UNIVERSITY OF CALIFORNIA

Los Angeles

Spatial and Temporal Dynamics of Relativistic Plasma

Beat Waves

A dissertation submitted in partial satisfaction of the

requirements for the degree

Doctor of Philosophy in Electrical Engineering

by

Amit Kumar Lal

1996

© Copyright by

Amit Kumar La!

1996

The dissert at.ion of Amit Kumar Lal is approved.

V John Dawson

I(

Eli Yablonovitch

/; <,

L00r~Warren Mori

\'

/ I '".JJ J .) )·1 iii, l,~v- LWIG VI ~_--~

Chan Joshi, Committee Chair

University of California, Los Angeles

1996

11

DEDICATION

To my wife Rachna,

and my parents Ravi and Usha La!.

iii

Contents

DEDICATION

List of Figures

ACKNOWLEDGMENTS

VITA AND PUBLICATIONS

ABSTRACT

1 Introduction

1.1 Plasma Beat Waves.

1.2 Experimental Setup.

1.2.1 Laser ..

1.2.2 Plasma.

1.2.3 Electron Beam

1.2.4 Electron Diagnostics

1.2.5 Optical Diagnostics .

2 Electron Acceleration by Plasma Beat Waves

2.1 Cumulative Energy Spectrum of Accelerated Electrons

IV

HI

vii

xii

xiii

xvi

1

3

6

7

9

11

11

13

15

15

2.2 Single Shot Spectrum. 19

2.3 Cloud Chamber . . . . 23

2.4 Cherenkov Detector/PMT 25

2.5 Summary 25

3 Thomson Scattering from Relativistic Plasma Waves 26

3.1 Experimental Setup for Thomson Scattering 26

3.2 Time Resolved Thomson Scattering. 28

3.3 Space Resolved Thomson Scattering. 31

3.4 Discussion 35

3.5 Summary 40

4 Forward Scattering from a Relativistic Plasma Wave 41

4.1 Frequency Resolved, Time Integrated Results 42

4.1.1 Rotational Raman Signal. 44

4.1.2 Beat Wave Signals . . . . 46

4.2 Time and Frequency Resolved Forward Scattering 47

4.2.1 Experimental Setup for Time Resolved Forward Scatter. 48

4.2.2 Rotational Raman Data 50

4.2.3 Beat Wave Data 50

4.3 Summary .. . . . . . . 55

5 Back Scatter as a Plasma Diagnostic 56

5.1 Experimental Setup for Back Scatter 58

5.2 Experimental Results. 58

5.3 Summary 65

v

6 Evidence for Beat Heating of Plasmas via x-Ray Emission 66

6.1 Experimetal Setup 67

6.2 Results....... 71

6.2.1 Argon - Linear Polarization 71

6.2.2 Argon - Circular Polarization 72

6.2.3 Beat Wave Results 74

6.2.4 Metal Foils 77

6.3 Photons or Electrons? 79

6.4 Summary 80

7 Conclusions 82

A Laser Pulse Measurements 85

A.l Introduction. . . . . . . . . . . . . 85

A.2 Oscillator / 103 Pulse Measurements 88

A.3 Two Frequency Measurements. 91

A.4 Mars Laser Pulse Measurements 92

Bibliography 95

VI

List of Figures

1.1 Solution to 1-D Beat Wave Equation. . . . . . . . 4

1.2 1-D Model to Predict Wave Amplitude vs. Space. 5

1.3 Wave Amplitude vs. Space, based on the 1-D Model. 6

1.4 Overview of Experimental Setup. ... 7

1.5 Schematic of the MARS Laser System. 8

1.6 Plasma Production by Tunneling Ionization. 10

1.7 Electron Beam Line and Parameters. 12

1.8 Detailed Experimental Setup. . . . . 13

2.1 Cumulative Accelerated Electron Spectrum. 17

2.2 Schematic of Single Shot Electron Spectrometer 20

2.3 Photograph of Single Shot Spectrometer . . . . 21

2.4 Single Shot Electron Spectrum with 8 Surface Barrier Detectors. 22

2.5 Electron Tracks in Cloud Chamber. 24

3.1 Thomson Scattering Setup. . 27

3.2 (w,k) Spectrum. . __ . . . 29

3.3 Beat Wave resolved in time and frequency. 29

3.4 Spatial Calibration of Thomson Scattering System. 32

3.5 Plasma Wave Amplitude as a function of Space and Frequency. . 33

vii

3.6 Plasma Wave Amplitude as a function of Space and Time. 34

