Contents
Page
Acknowledgements
Certificate
1
2-3
Journal publications
4
Conference proceedings
5
Summary
6
List of figures
Chapter 1: Introduction
11
16-51
1.1 Development of lasers
17
1.2 Nonlinear phenomena arising through laser-plasma interaction
23
1.2(a) Self-focusing of laser beams in plasma
23
1.2(b) Wakefield generation
30
1.2(c) Laser pulse compression
34
1.3 Propagation of a laser beam in a plasma channel
40
1.4 Variational technique
46
1.5 Approach
48
1.6 Aim
50
Chapter 2: Simultaneous evolution of spot-size and length and induced chirping of short
laser pulses in a plasma channel
52-81
2.1 Evolution of spot-size and pulse length
52
2.2 Analytical solution for laser spot-size and pulse length
63
2.3 Numerical solution for simultaneous evolution of laser spot-size and pulse length
69
2.4 Group velocity dispersion and self-phase modulation induced chirping of short laser
pulses in a plasma channel
74
Chapter 3: Propagation of ultrashort chirped laser pulses in a plasma channel
82-102
3.1 Analysis of wave equation
82
3.2 Effect of chirp on matched laser pulse propagation
90
3.3 Variation of laser frequency across chirped laser pulses
96
Chapter 4: Wakefield effects on the evolution of symmetric laser pulses in a plasma
channel
103-122
4.1Wave dynamics
103
4.2 Density perturbation
106
4.3 Evolution of laser spot-size and pulse length
111
4.4 Graphical description
118
2
Chapter 5: Conclusions
123-128
5.1 Conclusions
123
5.2 Recommendations for future work
127
References
129
3
Acknowledgements
First and foremost it is my pleasure to thank my supervisor, Prof. Pallavi Jha,
Department of Physics, University of Lucknow, for all her support and enthusiasm during
the course of my Ph. D. She is an inspirational teacher and tireless campaigner for the
cause of scientific rigour. She has been a fantastic supervisor, who maintains an
overwhelming energy throughout her work and carries that through to all around her. I
am very fortunate to have had the chance to work with her so closely.
I express my gratitude to Prof. U.D. Misra, Head, Department of Physics,
University of Lucknow, for being supportive.
My special thanks to Dr. Ajay Kumar Upadhyay, Dr. Gaurav Raj and Dr. Rohit
Kumar Mishra for their enthusiastic help and timely advice towards completion of the
thesis. I am thankful to Mr. Ram Gopal Singh, Mr. Vijay Singh, Ms. Akanksha Saroch
and Mrs. Anjani Singh for creating a congenial work environment in our Plasma
Electrodynamics Group at the University.
I am specially indebted to my Parents Mrs. Pratima Malviya and Mr. Ram
Krishna Malviya, brother Mr. Anand Malviya and entire family for their unflinching and
whole-hearted support throughout the endeavour.
The thesis in the present form would not have been possible without the blessings
of my grand parents Mr. Ram Shankar Malviya and Late Mrs. Chanda Malviya.
4
Journal Publications
1.
Simultaneous evolution of spot-size and length of short laser pulses in a
plasma channel
Pallavi Jha, Amita Malviya, Ajay K. Upadhyay and Vijay Singh
Plasma Phys. Control. Fusion, vol. 50, 015002 (2008).
2.
Propagation of chirped laser pulses in a plasma channel
Pallavi Jha, Amita Malviya and Ajay K. Upadhyay
Physics of Plasmas, vol. 16, 063106 (2009).
3.
Wakefield effects on the evolution of symmetric laser pulses in a plasma channel
Pallavi Jha, Amita Malviya and Ajay K. Upadhyay
Laser and Particle Beams, vol. 28, 245 (2010).
5
Conference Proceedings
1.
Simultaneous evolution of pulse length and spot of laser beam in homogeneous
plasma
Amita Malviya, Ajay K.Upadhyay, and Pallavi Jha
22nd National Symposium on Plasma Science and Technology-2007 (Ahmdabad).
2.
Group velocity dispersion and self-phase modulation induced chirping of short
laser pulses in a plasma channel
Amita Malviya, Ajay K. Upadhyay and Palllavi Jha
24th National Symposium on Plasma Science and Technology-2009 (Hamirpur).
6
Summary
Chirped-pulse amplification technique has permitted the production of laser
pulses with peak powers ranging from 100 terawatt (TW) to 1 petawatt (PW) by
stretching, amplifying and then recompressing. High power is achieved due to very short
pulse duration. Shorter pulses would be of interest because they could lead to higher peak
powers without increasing the size and cost of the laser system. These pulses find
applications in ultrahigh intensity laser-plasma interaction such as particle acceleration,
high harmonic generation and X-ray lasers. For all these applications long distance
propagation of intense lasers in plasma is desirable. However, since diffraction will
spread the laser beam after the focus, the effective interaction length is limited by the
Rayleigh length Z R
 k r
2
0 0
2 , where k0 and r0 are the laser wavenumber and beam
waist respectively). A plasma channel having a parabolic radial density variation prevents
diffraction of the laser pulse. While propagating in a plasma channel the laser spot
undergoes betatron oscillations which allows the pulse to remain confined close to the
channel axis. For a moderately intense laser pulse propagating in a pre-formed plasma
channel, focusing will be brought about due to the channel as well as relativistic selffocusing. On account of focusing of the laser beam, its pulse length is affected as the
laser pulse propagates in plasma. Thus the evolution of the spot-size and pulse length are
coupled. Therefore it is important to study the simultaneous evolution of the laser spotsize and pulse length for most laser-plasma applications. Also for short laser pulses,
wakefields will be generated and will affect the propagation characteristics in plasma.
7
The variation of pulse length and spot-size of the laser beam in turn affects the wakefield
generation.
The present thesis is a theoretical analysis of the evolution characteristics of short
laser pulses propagating in a plasma channel. The effects of group velocity dispersion,
induced chirp, as well as relativistic and ponderomotive nonlinearities have been
considered, to study the evolution dynamics. The wave equation for the laser field
amplitude is set up and an appropriate Lagrangian is defined. A variational technique is
used to analyze the laser-plasma system.
Chapter 1 presents the basic properties of plasma and condition required for the
propagation of electromagnetic waves inside the plasma. A brief summary of the
development of lasers and the properties of laser beams is presented. Nonlinear
phenomena such as self-focusing of a laser beam in plasma, wakefield generation and
laser pulse compression dynamics arising due to interaction of laser pulses with plasma
have been discussed briefly. The utility of a plasma channel for the purpose of guiding
laser beams has been shown. The method of using the variational technique as a
