical ﬁeld. This diﬀerential equation canbe solved in terms of sinusoids, with the boundary condition q(t = −∞) = 0.
q(t) =ti
exp(iω0 t)
dt I(t ) exp(iω0 t ) + c.c.
ω0
−∞

(5.4)

76
If we consider times after the driving ﬁeld has turned oﬀ, the integral becomes a
Fourier transform. From this it is evident that the eﬃciency of excitation is proportional
to the Fourier component of the driving envelope at the vibrational frequency. For
2example, we can consider a Gaussian driving pulse I(t) = I0 τF WTHM exp(−4 log 2 τF Wt HM ).
This will allow me to investigate the excitation of the vibration with respect to thedriving pulse duration τF W HM . We have deﬁned the amplitude of the driving pulse
such that the energy in the pulse is constant as the duration is varied.

BI0 T
q(t) = −
2ω0





τ2
π
ω2
exp − F W HM 0log 2
16 log 2

(5.5)

All of the dependence on pulse length is in the exponential and this is plotted inﬁgure 5.2. A driving pulse duration less than half of the vibrational period is needed toexcite the mode eﬃciently.
The vibrational excitation can be modeled quantum mechanically similarly tothe quantum mechanical model for the rotational excitation. This quantum mechanicalvibrational model is mainly useful for predicting the population in the ﬁrst excited state.

5.3

Observation of Vibrational Motion in SF6 using HHG
Figure 5.3 shows the intensity of the 39th harmonic generated in SF6 as a functionof pump-probe time delay. The red(black) curve shows the signal of the 39th harmonic
generated with(without) the presence of the pump pulse. When the pump and probe
pulses are overlapped, a coherent artifact appears due to the superposition of the twopulses. At positive delay times, the harmonic intensity is periodically modulated with
17 oscillation periods recorded. The amplitude of the oscillations decays as the delay
time is increased. At very small delay times, the baseline is lowered and then recovers
within 300fs.
Figure 5.4a shows a discrete Fourier transform (DFT) for the time interval from

77

τ

ω

π

τ

Figure 5.2: Relative excitation eﬃciency as a function of driving pulse duration relative
to vibrational period. A duration less than half the vibrational period is needed foreﬃcient excitation.

Figure 5.3: Red curve: Intensity of 39th harmonic generated from vibrationally excited
SF6 as a function of time delay between the pump pulse and the EUV generating probe
pulse. Black curve: high harmonic emission without the pump pulse present.

78

Figure 5.4: a) Discrete Fourier Transform of the 39th harmonic emission from Figure
5.3, showing the three Raman-active modes of SF6 that are excited by our impulsive
stimulated Raman scattering (ISRS) pump pulse. b) Stimulated anti-Stokes Raman
scattering of a 400 fs probe pulse centered at 400nm from SF6 , after excitation by
the same ISRS pump pulse used to excite vibrations in Fig. 5.3. c) Discrete Fourier
Transform of the harmonic emission data from Figure 5.3, for 0.3 ps time intervals
centered at diﬀerent times after the pump pulse (0.45 ps, 0.75 ps and 1.05 ps).

79
0.3 ps to 1.2 ps in the data in ﬁgure 5.3. The red curve shows the spectrum of themodulated high-harmonic signal, while the black curve corresponds to the DFT of thenoise background. The vertical axis corresponds to the peak-to-peak percent amplitude
modulation of the signal. Three peaks are visible at 525 cm-1 , 643 cm-1 , and 775 cm-1 ,corresponding to oscillation periods of 43 fs, 52 fs, and 63 fs, respectively. The Fourier
spectrum was normalized such that the spectral amplitude is the percentage modulationof the high harmonic signal in the time domain. These peaks can be assigned to thethree Raman-active vibrational modes of SF6 [28] ν 5 , ν 2 , and ν 1 , respectively. The
mode ν 1 is totally symmetric, whereas ν 2 and ν 5 are asymmetric and doubly andtriply degenerate, respectively.
The vibrational spectrum in ﬁgure 5.4a demonstrates that the HHG probe is sensitive to all three Raman-active vibrational modes in SF6 , regardless of their symmetry
or degeneracy. In contrast, when the eﬀect of the vibrational motion on the index ofrefraction of SF6 is studied, we could only observe the symmetric breathing mode, ν 1 .
The vibrational wave packet is created by the same pump pulse as in the HHG experiment; however, it is probed with a longer pulse of 400 fs duration and where the central
wavelength is 400 nm. The oscillations produce sidebands in the detected spectrum of
the probe pulse, as depicted in ﬁgure 5.4b. This result is consistent with earlier measurements by Bartels and Weinacht[11] as well as Korn[71] , where the latter used a 10
fs probe pulse.
The amplitude of the modulation in ﬁgure 5.3 is not constant in time. We studied
the time dependence of the amplitude by taking the DFT of the 39th harmonic at threetime intervals: 0.3 ps to 0.6 ps, 0.6ps to 0.9 ps, and 0.9 ps to 1.2 ps. The results areshown in ﬁgure 5.4c. Because of the shorter time interval, the resolution of the DFT iscoarse and only the modes ν 1 and ν 5 are visible. The amplitude of the symmetric modeν 1 is constant for all three time intervals; however, the amplitude of the asymmetricmode ν 5 decreases by half as the delay time is increased.

