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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