3
Journal Publications
1. Self-focusing of intense laser beam in magnetized plasma
Pallavi Jha, Rohit K. Mishra, Ajay K. Upadhyay and Gaurav Raj
Physics of Plasmas, 13, 103102 (2006)
Also published in ‘Virtual Journal of Ultrafast Science’, 5,
Issue 10 (2006).
2. Second harmonic generation in laser magnetized-plasma interaction
Pallavi Jha, Rohit K. Mishra, Gaurav Raj and Ajay K. Upadhyay
Physics of Plasmas, 14, 053107 (2007)
Also published in ‘Virtual Journal of Ultrafast Science’, 6,
Issue 5 (2007).
3. Spot-size evolution of laser beam propagating in plasma embedded in axial
magnetic field.
Pallavi Jha, Rohit K. Mishra, Ajay K. Upadhyay and Gaurav Raj
Physics of Plasmas, 14, 114504 (2007).
4
Conference proceedings
1. Interaction of laser pulses with magnetized plasma
Rohit K. Mishra, Ajay K. Upadhyay, Gaurav Raj and Pallavi Jha
Presented at ‘20th National Symposium on Plasma Science and
Technology’ Cochin (2005).
2. Modulation instability of a laser beam in a transversely magnetized plasma
Rohit K. Mishra, Ajay K. Upadhyay, Gaurav Raj and Pallavi Jha
Presented at ‘21st National Symposium on Plasma Science and
Technology’ Jaipur (2006).
3. Spot-size evolution in axially magnetized plasma
Rohit K. Mishra. Ajay K. Upadhyay, Gaurav Raj and Pallavi Jha
Presented at ‘6th National Laser Symposium’ Indore (2007).
4. Magnetic field detection via second harmonic generation
Rohit K. Mishra, Ram G. Singh and Pallavi Jha.
Presented at ‘22nd National Symposium on Plasma Science and
Technology’ Ahmedabad (2007).
5
Summary
The interaction of high power laser fields with ionized plasma is important
for many applications including laser fusion, laser wakefield acceleration, harmonic
generation and X-ray lasers. At high intensities the interaction between laser beams
and plasma becomes nonlinear. This leads to many interesting phenomena such as
self-focusing, wakefield generation, magnetic field generation and other parametric
instabilities.
The interaction of intense laser beams with magnetized plasma is an
important and relatively a new area of study. It has been experimentally and
theoretically shown that intense magnetic fields are generated when an intense laser
beam interacts with plasma. For example, in the fast ignition scheme in inertial
confinement fusion (ICF), quasi-static, self-generated magnetic fields are present in
the underdense corona region, close to the critical surface of the ignition pulse.
These fields affect the propagation characteristics of laser pulses since the
canonical momentum for magnetized plasma interacting with radiation is not
conserved as in the case of unmagnetized plasma.
The present thesis is devoted to a theoretical analysis of intense laser-
plasma interaction, in the presence of a uniform external magnetic field. The thesis
presents the effect of external magnetic fields on the self-focusing property of laser
beams. Modulation instability arising due to the propagation of the laser beam in
6
magnetized plasma has also been studied. Further, generation of second harmonic
frequency of the laser due to the presence of magnetic field has been shown.
Chapter 1 is devoted to the study of the basic properties of plasma and
conditions required for the existence of plasma. Plasma frequency has been defined
and the distinction between underdense, critically dense and overdense plasma has
been stated. A brief survey of theoretical, simulation and experimental studies for
various nonlinear phenomena has been presented. Regimes for nonrelativistic,
mildly relativistic and ultrarelativistic interactions have been defined.
Chapter 2 presents an analytical study of the evolution of the laser spot in
magnetized plasma. Self-focusing properties of (a) a linearly polarized laser beam
propagating in transversely magnetized plasma and (b) a circularly polarized laser
beam propagating in axially magnetized plasma, have been studied in detail. The
results are compared with the unmagnetized case. For both ((a) and (b)) cases,
expressions for laser spot-size have been obtained for a Gaussian laser profile,
using source dependent expansion (SDE) method. For a linearly polarized laser
beam the self-focusing property enhances in presence of the transverse magnetic
field while critical power required for self-focusing is reduced. Axial magnetic field
improves self-focusing property of a left circularly polarized beam, while the same
is reduced for a right circularly polarized laser.
Chapter 3 deals with the study of modulation instability of a circularly
polarized laser pulse propagating in axially magnetized plasma. Since the presence
of the magnetic field modifies the transverse current density, the modulation
7
instability of the laser pulse is expected to be affected. Growth rate of modulation
instability for left as well as right circularly polarized light have been studied in the
one-dimensional limit and compared with the unmagnetized case. Stability
boundaries for the left and right circularly polarized light have also been
graphically obtained. It has been shown that axial magnetization increases
(decreases) the growth rate of modulation instability for a left (right) circularly
polarized beam.
In Chapter 4, the possibility of second harmonic generation when a linearly
polarized laser beam propagates in homogeneous plasma in the presence of a
transverse magnetic field has been pointed out. Earlier workers have shown second
harmonic generation due to laser beams propagating in inhomogeneous plasma.
The present study shows that an intense, linearly polarized laser beam interacting
with homogeneous plasma embedded in a transverse magnetic field, sets up
transverse current density, oscillating with a frequency twice that of the laser field.
This current density oscillation leads to second harmonic generation. Linear
fundamental and second harmonic dispersion relations have been derived.
Expression for conversion efficiency has been obtained and graphically analyzed. It
has been shown that maximum conversion efficiency increases with the applied
magnetic field.
Conclusions from the present research and recommendations for future
work are given in Chapter 5.
8
List of figures
Page
Fig 1.1: Converging of laser wavefront along the propagation direction
and refractive index variation.
Fig 1.2: Normalized laser spot-size 0rrs against normalized
propagation distance RZz in vacuum (dashed curve) and in
the presence of plasma (solid curve) for 2.020 a with
mp 15 and mr 200 .
Fig 1.3: Growth rate of modulation instability of a laser beam
propagating in plasma for 141088.1 p s-1, mr 150 and
150 1088.1 s-1 considering finite pulse length effects.
.
Fig 1.4: Growth rate of modulation instability of a laser beam propagating
in plasma for 141088.1 p s-1, mr 150 and
150 1088.1 s-1, neglecting finite pulse length effects.
15
23
29
30
9
Fig 2.1: Variation of 0rrs with RZz for (a) unmagnetized plasma (b)
0c = 0.2 and (c) 0c = 0.4, with 271.00 a ,
150 1088.1 s-1 and 0 p = 0.1.
Fig 2.2: Variation of 0rrs with 0c at RZz = 0.3 for 0a 0.271,
150 1088.1 s-1 and 0 p = 0.1.
Fig 2.3: Variation of cmP with 0c for 271.00 a , 150 1088.1 s-1 and
0 p = 0.1.
Fig 2.4: Variation of 0rrs with RZz for (a) 0c =0, (b) 0c =
0.15; =-1 and (c) 0c =0.15; =+1 with 271.00 a and
150 1088.1 s-1.
Fig 2.5: Variation of 0rrs with 0c for right circularly polarized laser
beam having RZz = 0.3, 0a 0.271, 150 1088.1 s-1 and
0 p = 0.1.
63
64
65
73
74
10
Fig 2.6: Variation of 0rrs with 0c for left circularly polarized laser
beam having RZz = 0.3, 0a 0.271, 150 1088.1 s-1 and
0 p = 0.1.
Fig 3.1: Variation of modulation instability growth rate for right (curve a),
and left (curve c) circularly polarized laser beam propagating in
magnetized plasma and for laser beam propagating in
unmagnetized (curve b) plasma, with normalized wave number
with mr 150 , 271.00 a , 1150 1088.1 s , 1.00 p
and 05.00 c (curves a and b).
Fig 3.2: Stability boundary curves showing the variation of normalized
laser power mmP 20ˆˆ with k for right (curve a), left (curve c)
circularly polarized laser beam propagating in magnetized
plasma and unmagnetized case (curve b). The parameters used
are 1150 1088.1 s , 271.00 a , 1.00 p and
05.00 c .
75
88
89
11
Fig. 4.1: Variation of conversion efficiency ( ) with the propagation
distance z, for 0c = 0.1= 0 p , 21a =0.09 and 0 = 1.88
×1015 s-1.
Fig 4.2: Variation of maximum conversion efficiency ( max ) with
0c for 0 p = 0.1, 21a = 0.09 and 0 = 1.88 ×1015 s-1.
101
102
12
CHAPTER 1
INTRODUCTION
Charged particles in an ionized gas exhibit plasma like behaviour when the
linear dimension occupied by the gas is large compared to the Debye length
210
24 neTkD B . Here Bk is the Boltzman constant, T is the absolute
temperature of the plasma, 0n is the plasma electron density and -e is the electronic
charge. Another criterion for the existence of plasma is that the number of electrons
within a sphere of radius equal to the Debye length must be greater than unity
1)3/4( 03 nD . The third criterion for the existence of plasma is its quasi-
neutrality which implies that the ion density must be equal to the electron density.
The above conditions hold for steady state plasma. However these steady
state conditions are not sufficient to represent collective plasma motion. One of the
most important aspects of collective motion is bulk oscillation of plasma electrons
with respect to the ions. The plasma electrons may be expected to oscillate about
the much more massive ions under the collective restoring force provided by the
ion-electron Coulomb attraction. The collective oscillations are damped due to
collisions between electrons and ions. When the collision frequency is less than the
plasma frequency
2100
24 mnep , 0m being the rest mass of the plasma
electron, the plasma is said to be weakly coupled and collisions do not interfere
13
seriously with the plasma oscillations. On the other hand, in strongly coupled
plasma, the collision frequency is greater than the plasma frequency and collisions
effectively prevent plasma oscillations.
When an intense laser beam propagates through collisionless plasma, a
number of interesting, nonlinear phenomena occur. The propagation of the laser
light through pre-ionized plasma is governed by the dispersion relation
220
20
2pkc , where 0 and 0k are free space laser frequency and propagation
constant respectively. Interaction of laser beams with plasma depends on the
relative values of the laser and plasma frequency. For studying the interaction
processes, three categories have been defined: underdense 0 p , critically
dense 0 p and overdense 0 p plasma. Depending on the relative values
of p and 0 the propagation constant will be real, zero or imaginary.
In the presence of a laser field, a plasma electron oscillates with a quiver
velocity which depends on the amplitude ( 0E ) of the laser electric field.
Relativistic effects come into play when the quiver velocity tends to become equal
to the velocity of light. The dimensionless amplitude 0a 000 cmeE serves as a
parameter which determines the strength of interaction. In terms of the laser
intensity 0I and wavelength , the laser strength parameter is given by
20
100 10544.8 cmWIma . Depending on the value of 0a the laser-
plasma interaction may be non-relativistic 10 a , mildly relativistic 10 a or
14
ultra-relativistic 10 a . The basic mechanism of intense laser-plasma interaction
(in the relativistic regime) involves a number of nonlinear processes. Some of the
interesting phenomena arising due to interaction of intense laser beams with plasma
are self-focusing [1-6], modulation instability [7-11], harmonic generation [12-17]
and magnetic field generation [18-23].