mathematical tool for describing the evolution of laser pulses in plasma has been
presented. This technique has been used for all analyses presented in the thesis.
In Chapter 2, nonparaxial, nonlinear propagation of an intense, short, Gaussian
laser pulse in a plasma channel is studied in a particular regime in which relativistic
nonlinearity dominates over ponderomotive effects. Also the parameters are limited to a
regime in which the group velocity dispersion effect dominates over finite pulse length
effect. Using the variational technique, simultaneous equations describing the evolution
of the laser spot-size and pulse length have been set up. An analytical solution is obtained
8
for the spot-size (pulse length) by considering the pulse length (spot-size) to be constant.
Further, numerical solutions for simultaneous evolution of the laser spot-size and pulse
length are obtained and graphically analyzed and the condition for achieving pulse
compression has been highlighted. Simultaneous solutions show a higher rate of
compression of pulse length as compared with the constant spot-size case. Also, the
focusing effect increases for the evolving pulse length case. It is seen that the presence of
the channel leads to compression of the pulse length whereas in the absence of the
channel, a broadening effect is observed. It is also observed that a laser spot satisfying the
beam matching condition, when pulse compression effects are neglected, does not remain
matched when the laser spot and pulse length evolve simultaneously. The chirp induced
over the laser pulse due to group velocity dispersion and self-phase modulation is
evaluated. These studies point towards the possibility of generating high intensity laser
pulses via pulse compression and production of chirped laser pulses. The present study
reveals that an unchirped pulse becomes chirped as it propagates in plasma. This is due to
group velocity dispersion and self-phase modulation.
The propagation of an initially chirped ultrashort laser pulse in a pre-formed
plasma channel is presented in Chapter 3. For ultrashort laser pulses, relativistic and
ponderomotive nonlinearities cancel each other, hence a linear analysis can
approximately describe the propagation dynamics. Evolution of the laser spot-size and
phase shift with propagation distance of the laser pulse is analyzed. The effect of initial
chirp on the laser pulse length and intensity of a matched laser beam propagating in a
plasma channel has been presented. The effective pulse length and chirp parameter of the
laser pulse due to its interaction with plasma have been obtained and graphically
9
depicted. It is observed that for a positively chirped laser, the pulse length is compressed
up to a certain distance and then starts to broaden. However, the length of a negatively
chirped pulse broadens continuously and broadening is more rapid in comparison to an
unchirped pulse. For a positively chirped pulse, intensity of the matched beam increases
in the region where pulse compression occurs. The intensity of a negatively chirped pulse
is damped in comparison to its initial value. Also, after the pulse has traversed a certain
critical distance, the positively chirped laser pulse becomes unchirped. Beyond this
critical distance, chirp reversal occurs for a positively chirped laser pulse. This reversal
occurs because the group velocity induced negative chirp and its coupling with initial
chirp becomes more effective in comparison to the initial positive chirp. If the distance
traversed by the laser pulse is less than the critical distance, reversal of chirp does not
occur. Also chirp reversal does not occur for negatively chirped pulses. The resultant
variation in the laser frequency across the laser pulse is evaluated.
Chapter 4 deals with the combined effect of wakefields (inside the laser pulse)
and relativistic nonlinearity on the propagation of a laser pulse in a parabolic plasma
channel. Quasi-static approximation is used to obtain density perturbations. Considering a
wide channel, the radial dependence of the plasma frequency has been neglected. The
variation in density of the transverse and axial ponderomotive nonlinearity is evaluated
and coupled equations describing the evolution of the laser pulse length and spot-size are
set up. In the presence of wakefields, the evolution equation for pulse length and spotsize exhibit singularity if the pulse length approaches the plasma wavelength. Therefore
in this limit, the density perturbation is obtained using l’ Hospital’s rule and the evolution
equations for the pulse length and spot-size of the laser are derived separately. The laser
10
propagation characteristics have been analyzed with the help of numerical solutions of
these equations. The present study shows that the laser propagation dynamics is
extremely sensitive to the ratio of pulse length and plasma wavelength and show contrary
behavior for  (  L0  P where L0 and  P are the initial pulse length and plasma
wavelength) less than or greater than unity. This result will be significant for operation of
laser-plasma based accelerator systems.
Conclusions from the present research and recommendations for future work are
presented in Chapter 5.
11
List of figures
Page
Fig 1.1: Chirped pulse amplification concept.
19
Fig.1.2: History of light sources over the last century. Each advance in laser
intensity enables a new regime of optics.
20
Fig.1.3: Normalized pulse length ( L L0 ) against normalized propagation
distance ( z Z R ) in the absence of nonlinearity.
38
Fig.1.4: Normalized pulse length ( L L0 ) against normalized propagation
distance ( z Z R ) in the presence of nonlinearity.
39
Fig.1.5: Evolution of normalized spot size versus normalized propagation
distance
for a) vacuum diffraction
b) homogeneous plasma
c) a plasma channel.
Fig.2.1: Variation of
45
normalized pulse length L L0  with propagation
distance z Z R  in the absence of (curve a) and in the presence
(curve b) of the plasma channel for f 02  0.03,   1m, r0  33m,
L0  14m, P  10m, n  2  1017 and rch  50m .
Fig.2.2: Simultaneous variation of normalized spot size rs r0  (curve a)
12
70
and the normalized pulse length L L0  (curve b) with normalized
propagation distance
z
Z R  . Evolution of the spot-size keeping the
pulse length constant (curve c). Evolution of the pulse length keeping
the spot size constant (curve d) for f 02  0.03,   1m, r0  33m,
L0  14m, P  10m, n  2  1017 cm 3 and rch  50m .
72
Fig.2.3: Simultaneous variation of normalized spot-size rs r0  (curves a, b
and c) and the normalized pulse length L L0  (curves d, e and f ) with
normalized propagation distance z Z R  , f 02  0.03 , nc  1.7  1017