80

Figure 5.5: Peak to peak amplitude of the observed modulation for the three vibrational
peaks shown in Figure 5.4a, as a function of harmonic order; ν5 (red) , ν2 (green) , andν1 (blue).

81
For each of the three observed vibrational modes, ﬁgure 5.5 shows the modulationof the high harmonic signal as a function of harmonic order for orders 19 to 47, whichcorresponds to photon energies 30 eV to 75 eV. The amplitude of both asymmetricmodes ( ν 5 and ν 2 ) increases with harmonic order, with the signal of the mode ν 2
being above the noise level only for harmonic orders greater than 37. However, thesignal of symmetric vibration ν 1 shows no such trend.

5.4

Sensitivity to Vibrational Motion
There are three interesting and new aspects to the data presented in ﬁgures 5.3 -5.5. First, high harmonic generation as a probe of vibrational dynamics is sensitive tosmall-amplitude oscillations of all the Raman-active vibrational modes of the molecule.
(The other IR-active modes are not excited in this experiment.) Because the time scaleof the vibrations, at tens of femtoseconds period, is much longer than the sub-opticalcycle time scale of HHG, the nuclear motion is frozen during the HHG probing process.
Furthermore, the HHG process is sensitive to small changes in shape of the molecule,we estimate that the S-F coherent bond length change is approximately 5% based on a
10% population in the ﬁrst excited vibrational state of the symmetric mode, ν 1 . This
5% change in bond length increases the intensity of the HHG by 5%.

5.5Comparison with Other Experiments
A second interesting aspect of this work is that the observed intensities of thevibrational modes using high harmonic emission as a probe diﬀer from the spectra
obtained using either traditional Raman spectroscopy or stimulated Raman scattering
as a probe. In traditional Raman spectroscopy, the same three modes are observed andthe symmetric vibration, ν 1 , is most intense[17]. Using high harmonic emission as aprobe, all three modes have comparable amplitude; however, the asymmetric vibration,
ν 5 , is the most intense. Additionally, as we have shown in ﬁg. 5.4b, ISRS experiments

82

1.5

47
45

1.4

Harmonic Order

1.3

41
1.2

39
37

1.1

35
33

1

31

0.9

29
0.8

27
25

Normalized HH S ignal

43

0.7

23
-0.25

0.00

0.25

0.50

0.75

1.00

0.6

T ime (ps)1500

W avenumber (cm-1)

1300

20

1200
1100
1000

15

900
800
700

10

600
500
400

5

300
200
100

25

27

29

31

33

35

37

39

41

43

45

47

0

P eak-to-P eak Amplidute (% )1400

Harmonic Order

Figure 5.6: (Upper) High harmonic signal for harmonic orders 23 - 47 as a functionof time delay. The high harmonic signal without the pump pulse present has been
subtracted in order to show the modulations in the high harmonic signal for all harmonicorders. (Lower) Amplitude of the high harmonic modulation by the excited Ramanactive modes as a function of harmonic order. The amplitude of all modes signiﬁcantly
increases with harmonic order. The modulation of the high harmonic signal due to theν2 mode is above the noise level only for harmonic orders greater than 37.

83
have only observed the symmetric mode, ν 1 . This suggests that using high harmonic
emission as a probe is more sensitive to asymmetric modes than is spontaneous Raman
spectroscopy or ISRS.
There are several factors which inﬂuence the observed vibrational intensities. In
the impulsive excitation, the pulse length relative to the vibrational period as well as
the diﬀerential polarizability determines the population of the vibrational excited states.
The vibrational wave packet can modulate the harmonic intensity either through theionization rate[59] or the recombination integral[36]. The vibrational intensity varies
signiﬁcantly with harmonic order as can be seen in ﬁg. 5.5 . This is evidence that therecombination integral is important in determining the vibrational intensity, because achange in ionization rate would have the same eﬀect on all harmonic orders.
Using HHG as a probe of vibrational wave packets is quite sensitive in termsof the number of molecules needed to observe the vibrational spectrum. The column
number density is the number density of molecules multiplied by the interaction length.
For the HHG experiment it is greater than 1017 molecules/cm2 and for the spontaneousand ISRS experiments the column density is a factor of 1000 larger.

5.6

Decay of Oscillations
The third interesting aspect of the data is that the amplitude of the modulationsin the data of Figure 5.4 decay over the picosecond time window shown. To analyze
this decay, we applied a Fourier transform to the data at early times (0.3 ps to 0.6 ps),
middle times (0.6 ps to 0.9 ps), and at late times (0.9 ps to 1.2 ps) shown of ﬁgure
5.4c. We then compared the magnitude of the symmetric and asymmetric vibrationalmodes ν 1 and ν 5 in the three time windows. We found that the asymmetric mode
was signiﬁcantly stronger (x2) than the symmetric mode initially, but then decayed
to a comparable intensity by the late time interval. In contrast, the amplitude of thesymmetric mode was relatively unchanged. We speculate that rotational dephasing may

84
explain the rapid decay of the asymmetric mode. The excitation of the Raman modes isdone using linear polarized light. The polarized light excites a subset of the degenerate
set of modes corresponding to the orientation stretched in the direction of the laser
polarization. Subsequent random rotational motion of the molecules redistributes theorientation of this excitation, but only for asymmetric modes.