1.1 Self-focusing of a laser beam in plasma
For a laser beam having a Gaussian radial profile, the intensity is peaked
on-axis 0 rI causing the plasma electrons to be repelled away from the axis.
Therefore, the refractive index tends to maximize along the axis 0 rr .
Due to this refractive index gradient the phase velocity of the laser wavefront
increases with the radial distance, causing the wavefronts to curve inwards and the
laser beam to converge (Fig. (1.1)). When the focusing force is strong enough to
counteract the diffraction effects the laser beam can propagate over a long distance
while maintaining a small cross-section.
Self-focusing can occur as a result of two effects: (1) the relativistic
modification of electron mass in the laser field and (2) the reduction of electron
density due to expulsion of electrons by the ponderomotive force (self-channeling)
of the laser beam. For an intense laser beam propagating in plasma, the refractive
15
Fig. 1.1: Converging of laser wavefront along the propagation direction and
refractive index variation.
r
r z
16
index is given by
21
020
2
1
rnrnr p
. Therefore the radial profile of the
refractive index can be affected either through the relativistic factor r or the
radial dependence of the plasma density rn . The laser ponderomotive force
expels the electrons from the axis (the ions are considered to be immobile because
of their greater mass) and prevents their return, despite the Coulomb force, which
arises from charge separation.
Self-focusing of a laser beam due to relativistic effect was first considered
by Litvak [24] and Max et al [25]. For long laser pulses (pulse length L > plasma
wave length p ) self-focusing occurs when the laser power P exceeds the
critical power for relativistic self-focusing WattP pC 4.17 220 [26].
However, for short laser pulses (L p ) [27] relativistic self-focusing does not
occur even when the laser power exceeds the critical power ( CPP ). This is due
to the fact that the index of refraction becomes modified by the laser pulse on the
plasma frequency time scale, not the laser frequency time scale.
Experiments on relativistic self-focusing and ponderomotive self-
channeling have been performed for laser pulses propagating in homogeneous
plasma. Monot et al [28, 29] and Chiron et al [30] have reported the propagation of
a 1µm, 15 TW, 400fs laser pulse through a pulsed hydrogen gas jet having electron
density 3190 10 cmn . In vacuum the focal spot radius was mr 150
17
(Rayleigh length mZR 700 ) giving a peak intensity near 4×1018W/cm2.
Propagation was studied by measuring the Thomson side-scattered laser light at an
angle of 90 with respect to the propagation axis. For CC PPP 5 the laser pulse
was observed to propagate through the entire 3.5 mm length RZ5 of the gas jet.
Relativistic as well as ponderomotive self-focusing has proved to be an efficient
way for guiding laser pulses over distances much longer than the Rayleigh
diffraction length RZ .
1.1(a) Analytical theory of self-focusing
Consider a laser beam having electric field E
and magnetic field B
,
propagating in a homogeneous plasma. The wave equation governing the evolution
of the electric vector of the radiation field is given by
tJ
cE
tc
22
2
22 41 . (1.1)
If n, -e and v are plasma electron density, charge and velocity respectively, the
plasma current density is given by
vneJ . (1.2)
18
The equations governing the relativistic interaction between the
electromagnetic field and plasma electrons are the Lorentz force equation
Bvcm
eEme
dtvd
00
(1.3)
and the continuity equation
0. vn
tn . (1.4)
If the laser beam is considered to be propagating along the z-direction and is
linearly polarized, its electric vector is given by
..,ˆ21
000 ccetrEeE tzki
x , (1.5)
where trE ,0 , 0k and 0 are the amplitude, wave number and frequency of the
radiation field respectively.
In order to obtain the source driving the laser beam in plasma, Eqs. (1.3)
and (1.4) are simultaneously solved. With the help of Eq. (1.2) the wave equation
(1.1) is given by
19
aakatc p
4
11 22
02
2
22 . (1.6)
The first term on the right hand side of Eq. (1.6) represents the linear source while
the second term (nonlinear source) arises due to relativistic mass correction effects.
In deriving Eq. (1.6) the mildly relativistic regime 100 cmEea has been
considered. Substituting Eq. (1.5) into Eq. (1.6) and reducing the resulting wave
equation into paraxial form gives
zraakzraz
ik p ,4
1,2 0
22
0002
. (1.7)
In order to study the evolution of the laser spot-size, the laser field
amplitude is assumed to be axisymmetric and is expanded in terms of a complete
set of Laguerre-Gaussian functions, i.e., source dependent modes. The dynamics of
the laser beam can be adequately described by the behavior of a single source
dependent mode, in particular, the fundamental Gaussian mode as
21expˆ, 00 sizazra (1.8)
20
where za0ˆ is the complex amplitude, 222 srr , Css Rrkz 220 , zrs is
the laser spot-size and CR is the radius of curvature associated with the wavefront.
In order derive the analytical expression for the envelope equation
describing the evolution of the fundamental mode, it is assumed that coupling to, as
well as amplitude of, the higher order source dependent expansion (SDE) modes
are small. To proceed with SDE analysis, Eq. (1.8) is substituted into Eq. (1.7),
differential operations are performed, both sides are multiplied by
2/1exp si and integrated over from 0 to ∞. The resulting equation for
0a is given by,
000 ˆ iFaAz
(1.9)
where
21
20
2
0s
s
ss
s
s
s
s
rr
rki
rrA
(1.10a)
and
00
2
20
200
0 2/1expˆ4
12
sp izaa
kk
dkF . (1.10b)
The dot (.) denotes the operator z .
21
Eq. (1.9) determines the evolution of the fundamental Gaussian source
dependent mode. Substituting Eq. (1.10a) into Eq. (1.9), setting ss iaa expˆ0
(where sa and s are real) and comparing the real and imaginary parts gives the
differential equation describing the evolution of the laser spot along the axis of
propagation as,
32
14 20
20
20
320
2
2 ark
rkzr p
s
s . (1.11)
The first term on the right hand side of Eq. (1.11) represents vacuum diffraction
while the second term is responsible for self-focusing of the laser beam. For initial
conditions 0rrs ( 0r being the laser spot-size at the focus) and 0 zrs at z = 0,
the solution of Eq. (1.11) is given by
2
220
20
20
20
2
3211
R
ps
Zzark
rr
(1.12)
where 2200rkZR represents the Rayleigh diffraction length. From Eq. (1.12) it
is observed that the laser spot focuses for 13220
20
20 rak p , remains guided for
13220
20
20 rak p and diffracts for 1322
020
20 rak p .
22
Analysis of self-focusing effects has been shown in Fig. (1.2), by plotting a
graph between nomalized laser spot 0rrs and normalized propagation distance
RZz , for 2.020 a , mp 15 and mr 200 . The dotted curve depicts
vacuum diffraction while the solid curve represents the spot-size evolution in
presence of relativistic nonlinearity in plasma.
1.2 Laser-plasma interaction instabilities
Several instabilities arise when an intense laser beam propagates in plasma.
These include stimulated Raman scattering (SRS) [31], filamentation [32-34],
modulation instability (MI) [35-37] and stimulated Brillouin scattering (SBS) [38].
These instabilities dominate under appropriate conditions defined on the basis of
the critical plasma electron density crn (for which the plasma frequency is equal to
the laser frequency). Filamentation, MI and SBS become important for crnn 0 ,
while, in particular, for 40 crnn , SRS produces a pair of electromagnetic (stoke
and anti-stoke) and Langmuir waves.
SRS is a process in which the pump electromagnetic wave is scattered off a
Langmuir wave mode. Since the frequency of Langmuir and scattered (sideband)
electromagnetic waves are both greater than p , SRS occurs when p 20 . An
electron plasma wave (Langmuir wave) can have a very high phase velocity (of the
23
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5
Fig. 1.2: Normalized laser spot-size 0rrs against normalized propagation
distance RZz in vacuum (dashed curve) and in the presence of
plasma (solid curve) for 2.020 a with mp 15 and
mr 200 .
24
order of velocity of light) and so can produce energetic electrons when it damps.
Since such electrons can preheat the fuel in fusion applications, the study of SRS
instability is important.
In SRS process if the scattered beam is inclined at an angle with respect
to the pump wave vector, in particular, if the scattering angle is less than 180 and
greater than approximately 1º 1801 , side-SRS occurs. If 180 , back-
SRS (BRS) is excited in which the scattered wave vector propagates directly
backwards with respect to the pump wave vector. Both side-SRS and BRS are three
wave processes. However, in stimulated Raman forward scattering (SRFS) 0
both, the stokes and the anti-stokes (scattered) waves are driven resonantly.
Consequently SRFS is a four wave process.
The process of stimulated Brillouin scattering (SBS) can be described as a
nonlinear interaction between the pump and stoke waves through an ion-acoustic
wave. The pump field generates an ion-acoustic wave which in turn modulates the
refractive index of the plasma. The physics of SBS is analogous to Raman
scattering except that for SBS the density perturbations which couple with the
pump and scattered light waves are those due to low frequency acoustic waves.
In filamentation instability the laser beam amplitude is transversely
modulated. Filamentation can be driven by ponderomotive force, thermal force and
relativistic effects. Filamentation occurring in presence of free electrons only is
known as relativistic filamentation instability (RFI) while that occurring in
25
presence of both free and bound electrons is known as atomic filamentation
instability (AFI).
Modulation instability is a process in which the pump wave amplitude gets
modulated in space or time. An electromagnetic wave at frequency 0 propagating
in plasma, decays into two forward moving electromagnetic sidebands at
frequencies 10 (the stoke wave) and 10 (the anti-stoke wave). The
frequency 1 corresponds to the modulation of the index of refraction (the
corresponding wave vector being 1k
). When 0011 kk and 1k
|| 0k
, the
perturbation propagates with a phase velocity equal to the group velocity of the
pump wave, leading to amplitude modulation of the pump wave. A small
modulation in the amplitude of the pump wave exerts a ponderomotive force on the
electrons along the direction of 0k
. This leads to a modified density which
modulates the group velocity of the pump laser leading to the build up of amplitude
modulation. In MI the daughter light waves grow at the cost of the parent wave
leading to amplitude modulation along the direction of propagation. MI can be
characterized as relativistic modulation instability (RMI) which is excited in
completely ionized plasma due to relativistic effects while atomic modulation
instability (AMI) occurs in partially stripped plasma (presence of bound atomic
electrons).
26
1.2 (a) Growth rate of modulation instability
In order to study the modulation instability of laser pulses interacting with
plasma, consider a linearly polarized laser beam ( ..,ˆ21
000 ccetraea tzki
x )
propagating in homogeneous plasma. The nonlinear contribution of the plasma
current density is initially neglected and the Fourier transform of the wave equation
(1.6) is taken, to give,
,ˆ,ˆ 2
02
22 rakra
c p
, (1.13)
where,
),21(,ˆ dtetrara ti represents the Fourier transform of the
normalized electric field tra , .