cm 3 , 0.04 nc  1.4  1017 cm 3  and 0.05 nc  1.2  1017 cm 3  ,
  1m , r0  33m, L0  14m, P  10m and rch  50m .
Fig.2.4: Variation
n
n
of
intensity
for
the
initial
values f 02  0.03
 0.04 n  1.4 10 cm 
. Other laser plasma parameters
c
 1.7  1017 cm 3 ,
c
 1.2  1017 cm 3
17
73
3
c
and
0.05
are   1m ,
r0  33m, L0  14m , P  10m and rch  50m.
75
Fig.2.5: Variation of normalized frequency  0  with z Z R . Solid (dotted)
curve a (curve d) , curve b (curve e ), curve c (curve f ) are due SPM,
GVD and combined effect of
SPM and GVD at front (back)
  0.5 - 0.5 of the pulse for f 02  0.03,   1m, r0  33m,
L0  14m, P  10m, n  nc  1.7  1017 cm -3 and rch  50m.
13
77
Fig 2.6: Variation of normalized chirp  0  across the laser pulse after
traversing distance z Z R  3 (curve a) and z Z R  6 (curve b) for
f 02  0.03,   1m, r0  33m , L0  14m, P  10m, rch  50m
and n  nc  1.7  1017 cm -3 .
Fig.2.7: Variation of intensity f
2
79
z,  
with  after pulse has traversed
a distance z Z R  0 (curve a), z Z R  3 (curve b) and z Z R  6
(curve c)
r0  33m,
  1m,
f 02  0.03,
for
L0  14m,
P  10m , n  nc  1.7  1017 cm -3 and rch  50m .
Fig.3.1: Variation of effective pulse length
Le0
L0  with
80
propagation
distance z Z D 0  for a positively chirped pulse ((curve a),   0.3 ),
unchirped
pulse
(curve b)
and
negatively chirped
pulse
((curve c),   0.3 ).
94