5.7

Calculation of Modulation Amplitudes
Zach Walters has worked to model the high harmonic generation process inmolecules, and has modeled the high harmonic intensity from SF6 [67]. The calculation involves expanding each step in high harmonic generation to the ﬁrst order in thevibrational coordinate. The product of the modulation amplitude for each step predicts the total intensity modulation of the high harmonic intensity. The calculation
also includes the eﬀect of the potential on the recombination amplitude. One aspectof the calculation is the connection between high harmonic intensity modulation andthe shifting of population between the ground and excited states. The same matrix
coeﬃcient in the high harmonic Hamiltonian causes both the intensity modulation andthe transfer of population between the vibrational states. This is illustrated in ﬁg. 5.7.
A comparison between the calculated modulation amplitudes and experimental
results is shown in ﬁg. 5.8. The modulation amplitudes agree to within an order ofmagnitude which is an encouraging result considering that the this theory does not take
into account macroscopic eﬀects, such as phasematching, absorption, etc.

5.8

Vibrations from Other Molecules
We have observed vibrational wave packets using HHG in two other molecules,

CClF3 and N2 O4 . In CClF3 all of the modes are Raman active due to the lack ofcentrosymmetric symmetry; however, we have observed only two of the Raman activemodes. The other modes are not visible most likely due to the noise being too large

85

a0

a1e-i
M
I
N

e-

R
d0

d1

Figure 5.7: A cartoon illustrating the steps involved in generating high harmonics froma system with two vibrational states. Every step can step can shift population between
the vibrational states.

868
Experiment 1
Experiment 2
Phase information included
Phase information excluded-1

T 2g mode, 525 cm

Peak-to-Peak Modulation-1

A 1g mode, 775 cm
0.1-1

E g mode, 642 cm
0.05

0
500

550

600

700
650
-1
Wavenumbers (cm )

750

800

Figure 5.8: A comparison of calculated vibrational amplitudes with experimentallymeasured amplitudes.

0.0

-1

e fundamental
a1 fundamental Cl37
a1 fundamental Cl35
e fundamental
a1 fundamental
a1 fundamental
e fundamental

Period (fs)
95
71
70
60
43
30
27

1.5

2.0

Spontaneous Raman spectrum of CClF3 Claassen, J. Chem. Phys. 22, 50

Wavenumber (cm )
350
X
468
X
476
X
560
X
781
X
1106
X
1217
X

1.0

Time delay (ps)0.5

Vibrational modes of CClF3

-0.5

0

200

400

600

800

1000

1200

0

2

4

6

8

10

200

400

800

1000

X2, 781 cm-1

1200

X3
475 cm-1
70.2 fs
non-degenerate

Wavenumber (cm-1)

600

X3, Cl35, 476 cm-1

X3, Cl37, 468 cm-1

X2
782 cm-1
42.6 fs
non-degenerate

Peak-to-Peak ampiltude (%)
41st Harmonic

1400

Figure 5.9: The intensity modulation and DFT of the 41st harmonic from vibrationally excited CClF3 . Two vibrational modes are visible
as well as the isotopic splitting of the ν3 mode.

41st Harmonic Intensity (arb.)

1400

87

88
and also because for some modes the pulse length is not suﬃciently short for a strong
excitation. We also have enough resolution to distinguish the isotopic variation in
frequency of ν3 . This is shown in ﬁg. 5.9. This mode is dominated by movement of thechlorine atom which is naturally abundant in two common isotopes, 35 Cl - 75.8% and
37 Cl - 24.2%.
N2 O4 is dimer of two NO2 molecules which forms at a temperature slightly
below room temperature. A bond between the two nitrogen atoms forms the dimer andis Raman active. The intensity modulation and frequency spectrum of the modulation
are shown in ﬁgure 5.10. The amplitude of the modulation is quite large when comparedwith the modulation observed in SF6 . This large modulation is due to two causes. First,
the excitation pulse is much shorter than the vibrational period of 128 fs, leading tomuch more eﬃcient excitation compared with the SF6 case. Second, there is evidence
that the vibration greatly increases the Franck-Condon overlap between the groundstate and the cation. This causes the vibration to modulate the ionization rate strongly
and consequently the high harmonic intensity.

5.9

OutlookThis work demonstrates the potential for a new kind of vibrational spectroscopy

with unique advantages. First, the high harmonic emission as a probe of internal dynamics in a molecule is sensitive to all Raman-active vibrational modes in a molecule,and it is possible that the infrared-active and Raman-active modes could be observed
simultaneously. This technique should thus be generally useful for observing dynamics in polyatomic molecules, and could be used to investigate energy redistribution
among vibrational modes in molecules, or coherences between vibrational modes. Second, this technique is useful for observing chemically important ground state dynamics
in molecules, and since the vibrations observed herein are small-amplitude motion, theutility of this technique to monitor (generally much larger) conformational changes inTime Delay

Harmonic Order

0

200

400

600

800

1000

1200

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0

0

100

200

2000

300

400

259 cm-1

3000

500

Time delay (fs)Wave number (cm-1)

1000

600

700

21S

4000

21S

Figure 5.10: The intensity modulation of the high harmonic signal in vibrationally excited N2 O4 .