In order to introduce the role of the laser spot-size for a Gaussian beam, 204 r is
added and subtracted on the left hand side of Eq. (1.13). Now on substituting the
value of ,ˆ ra the wave equation becomes
0,ˆ22 002
000
20
2
02
22
ra
rkkk
zik
z (1.14)
27
where 00 ,ˆ ra is the Fourier transform of the normalized amplitude tra ,0
and 21220
222 41 rcc p is the mode propagation constant. If
0k then 0020
2 2 kkk .
The nonlinear contribution to the plasma current density originates from plasma
waves and relativistic effects. The plasma waves (wakefields) can be generated by
the radiation pressure (ponderomotive force) associated with the electromagnetic
field envelope. However, in the long pulse limit wakefield generation can be
neglected. The relativistic contribution to the plasma current density is due to
relativistic changes in mass of the oscillating electrons. Substituting Taylor’s series
expansion of about 0 [39] up to the second order dispersion term 2 ,
taking inverse Fourier transform, introducing nonlinear current density term and
changing variables from tz, to ,z ( tvz g , gv being the group velocity),
the transformed equation for the wave amplitude is given by
0,ˆ,2212 0
222
2
22
20
zazakz
vz
ik NLg (1.15)
where 222 4ck pNL . By substituting perturbed equilibrium amplitude
RZzPiaza 0000ˆ2exp),(ˆ + RZzPiza 010
ˆ2exp),( , where
ikzaa exp10 + ikza exp , CPPP 0 is the normalized laser power
28
and assuming a to vary with z as iKzexp in the Eq. (1.15), a dispersion
relation is obtained and solved to get the growth rate of modulation instability as
420
220
422
420
2202
ˆˆ25.0ˆˆˆˆˆˆˆˆˆˆ75.0ˆ14 kPkPkkPkPk
(1.16)
where CPPP 0 is the normalized laser power, 0ˆ kkk and
220
22 81ˆ Rg Zkv . In the absence of finite pulse length effects, the growth rate of
modulation instability is given by
212202
ˆˆˆˆˆ4 kPk . (1.17)
It may be observed from the figures (1.3) & (1.4) that growth rate of modulation
instability increases with k . In the presence of finite pulse length effects the peak
value of the growth becomes much larger as compared to the case where finite
pulse length effects are neglected. The parameters used to plot the graphs are:
1141088.1 sp , 1151088.1 sc and mr 150 .
29
0
0.1
0.2
0.3
0.4
0.5
0 0.05 0.1 0.15 0.2k
Fig. 1.3: Growth rate of modulation instability of a laser beam propagating in
plasma for 141088.1 p s-1, mr 150 and 150 1088.1 s-1
considering finite pulse length effect.
30
0
0.01
0.02
0.03
0.04
0.05
0 0.05 0.1 0.15 0.2
k
Fig. 1.4: Growth rate of modulation instability of a laser beam propagating in
plasma for 141088.1 p s-1, mr 150 and 150 1088.1 s-1,
neglecting finite pulse length effects.
31
1.3 Generation of harmonic frequencies
Generation of harmonic radiation is an important subject of laser-plasma
interaction and attracts great attention due to wide range of applications. High order
harmonics can provide X-ray sources [40, 41]. Harmonic generation has been
studied theoretically [42-44] as well as experimentally [45-48]. Gibbon [16] has
studied even as well as odd harmonic generation from plasma electrons oscillating
in pre-ionized matter irradiated by short intense laser pulses.
In most laser interactions with homogeneous plasma, odd harmonics of
laser frequency are generated [49, 50]. Sprangle et al [51] have reported stimulated
back-scattered harmonic radiation generated by the interaction of an intense laser
field with an electron beam or plasma using a fully nonlinear, relativistic fluid
theory valid to all orders in the pump laser amplitude. The back-scattered radiation
occurs at odd harmonics of the Doppler-shifted incident laser frequency. This
mechanism may provide a practical method for producing coherent radiation in
XUV regime. Mori et al [43] have investigated the relativistic third harmonic
content of large amplitude electromagnetic waves propagating in underdense
plasma, using perturbative procedure of Montgomery and Tidman [52]. Yang et al
[48] have experimentally observed a strong third order harmonic (TH) emission
with conversion efficiency higher than 10-3 from a plasma channel formed by self
guided femtosecond laser pulses propagating in air.
32
Second harmonics have been observed in the presence of plasma density
gradients [53, 47]. This is due to laser induced quiver motion of electrons across
the density gradient, which gives rise to perturbation in the electron density at the
laser frequency. This density perturbation coupled with the quiver motion of
electrons produces a source current at the second harmonic frequency. Second
harmonic generation has also been related with filamentation [54, 55]. In this case
second harmonic radiation is emitted in a direction perpendicular to the propagation
direction of the laser beam, from filamentary structures in the underdense target
corona.
Second harmonic generation has been reported by Singh et al [56] in the
reflected component of a high intensity laser incident obliquely on a vacuum
plasma interface. A laser beam of frequency 0 induces an oscillatory electron
velocity 1v at ( 0 , 0k
) and exerts a ponderomotive force 11 Bv
on electrons at
( 02 , 02k
). The ponderomotive force and the self consistent field 2E
, induce
oscillatory electron velocity 2v at ( 02 , 02k
) which couples with 0n to produce
second harmonic current density 2n at ( 02 , 02k
). The current at second harmonic
creates space charge oscillations on the plasma surface and gives rise to second
harmonic electromagnetic radiation in the reflected component.
Second harmonic radiation has been experimentally detected in laser
induced gas plasma [57-59]. The plasma was created by a laser pulse focused on a
solid target. Being expanding plasma it was highly nonuniform and the second
33
harmonic was generated in the bulk of the plasma. In an experiment by V. Malka et
al [47], at Rutherford Appleton Laboratory, with the Vulkan Nd: Glass laser,
operating at 1.054 m in the chirped-pulse amplification mode, forward Raman
scattering was observed to be accompanied by second harmonic light, with a
conversion efficiency up to 0.1%. Moreover this radiation was phase modulated by
the co-propagating relativistic electron plasma wave produced by forward Raman
scattering of the main laser beam. It has also been shown that ionization induced
density gradient is a cause for generation of second harmonics.
Generation of higher order harmonics in laser induced plasma has also been
observed [60]. When a high energy CO2 laser pulse was focused on a target (for
example a 100J, 1ns pulse focused to 8×1014W/cm2), harmonics as high as 46th
were observed in the output.
1.3(a) Analysis of second harmonic generation
In order to study the generation of second harmonic frequency in plasma,
consider a laser beam propagating along the z-axis. The wave equation governing
the propagation of laser beam through plasma is represented by Eq. (1.1). The
current density J
can be obtained with the help of Lorentz force (Eq. (1.3)) and
continuity (Eq. (1.4)) equations. Successive approximations can be used to find J
34
as a function of E
by considering the perturbative expansion of the plasma current
density, electron density and velocity as,
.........321 JJJJ
, (1.17a)
....210 nnnn (1.17b)
and
.........321 vvvv . . (1.17c)
Thus 10
1 venJ , 112
02 venvenJ
and so on. In the non-relativistic
limit the Lorentz force equation gives
Emevi
tv
1
0
1 (1.18a)
and
Bvmcevvvi
tv
11120
2.2 (1.18b)
while the continuity equation gives
10
10
1. vnni
tn
(1.18c)
35
The first order density perturbation generates an electrostatic field governed by the
Gauss’ law as
14. enEs
. (1.18d)
The second order current density oscillating at the second harmonic frequency is
then given by
EEmieBE
cmieEE
meienJ s
.
4.
22
002
2
20
2
2
0
00
2
. (1.19)
Substituting the value of EEnm
eEEp
s
2
020
20
2
1.4.
, with EicB
0
and EEEEEE
.21. , the current density equation (1.19) may be
written as
EEn
mieEE
mnieJ
p
2
020
30
2
3
30
20
3
02
1..
42
(1.20)
Eq. (1.20) shows that second harmonic generation will occur if the plasma is
inhomogeneous 00 n . However, for uniform plasma 00 n the second
36
term on the right side of Eq. (1.20) is zero while the first term leads to current
density along the direction of beam propagation. Since an oscillating current cannot
radiate longitudinally, no coherent second harmonic radiation is expected along the
axis of beam propagation from the bulk of a uniform plasma.
1.4 Magnetic field generation
Magnetic field generation by the interaction of linearly as well as circularly
polarized intense laser beams with plasma, has been studied. Interaction of a
linearly polarized laser pulse, having an axisymmetric envelope, with underdense
plasma generates an azimuthal magnetic field [61]. The structure of this magnetic
field inside the laser pulse body depends on the pulse shape.
The generation of axial magnetic field in plasma by a circularly (or
elliptically) polarized laser is often referred to as inverse Faraday effect. It was first
reported by Pitaevskii [62], J. Deschamps et al [63] and Steiger and Woods [64]
and results from the features of electron motion in a circularly polarized
electromagnetic wave. Berezhiani et al [65] analytically studied the generation of
quasi-static magnetic field for a circularly polarized laser pulse propagating in
underdense plasma. The mechanism involves the rotation of the polarization vector
of the external radiation field. The basic approach utilized a relation describing the
conservation (at each point) of the generalized velocity and then calculating the low
frequency drag current excited by electromagnetic radiation. It has also been shown
37
that quasi-static magnetic fields are generated [66] due to strong inhomogeneity
caused by the intense laser beam itself. Since electron distribution is determined
completely by the pump wave intensity, the generated magnetic field is negligibly
small for non-relativistic laser pulses but increases rapidly in the ultra-relativistic
case. Due to the possibility of cavitation for narrow and intense laser beams, the
increase in generated magnetic field amplitude slows down as the beam intensity is
increased. The structure of the magnetic field closely resembles that of the field
produced by a solenoid: the field is maximum and uniform in the cavitation region,
then it falls, changes polarity and vanishes. Inverse Faraday effect has been
measured in several experiments [67-71]. It does not occur for linearly polarized
laser pulses due to absence of angular momentum of photons.
Experiments have been performed for measuring axial as well as azimuthal
magnetic fields generated by propagation of intense laser beams in plasma. A
recent experiment [71] reported about 2MG axial magnetic field generation by
propagation of a circularly polarized laser of intensity 2180 1067.6 cmWI ,
transverse beam radius 100 r m 05.1 , and pulse duration 0.9-1.3ps in
uniform plasma of density 3190 108.2 cmn . Fuchs et al [66] have measured
azimuthal magnetic field of about 35-70MG produced in plasma with
3200 100.2 cmn , due to propagation of a linearly polarized laser beam
( 0I = 218107.4 cmW , 40 r , m 05.1 and pulse duration 0.6ps). These self
38
generated magnetic fields are expected to affect the propagation characteristics of
laser beams in plasma.