Fig 3.2: Variation of normalized intensity a 2

f 02 with  L 0 for the initial
laser pulse z Z D 0  0 , dotted curve) and after the laser pulse has
traversed a distance z Z D0  0.275 (critical distance) for chirp
parameter   0.3 (curve a),   0 (curve b) and   0.3 (curve c).
Fig.3.3: Variation
of
normalized
frequency
 0 
97
with  L 0 at
z Z D 0  0.1 for chirp parameters   0.3 (line a),   0 (line b) and
  -0.3 (line c) for r0  25m , L0  10m ,   1m and p  40m.
14
99
Fig.3.4: Variation
of
normalized
z Z D 0  0.275
frequency
for chirp
parameters
 0 
with  L 0 at
  0.3 (line a),   0
(line b) and   0.3 (line c) for r0  25m , L0 = 10 m ,   1m
and p  40m.
100
Fig. 3.5: Variation of
normalized
frequency
z Z D 0  0.45
for chirp
parameters
 0 
with  L 0 at
  0.3 (line a),   0
(line b) and   0.3 (line c) for r0  25m , L0 = 10 m ,   1m
and p  40m.
Fig.4.1: Variation
when
101
of normalized pulse length with propagation distance
  1 curve a,
  1 curve b
and   1 curve c for
L0  16m , 9m and 11m , r0  25m,   1m ,
f 02  0.015
P  11m , rch  40m and n  nc  4.3  1017 cm 3 .
120
Fig.4.2: Variation of normalized spot-size with propagation distance when
  1 curve a,   1 curve b and   1 curve c for L0  16m ,
9 m and 11m , r0  25m,
f 02  0.015 ,   1m , P  11m ,
rch  40m and n  nc  4.3 1017 cm 3 .
121
Fig.4.3: Variation of laser pulse intensity with propagation distance when
  1 curve a,   1 curve b and   1 curve c for L0  16m ,
9 m and 11m , r0  25m, f 02  0.015 ,   1m , P  11m ,
rch  40m and n  nc  4.3 1017 cm 3 .
15
122
Chapter 1
Introduction
Plasma is a quasi-neutral gas of charged and neutral particles which exhibits
collective behaviour. It is a phase of matter distinct from solids, liquids and normal gas.
Since plasma electrons are of very small mass, their motion controls the plasma
16
behaviour in fast processes. In fully ionized plasma, electron-electron collisions are
negligible and ionic thermal motion is also negligible. In plasma only electrons are
mobile, ions provide a neutralizing background of positive charges. Irving Langmuir, the
Nobel laureate who pioneered the scientific study of ionized gases, gave this fourth state
of matter, the name plasma. Any ionized gas cannot be called plasma. For an ionized gas
to be categorized as plasma, the existence of the following conditions must be fulfilled:
(i) the Debye length (  D  k B T 4n0 e 2 where k B , T , n0 , and e represent the
Boltzman constant, absolute temperature, plasma electron density and electronic charge
respectively), must be smaller than the plasma size, (ii) the number of electrons within a



sphere which has volume 43D 3 must be greater than unity 43D 3  1 , and (iii)
quasi-neutrality of plasma implies that the ion density must be equal to the electron
density. If the quasi-neutrality of plasma is disturbed by some external force, the charge
separation between ions and electrons gives rise to electric fields and the flow of plasma
electrons gives rise to current and magnetic fields. These electrons begin to accelerate in
an attempt to restore the charge neutrality. Due to their inertia they will move back and
forth about the equilibrium position, resulting in fast collective oscillations around the
more massive ions. The frequency of these oscillations is known as plasma frequency