N2O4 structure showing the observed mode

19

21

23

25

Normalized Flux
FFT amplitude

33
31
29
27

89

90
molecules is very likely. Third, HHG is a coherent ultrafast process that occurs on a
sub-cycle timescale. Thus, it should be possible to monitor ultrafast electronic redistribution within the molecule. And ﬁnally, by combining this technique with molecularalignment techniques, structural information about the molecule can be retrieved as afunction of time as both vibrational and rotational wave packets evolve, or as molecules
are highly excited and dissociate.

Chapter 6

Summary

In this work we have used HHG as a probe of molecular wave packets. We haveobserved rotational wave packets in molecules. Because CO2 can easily be aligned,
rotational wave packets in CO2 exhibit several unique features such as the bump at
maximum alignment. These features can be explained as a π phase shift in the HHG
emission, which was also measured experimentally. Furthermore, a ﬁtting method waspresented which extracts the complete orientational dipole from the modulated intensityand phase. The general features of the experimental orientational dipole are found to bein agreement with the two center emission model. Future experiments could extract amore detailed orientational dipole and show where the two-center emission model breaks
down.
Additionally, we have excited vibrational wave packets in SF6 . All of the Ramanactive modes of SF6 were observed, in contrast to a comparable experiment using an
optical probe. We also observed the rotational dephasing of non-spherically symmetricvibration modes in SF6 . The probing of vibrational wave packets using HHG has been
extended to additional molecules: CClF3 and N2 O4 where a very large modulation has
been observed.

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13(11):984–986, 1988.
[73] K. Yamane, Z. G. Zhang, K. Oka, R. Morita, M. Yamashita, and A. Suguro.
Optical pulse compression to 3.4 fs in the monocycle region by feedback phasecompensation. Optics Letters, 28(22):2258–2260, 2003.
[74] X. H. Zhang, A. L. Lytle, T. Popmintchev, X. B. Zhou, H. C. Kapteyn, M. M.
Murnane, and O. Cohen. Quasi-phase-matching and quantum-path control of highharmonic generation using counterpropagating light. Nature Physics, 3(4):270–275,
2007.

Appendix A

Lewenstein Model of High Harmonic Generation

The Lewenstein model is a semi-classical model of high harmonic generation that
captures the important details of the process in atoms, and is a good place to start a
discussion of high harmonic generation in molecules.
Let us start with the time-dependent Schrödinger equation. This formalism isdeveloped in the single active electron approximation and the interaction with the electromagnetic ﬁeld is limited to the dipole approximation.


i



1
∂
|Ψ(r, t) = − ∇2 + V (r) − E(t) · r |Ψ(r, t)
∂t
2

(A.1)

This analysis ignores any bound states above the ground state and ignores depletion of
the ground state. Furthermore we will be making the strong-ﬁeld approximation(SFA)
which assumes that when the electron is in the continuum it is not aﬀected by thepotential of the atom V (r). Another way of stating the SFA is 2Up > Ip .
Now, we divide the wavefunction into a part which stays in the ground state anda part that ionizes into the continuum. The part in the continuum is expressed in termsof momentum eigenstates.
|Ψ(r, t) = eiIp ta(t) |0 +3d vb(v, t) |v



(A.2)

Here |0 is the ground state of the atom and |v is a continuum state with momentum
v. We have also factored out the free oscillations of the ground state, eiIp t . Factoring99
out this oscillation can be thought of as setting the zero-energy to be the energy of theground state.
Now, we insert the wavefunction into the the Schrödinger equation. First, wework with the time derivative on the left hand side of the equation. Here, we ignore theground state depletion by a(t) ∼
= 1.

i

∂
|Ψ(r, t) = −Ip |Ψ(r, t) + ieiIp t∂t

d3 vḃ(v, t) |v

(A.3)

Now, we insert the wave function into the right side of the Schrödinger equation,
remembering that eiIp t |0 is an eigenstate of the Hamiltonian (without the electric ﬁeld),with energy −Ip . Similarly, |v is an eigenstate with energy v2 /2 + Ip . The addition of
Ip comes from the phase we have factored out of |v.

H |Ψ(r, t) = −Ip eiIp t a(t) |0+ E(t) · reiIp t a(t) |0


iIp t



3

d v

+e


iIp t

+e



v2
+ Ip b(v, t) |v
2d3 vb(v, t) |v

V (r)+ eiIp t E(t) · r

d3 vb(v, t) |v

(A.4)

Next, we simplify the 3rd term using the SFA. Basically, the potential only has
a zero-order eﬀect on the continuum state, V (r) |v = −Ip |v. Now the 1st and 4th
terms can be combined.