3.7 Maximum Energy Gain for an Injected Electron vs. Time. 36

3.8 Comparison between Experimental Data and Simulations in WAVE. 39

4.1 Theoretical Stokes power vs. plasma wave amplitude-length product. 43

4.2 Theoretical Electron energy gain vs. plasma wave amplitude-

length product. . . . . . . . 43

4.3 Time Integrated Forward Scattered Spectra. 48

4.5 Time Resolved Forward Scattering Setup . . 49

4.6 Time Resolved Forward Scattering Data in Vacuum. . 51

4.7 Time Resolved Forward Scattering Data from a Plasma Beat Wave. 52

4.8 Pump Lasers and Sidebands vs. Time. 54

4.9 Lineout of 4 Beams vs. Time ..... 54

5.1 k-rnatching Diagrams for Beat Wave and Back Scatter 56

5.2 Raman Back Scatter Frequency Spectra from the Gas Jet, for three

backing pressures. . . . . . . . . . . . . . . . . . . . . . . . . . .. 59

5.3 Back Scatter from a Static Fill of Hydrogen, Single Frequency Laser. 61

5.4 Back Scatter Frequency Spectrum from a Static Fill of Hydrogen,

Plasma Beat Wave. . . . . . . . . . . . . . . . . . . . . . . . . .. 62

5.5 Back Scatter Signal at 1O.9J.Lmvs. Signal on Surface Barrier De­

tector at 3.5 MeV. . . . . . . . . . . . . . . . . . . . . . . . . .. 63

5.6 Back Scatter Frequency Spectrum from a Static Fill of Hydrogen,

Two Frequency Laser, with No Plasma Beat Wave. . . . . . . .. 64

6.1 Laser Intensity Profile, and Theoretical Plasma Density and Ve-

locity Distributions for Hydrogen and Argon Plasmas. . . . . . .. 68

VIll

6.2 X-Ray CCD Camera. . . . .. 69

6.3 Theoretical Number of X-Ray Photons Detected by CCD 70

6.4 Split Pixel Correction. . . . . . . . . . . . . . 71

6.5 X-Ray Spectrum from an Argon Plasma, Laser Linearly Polarized. 72

6.6 X-Ray Spectrum from an Argon Plasma, Laser Circularly Polarized. 73

6.7 X-Ray Spectrum from a Hydrogen Plasma, 250 mTorr. 73

6.8 Transmission of beryllium windows. . . . . . . . 75

6.9 X-Ray Spectrum from an Beat Excited Plasma 76

6.10 Temperature Measurement from a Beat Wave Shot 76

6.11 Temperature Measurement from a Weaker Beat Wave Shot 77

6.12 X-Ray Spectrum from a Beat Wave Plasma, with Various Metal

Foils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78

A.l Transmitted Oscillator Energy vs. Hot CO2 Pressure at 440°C. . 87

A.2 Single Pass Absorption vs. Hot CO2 Pressure at 440°C. . . . 87

A.3 Oscillator Power vs. Time, at 35 Torr hot CO2 gas pressure. 88

A.4 Laser Pulse Duration vs. Hot CO2 Pressure ... 89

A.5 Peak Signal on Star 1 CCD vs. Hot CO2 Pressure 90

A.6 Peak Signal on Star 1 CCD vs. 103 Gain (or CO 2 Flow Rate) 90

A.7 Pulse Duration (FWHM) of Laser Pulse vs. 103 Gain. . 91

A.8 Pulse Duration (FWHM) of Laser Pulse vs. 103 Gain. . 92

A.9 Streak image of the Oscillator/l03 Output. . 93

A.lO Power of Mars Laser vs. Time.. . . . . . . . 94

IX

ACKNOWLEDGMENTS

Thanks to my advisor, Chan Joshi, for his guidance and his valuable help

with the experiments, theory, and publications. Thanks also to Ken Marsh and

Chris Clayton for their instruction and assistance in the Mars Lab, and to Warren

Mori for his help with theoretical problems. Thanks also to Frank Chen for many

useful discussions and insights. I also want to thank Maria Guerrero and Dorothy

Tarkington for their administrative help.

Many students have also helped a great deal in my research, including Matt

Everett, Dan Gordon, David Blackwell, Pepe Davis, Rob Brogle, and Ken Whar­

ton.