P

 4n0 e 2 m

12

where m is the electron mass .
The propagation of electromagnetic waves through pre-ionized plasma is
governed by the dispersion relation c 2 k 02   02   P2 , where  0 and k 0 (  2 0 ,  0
being the radiation wavelength) are
the wave frequency and wavenumber. An
electromagnetic wave can propagate through plasma if its propagation constant is real i.e.
its frequency is greater than the plasma frequency. Such a plasma is known as underdense
17
plasma. As the plasma density increases  P becomes equal to  0 and the plasma is said
to be critically dense. With further increase in density, the wavenumber becomes
imaginary and the electromagnetic wave is unable to propagate. This plasma is known as
overdense plasma.
The interaction of electromagnetic waves with matter becomes interesting as well
as important if the intensity of the wave becomes high such that the index of refraction
becomes nonlinear i.e. it becomes a function of the radiation intensity.
1.1 Development of lasers
Since 1960, lasers [1-5] have evolved in peak power by a succession of leaps,
each three orders of magnitude. These advances were produced each time by decreasing
the pulse duration accordingly. Initially laser sources emitted a peak power in the
kilowatt (KW) range. In 1962, modulation of the laser cavity quality factor enabled the
same energy to be released on a nanosecond (ns) timescale, a thousand times shorter, to
produce pulses in the megawatt (MW) range. In 1964, locking the longitudinal modes of
the laser (mode locking) enabled the laser pulse duration to be reduced by another factor
of a thousand down to the picosecond (ps) level, thus increasing the peak power to the
gigawatt (GW) level.
Presently, chirped pulse amplification (CPA) [6-10] technique has led to the
production of femtosecond (fs) laser pulses generating multiterawatt (TW) and even
petawatt (PW) of power. In CPA, the pulse is first stretched by a factor of a thousand to a
hundred thousand. This step does not change the input pulse energy (input fluence), and
18
therefore the energy extraction capability, but it does lower the input intensity by the
stretching ratio. The pulse is then amplified by 6 to 12 orders of magnitude i.e. from the
nanojoule (nJ) to the millijoule-kilojoule (mJ-kJ) level and is finally recompressed by the
same stretching ratio back to a duration close to its initial value. The power production
through CPA is 10 3  10 4 times more than that produced by dye or excimer systems of
equivalent size. (Fig. 1.1)
At present, about fifteen petawatt lasers are being built around the world and plans
are afoot for new, even higher power lasers reaching values of exawatt (EW) or even
zetawatt (ZW) powers. Fig. 1.2 illustrates the increase in achievable laser intensity since
1960. High-intensity laser radiation may now be applied in many traditional areas of
science usually reserved to nuclear accelerators and reactors.
Fig. (1.1) Chirped pulse amplification concept
19
Fig.(1.2) History of light sources over the last century. Each advance in laser
20
intensity enables a new regime of optics. (Ledingham and Galster
New J. Phys. 12, 045005, (2010)).
As the laser intensity and associated electric field is increased the electron quiver energy,
(the energy a free electron has in the laser field) increases accordingly as illustrated in
Fig.1.2. When the laser radiation is focused onto solid and gaseous targets at
intensities  1018 W cm -2 , electron quiver with energies greater than their rest mass
energy (0.511MeV) are achieved, creating relativistic plasma.
In most laser applications it is necessary to know the propagation characteristics
of the laser beam. The simplest and most appropriate type of beam provided by a laser
source is a Gaussian beam. It is the lowest order ( TEM 00 ) mode of the laser optical
resonator. The mathematical function that describes the Gaussian beam is a solution to
the paraxial form of the Helmholtz equation. The amplitude of a fundamental Gaussian
laser beam as a function of the transverse and axial coordinates propagating along the zdirection is given by

ik 0 r 2
r2 

E r , z   E0 exp  i z  
 2 


2
R
z
rs z  

21
(1.1)
where rs z  is defined by the transverse distance at which the field amplitude drops to
1 e of its axial (peak) value. Rz  and  z  are the radius of curvature and phase shift of
the laser beam respectively. The characteristic laser beam parameters (evolving with
propagation distance) are therefore given by rs z  , R z  and  z  [11].
In vacuum the evolution of the spot-size along the direction of propagation  z  is
given by
  z
rs z   r0 1  
  ZR