1.5 Propagation of laser beams in magnetized plasma
When the plasma is immersed in a uniform, static magnetic field, the behavior
of propagated waves can be considerably more complicated than in the absence of
magnetic field. The critical factor in determining the role of magnetic field is the
direction of wave propagation and polarization with respect to the magnetic field.
In magnetized plasma electrons and ions cannot move freely, perpendicular to the
magnetic lines of force. The path of each particle then becomes helical; the axis of
the helix being parallel to the magnetic field and in that state plasma becomes
highly anisotropic.
Earlier workers have shown that laser-plasma interactions are affected by the
presence of magnetic field. Recently, Hur et al [72] have shown that an externally
applied magnetic field enhances the particle trapping in laser wakefield
acceleration. When a static magnetic field is applied along the propagation
direction of a driving laser pulse it has been shown from two dimensional particle
in cell simulations that the total charge of the trapped beam and its maximum
energy increases. Ren and Mori [73] have studied the effects of external magnetic
fields on wake excitation and its reaction on nonlinear evolution of laser pulses.
Jha et al [11] have studied modulation instability of a linearly polarized laser pulse
39
propagating through transversely magnetized underdense plasma. Gupta et al [74]
have studied the transient self-focusing of an intense short pulse laser in
magnetized plasma. The laser with Gaussian radial distribution of intensity exerts a
ponderomotive force on electrons and sets in ambipolar diffusion of plasma. The
ambient magnetic field, however, strongly inhibits the process, when the electron
Larmor radius is comparable to or shorter than the laser spot-size. As the plasma
density is depleted, the laser beam becomes more and more self-focused.
Wadhwani et al [75] have studied the dispersion of incident radiation and its
harmonics for a linearly polarized laser beam propagating through cold underdense
plasma in the presense of constant magnetic field applied perpendicular to both the
electric vector and the direction of propagation. Yoshii et al [76] have analyzed
the Cerenkov wakes excited by a short laser pulse in a perpendicularly magnetized
plasma.
H. Parchamy et al [77] have observed radiations in microwave frequency range
from a tightly focused, highly intense, ultrashort laser pulse interacting with weakly
magnetized plasma. To investigate the microwave radiation produced by the laser-
plasma interaction, a mode locked Ti: Sapphire laser beam of wavelength 800nm,
pulse width 100fs (Full width at half maximum), a maximum energy of 100mJ per
pulse and a repetition rate of 10Hz were employed. D. Dorranian et al [78] have
observed the generation of short pulse radiation from magnetized wake in gas-jet
plasma and laser radiation having the same parameters as mentioned above. Gas-jet
radiation is used to generate sharp boundary plasma. Strength of the applied
40
external dc magnetic field normal to the direction of laser pulse propagation varied
from 0 to 8KG in the interaction region. Radiation was observed in the forward
direction due to the axial component of the magnetized wakefield and in the normal
direction due to the radial component of the magnetized wakefield, both
perpendicular to the direction of applied magnetic field.
1.6 Aim
Intense laser-plasma interactions have been widely studied for applications such
as plasma-based accelerators, inertial confinement fusion (ICF) and new radiation
sources. Most of the earlier studies consider the plasma to be unmagnetized. The
analysis of interaction of laser beams with magnetized plasma is relatively a new
area of study. Magnetic fields play an important role in many interesting
phenomena such as radiation from Cerenkov wakes and fast ignitor concept in ICF
where either self generated or external magnetic fields may be present.
The present thesis is aimed at a detailed theoretical study of the propagation
characteristics and instabilities arising due to propagation of intense laser beams in
plasma embedded in a magnetic field. The self-focusing of (a) a linearly polarized
laser beam in transversely magnetized plasma and (b) a circularly polarized laser
beam in axially magnetized plasma has been studied. The growth of modulation
instability for a circularly polarized laser beam propagating in plasma embedded in
41
an axial magnetic field has been analyzed. Also the possibility of the second
harmonic generation in homogeneous magnetized plasma has been pointed out.
1.7 Approach
In the present thesis the propagation of an intense laser beam in magnetized
plasma has been studied. Some important nonlinear parametric processes such as
self-focusing, modulation instability and harmonic generation have been analyzed
for plasma embedded in a magnetic field. The wave dynamics of the laser beam
propagating through underdense plasma is completely determined by the set of
three equations namely the wave equation, continuity equation and the Lorentz
force equation. The analysis has been done on the basis of the following
assumptions: the plasma is considered to be cold, homogeneous and neutral and the
laser interaction with plasma is considered to be in the mildly relativistic regime. In
order to obtain the source current density driving the laser field, the electron
velocity and the plasma electron density are perturbatively expanded (in orders of
the radiation field). The applied magnetic field has been considered to be a zeroth
order quantity. The resulting wave equation governing the evolution of the laser
amplitude is set up.
For studying the laser spot evolution in magnetized plasma, the wave equation
is reduced to its paraxial form, by neglecting finite pulse length and group velocity
dispersion effects. Assuming the laser field amplitude to have a Laguerre-Gaussian
42
form the differential equation for laser spot-size, curvature and phase shift are
obtained, using source dependent expansion technique. Graphical analysis for the
variation of normalized laser spot with normalized propagation distance in the
presence as well as absence of the magnetic field is given and critical power
required for nonlinear self-focusing of the laser beam in presence of the magnetic
field is obtained.
Modulation instability of a laser beam propagating in transversely magnetized
plasma has been studied. A non-paraxial wave equation is set up and an algebraic
transformation is performed from ,, ztz where tvz g . The non-
paraxial wave equation is solved (in one-dimensional limit) to yield the
unperturbed laser beam amplitude. Perturbed wave amplitude, due to spatially
growing modulation instability, is assumed to have the same form as the
unperturbed wave amplitude. Substituting the total wave amplitude (superposition
of the unperturbed and perturbed wave amplitudes), in the wave equation, the
nonlinear dispersion relation for spatially modulated laser beam amplitude is
obtained. The dispersion relation is then solved to give the spatial growth rate of
modulation instability.
Second harmonic generation for a linearly polarized laser beam propagating in
transversely magnetized plasma is studied. Expression for nonlinear current density
and dispersion relations for fundamental as well as second harmonic frequency are
obtained. In order to obtain the normalized wave amplitude of the second harmonic
43
za2 and its conversion efficiency, it is assumed that the distance over which
za 2 changes appreciably is large compared to the wavelength and that the
amplitude of the fundamental ( 1a ), changes very slowly with z. Graphical analysis
of maximum conversion efficiency with respect to the magnetic field is given.
44
CHAPTER 2
SELF-FOCUSING OF INTENSE LASER BEAMS PROPAGATING IN
MAGNETIZED PLASMA
In this chapter, self-focusing of an intense laser beam propagating in plasma
embedded in a uniform magnetic field has been presented. The plasma is assumed
to be cold, underdense and homogeneous. The spot size evolution of (a) a linearly
polarized laser beam propagating in transversely magnetized plasma [79] and (b) a
circularly polarized laser beam propagating in axially magnetized plasma [80] has
been analyzed. The study is motivated by the fact that intense magnetic fields are
generated via laser-plasma interaction and in many applications, modification of
the propagation characteristics of the laser beam due to presence of these fields
become important.
For studying the laser spot evolution in magnetized plasma, a nonlinear
wave equation is set up. The plasma current density driving the laser field is
obtained (in the mildly relativistic limit) using perturbative technique. The source
dependent expansion (SDE) technique is used to obtain the equation governing the
spot-size evolution. The effect of magnetic field on the self-focusing property of
the laser beam is discussed and the expression for the critical power required for
self-focusing of the laser beam is obtained and compared with the unmagnetized
case.
45
2.1 Wave Dynamics
The basic equations governing the propagation of a laser beam through pre-
ionized plasma are the Maxwell’s time dependent equations
tB
cE
1 (2.1)
and
tE
cJ
cB m
14 . (2.2)
E
and B
are the electric and magnetic vectors of the radiation field respectively,
mJ
is the electron current density and c is the velocity of light in vacuum. With the
help of Eqs. (2.1) & (2.2) and using Coulomb gauge 0. E
the evolution of
electric field is given by
t
Jc
Etc
m
22
2
22 41 . (2.3)
The equations describing the relativistic interaction between the
electromagnetic field and plasma electrons are the Lorentz force equation
46
bBv
cmeE
me
dtvd
mmm
00
(2.4)
and the continuity equation
0.
mmm vnt
n , (2.5)
where 21221
cvmm is the relativistic factor, mv is the velocity of plasma
electrons, mn is the plasma electron density and b
is the applied magnetic field.
Here, subscript m denotes the physical quantities in presence of external magnetic
field.
Using perturbative technique all quantities can be expanded simultaneously
in orders of the radiation field. Thus
321mmmm vvvv
, (2.6a)
210mmmm (2.6b)
and
210mmmm nnnn . (2.6c)
47
The plasma is considered to be cold so that initially the plasma electrons are
assumed to be at rest 00 v and 00 nn is the ambient plasma electron
density. Expanding the relativistic factor up to the second order and comparing
similar order terms gives
10 m , (2.7a)
01 m (2.7b)
and
212
2
21
mm vc
. (2.7c)
Substituting Eqs. (2.6a) & (2.6b) in Eq. (2.4) and using convective derivative
.mvtdt
d , the first, second and third order equations of motion for
plasma electrons are
01
00
1
bvcm
eEme
tv
mm
, (2.8a)
02
0
11
0
112
. bvcm
eBvcm
evvt
vmmmm
m
(2.8b)
48
and
1221123
.. mmmmmmm vvvvv
ttv
03
0
12
0
bvcm
eBvcm
emm
(2.8c)
The magnetic field has been considered to be of order zero. Similarly substituting
Eq. (2.6c) into Eq. (2.5), the first and second order continuity equations are given
by
0. 10
1
mm vnt
n (2.9a)
and
0.. 1120
2
mmmm vnvnt
n . (2.9b)
The current density can now be obtained by using perturbed velocities (Eq.
(2.6a)) and plasma densities (Eq. (2.6c)) as,
122130
10 mmmmm vnvnvnvneJ
. (2.10)
49
The nonlinear current density is represented by the second, third and fourth terms
in Eq. (2.10). It may be noted that the presence of the magnetic field modifies the
plasma electron velocities (Eq. (2.8)) and densities (Eq. (2.9)). Subsequently the
plasma current density also becomes a function of the magnetic field. These
contributions of the magnetic field to the current density lead to modification of the
nonlinear refractive index and will therefore affect the propagation characteristics
of the laser beam in plasma.
2.2 Linearly polarized laser beam propagating in transversely
magnetized plasma
2.2.1 Formulation
Consider a linearly polarized laser beam propagating in plasma embedded in a
uniform, transverse magnetic field yebb ˆ
. The normalized electric field vector
00 cmEea
of the radiation field propagating along the z-direction is
represented by
..,21ˆ 00
0 ccezraea tzkix . (2.11)
50
Using Eq. (2.8a) the first order equations for transverse and longitudinal velocities
are respectively given by
111
mzcmxmx vE
me
tv
(2.12a)
and
11
mxcmz vt
v
, (2.12b)
where cmebc 0 is the cyclotron frequency of the plasma electron.