H |Ψ(r, t) = −Ip |Ψ(r, t) + eiIp t E(t) · ra(t) |0


iIp t

+e



3

d v

+ eiIp t E(t) · r





v2
+ Ip b(v, t) |v
2d3 vb(v, t) |v

(A.5)

100
Now put the left and right sides of the equation together and cancel terms.

id3 vḃ(v, t) |v = E(t) · ra(t) |0




+

3

d v

+E(t) · rv2
+ Ip b(v, t) |v
2d3 vb(v, t) |v

Now multiply both sides of the equation by(A.6)

 3  d v v | and express the position

operator r in momentum space, i∇v . And then, drop the integral with respect to v
from every term.
iḃ(v, t) = − v |E(t) · r| 0 ++ iE(t) ·v2+ Ip b(v, t)2d3 v v |∇v b(v, t)| v



(A.7)

Looking at the fourth term,

d3 v v |∇v b(v, t)| v









= (∇v b(v, t))

d3 v v | v+ b(v, t)

d3 v v |∇v | v

According to Sakurai pg. 54, eqn. 1.7.18,







(A.8)

 3  d v v |∇v | v = 0. So, the diﬀerential

equation for b(v, t) isv2+ Ip b(v, t) − E(t) · ∇v b(v, t) + i v |E(t) · r| 0ḃ(v, t) = −i
2

(A.9)

This diﬀerential equation is the Volkov equation. It can be solved exactly andthe solution is stated in the Lewenstein paper.

 tb(v, t) = i
× exp −i

0



 t
t



dt E(t ) · d v + A(t) − A(t )2(v + A(t) − A(t )) /2 + Ip



(A.10)

101
where d(v) = v |r| 0 and A(t) is the vector potential in the radiation gauge, E =−dA/dt.
The argument of the exponential is the semi-classical action,
 t

S=

t



dt (v + A(t) − A(t ))2 /2 + Ip



(A.11)The integral is a Feynman path integral, where S is the classical Lagrangian fora free particle in an electric ﬁeld. Note that the potential of the atom is approximatedas Ip . The classical electron trajectories associated with this Lagrangian are ∇v S = 0.
Now that we know the wave function for the ionized electron, we can calculate thetime-dependent dipole. The high harmonic emission’s electric ﬁeld will be proportional
to the second time-derivative of the dipole.

Ψ(r, t) |r| Ψ(r, t) = 0 |r| 0 +


+


+





d3 vd3 v b∗ (v , t) v |rb(v, t)| v

d3 v 0 |rb(v, t)| v
d3 vb∗ (v, t) v |r| 0

(A.12)

The ﬁrst term is zero. The second term will be neglected because it represents
continuum-continuum transitions not transitions back to the ground state. The thirdand fourth terms are complex conjugates.

ξ(t) = Ψ(r, t) |r| Ψ(r, t) ∼=

d3 vb∗ (v, t) v |r| 0 + c.c.

(A.13)

Or,

ξ(t) ∼=

d3 vb(v, t)d(v) + c.c.

Now, we insert the solution for b(v, t).

(A.14)

102



d3 v

ξ(t) = i t
0dt E(t ) · d(v + A(t) + A(t )) exp(−iS(v, t, t ))d∗ (v)(A.15)

Next, we introduce a change of variables, p = v + A(t). p is the canonicalmomentum and is a constant of the motion. However, it is not the physical momentumof the electron.

 tξ(t) = i0dt E(t ) ·d3 pd(p + A(t )) exp(−iS(p, t, t ))d∗ (p − A(t))

 t

S=

t



dt (p − A(t ))2 /2 + Ip

(A.16)

(A.17)

Next, we use the stationary-phase approximation to do the integral over p. The
integral over p has the following formd3 pf (p) exp(−iS(p))(A.18)

Because exp(iS(p)) oscillates rapidly, it averages to zero except near the extreme
or when ∇p S(p) = 0. So, the contributions to the integral are dominated by areas
around the roots of ∇p S(p) = 0. These roots deﬁne the classical electron trajectories.
The function S(p) can be expanded in a Taylor series around the roots ps .

1
S(p) ∼= S(ps ) + ∇2p S|ps (p − ps )22(A.19)

Now, we also assume that f (p) varies slowly compared to the phase of the exponential. This allows us to pull f (ps ) out of the integral and the integral becomesf (ps ) exp(−iS(ps ))1
d3 p exp(−i ∇2p S|ps (p − ps )2 )2

(A.20)

This integral in 3 dimensions can be converted to the product of 3 one dimensional
integrals.

103



f (ps ) exp(−iS(ps ))1
d3 px exp(−i ∇2p S|ps (px − psx )2 )21d3 py exp(−i ∇2p S|ps (py − psy )2 )21
3d pz exp(−i ∇2p S|ps (pz − psz )2 )2

(A.21)

Now we can perform a change of variables πζ 2 /2 = a(px − psx )2 for the integral.
πf (ps ) exp(iS(ps ))2∇p S|ps
This is now a Fresnel integral,3/2 

∞−∞3dζ exp(−iπζ 2 /2)(A.22)√∞22−∞ (cos(πx /2) − i sin(πx /2))dx = 1 − i = i

i.