Finally, thanks to my wife Rachna and my parents for their patience and

encouragement during graduate school.

x

June 5, 1966

1988

1991

1989-96

VITA

Born, Huntsville, Alabama

B.S.E.E., Electrical Engineering, Princeton University

M.S., Electrical Engineering, University of California,

Los Angeles

Graduate Student Researcher, Departments of Physics

and Electrical Engineering, University of California, Los

Angeles

PUBLICATIONS

A.K. Lal et al., "Resonant Four Wave Mixing in Plasmas" , paper presented at

the 1990 American Physical Society, Division of Plasma Physics, Cincinnati,

Ohio, November, 1990.

A.K.Lal and C.Joshi, "Resonant Four Wave Mixing of Laser Radiation in

Plasmas", Journal of the Optical Society of America 8, 10, p.2148 (1991).

Xl

C.Joshi, Y.Kitigawa, and A.K.Lal, "Degenerate and Resonant Four Wave

Mixing in Plasmas", International Journal of Nonlinear Optical Physics 1, 1,

p.l (1992).

A.K.Lal et al., "Raman Backscatter from Tunnel Ionized Underdense Plasmas",

paper presented at the 1992 American Physical Society, Division of Plasma

Physics, Seattle, Washington, November, 1992.

C.E.Clayton et al., "Ultrahigh Gradient Acceleration of Injected Electrons by

Laser-Excited Relativistic Electron Plasma Waves", Physical Review Letters

70, 1, p.37 (1993).

A.K.Lal et al., "Studies of Single and Dual Frequency Laser Heating in Tunnel

Ionized Plasmas" , paper presented at the 1993 American Physical Society,

Division of Plasma Physics, St.Louis, Missouri, November, 1993.

A.K.Lal et al., "Experiments on Forward Brillouin Instability of Crossed Laser

Beams in an Underdense Plasma", paper presented at the 1994 American

Physical Society, Division of Plasma Physics, Minneapolis, Minnesota,

November, 1994.

M.J .Everett, A.K.Lal et al., "Trapped Electron Acceleration by a Laser-Driven

Relativistic Plasma Wave", Nature 368, 6471, p.527 (1994)

xii

C.E.Clayton, A.K.Lal et a!., "Acceleration and Scattering of Injected Electrons

in Plasma Beat Wave Accelerator Experiments", Physics of Plasmas 1, 5,

p.1753 (1994)

M.J.Everett, A.K.Lal et a!., "Evolution of Stimulated Raman into Stimulated

Compton Scattering of Laser Light via Wave Breaking of Plasma Waves",

Physical Review Letters 74, 8, p.1355 (1995)

M.J.Everett, A.K.Lal et a!., "Coupling Between High Frequency Plasma Waves

in Laser-Plasma Interactions", Physical Review Letters 74, 12, p.2236 (1995)

A.K.Lal et al., "Measurements of the Beatwave Dynamics in Time and Space" ,

Proceedings of the 1995 Particle Accelerator Conference, Dallas, Texas (1995)

A.K.Lal et a!., "Exact Forward Scattering of a CO2 Laser Beam from a

Relativistic Plasma Wave by Time Resolved Frequency Mixing in AgGaS2" , to

be published in Review of Scientific Instruments, 1997

A.K.Lal et al., "Transient Filamentation of a Laser Beam in a Thermal Force

Dominated Plasma" , to be published in Physical Review Letters, 1997

Xlll

ABSTRACT OF THE DISSERTATION

Spatial and Temporal Dynamics of Relativistic Plasma

Beat Waves

by

Amit Kumar Lal

Doctor of Philosophy in Electrical Engineering

University of California, Los Angeles, 1996

Professor Chan Joshi, Chair

Recent experiments at UCLA have demonstrated that relativistic plasma

waves, excited by two collinear electromagnetic waves beating at the plasma

frequency, can be used to accelerate electrons to high energies. The accelerating

electric field of this plasma beat wave is proportional to the plasma wave density

modulation, nJ!no, and the total energy gain is proportional to this wave ampli­

tude and the spatial length of the wave. In this dissertation, experimental results

are presented of the accelerated electron energy spectrum, and of the proper­

ties of the plasma beat wave. In particular, the wave amplitude as a function of

time, space, frequency, and wavenumber have been determined through Thomson

scattering and forward scattering, In addition, Raman/Compton back scattering

is developed as a diagnostic for both the plasma and the beat wave. Also, the

heating of the plasma by the beat wave has been measured by monitoring the

x-ray emission spectrum with a CCD camera used as a single photon counter.