Differentiating Eq. (2.12a) with respect to ‘t’ and substituting Eq. (2.12b), gives
..2
000
20
122
12
cceacivtv tzki
mxcmx
. (2.13a)
Again differentiating Eq. (2.12b) with respect to ‘t’ and substituting Eq. (2.12a), the
differential equation for the first order longitudinal plasma electron velocity is
given by
..21
0000
122
12
cceacvtv tzki
cmzcmz
. (2.13b)
51
It may be noted that the first order transverse and longitudinal velocities are driven
by forces oscillating with the laser frequency. The solutions of Eqs. (2.13a) and
(2.13b) are respectively given by
..
200
220
2001 cceicav tzki
cmx
(2.14a)
and
..
200
220
001 ccecav tzki
c
cmz
. (2.14b)
The presence of the magnetic field increases the transverse quiver velocity
(Eq. (2.14a)) and also leads to the generation of a longitudinal velocity component
(Eq. (2.14b)), due to bvm
force acting on the plasma electrons. This leads to an
increase in and hence the relativistic mass of the plasma electrons and results in
the modification of the refractive index.
The same procedure is used to obtain the second order differential equation
for the electron velocity. These equations can be obtained from Eq. (2.8b). With the
help of the first order velocities (Eq. (2.14)) the second order equations are given
by
..4
4002
2220
220
20
200
222
2
22
cceakicvtv tzki
c
ccmxc
mx
(2.15a)
52
and
..4
442002
2220
4220
400
200
222
2
22
cceakcvtv tzki
c
ccmzc
mz
. (2.15b)
The solutions of Eqs. (2.15a) and (2.15b) are respectively given by
..44
4002
220
2220
220
20
200
22 cceakicv tzki
cc
ccmx
(2.16a)
and
..44
442002
220
2220
4220
400
200
22 cceakcv tzki
cc
ccmz
. (2.16b)
The second order, high frequency, x-component of plasma electron velocity is
generated due to the uniform magnetic field and reduces to zero in its absence.
However, the second order z-component of velocity is due to the magnetic vector
of the radiation field as well as the external magnetic field.
Similarly, the third order equation for the x-component of velocity,
neglecting harmonics, is given by
53
220
3220
40
220
40
220
20
230
322
32
446115
2cc
ccmxc
mx kccaivtv
0..
8
32300
3220
4220
40
40
cce tzki
c
cc
. (2.17)
Eq. (2.17) is solved to get
220
4220
40
220
40
220
20
230
3
446115
2cc
ccmx
kccaiv
0..
8
32300
4220
4220
40
40
cce tzki
c
cc
. (2.18)
Density perturbations introduced in the plasma due to interaction with the
laser beam can be obtained with the help of Eqs. (2.9). Substituting the value of 1mzv
(Eq. (2.14b)) in Eq. (2.9a) and using the transverse Coulomb gauge 0. E
, the
first order electron density perturbation is given by
..
200
220
0001 cceacknn tzki
c
cm
. (2.19a)
54
The first order density perturbation arises due to the presence of the external
magnetic field and reduces to zero in its absence. The second order density
response is calculated with the help of Eq. (2.9b) as
..42
4002
220
2220
220
20
20
20
202 cceakcnn tzki
cc
cm
. (2.19b)
The current density equation is obtained by substituting first, second and
third order quantities in Eq. (2.10) as
22
0
4220
40
220
20
2
4220
2022
0
20
00 4573
21
21
c
ccc
ccmx
kcacaienJ
220
3220
220
220
20
24220
40
40
44
44
323
cc
cccc kc
..
42
400
220
3220
220
40
20
2ccekc tzki
cc
c
. (2.20)
The first term on the right side of Eq. (2.20) is the linear current density.
The presence of the magnetic field modifies the second and fourth terms, while the
third term arises solely due to first order electron density oscillations set up by the
magnetic field. In deriving Eq. (2.20) all harmonics have been neglected.
55
Substitution of the linear current density into Eq. (2.3) leads to the linear
dispersion relation for a linearly polarized laser beam propagating in transversely
magnetized plasma as
220
2202
020
2
c
pkc
(2.21)
In the absence of the magnetic field 0c , Eq. (2.21) reduces to the well known
dispersion relation for a laser beam propagating in unmagnetized plasma.
2.2.2 Wave equation
Nonlinear propagation of the laser beam in transversely magnetized plasma
can be described by substituting the current density (Eq. (2.20)) into the wave
equation (2.3) as
aNakatc c
p
2022
0
202
02
2
22 1
(2.22)
56
where
220
3220
4220
40
20
20
2
220
4220
4220
40
220
20
2
44492
446115
cc
cc
cc
ccc kckcN
422
0
4220
40
40
8
323
c
cc
. The second term on the right hand side of Eq. (2.22)
includes nonlinear perturbations due to relativistic effects, density fluctuations and
coupling of the radiation field with magnetic field. Substituting Eq. (2.11) into Eq.
(2.22) gives
zraNakzraz
ikz c
p ,,2 02022
0
202
0002
22
. (2.23)
Assuming the radiation amplitude to be a slowly varying function of z
zkz 022 2 , Eq. (2.23) reduces to
zraNakzraz
ikc
p ,,2 02022
0
202
0002
. (2.24)
Eq. (2.24) represents the paraxial form of the wave equation describing the
evolution of the laser field amplitude for which higher order diffraction effects have
been neglected.
57
2.2.3 Spot-size evolution
In order to study the evolution of the laser spot-size, the laser field
amplitude is assumed to be axisymmetric and is expanded in terms of a complete
set of Laguerre-Gaussian functions, i.e. source dependent modes as
p
spp
iLzazra2
1expˆ,0
, (2.25)
where p = 0,1,2,3……, )(ˆ za p is the complex amplitude, 222 srr , rs(z) is the
spot size, Css Rrkz 2)( 20 , (RC ) is radius of curvature associated with the wave
front and Lp (χ) is a Laguerre polynomial of order p. The dynamics of the laser
beam can be adequately described by the behaviour of a single source dependent
mode, in particular, the fundamental Gaussian (p = 0) mode.
To obtain the analytical expression for the envelope equation describing the
evolution of the fundamental mode, it is assumed that coupling to as well as
amplitude of the higher order source dependent expansion (SDE) modes are small.
To proceed with SDE analysis Eq. (2.25) is substituted into Eq. (2.24), differential
operations are performed and both sides are multiplied by
21exp sp iL and integrated over from 0 to . The resulting
equation for pa is given by
58
ppppp iFaBpiaipBaAz
11 ˆ1ˆˆ , (2.26)
where
21
12 20
2s
s
ss
s
s
s
sp r
rrk
pirrA
, (2.27a)
2
02
0
2 22
1
s
s
s
ss
s
s
s
ss
rkrri
rkrrB
(2.27b)
and
21
expˆ4
12 0
40
20
200
sp
pp
iLzaakk
dkF . (2.27c)
The dot (.) denotes the operator z and the asterisk (*) denotes the complex
conjugate.
An optimal choice for B(z) can be obtained from Eq. (2.26) by requiring
that the higher order SDE modes are small. Assuming 22
0 ˆˆ paa for 1p , the
optimal choice for B(z) is given by
0
1
aFB . (2.28)
Substituting Eq. (2.28) into Eq. (2.26) gives
59
000 ˆ iFaAz
. (2.29)
Eqs. (2.28) and (2.29) completely determine the evolution of the fundamental
Gaussian source dependent mode. Substituting Eqs. (2.27) into Eqs. (2.28) and
(2.29) and setting ss iaa expˆ0 , where sa and s are real, the comparison of
real and imaginary parts gives
0)(
ss raz
, (2.30a)
)1(4 2032
02
2
Hrkrkz
rs
s
s , (2.30b)
22
02
0 ss
C
ss
rrkRrk
. (2.30c)
and
GHrk s
s 20
2 , (2.30d)
where,
2
2
220
20
20
20 Na
kk
G s
c
p
(2.31a)
and
60
0
220
8kNak
H sp . (2.31b)
Eq. (2.30a) shows that the total laser power is conserved (independent of z);
therefore 20
20
22 rara ss . The evolution of the laser spot is determined from
Eq. (2.30b) which may be explicitly written as
N
rakrkz
r p
s
s
814 2
020
20
320
2
2
. (2.32)
The first term on the right hand side of Eq. (2.32) represents vacuum diffraction. If
the second term is positive, it can lead to nonlinear self-focusing of the laser beam.
Multiplying both sides of Eq. (2.32) by
0rr
zs and normalizing sr by the
minimum laser spot-size 0r gives
0
20
20
20
30
330
200
2
2
0 814.
rr
zN
rakrrrkr
rzr
rz
sp
s
ss . (2.33)
Integrating Eq. (2.33) with respect to ‘t’ and applying the initial conditions that at
0z , 0rrs and 0dzdrs gives
61
2120
20
20
320
21
20
2
00 81411
N
rakrkr
rrrr
rz
p
s
s
s
s . (2.34)
Again, integrating Eq. (2.34) with respect to z and applying the same initial
conditions as in Eq. (2.33) leads to
2
2
20
2
11Rcm
s
Zz
PP
rr
(2.35)
where 820
20
20 NrakPP pcm is the normalized laser power and RZ is the Rayleigh
length. It may be noted that in the absence of the magnetic field 0c Eq. (2.35)
reduces to spot-size evolution of a laser beam propagating in unmagnetized plasma
[27]. cmP defines the critical power for nonlinear self-focusing of a laser beam in
magnetized plasma and its value is given by
NekmcP
pcm 222
0
20
522
. (2.36)
In the absence of the magnetic field 0c Eq. (2.36) reduces to the critical
power required for self-focusing of a laser beam propagating in unmagnetized
plasma [27].
62
The variation of normalized spot-size 0rrs of a laser beam having
intensity 1017 W/cm2 is plotted against the normalized propagation distance
RZz in Fig. (2.1) for unmagnetized (curve (a)) and magnetized (curve (b) for
0c = 0.2 and curve (c) for 0c = 0.4) plasma. The parameters used are
271.00 a , 150 1088.1 s-1 and 0 p = 0.1. The self-focusing property of the
laser spot is seen to enhance due to magnetization of the plasma. This is because of
the additional plasma current density, which arises due to presence of the external
magnetic field.
In order to study the effect of increasing magnetic fields on the laser spot,
0rrs is plotted against 0c in Fig. (2.2) at RZz = 0.3 for 271.00 a . The
spot-size is seen to decrease with increase in the magnetic field. Thus the beam
becomes more focused as the magnetic field is increased. The critical power ( cmP )
required for self-focusing of the laser beam, is plotted against 0c in Fig. (2.3).
It may be noted that an increase in magnetic field leads to a significant decrease in
critical power required for self-focusing the laser beam. The parameters used are
same as in Fig. (2.1).