Additionally, ∇2p S|ps = t − t . Inserting this, we getπi−if (ps ) exp(iS(ps ))t − t

3/2(A.23)

Now we can put the result of the integral into the equation for the dipole, ξ(t).

 t

ξ(t) =

0

dt



πi
t − t3/2

E(t ) · d(ps + A(t ))× exp(−iS(ps , t, t ))d∗ (ps − A(t))(A.24)

where ps is deﬁned by∇p S(ps , t, t ) = ttdt (ps − A(t )) = 0(A.25)

This is the main result: the dipole as a function of time in the generating pulse.
Taking the 2nd time derivative and Fourier transforming will give the high harmonic
spectrum for a particular pulse shape.
Usually the optical ﬁeld is sinusoidal in which case the integration over the ionization time can be carried out. This integral is simpliﬁed using the stationary phaseapproximation. For a sinusoidal ﬁeld there are two stationary phase points for every

104
cycle of the pulse. These two solutions are the long and short trajectories. Furthermore
due to the spatial spread of the continuum electron wave packet, only ionizion times
from the immediate cycle need to be considered. Using the stationary phase approximation reduces the integral to a sum with two terms: one for the long trajectory andone for the short trajectory.
Each term is made up of three complex amplitudes. First, the ionization rate,
where the ADK ionization rate is usually employed. Second is the propagation, which
determines the energy of the harmonic because the phase accumulation of the returning
electron determines how fast the dipole oscillates. Third is the recombination amplitude,
which in the plane wave approximation is a Fourier transform of the highest orbital andthe the dipole operator.

Appendix B

Self-compression of Ultrashort Pulses in an ionizing medium

The propagation of intense femtosecond duration light pulses in materials, gases
and plasmas is important for many areas of high ﬁeld science, such as high-harmonic
generation[40][39], laser-particle acceleration[61], and LIDAR based on atmosphericpropagation of laser beams[70]. Nonlinear propagation has also been employed extensively in ultrafast technology for temporal compression of pulses to even shorter
duration. Nonlinear self-phase modulation (SPM) experienced by a pulse as it travels
through a medium will spectrally broaden it. This broadened pulse bandwidth canthen be compressed by applying appropriate dispersion to the pulse subsequent to thespectral broadening. Typically, a pulse is self-phase modulated by propagation throughan optical ﬁber[58][43] or a gas-ﬁlled hollow optical waveguide[44], and dispersion compensation is provided by pairs of prisms or gratings, or using chirped mirrors to inducea negative chirp. These schemes have been employed to compress pulses to < 4 fs
pulse duration[73]; however, they are also relatively complex and lossy. Furthermore,
all previously demonstrated schemes are limited to compression of pulses at nanojoule
to sub-millijoule energy, limited by the characteristics of the ﬁber or waveguide propagation, and by the damage threshold of the nonlinear material. SPM-based techniques
also tend to generate pulses with large spectral modulations and complex nonlinear
chirps that are diﬃcult to compensate for, resulting in large temporal pedestals on thepulse. A variety of other schemes for pulse compression have been demonstrated or106
proposed; for example, phase modulation of a light pulse induced by molecular phasemodulation[71] [12] [54] using vibrational or rotational motion. However, these schemes
are also limited to laser intensities below the dissociation threshold of the molecule. The
use of ionization to blue-shift and phase modulate a pulse for compression has been previously considered [57] [4] [5]; however, this scheme also depended on the use of external
dispersion compensation, and is distinct from the mechanism seen in this work.
In this work, we discovered a new and unanticipated regime of ultrashort pulse
compression that operates at high laser intensities, where the pulse is ionizing the
medium[65]. We show that an intense 800 nm wavelength pulse propagating througha short gas-ﬁlled hollow waveguide, can reshape and self-compress in time without theneed for subsequent dispersion compensation. Theoretical models show a new mechanism for pulse compression where spatio-temporal reshaping of the pulse occurs byionization-induced spectral broadening, plasma-induced refraction, and guiding in thehollow waveguide. Therefore, this new technique is fundamentally diﬀerent from previous mechanisms that broaden the spectrum of a short pulse in a waveguide, and thensubsequently compress it using dispersion. This high ﬁeld pulse compression is uniquein its ability to shorten the duration of intense light pulses with high eﬃciency and withexcellent pulse ﬁdelity. Because this new mechanism can operate at intensities thatare signiﬁcantly higher than in other schemes, it may be scalable to much higher pulse
energies, with implications for the design of many high ﬁeld science experiments.
In our experiment, pulses from a Ti:sapphire ampliﬁer with ∼ 30 fs near transformlimited pulse duration and with energy of 2.2 mJ, at repetition rate of 1 kHz[6] werefocused into a 2.5 cm long, 150 μm inner-diameter, hollow waveguide ﬁlled with lowpressure argon. The spot diameter of the laser at the entrance to the waveguide was
50 μm FWHM. The energy throughput of the waveguide is ∼ 60% with no gas present,
adjusted for other losses. The focusing region before and after the waveguide was held
under vacuum to limit the interaction region and to maintain a constant pressure[49].