XIV

Chapter 1

Introduction

Since the first electron accelerator was built in the 1930s, the maximum electron

energies have increased from 1 MeV up to 50 GeV. The maximum acceleration

gradient in conventional accelerators, however, is limited to 10-100 MeV[ia, due

to RF breakdown in the walls of the structure. Therefore, achieving higher elec­

tron energies requires larger and more expensive accelerators.

One method to overcome this limit of the acceleration gradient is to use a

plasma as the accelerating structure. Since the plasma is already broken down, it

can sustain much higher electromagnetic fields than conventional RF accelerators.

In 1979, Tajima and Dawsonj l] proposed several schemes to accelerate particles

from the longitudinal electric fields of relativistic plasma waves driven by intense

laser beams. One particular scheme, the plasma beat wave accelerator (PBWA),

utilizes two co-propagating laser beams such that the frequency difference of the

lasers matches the natural frequency wp of the plasma. Several groups around

the world have reported the generation of plasma beat waves [2, 3, 4, 5], as well

as successful electron acceleration [6, 7, 8J.

1

This thesis describes the latest developments in the PBWA experiments car­

ried out at UCLA under the supervision of Professor Chan Joshi. In particular,

several experiments are discussed that characterize the details of the electron ac­

celeration and plasma wave dynamics. Chapter 1 begins with an introduction to

relativistic plasma beat waves and the experimental setup. Chapter 2 describes

the successful acceleration of injected electrons up to 30 MeV, and the measure­

ment of the accelerated electron energy spectrum. A single shot, eight channel

spectrometer, has been developed and implemented.

In Chapter 3, Thomson scattering is used as a diagnostic for the relativistic

plasma beat waves responsible for electron acceleration. Thomson scattering is

used to determine the plasma beat wave amplitude as a function of time, fre­

quency, space, and wavenumber. When the plasma wave amplitude and phase

are known as a function of time and space, the expected energy gain of an injected

electron can be estimated. In Chapter 4, time and frequency resolved forward

scattering is used as a second independent diagnostic of the plasma wave ampli­

tude and length. The intensity of the forward scattered light shifted in frequency

by wp is proportional to (J nI/n odl )2, where nI/n o is the plasma wave amplitude

per unit length dl. Chapter 5 discusses the data from experiments on the back

scattered light from the plasma. The back scatter gives direct information about

non-relativistic plasma waves, such as those produced from Stimulated Raman

and Brillouin Scattering, and indirect information about the plasma beat wave

since coupling between the 'fast' (relativistic) and 'slow' (non-relativistic) waves

will appear in the back scatter. It has been shown previously[9] that the presence

of a large amplitude plasma wave results in the heating of the plasma electrons.

2

In Chapter 6, this heating is measured by monitoring the x-ray radiation spec­

trum from the plasma. Chapter 7 summarizes the results of the current UCLA

experiments, and discusses the implications on future PBWA work.

This thesis describes the continuation of the work carried out by the present

author that builds on the previous work of other students. The reader should

refer in particular to Ph.D. theses by W.P. Leemans[10] "Topics in High Intensity

Laser-Plasma Interactions" (UCLA, 1991) and M.J. Everett [9] "The Physics of

Beat-Excited Plasmas" (UCLA, 1994) for completeness.

1.1 Plasma Beat Waves

In the plasma beat wave accelerator (PBWA), two electromagnetic waves (wo,ko)

and (WI, kd excite an electrostatic plasma wave (W2, kz) that obeys the relations [11]

(1.1)

where the plasma density no has been chosen so that Wz equals the plasma fre­

quency wp = 41fnoe2/m. Here e is the electron charge, and m is the electron

mass. The phase velocity of the plasma wave is equal to the group velocity of the

light waves in the limit WO,l »w p . This ensures that the particles trapped in the

plasma wave will travel in near synchronism with the light pulses. For the UCLA

PBWA experiments, the laser frequencies are W o 21fC/l0.6/lm = 1.78 x 10 14 S-1

and WI = 21fc/lO.3/lm = 1.83 x 1014 S-1 while wp ~101Z s-\ so this requirement

is satisfied.