63
0
1
2
3
0 0.5 1 1.5 2 2.5
Fig. 2.1 Variation of 0rrs with RZz for (a) unmagnetized plasma (b)
0c = 0.2 and (c) 0c = 0.4, with 271.00 a , 150 1088.1 s-1
and 0 p = 0.1.
64
1
1.01
1.02
1.03
1.04
0 0.1 0.2 0.3 0.4
Fig. 2.2 Variation of 0rrs with 0c at RZz = 0.3 for 0a 0.271,
150 1088.1 s-1 and 0 p = 0.1.
65
0
5
10
15
20
0 0.05 0.1 0.15 0.2
Fig. 2.3: Variation of cmP with 0c for 271.00 a , 150 1088.1 s-1 and
0 p = 0.1.
0c
66
2.3 Spot-size evolution of a circularly polarized laser beam propagating in
axially magnetized plasma
2.3.1 Formulation
Consider the propagation of an intense circularly polarized laser beam along
the z-direction, in the presence of homogeneous plasma embedded in a longitudinal
magnetic field zebb ˆ
. The normalized electric field of the laser is given by
..ˆˆ,21
000 cceeezraa tzki
yx (2.37)
where zra ,0 , 0k and 0 are the normalized amplitude, wave number and
frequency of the radiation field, respectively. takes values 1 for right or left
circularly polarized radiation, respectively.
The wave equation governing the propagation of the laser beam through
plasma is given by Eq. (2.3). Relativistic interaction between the electromagnetic
field and plasma electrons is governed by the Lorentz force (Eq. (2.4)) and
continuity (Eq. (2.5)) equations. In the mildly relativistic regime, all parameters can
be expanded in orders of the radiation field amplitude. Using Eq. (2.4), the first
order expansion leads to
67
110
1
mycxmx vact
v
, (2.38a)
110
1
mxcymy vact
v
(2.38b)
and
01
tvmz . (2.38c)
Simultaneous solution of Eqs. (2.38) leads to the first order transverse plasma
electron velocity as
..ˆˆ,
20
0220
001 cceeiezraicv tzkiyx
c
cm
o
(2.39)
Eq. (2.39) shows that the quiver velocity of the plasma electrons increases
significantly due to axial magnetization for left circular polarization 1 . This
increases the relativistic factor and hence the relativistic mass of the plasma
electrons. However, for right circular polarization, the velocity decreases.
The second order velocities are found to be zero. The third order expansion
of Eq. (2.4) leads to
68
3123
mycmxmmx vv
ttv
, (2.40a)
3123
mxcmymmy vv
ttv
(2.40b)
and
03
tvmz . (2.40c)
Using Eq. (2.39) and simultaneously solving Eqs. (2.40) yields
..ˆˆ,
2003
04220
40
403 cceeiezraicv tzki
yx
c
cm
. (2.41)
It may be noted that the third order velocity also increases or decreases for 1 ,
respectively and that the longitudinal velocity perturbations are zero. Since the
presence of magnetic field changes the plasma electron velocities, the refractive
index is also modified. The propagation characteristics of the laser beam will
therefore be affected. Perturbative expansion of the continuity equation (2.5) and
substitution of Eq. (2.39) yields 021 mm nn . With the help of the perturbed
plasma electron velocties and densities, the transverse current density is given by,
69
3220
30
302
0220
000
2,1
2c
c
c
cm zracienJ
..ˆˆ, 000 cceeiezra tzki
yx . (2.42)
By substituting the linear part of the current density (first term on right
hand side of Eq. (2.42)) into Eq. (2.3), the linear dispersion relation is obtained as
220
002
20
20
2
c
cpkc
(2.43)
In the absence of the magnetic field 0c , Eq. (2.43) reduces to the well known
dispersion relation of a laser beam propagating in unmagnetized plasma.
2.3.2 Wave Equation
Substituting Eq. (2.42) into Eq. (2.3) leads to the wave equation governing
nonlinear propagation of the laser beam in plasma as
zraSzrakzra
tc c
cp ,,,1 2
0220
00202
2
22
(2.44)
where
70
422
0
40
40
2 c
cS
. (2.45)
Substituting Eq. (2.37) in to Eq. (2.44) leads to
zraSzrakzra
zik
z c
cp ,,,2 0
2022
0
0020002
22
.
(2.46)
Assuming the radiation amplitude to be a slowly varying function of z, the higher
order diffraction terms are neglected. The paraxial form of the wave equation is
thus given by
zraSzrakzra
zik
c
cp ,,,2 0
2022
0
002000
2
. (2.47)
2.3.3 Spot-size evolution
In order to study the evolution of the laser spot, the amplitude is assumed to
be axisymmetric and is expanded in terms of a complete set of Laguerre-Gaussian
functions. SDE method (as in Section 2.2.3) is applied to obtain the evolution
equation for the laser spot as
71
Srak
rkzr p
s
s8
14 20
20
20
320
2
2 (2.48)
Solution of Eq. (2.46) is obtained as
2
2
20
2
11Rcm
s
Zz
PP
rr
, (2.49)
where P is the laser power and SekmckP pcm22
020
520 2 defines the critical power
required for nonlinear self-focusing of the circularly polarized laser beam in axially
magnetized plasma. It may be noted that the critical power required for self-
focusing a left (right) circularly polarized laser beam decreases (increases) as the
external magnetic field in increased.
A plot for normalized spot-size versus normalized propagation distance is
shown in Fig. (2.4) for (a) 0c =0 (b = 0), (b) 0c = 0.15; =-1 and (c)
0c =0.15; =+1 with 271.00 a and 150 1088.1 s-1. The graph shows
that the magnetic field reduces the diffraction of the beam for left circular
polarization and enhances the self-focusing property of the laser beam. However,
for right circular polarization, the magnetic field slightly increases the diffraction
and hence reduces the self-focusing property of the beam. Changes in the spot-size
72
as observed in the left circularly polarized laser beam are more effective as
compared to the case of right circularly polarized laser beam.
The variation of normalized spot-size of a right circularly polarized laser
beam with normalized cyclotron frequency (magnetic field ) is shown in
Fig. (2.5) for RZz = 0.3, 0a 0.271, 150 1088.1 s-1 and 0 p = 0.1. The
laser spot-size is seen to increase with magnetic field. Therefore increasing the
magnetic field leads to defocusing of the laser beam. A similar graph for a left
circularly polarized laser beam is shown in Fig. (2.6). It is seen that on increasing
the magnetic field the spot-size initially remains constant. However, further
increase in magnetic field brings about a reduction in the laser spot-size.
73
0
1
2
3
4
5
0 1 2 3 4 5 6
Fig. 2.4: Variation of 0rrs with RZz for (a) 0c =0, (b) 0c = 0.15;
=-1 and (c) 0c =0.15; =+1 with 271.00 a and
150 1088.1 s-1.
74
1.02
1.03
1.04
1.05
0 0.1 0.2 0.3 0.4 0.5 0.6
Fig. 2.5 Variation of 0rrs with 0c for right circularly polarized laser
beam having RZz = 0.3, 0a 0.271, 150 1088.1 s-1 and
0 p = 0.1.
75
0
0.5
1
1.5
2
0 0.1 0.2 0.3 0.4 0.5 0.6
Fig. 2.6 Variation of 0rrs with 0c for left circularly polarized laser
beam having RZz = 0.3, 0a 0.271, 150 1088.1 s-1 and
0 p = 0.1.
76
CHAPTER 3
MODULATION INSTABILITY OF LASER PULSES IN AXIALLY MAGNETIZED PLASMA
In this chapter, modulation instability of a circularly polarized laser pulse
propagating through axially magnetized, cold and underdense plasma has been
studied. Since the presence of a uniform axial magnetic field modifies the source
driving a laser beam in plasma (Chapter 2), the modulation instability of the laser
pulse interacting with magnetized plasma is expected to be affected. The
nonparaxial form of the wave equation has been considered and the spatial growth
rate of modulation instability for left as well as right circularly polarized laser
beams propagating through axially magnetized plasma has been obtained. The
results are compared with the growth rate obtained for a laser beam propagating in
unmagnetized plasma. The range of wave numbers over which the instability
occurs has also been evaluated.
3.1 Formulation
Consider a circularly polarized laser beam propagating through uniform
plasma. The normalized electric vector of the radiation field is given by
..ˆˆ,21
000 cceeetraa tzki
yx , (3.1)
77
where tra ,0 , 0k and 0 are the amplitude, wave number and frequency of the
radiation field, respectively. takes values 1 for right or left circularly polarized
radiation, respectively. The plasma is embedded in a constant axial magnetic field
beb zˆ
.
The wave equation governing the laser plasma interaction dynamics is
given by
tJ
cmea
tcm
03
02
2
22 41
. (3.2)
The plasma current density, in the presence of the magnetic field is given by
mmm venJ , (3.3)
where nm is the plasma electron density, -e is the electronic charge and mv is its
velocity. Quantities with subscript m have been evaluated in the presence of the
magnetic field. Expanding plasma current density in orders of the radiation field
10 a gives
10
1mm venJ
(3.4a)
78
1120
2mmmm venvenJ
(3.4b)
and
122130
3mmmmmm venvenvenJ
. (3.4c)
The velocities and densities may be obtained by perturbative expansion of
the Lorentz force equation
bBvcm
eacvdtd
mmm
00 (3.5a)
and the continuity equation
0.
mmm vnt
n . (3.5b)
Simultaneous solution of various orders of Eqs. (3.5a) and (3.5b) gives the first and
third order velocities as
..ˆˆ,
20
0220
001 cceeietraicv tzkiyx
c
cm
o
(3.6a)
and
79
..ˆˆ,
2003
04220
40
403 cceeietraicv tzki
yx
c
cm
. (3.6b)
The second order velocity is found to be zero. Using Eq. (3.5b) the higher order
density perturbations are found to be zero, i.e. 021 nn . The time derivative of
the total current density is thus given by
3220
30
302
0220
0200
2,1
2c
c
c
cm tracent
J
..ˆˆ, 000 cceeietra tzki
yx . (3.7)
Wave equation
Substituting Eq. (3.7) into Eq. (3.2) gives
traa
ctra
tcc
c
c
cp ,42
,1422
0
40
402
0220
002
2
2
2
22
(3.8)
The first term on the right side of Eq. (3.8) is the linear source term driving the
laser amplitude while the second term is nonlinear. Now, considering only the
linear source term and taking Fourier transform of both sides of Eq. (3.8) gives
80
0,ˆ
2 220
002
2
2
22
ra
cc c
cp , (3.9)
where ,ˆ ra is the Fourier transform of tra , . To introduce the role of the laser
spot-size for a Gaussian beam, 204 r is added and subtracted on the left hand side
of Eq. (3.9), to give
0,ˆ40
02
2
2
20
2
zik
Lm eracr
, (3.10)
where 00 ,ˆ ra is the Fourier transform of the slowly varying amplitude
tra ,0 and
21
220
002
2
220
2
241
c
cpLm r
c
is the linear part of the total
refractive index having contributions due to vacuum, finite spot-size of the laser
radiation and presence of magnetized plasma, respectively. Defining mode
propagation constant Lmm c)( and substituting in Eq. (3.10) gives
0,ˆ22
2 02000
20
2
02
ra
rkkk
zik m . (3.11)
81
In the limit that the mode propagation constant is close to the unperturbed wave
number 0k , Eq. (3.11) may be written as
0,ˆ22 0200
002
rark
kz
ik m . (3.12)
Using Taylor series expansion the frequency dependent function may be
expanded about 0 as
nmn
mmm n 02
20100 !