107
The pulse was characterized before and after the waveguide using second-harmonic
Frequency Resolved Optical Gating (SHG-FROG)[19]. All FROG measurements were
deconvolved using commercial software from Femtosoft Technologies (Version 3.03) toan error of < 1% at for a 256 × 256 grid. After exiting the waveguide, the pulse passes
through a 250 μm sapphire window and a 1 mm thick fused silica beamsplitter in the
FROG apparatus; the dispersion of these elements corresponds to an additional pulse
chirp that stretches the measured pulse duration of at-most 1-3 fs, and therefore does
not aﬀect our ﬁndings. Therefore, our pulsewidth measurements accurately determine
the pulse duration at the exit of the waveguide. No eﬀort was made to further optimize
the measured pulsewidths. If there is no gas in the waveguide, the duration of the pulse
going into the waveguide is equal to that after the waveguide to within our measurement
resolution.
The intensity in the waveguide, at approximately 1015 W cm-2 , is high enoughthat the argon gas is fully ionized on the leading edge of the pulse. This rapidlychanging ionization during the leading edge of the pulse results in an index of refractionthat decreases rapidly with time, causing an overall blue shift in the laser spectrum oftens of nanometers, which has been observed previously[72]. Figure B.1 shows the pulse
spectra at the exit of the waveguide with and without gas in the guide. Pressures as low
as 1 torr result in a signiﬁcant blue shift and broadening. As the pressure is increased,
the pulse width also decreases to a minimum value of 13.3 fs at pressures around 4 torr.
The compressed pulse width then remains relatively constant for increased pressures up
to 9 torr, at which point the laser mode begins to break up. Both the input pulse andthe compressed ﬁnal pulse have pulse width within 20% of the time-bandwidth product,as shown in Fig. B.2. Figure B.2(a) shows the temporal envelope of the pulse with andwithout gas in the waveguide. The initial pulse width of 28.9 fs has been dramaticallyreduced in duration to 13.3 fs at a pressure of 4 torr. The measured pulse spectra andthe deconvolved spectra from the FROG are shown in Fig. B.2(b), and correspond to a

108

1.1torr
1.9torr
2.8torr
3.9torr
650

700

750

800

850

900

Wavelength (nm)Figure B.1: Spectra of pulses emerging from a waveguide ﬁlled with argon at pressures
starting at 4 torr decreasing to 0 torr, front to back. Ionization induces a spectral
blueshift and broadening of the pulse.

109transform limit of 11 fs. The excellent agreement between the measured and retrieved
spectra indicates that the measurement has high accuracy and that it is self-consistent.
At a pressure of 4 torr, the energy loss from the compression process is only 5% of thetotal pulse energy i.e. the output energy drops by 5% at 4 torr, while at 8 torr theloss is 10%. We observe lower levels of compression (from 30 fs to 24 fs) using a larger
diameter, 230 μm, waveguide, and minimal amounts of compression by < 1 fs when the
waveguide is replaced by a large-diameter cell.
Conventional 1-D pulse propagation models, such as those used to explain selfphase modulation in optical ﬁbers, microstructured ﬁbers, or hollow waveguides, failto explain the observed compression phenomenon. In conventional pulse compression
techniques, spectral broadening is accomplished through self-phase modulation (SPM).
However, SPM alone broadens the pulse spectrum, but does not alter the temporalenvelope. Higher-order corrections due to self-steepening and other eﬀects can reshape
the trailing edge of the pulse, but will not shorten the pulse in time without subsequent
pulse compression. At high intensities however, signiﬁcant spectral and temporal eﬀects
are expected due to the rapid ionization on the leading edge of the pulse that leads to a
rapidly decreasing plasma index np =1 − (ωp2 /ω 2 ) where ωp is the plasma frequency

given by ωp2 = ne e2 /mo , e and m are the charge and mass of the electron, and ne
is the electron density. The rapidly-decreasing index due to the creation of a plasma
leads to a blue-shifting and broadening of the spectrum, that can then support a shorter
pulse[14]. However, this eﬀect alone cannot change the temporal envelope of the pulse.
Self-steepening will also occur due to the rapidly changing plasma index of refraction.
The sign of this self-steepening term is same as self-steepening in a normal material, andtherefore it also leads to a steep trailing edge on the pulse, but should not decrease pulse
duration. Table B.1 summarizes the potential 1-D pulse-shaping mechanisms and thecharacteristic propagation length over which these eﬀects becomes signiﬁcant. From
Table B.1, it is clear that the plasma-induced phase modulation and self-steepening

110
1.0
Output 13.3fs
Input
28.9fsIntensity

0.8
0.6

13.3 fs
0.4
0.2
0.0
-40

-20

0
Time (fs)

20

40

a)

MeasuredReconstructed

3000

Intensity

2500
2000
1500
1000
500
0
650

700

750

800

850

900

Wavelength (nm)b)

Figure B.2: Input and output pulse characteristics at the 4 torr optimum pressure
for compression of pulse with an input laser intensity of 1015 Wcm-2 . (a) Temporal
proﬁle and phase of the input pulse (29 fs) and the ﬁnal compressed pulse (13.3 fs),(b) Measured spectrum of the compressed pulse, compared with the spectrum of thepulse reconstructed from the time-domain second-harmonic FROG data. These data
demonstrate the ﬁdelity of the compressed pulse measurement.

111
will be dominant, with some contribution due to conventional SPM. However, no selfcompression is possible due to any of the eﬀects listed in Table B.1.
Mechanism
Waveguide GVD
Gas GVD
Plasma GVD
Gas SPMGas self-steepening
Plasma blue-shift
Plasma self-steepening

Length Scale
T 2 a2 ω 3
8 log(2)μ2mn c2πc2
T24 log(2)λ3 d2 n/dλ2
T 2 cω ω 2
4 log(2) ωp2
c
ωn2 I
cT
(g)
n2 I
2ωc
ωp2
2
2T c ωω2pDistance
1m
1.5 km
60 cm
10 cm
3m
2.3 mm
6.5 cmTable B.1: Table of possible pulse shaping mechanisms that involve only 1-D propagation eﬀects in Ar at a pressure of 4 torr (i.e. not spatio-temporal pulse evolution eﬀects)
and the length of propagation over which they become signiﬁcant[14]. T = pulse duration; GVD = group velocity dispersion; n2 = nonlinear index of Ar; n = linear index of
Ar; I = laser intensity; ω = laser frequency; c = speed of light; unm = waveguide mode
index; a = waveguide radius. The calculated gas terms are lower limits calculated forneutral Ar.