In Lagrangian coordinates, the evolution of the plasma wave can be modeled

3

by the 1-D differential equation[12]

(1.2)

Here I; is the electron displacement, 0<1,2 = eE 1,2/ mW1,2c are the normalized laser

intensities, ,6,w is the laser difference frequency, ,6,k is the laser wavenumber

difference, and the damping of the wave (which is on the timescale of its satura­

tion) has been neglected. A solution of this equation is shown in Figure 1.1, for

0<1 = 0.17, 0<2 = 0.07 (150 ps risetime, 300 ps FWHM), and assuming the initial

density is 10% above the theoretical resonant density (where wp equals W o - Wl),

with the density decreasing at the experimentally measured rate of 20% per 100

ps. (Section 3.2). The initial density must be set ~10% above the resonant den-

12040 80

Time (ps)

o.0 0 L-L~:c:::::::i_--l----l_-.J

o

0.50

0.25

oc-­~cCD

"0:::l:=:0..E«CD

~S

Figure 1.1: Solution to Equation 1.2 for the following parameters: 0<1 = 0.17,0<2 = 0.07, ninit=1.1 n res

sity to compensate for hydrodynamic expansion of the plasma and any plasma

blowout induced by the laser[13] or the plasma wave[14].

4

This predicts a wave reaching a peak amplitude of approximately 35%, and

reaching this peak before that of the laser pulses. This equation, which applies to

one point in space, can be applied to all points along the laser axis, treating each

point independently, but with a slightly different laser and density evolution due

to the laser propagation[7] (See Figure 1.2). For example, one Rayleigh length 2 0

o 100 200Time (ps)

0 1.2,.........,...-r-r--,.........,... ........

c:;c0.8o r m+-.m;.; ..m..;m.m...+==~-. 0.4 r-+---;-;---i;T"'­

QIc: 0.0 r-+--;e.-.

-o.4......................~J.....w....... .......,o 100 200Time (ps)

- -Laser Ium Densit?-n/n"

z=o.o/

o 100 200Time(ps)

Focusing ~

Figure 1.2: I-D Model to Predict Wave Amplitude vs. Space.

before the best focus of the laser, the peak laser intensity is reduced by a factor

of two, and the laser arrives about 30 ps earlier than it does at best focus. The

plasma formation and the beat wave evolution are determined from this laser

profile. A similar situation can be seen one Rayleigh length after best focus,

except that the events are occuring about 60 ps later. It is important to note

that although the wave amplitude in Figure 1.2 remains high for the duration of

the laser pulse (since there is no damping included), the phase coherence of the

5

i , - t=60ps

! I I - - t e toocsI

1 1

I \ I

!...'/I r-, \

i <,

wave is lost after the first saturation. Also, ion effects, which were not included

here, become important after a few ion plasma periods (lJpi "" 50 ps). Therefore,

only the data in approximately the first 100 ps is an accurate representation of

the beat wave evolution.

From this model, one can construct the wave amplitude as a function of space

and time, as shown in Figure 1.3. Early in time (t = 60 ps), the wave has not

o

"<,

"I:: 0.3w'g.w 0.2."~

j 01w>~ 0.0

-3 -2 -1 0 2 3Space (em)

Figure 1.3: Wave Amplitude vs. Space at t=60 and t=lOO, based on the I-DModel.

reached the best focus yet, and has a smaller amplitude due to the reduced laser

intensity. At t = 100 ps, the maximum wave amplitude has been reached, and

extends for ~l cm in space (FWHM).

1.2 Experimental Setup

The experiment consists of five main components (See Figure 1.4) : a) CO2 Laser,

b) Plasma, c) Electron Beam, d) Electron diagnostics, and e) Optical diagnostics.

6

Overviewof beatwaveexperiment• Experiment consists of five major components ...

1. Two-frequencylaser beam

\3. Electron

linac

~ ...JIZI

/Electron

lens

2. Plasma

5. Opticaldiagnostics

4. Electrondiagnostics

Figure 1.4: Overview of Experimental Setup

1.2.1 Laser

The CO 2 laser system (discussed in more detail in Appendix I) is designed to

produce a two frequency laser beam (10.6, 10.3 /.lm) with an energy of ~ 100J

and a pulse width of ~ 300 ps (FWHM). The laser system begins with a hybrid

CO 2 oscillator (See Figure 1.5). The high pressure oscillator (Lumonics 280, 1.5

ATM) produces a multimode pulse with an energy of approximately 200 mJ and

a pulsewidth of 150 ns (FWHM). The oscillator cavity is seeded by a low pressure

laser (22 Torr) to produce a single longitudinal mode. Also within the main cavity

is a small gas cell which can be filled by various absorber gases (SF6, Freon 115,

etc) to adjust the gain of the various laser lines (10.6 psu; 10.3 ux«, and 9.6 /.lm).