1..............21
,
(3.13)
where 0
nm
nnm dd . In Eq. (3.13) m2 is related to the group
velocity dispersion (GVD). Substituting Eq. (3.13) into Eq. (3.12) gives
mrkk
zik 102
00000
2 22
0,ˆ........21
022
0
ram
. (3.14)
82
Taking inverse Fourier transform of Eq. (3.14), retaining terms up to m2
( 021 gg vv , where gv is the group velocity) and introducing nonlinear
current source term on the right hand side gives the nonlinear, nonparaxial wave
equation as follows,
tratt
irk
kz
ik mm ,
222 02
22
1200
0002
trac
ac
cp ,4
02220
40
40
2
220
. (3.15)
Growth rate of modulation instability
In order to study spatial modulation instability, it is convenient to carry out
transformation from spatial and temporal coordinates tz, in laboratory frame to
spatial coordinates ,z in pulse frame. The transformation is achieved by
substituting tvz g and zz , the differential operators in Eq. (3.15) may be
written as: zz and . gvt Substituting the nonlinear
parameter 4220
240
40
22 4 ccpNLm c , setting 00 k , gm v11
and neglecting 22 z in comparison to zk 02 , Eq. (3.15) may be written, in the
1-D limit, as
83
0,2212 0
20
22
2
22
20
raa
zv
zik NLmgm
. (3.16)
In the long pulse limit, variation of the laser amplitude with respect to the
coordinate may be considered to be a perturbation on the equilibrium. Thus the
zeroth order ( independent) solution of Eq. (3.16) may be written as
Rm Z
zPiaza 0000ˆ2exp)( , (3.17)
where 00a is the initial normalized peak amplitude and mP0 is the normalized laser
power in presence of axial magnetic field. Assuming the first order contribution to
the pulse amplitude, obtained due to variations, to be of the same form as that of
the unperturbed amplitude (Eq. (3.17)), the total amplitude may be written as
Rm
Rm Z
zPizaZzPiaza 0100000
ˆ2exp),(ˆ2exp),( . (3.18)
where ,10 za is the complex perturbed beam amplitude. Substituting ),(0 za
from Eq. (3.18) into Eq. (3.16) gives
84
0)(ˆ21ˆ2
21 *
1010010
2
0
100
0210
22
210
aa
ZP
za
kaP
Zkiav
zai
R
mm
Rgm
(3.19)
Considering 10*10 aa , Eq. (3.19) reduces to
0ˆ21
21ˆ4
100
02
02
22
20
a
ZkP
zi
zkv
ZP
R
mgm
R
m
. (3.20)
Operating the left hand side of Eq. (3.20) (from the left) by
R
mgm
R
m
ZkP
zi
zkv
ZP
0
02
02
22
20
ˆ2121ˆ4
gives,
2
2
0
22
2
2
220
20
2
22
0
022
4
20
4
4422 ˆ4ˆ814
k
vZk
PzzZk
Pzk
v gm
R
m
R
mgm
0ˆ2
102
22
20
av
ZP
gmR
m
(3.21)
The exponentially varying perturbed amplitude may be taken to be of the form
ikzaikzaza expexp),(10 , (3.22)
85
where k is the propagation wave number of the perturbed wave amplitude.
Substituting Eq. (3.22) into Eq. (3.21) yields
222
04422
0
222
02
2
20
2 ˆ241ˆ81 kv
ZPkv
zkkkv
ZP
zkk
gmR
mgmgm
R
m
0ˆ4
20
2
2
20
za
kk
ZP
R
m
(3.23)
Taking za to vary with z as Kzexp , where K is the modulation wave
number, we get the dispersion relation for one-dimensional modulation instability
as
0ˆ4
ˆˆˆˆˆ16ˆˆˆˆˆ8ˆˆ1 22
020
22
220
22
kPPkKkkPKk m
mmmmm , (3.24)
where 0ˆ kkk , KZK Rˆ and mRgm Zkv 2
222 81ˆ are normalized
dimensionless quantities. Eq. (3.24) represents a quadratic in K , having
roots
.)ˆ1(4
ˆˆˆˆˆ)ˆˆˆ()ˆˆˆ(ˆ1
ˆ4ˆ 22
020
222
2220
2202
kPPkkPkP
kkK m
mmmmmmm
(3.25)
86
Modulation instability is excited provided m2 is sufficiently
negative 4ˆ3ˆ02 mm P , so that K can be complex. Consequently the range of
unstable wave numbers for which the instability exists is given by
2
20
20
02
ˆ2
ˆ
ˆ4
ˆ3ˆˆ
mm
mm
m
P
PPk
. (3.26)
The growth rate of modulation instability for the laser beam propagating through
transversely magnetized plasma is given by the imaginary part of Eq. (3.25) as
2
4202
2042
24
2022
0
ˆ14
ˆˆˆˆˆˆˆˆˆˆˆˆ434
k
kPkPkkPkP mmmmmmm
m
. (3.27)
The spatial growth rate of modulation instability for right (curve a) as well
as left (curve c) circularly polarized laser beam propagating in magnetized and
unmagnetized (curve b) plasma as a function of normalized wave number k ,
using Eq. (3.27), is plotted in Fig. (3.1). The parameters used are mr 150 ,
271.00 a , 1150 1088.1 s , 1.00 p and 05.00 c (b = 5.35 MG). It
may be noted that the curves for the spatial growth rate of modulation instability
87
are identical for left as well as right circularly polarized light, in the absence of
magnetic field. Due to the presence of the magnetic field, the peak growth rate of
modulation instability for right circularly polarized laser beam is reduced by about
18% while for left circularly polarized laser beam, it increases by about 22% as
compared to the unmagnetized case.
Using Eq. (3.26), stability boundary curves are plotted in Fig. (3.2) showing
the variation of normalized laser power mmP 20ˆˆ with k for right (curve a) and
left (curve c) circularly polarized laser beam propagating in magnetized plasma and
compared with the unmagnetized case (curve b). The curves are plotted for the
same parameters as used for plotting Fig. (3.1). It is observed that the area bounded
by the stability curve representing unstable interaction in parameter space 0,ˆ Pk is
significantly reduced for a right circularly polarized laser beam (curve a)
propagating in axially magnetized plasma as compared to the unmagnetized (curve
b) case while in the case of left circularly polarized laser beam (curve c) the area
bounded by the stability curve increases. The reduction in the area bounded by the
stability curve denotes the reduction in the cutoff power above which the laser
pulse is stable. Thus for right circularly polarized laser beam the cutoff power is
reduced while for left circular polarization the cutoff power is increased.
88
0
0.2
0.4
0.6
0.8
1
1.2
0 0.05 0.1 0.15 0.2 0.25 0.3
Fig. 3.1: Variation of modulation instability growth rate for right (curve a),
and left (curve c) circularly polarized laser beam propagating in
magnetized plasma and for laser beam propagating in
unmagnetized (curve b) plasma, with normalized wave number
for mr 150 , 271.00 a , 1150 1088.1 s , 1.00 p
and 05.00 c (curves a and b).
89
0
0.4
0.8
1.2
1.6
0 0.4 0.8 1.2
Fig 3.2: Stability boundary curves showing the variation of normalized laser
power mmP 20ˆˆ with k for right (curve a), left (curve c)
circularly polarized laser beam propagating in magnetized plasma
and unmagnetized case (curve b). The parameters used are
1150 1088.1 s , 271.00 a , 1.00 p and 05.00 c .
90
CHAPTER 4
SECOND HARMONIC GENERATION IN LASER MAGNETIZED PLASMA INTERACTION
In Chapter 2 it has been shown that when an intense laser beam interacts
with homogeneous plasma embedded in a transverse magnetic field, second order
transverse plasma electron velocity oscillating with frequency twice that of the
laser field is set up (Eq. 2.16). This plasma electron velocity couples with the
ambient plasma density leading to a transverse plasma current density oscillating at
the second harmonic frequency. Also, a first order density perturbation oscillating
at the laser frequency (Eq. 2.19a) arises due to the presence of the external
magnetic field. This density perturbation couples with the fundamental transverse
quiver velocity (Eq. 2.14a) to give a transverse plasma current density oscillating at
twice the laser frequency. These two transverse current density contributions point
towards the possibility of generation of second harmonic frequency of the laser,
due its propagation in transversely magnetized plasma [81].
Formulation
Consider a linearly polarized laser beam propagating along the z-direction
in cold, underdense plasma. The plasma is embedded in a transverse magnetic field
ybb ˆ
. The electric component of the laser field is given by
91
..ˆ21
000 cceEeE tzki
xl
, (4.1)
where 0 is the frequency and 0k is the propagation constant of the laser. As the
beam propagates through transversely magnetized plasma, transverse current
density at frequency 02 arises [80] and acts as a source for second harmonic
generation. Corresponding to the frequencies 0 and 02 , the electric field is
assumed to be given by
..ˆ21
0111 cceEeE tzki
x
(4.2)
and
..ˆ21
02 222 cceEeE tzki
x
, (4.3)
respectively. The amplitudes 1E and 2E are assumed to be z-dependent. 1k
and 2k
represent propagation vectors at frequencies 0 and 02 respectively and their
values are given by
10
1 c
k (4.4)
and
92
20
22
c
k . (4.5)
Here 1 and 2 represent the corresponding wave refractive indices.
The wave equation governing the propagation of the laser pulse through
plasma is given by
t
Jc
Etc
m
22
2
22 41 . (4.6)
where 21 EEE
.
The plasma current density is given by
mmm venJ , (4.7)
where mv and nm are the plasma electron velocity and density, in presence of the
transverse magnetic field, respectively. The equations governing relativistic
interaction between the electromagnetic field and plasma electrons are given by the
Lorentz force equation
)()(
00
bBvcm
emEe
dtvd
mmm
, (4.8)
93
and the continuity equation
0).(
mmm vnt
n . (4.9)
m is the relativistic factor and B
is the magnetic vector of the radiation field. The
plasma is assumed to be cold so that initially the plasma electrons are at rest and
the external magnetic field does not exert a force on them.