Since 1-D nonlinear wave propagation cannot explain this phenomenon, and since
experimentally we found the waveguide to be essential, we turned to a 3-D model toexplain the results. Propagation of the laser in the waveguide was modeled by a numerical solution of a 3-D scalar wave equation for the laser ﬁeld, as reported previously[14].
The model takes into account dispersion due to the transient plasma, power dissipation
due to ionization, and dispersion due to the neutral gas, at each radial position in theguide as a function of time. Also, the 3-D diﬀraction experienced by the beam in the
waveguide automatically includes the correct value of negative dispersion due to the
waveguide. Finally, waveguide loss was measured experimentally and incorporated into
the model.
The theoretically predicted pulse durations as a function of pressure are shownin Fig. B.3(a), together with the experimentally measured values for comparison (Fig.
B.3(b)). The model is in excellent agreement with experimental data, predicting that

112

9 torr
8 torr
6 torr
4 torr
2 torr0 torr
-40

-20

0

20

40

60

80 100 120

Time (fs)

9 torr
8 torr
6 torr
4 torr
2 torr0 torr
-40

-20

0

20

40

60

80 100 120

Time (fs)Figure B.3: Comparison between the experimentally observed (reconstructed FROG)
and theoretically predicted pulse envelopes as a function of pressure for 0, 2, 4, 6, 8
and 9 torr. (a) Experimentally measured pulse shapes (b) Theoretically predicted pulse
shapes. The pulse compression and pulse splitting behavior are seen both experimentallyand theoretically, with excellent qualitative and good quantitative agreement between
experiment and model.

113
the pulse undergoes temporal reshaping that reduces the pulse width from 30 fs toapproximately 13 fs with increasing pressure, and that the minimum pulse duration is
obtained around 4 torr. The leading edge of the pulse also steepens in time, accompanied
by a longer trailing edge that develops a shoulder as the pressure is increased- again
in very good agreement with experiment. The model also predicts less compression forlarger diameter waveguides and little or none for the case of no waveguide, as is thecase experimentally. Finally, the corresponding pulse spectral blue shift and increasein bandwidth is due to the combined eﬀects of rapidly changing plasma index and selfphase modulation, also in good agreement with experiment. At high pressures above ∼8 torr, the pulse splits and the spectrum develops a large long-wavelength wing, leading
to the breakup also observed in experiment. The only parameter in our calculations
that was adjusted to obtain agreement is the input diameter of the focused laser mode
into the waveguide.
From our models, we can identify the mechanism behind the self-compression as a
spatio-temporal interplay of plasma-induced refraction and blue shifting, diﬀraction of
the laser beam, phase modulation due to the rapidly-changing plasma index, and ﬁnally
guiding by the hollow waveguide. The role of plasma defocusing[48] is very importantin the compression process because it is responsible for the spatio-temporal couplingat high, ionizing, laser intensities. The presence of the waveguide is also crucial, bothexperimentally and theoretically. Our models show that as the pulse begins to ionize
the gas, the trailing edge of the pulse refracts from the center of the waveguide due toplasma-induced defocusing. However, this portion of the beam is then reﬂected back into
the guide by reﬂection from the waveguide walls. This leads to a temporally compressed
pulse with a sharp leading edge and ﬂat temporal phase, in dramatic contrast to otherpulse compression schemes. The reshaping also leads to a small periodic oscillation in
the beam diameter and pulsewidth as the pulse propagates down the waveguide. This
high-ﬁeld pulse reshaping process has similarities with passive mode locking of lasers

114
using saturable absorbers - the trailing edge of the pulse is slowly depleted periodicallyalong the waveguide due to ionization-induced defocusing, where the role of a laser
resonator is replaced by the waveguide. Both experimentally and theoretically, thepulse compression process is sensitive to the initial mode launched into the waveguide.
The laser beam must be focused to a tighter focal spot ( 50 μm FWHM) than ideal(∼ 100 μm FWHM) for coupling into the lowest-order mode of 150 μm waveguide, inorder to launch a mode that gives rise to large ionization levels and a high degree ofplasma-induced refraction.
In conclusion, we demonstrate a new and unanticipated mechanism for pulse
compression that operates at laser intensities above the ionization threshold for theﬁrst time. By propagating intense, femtosecond, pulses inside a hollow-core waveguideﬁlled with low-pressure argon gas, we demonstrate an increased bandwidth as wellas temporal self-compression of a pulse without the need for dispersion compensation.
Theoretical models show a new mechanism for pulse compression where spatio-temporal
reshaping of the pulse occurs by ionization-induced spectral broadening, plasma-induced
refraction, and guiding in the hollow waveguide. This result represents a signiﬁcantsimpliﬁcation over other pulse compression techniques at laser intensities above theionization thresholds of atoms and molecules, and will be useful in many applications
in high ﬁeld science. This mechanism is likely to scale to higher intensities and pulse
energies, and likely also to the use of plasma-based waveguides.