By adjusting the gas cell mixture and by tuning the cavity length with a small

piezoelectric crystal, it is possible to produce approximately equal output on a

pair of CO 2 lines (10.6 and 10.3 us«, for example).

7

MARS Laser System

(4) Sparker

(5)Hot C0 2Cell

(1) - Oscillator: 150 mJ, 200 ns, A= 10.6 11m,multi-mode pulse(2) - Low Pressure Laser: selects a single longitudinal mode(3) - Absorber Cell: used for two frequency laser operation(4) - Sparker: creates fast cutoff ( < 50 ps) of pulse(5) - Hot CO2 Cell : transmits only a short pulse (75 Ill, 100 ps)(6) - Pre-Amplifier: amplifies to 1 mJ, some pulse broadening(7) - MARS Amplifier: produces 150 ps rise, 300 ps FWHM pulse

1O.61lm: 60J (vase/c = 0.17), 10.3 11m: 20J (vase/c = 0.10)Spot Radius: 150 11m, Rayleigh Range 2zo= 1.3 ern

Figure 1.5: Schematic of the MARS Laser System

8

This long (150 ns), single longitudinal mode pulse is focused through a spark

gap, which is triggered at the peak of the laser pulse. When the spark gap

fires, the strong fields of the laser quickly produce a plasma, which reflects the

remainder of the laser pulse. The result is a pulse with a very rapid ("" 50 ps)

cutoff. The chopped pulse then triple passes a three meter cell containing 35

mT of CO2 gas at 400°C. Through the process known as free induction decay

(described in Appendix I), the 100 ns pulse is converted to a 100 ps pulse with

approximately the same peak power. The pulse then double passes a 1 ATM

TEA pre-amplifier (Lumonics 103), which increases the energy by a factor of 10­

20, and also gain broadens the pulsewidth to approximately 250 ps. The pulse

then triple passes the main MARS amplifier (2.3 ATM), which amplifies the pulse

up to 100 Joules with a FWHM of 350 ps. This pulse then enters the vacuum

chamber through a 8" NaCI window and is focused by a f/11.5 parabolic mirror

to a near diffraction limited 300 /-lm spot diameter at the center of the vacuum

chamber. This results in a peak laser intensity of 5xl0 14 W/cm 2, or a normalized

quiver velocity ex= eE/mwc = 0.2.

1.2.2 Plasma

The plasma is produced by the laser itself by tunneling ionization[15] of a static

fill of hydrogen gas (140 mTorr). Tunneling ionization occurs when the strong

electric fields of the laser suppress the atom's Coulomb potential barrier, and

allow the electron to tunnel free (See Figure 1.6). The evolution of the plasma

density n(t) as a function of time can be determined from the following rate

9

<p(X)= Coulomb potential...laser field ...unperturbed

...Iaser perturbed

~_V""':-~Tunnelledelectron

1.0

Relativedensity

0.0o

F

20 40 60 80

Time (psec)

Plasma Parameters: Gas: HydrogenDensity: 8.6 x 1015 cm'Plasma Period: 1.2 psPlasma Wavelength: 360 11mLorentz Factor: 34

Figure 1.6: Plasma Production by Tunneling Ionization.

equations:

dn(t) = w(t)(no - n(t)) (1.3)dt

w(t) = 4~t (;J(exp [~2 (;:) 3/2 (] , (1.4)

where no is the initial neutral gas density, ( = EalE(t), E; = 5.21 X 1011 V1mis

the atomic unit of electric field, E(t) is the applied electric field, E, is the ioniza­

tion potential of the atom, and Eh is the ionization potential of hydrogen. For

our laser parameters, assuming atomic rather molecular hydrogen, the plasma is

fully ionized within the first 20-30 ps of the laser pulse. From tunneling ionization

theory, the plasma has a transverse temperature of approximately 75 eV, and a

longitudinal temperature of a few eV [16, 17].

10