Using perturbative technique all quantities can be expanded in orders of the
radiation field. With the help of Eq. (4.8) the first order equations for velocity along
x and z directions are respectively given by
cmzxxmx vEE
me
tv
)1(21
0
)1(
(4.10)
and
cmxmz vt
v)1(
)1(
, (4.11)
where cmebc 0/ is the cyclotron frequency of the plasma electrons. Using
Eqs. (4.2) and (4.3), Eqs. (4.10) and (4.11) can be simultaneously solved to give the
first order transverse and longitudinal velocities as
94
..42
20201 2
220
202
220
201)1( ccecaecaiv tzki
c
tzki
cmx
, (4.12)
and
..42
10201 2
220
0222
0
01)1( ccecaecav tzki
c
ctzki
c
cmz
, (4.13)
where 0011 cmeEa and 0022 cmeEa .
The second order equations for velocities are given by,
)2()1(
)1()1()1(
0
)2(
mzcmx
mzymzmx v
zvvBv
cme
tv
(4.14)
and
)2()1(
)1()1()1(
0
)2(
mxcmz
mzymxmz v
zvvBv
cme
tv
. (4.15)
Using Eqs. (4.12) and (4.13), the simultaneous solution of Eqs. (4.14) and (4.15)
leads to the second order transverse velocity
..4
44
012
220
2220
21
220
201
2)2( cceakciv tzki
cc
ccmx
. (4.16)
95
The first order plasma electron density is obtained from Eq. (4.9) as
..
4221
0201 222
0
20222
0
1011 cceanckeanckn tzki
c
ctzki
c
cm
. (4.17)
In Eq. (4.17) the density perturbation is generated due to the first order longitudinal
velocity of the plasma electrons oscillating at fundamental as well as second
harmonic frequency.
The transverse current density can now be obtained by perturbatively
expanding Eq. (4.7) and substituting the plasma electron density (Eq. 4.17) and
velocity (Eqs. (4.12) and (4.16)). Thus,
tzki
c
tzki
c
mxmmxmxmx
eaiecnecaienvenvenvenJ
0201 222
0
2200
220
2010
1120
10
4)(2
..44
3012
220
2220
21
220
201
20 cceakceni tzki
cc
cc
. (4.18)
The first term on the right hand side of Eq.(4.18) gives the current density
oscillating at the fundamental frequency while the second and the third terms
represent the current density at the second harmonic frequency which arises via (i)
transverse plasma electron velocity oscillating at the second harmonic frequency
96
and (ii) coupling of the electron density oscillation at the fundamental frequency
with the transverse electron quiver velocity also oscillating at the fundamental. The
latter contribution is attributed to the external magnetic field and provides the
source for generation of second harmonic radiation.
Substitution of the lowest order fundamental and second harmonic current
densities in Eq. (4.6) leads to the linear fundamental and second harmonic
dispersion relations given by
220
2202
021
2
c
pkc
(4.19a)
and
220
2202
022
2
44
4c
pkc
(4.19b)
respectively. In the absence of magnetic field ( c =0) Eqs.(4.19) reduce to the well
known linear, fundamental and second harmonic dispersion relations for a laser
beam propagating in plasma [16]. The refractive indices corresponding to the
fundamental and second harmonic frequencies are m1 = 21220
21 cp and
m2 = 21220
2 41 cp respectively.
97
4.2 Second harmonic generation
In order to obtain the amplitude of the second harmonic field, the current
density (Eq. 4.18) is substituted in the wave equation (4.6) and second harmonic
terms are equated, to give
tzkieac
kz
ikz
02 222
202
202
2 42
tzki
c
p ea 02 2222
0
220
4
4
tzki
cc
ccp eac
k0122
1220
220
220
2201
4
3
(4.20)
Assuming that the distance over which zza )(2 changes appreciably is large
compared with the wavelength ( 22
2 )( zza << zzak )(22 ) and that 1a depletes
very slowly (with z), so that the quantity 21a can be considered to be independent of
z, the evolution of the amplitude of the second harmonic is given by
)(22
022
0
220
220
2
1212
423 kzi
cc
ccp ekk
cia
zza
(4.21)
where 12 2kkk .
98
Integrating Eq. (4.21) and applying the initial condition that at z = 0, 02 za ,
gives
20
22
20
2211
20
2
20
2
20
2211
20
2
20
2
20
221
2
411
41
41
111
83
cccp
ccp
pcc
aza
k
zkzki
2.sin
2.exp (4.22)
The second harmonic conversion efficiency ( ) is defined as
21
22
1
2
a
a
. (4.23)
Substituting the value of )(2 za from Eq. (4.22) into Eq. (4.23) leads to
2
2
2
20
24
20
2
2
20
2
211
20
2
2
2
211
20
2
2
2
4
4
2
221 2.sin
411
1
41
41
11
169
kzk
ca
cc
c
c
c
p
c
c
p
c
pc
(4.24)
99
For a given value of k , the conversion efficiency is periodic in z. The minimum
value of z for which η is maximum, is given by
klz c . (4.25)
The length cl represents the length of plasma upto which the second harmonic
power increases. For z > cl the second harmonic power reduces again. The
maximum second harmonic efficiency (obtained after traversing a distance cl ) is
given by
22
20
24
20
2
2
20
2
211
20
2
20
2
211
20
2
20
2
40
4
2
221
max1
411
1
41
41
11
169
kca
cc
c
cp
cp
pc
.
(4.26)
The variation of conversion efficiency ( ) with z for 0 p
= 0c = 0.1 MGb 7.10 , for a laser beam of intensity 1017 W/cm2 and
wavelength 1 μm ( 21a = 0.09), propagating through transversely magnetized plasma
is shown in Fig. (4.1). The maximum value of is seen to be 0.093% and is
100
obtained after the laser beam traverses a distance of 0.006 cm in transversely
magnetized plasma. For a laser having minimum spot-size of 15 m , this distance
is equivalent to RZ085.0 . Fig. 4.2 shows the variation of maximum conversion
efficiency ( max ) with the normalized cyclotron frequency (magnetic field varying
from 0 to 12.84 MG), for 0 p = 0.1 and 21a = 0.09. It is seen that max initially
increases gradually with magnetic field ( 0c = 0.05) after which a sharp
increase in max is seen. The conversion efficiency increases upto 0.1% for
0c = 0.1.
101
0
0.03
0.06
0.09
0.12
0 0.01 0.02 0.03
Fig. 4.1: Variation of conversion efficiency ( ) with the propagation
distance z, for 0c = 0.1= 0 p , 21a =0.09 and 0 = 1.88
×1015 s-1.
102
0
0.02
0.04
0.06
0.08
0.1
0 0.02 0.04 0.06 0.08 0.1 0.12
0c
Fig. 4.2: Variation of maximum conversion efficiency ( max ) with 0c for
0 p = 0.1, 21a = 0.09 and 0 = 1.88 ×1015 s-1.
103
CHAPTER 5
CONCLUSIONS
5.1 Conclusions In the present thesis the propagation of intense laser beams in magnetized
plasma has been studied. The underdense plasma is assumed to be cold so that
initially the plasma electrons are at rest and the external magnetic field plays no
role. The interaction is studied in the mildly relativistic regime using the
perturbative expansion method. The nonlinear wave equation governing the
propagation of a laser beam in magnetized plasma is set up and is coupled with the
Lorentz force and continuity equations to study various phenomena..
The evolution of the spot-size of a linearly (circularly) polarized laser beam
propagating in transversely (axially) magnetized plasma is obtained with the help
of source dependent expansion method. A graphical analysis of the variation of
laser spot with propagation distance as well as external magnetic field shows that
the self-focusing of the laser beam varies due to the presence of the magnetic field.
It is seen that the critical power required for self-focusing the laser beam in plasma
also changes due to the presence of the magnetic field.
When a linearly polarized laser beam propagates in transversely magnetized
plasma, the force acting on the plasma electrons due to externally applied magnetic
field introduces changes in the relativistic mass and causes plasma electron density
perturbations. This leads to modification in propagation characteristics of the laser
104
beam. It is seen that transverse magnetization of plasma enhances the self-focusing
property of a linearly polarized laser beam. The critical power required to self-
focus the linearly polarized laser beam propagating in transversely magnetized
plasma is reduced. The above results are also valid for a left circularly polarized
laser beam propagating in axially magnetized plasma. However, if the laser beam is
right circularly polarized, the beam will be defocused and the critical power will
increase. Focusing of the right circularly polarized beam can be brought about by
reversing the direction of the external magnetic field. The theory can find
application in laser driven fusion scheme as well as laser wakefield accelerators.
The spatial growth rate of modulation instability for a circularly polarized
laser beam propagating in axially magnetized plasma is analyzed using a one
dimensional model. Magnetic fields alter the growth rate of modulation instability.
For a given set of parameters, the peak growth rate of modulation instability for a
left circularly polarized laser beam is found to increase by 22% as compared to the
unmagnetized case while for right circularly polarized beam the spatial growth rate
reduces by about 18% in the presence of the magnetic field as compared to its
absence. The stability boundary curve showing the variation of the normalized laser
power with normalized wavenumber for unmagnetized and magnetized cases are
plotted. It is seen that for left circularly polarized beam, the area representing the
unstable interaction is increased while that for left circularly polarized laser beam it
reduces.
105
Generation of second harmonic frequency of a linearly polarized laser beam
propagating in homogeneous plasma in presence of a transverse magnetic field has
been analyzed. The amplitude of second harmonic frequency has been derived and
hence its conversion efficiency has been obtained. It is seen that second harmonic
conversion efficiency oscillates as the wave propagates along the z-direction. It is
found that maximum conversion efficiency is zero in the absence of magnetic field
and increases as the magnetic field is increased. However, close to electron
cyclotron resonance the theory breaks down. The conversion efficiency also
increases with increase in intensity of the laser beam. Conversely, observation of
second harmonics in homogeneous plasma could point towards the possibility of
presence of a magnetic field, since second harmonics have so far been generated by
the passage of linearly polarized laser beams through inhomogeneous plasma.
5.2 Recommendations for future work
In the present thesis, interaction of laser radiation with magnetized plasma
has been studied. Nonlinear processes such as self-focusing, modulation instability
and second harmonic generation have been analyzed in the mildly relativistic
regime. Recent experiments and simulation studies have shown that self-generated
magnetic fields increase with the laser intensity. Therefore, the effects observed in
the case of mildly relativistic regime are expected to become more significant for
ultrarelativistic laser beams interacting with magnetized plasma. Thus the study of
106
evolution of the laser spot for ultraintense beams propagating in magnetized
plasma will be an interesting proposal for future work.
Instabilities (other than modulation instability) such as stimulated Raman
scattering, stimulated Brillouin scattering (SBS) and filamentation will also be
affected due to the presence of an external magnetic field. The growth of these
instabilities for a laser beam propagating in magnetized plasma can also be taken
up as future work.
Wakefields generated by laser pulses interacting with magnetized plasma
can also be taken up as future work because in the presence of a magnetic field the
self-focusing property of the laser beam is enhanced. This may play an important
role in the development of laser wakefield accelerators (LWFA).
The present work has been done in the underdense regime neglecting
thermal effects. The study of laser-plasma interaction, in presence of magnetic
fields, for plasma densities close to critical value (including thermal effects) can be
explored.
107
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