2




12
(1.2)
where Z R  k 0 r02 2  is the Rayleigh length. It is the distance at which the laser spot
becomes 2
times the beam waist r0  . At the beam waist the wavefront is
planar, R0  . As the beam propagates outwards, the wavefront gradually becomes
curved and the radius of curvature Rz  drops down to finite values. For distances well
beyond the Rayleigh length, the radius increases again. The analytical form of the radius
of curvature Rz  of the wavefront is given by,
Z2
R z   z  R
z
  for z  Z R

 2 Z R for z  Z R
 z for z  Z
R



.


(1.3)
The third beam parameter,  z  is known as the Guoy phase shift and is given by
22
 z
 ZR
 z    tan 1 

 .

(1.4)
This factor should be taken into account when the exact knowledge of the wavefront is
needed for involved applications.
1.2 Nonlinear phenomena arising through laser-plasma interaction
With the advent of high intensity laser systems, studies based on nonlinear laserplasma interaction became a frontier field of Physics. The rapid progress in the
technology of high power lasers based on chirped pulse amplification makes possible, the
exploration of relativistic Plasma Physics in the laboratory. In this regime, plasma
electrons oscillate at relativistic velocities leading to nonlinear interaction dynamics.
Nonlinear propagation of laser pulses in underdense plasma can change its temporal,
spatial and spectral properties. These changes give rise to various interesting phenomena
such as self-focusing, pulse compression, self-phase modulation and wakefield
generation. In this context, the possibility of development of a wide range of applications,
such as high energy particle acceleration [12-16], inertial confinement fusion [17-19],
and harmonic generation [20-24], are being explored.
1.2(a) Self-focusing of laser beams in plasma
A primary phenomenon that occurs when a laser beam having a Gaussian radial
profile (intensity peaked on-axis) passes through plasma, is self-focusing [25-33]. When
23
intense laser beams interact with plasma, the refractive index is modified and is given by
 r   1   P2 nr   02 n0 r  . The change in refractive index occurs due to relativistic
12
mass correction  r  and the radial variation nr  of plasma electron density due to
ponderomotive effects. Since the laser intensity is peaked on-axis, plasma electrons are
expelled radially outward, leading to decrease (increase) in on-axis density (refractive
index). Also, the relativistic factor is maximum on the axis. This contributes to a
maximum value of refractive index at the laser axis. Therefore, the phase velocity of the
laser wavefront increases with the radial distance causing them to curve inwards and the
laser focuses towards the axis.
Consider the propagation of a linearly polarized laser beam in homogeneous
plasma. The vector potential of the radiation field is given by

Ar , z , t 
A  xˆ
exp ik 0 z  i 0 t   c.c.
2
(1.5)
where Ar, z, t  is the field amplitude and k0 ( 0 ) is the wavenumber (frequency). The
wave equation governing the propagation of the laser field in plasma can be obtained
from Maxwell’s time dependent equations [34] and is given by,

 2 1 2  
4J
  2 2  A  
c
c t 

(1.6)
24


where J   nev  is the plasma current density. In deriving Eq. (1.6), Coulomb gauge
 .A  0

has been used. The current density may be obtained with the help of the
Lorentz force equation,
 
1   
   
2  1 a
   v    a 
  v .  p  mc 
c
 t

 c t



(1.7)
and the continuity equation given by
n  
 .nv  0
t
(1.8)


where p mv   is the momentum of plasma electrons. Eq. (1.7) shows that the fast
response of the plasma electrons to the laser field gives rise to the electron velocity





12

 


2
v  v 1  v  ca  where    1  a 2  is the relativistic factor, a  eA mc 2 is





the normalized laser strength parameter and   e mc 2

is the normalized scalar
potential. Thus the lowest order quiver velocity of the plasma electrons is given by


 2


v 1  ca and the perturbed part of plasma electron velocity is given by v  ca a 4

in the mildly relativistic regime a  1,   1  a
2
2


4 . Substituting the value of the


 
plasma current density J (=  e n0 v 1  n0v  v 1n  where n is the higher order
density perturbation) into Eq. (1.6), gives the wave equation describing the evolution of
the vector potential driven by relativistic and ponderomotive nonlinearities, as,
25
 a 2 n  
 2 2
1 2  
2
 a
   2  2 2  a  k P 1 
4
n0 
z
c t 



