university of california los angeles spatial … thesis/phd thesis/1996...amit kumar lal doctor of...
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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