(1.9)
where k P  P c is the plasma wavenumber. Substituting Eq. (1.5) into Eq. (1.9) and
transforming independent variables z, t to z ,  (  z   g ct , where  g  ck 0  0  is the
normalized group velocity of the laser pulse) gives
2
 2

2
2 
2 


2
ik

2

1



0
g
 
 ar , z,  
z
z
 2 z 2 



2

a
n 

 k 1
 ar , z,    k 02  P2  1 ar , z,  

4
n0 



2
P

(1.10)
where  P  0 ck 0 and  g  P  1 . If the laser pulse length L  is sufficiently greater
than the plasma wavelength P  and the laser beam is broad k P r0  1 and L  r0 ,
the electron density response n may be neglected in comparison to the relativistic
effect (Sec.1.2b). Thus Eq. (1.10) reduces to
2
 2

2
2 
2 


2
ik

2

1



0
g
 
 ar , z,  
z
z
 2 z 2 



2

a 

 k 1
ar , z,    k 02  P2  1 ar , z,   .

4 



2
P
26

(1.11)
Eq. (1.11) describes the nonparaxial wave equation governing the evolution of the laser
pulse amplitude. The first term on the left side of this equation represents transverse
amplitude variation, the second term shows the first order diffraction effect, the third
term represents finite pulse length effects, while the fourth and fifth terms represent
higher order finite pulse length (group velocity dispersion) and diffraction effects
respectively. Neglecting finite pulse length and higher order diffraction effects in Eq.
(1.11) the paraxial wave equation is given by
2

a 

 2
2
2
2
   2ik 0 z  ar , z ,    k P 1  4 ar , z,    k 0  P  1 ar , z,   . (1.12)




In order to study the evolution of the laser spot-size rs z  , several methods, such
as, the heuristic method [35-36], Variational method [37-51] or source dependent
expansion (SDE) method [52-56] may be used. For any of these techniques, a trial
function for the amplitude of the radiation field is assumed. For a Gaussian beam a
suitable trial function can be assumed as

ik r 2
r2 
a  f z  exp  i z   0  2  .
2 Rz  rs z  

(1.13)
Considering the heuristic method, the trial function (Eq. 1.13) is substituted into the
paraxial wave equation (1.12) to give
27
 4r 2
k 02 r 2
4ik 0 r 2
2ik 0 
4


 2 
 4
a
2
2
 rs z  R z  rs z Rz  rs z  Rz 
 1 f z   z  2r 2 rs z  ik 0 r 2 Rz 
i
 3

 2ik 0 
a
z
rs z  z
2R 2 z  z 
 f z  z
 k2
  P
 4


f


2

z 1 


2r 2  

  k P2  k 02  P2  1  a .
2
rs z  




(1.14)
Comparing imaginary and real parts and equating coefficients of r 2 and r 0 on both
sides of Eq. (1.14) gives the evolution of curvature, amplitude, spot-size and phase shift
of the laser as
R z   
rs
rs
z 
,
(1.15)
f 02 r02
f z   2 ,
rs
2
 2 rs
4
 2 3
2
z
k 0 rs
(1.16)
 k P2 f 02 r02 
1 
,
8


(1.17)
and
k P2


z 2k 0
 f 2 z   k 0 2
2

 1 
P 1  2
rs k 0
 4
 2


28
(1.18)
where r0 is the beam waist and f 0 (  eA0 mc 2 where A0 is the initial vector potential) is
the laser strength parameters. The first term on the right hand side of Eq. (1.17)
represents vacuum diffraction, whereas the second term leads to relativistic self-focusing.
The solution to Eq. (1.17) with initial condition rs z  0 at z  0 , is given by

rs2
P
 1  1 
2
r0
 Pc
where
 z2
 2
 ZR
(1.19)
  2r 2c5
P k P2 r02 f 02
0

. The laser power is given by P 

4
Pc
8

 e2c  
Pc   2  0
 2re   P

2


e2

 GW where re 
in
practical
units
2

mc


 m f0

 e0
2


 GW  and



is the critical power for
relativistic self focusing of the laser beam. Eq. (1.19) shows that the spot-size diffracts
for P  Pc , remains guided or matched rs  r0  for P  Pc and for P  Pc it focuses
initially but predicts catastrophic focusing after a certain value of z . This is due to the

approximation 1  a
2
2

1 2

 1 a
2

4 in the a  1 limit. In a relativistic situation,
2
however, higher-order nonlinearities will prevent the laser from focusing indefinitely. For
short laser pulses (pulse lengths less than a plasma wavelength) relativistic optical
guiding is significantly diminished even when the laser power exceeds the critical
power P  Pc  . 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.
29
1.2(b) Wakefield generation
When the laser pulses interacting with plasma are short k P L  1, L c  1 ps ,
density perturbations become important. The laser pulse will exert a significant radial and
axial ponderomotive force on the plasma electron. The radial ponderomotive force expels
electrons radially outward, while the front (back) of the laser pulse exerts a forward
(backward) axial force on the electrons. The ponderomotive force exerted by the laser
pulse on the plasma electrons can be obtained from the slow components of the Lorentz
force equation (1.7), as
 2

 2  v 1
v
2
 c  
t
2
(1.20)

where v represents the second order slow velocity. The first term on the right of Eq.
(1.20) is the space charge force while the second term represents the generalized

 v 1

force FP  m
 mc 2  2  .
2
2
ponderomotive
This
ponderomotive
force
is
responsible for generation of plasma waves. In order to analyze the excitation of plasma
waves, perturbation technique is used to expand the Lorentz force, continuity and
Poisson’s equations in various orders of the laser field amplitude. The zeroth order
describes the equilibrium plasma n  n0 ,  0   1 and  0   0 . The first order is the


electron quiver motion of the plasma electron in the laser field v 1  ca and
30
n 1   1   1  0. The higher order expansion of the momentum, continuity and
Poisson’s equations yields
2

1 v   2  a 
  

4 
c 2 t



(1.21)

n
 n0 .v  0
t

(1.22)
4e 2n
.
mc 2
(1.23)
and
 
 2  2  
Differentiating Eq. (1.22) with respect to time and substituting Eq. (1.21) and Eq.
(1.23)
gives the equation governing the evolution of the slow density perturbation as,
2
a
 2
 n
 2 2  k P2 
 2
.
4
 c t
 n0
(1.24)
In order to solve Eq. (1.24), the quasi-static approximation [57-61] is used. Under the
quasi-static approximation (QSA) it is assumed that the electron transit time (distance)
through the laser pulse is short in comparison with the characteristic laser pulse evolution
time (distance). This implies that, QSA is valid if  L   E L  Z R , where  L  L c 
31
is the laser pulse duration and  E is the laser pulse evolution time, which is on the order
of Rayleigh time Z R c  . Transforming independent variables   z  ct  and   t
(or z  z ) in the plasma fluid equation (1.24) and neglecting the derivative with respect to
 ( z ) (under QSA) gives,
 2
 2
2
2  n
 2  k P 
     2

 
 n0 
2
a

.
 4
(1.25)
Eq. (1.25) represents the governing equation for the generation of plasma waves
(wakefields) [62-71]. It has been shown [72] that if the laser pulse duration is
approximately equal to the plasma wave period a large amplitude plasma wave
(wakefield), with phase velocity close to the speed of light, is generated. This large
amplitude wakefield can be used for acceleration and/or focusing of electron bunches.
Accelerating gradients as high as a few tens of GV/m have been measured in
experiments. These gradients are approximately three orders of magnitude higher than
those achieved in conventional radio frequency linear accelerators.
The evolution of the density perturbation can be reduced under two separate
conditions viz 1) long pulse, and 2) ultrashort pulse regimes. For a sufficiently long laser
pulse such that    1  P and L  r0 , Eq. (1.25) reduces to
n
2
1 a
.
 2 2
n0 k P r0 4
(1.26)
32
It may be noted that for a broad beam k P r0  1 , the right side of Eq. (1.26) will be
much less than a
2
4 . Since a
2
4 represents the contribution of relativistic nonlinearity
in Eq. (1.9), it may be concluded that for a broad beam, relativistic nonlinearity will
dominate over pondermotive effects. However for a narrow beam
k P r0  1
ponderomotive effects will dominate over relativistic effects. For an ultrashort laser pulse
L  P  Eq. (1.25) reduces to
 2 n  2
2





 2 n0 
 2
2
a

.
 4
(1.27)
Further, if the laser spot is broad r0  L  the transverse ponderomotive effect is
negligible and n  a
2
4 . In this regime the effect of relativistic nonlinearity is exactly

cancelled by the axial ponderomotive effect n n0  a
2

4 in Eq. (1.9) [73]. Hence the
laser amplitude of an ultrashort pulse is driven by a linear source. However for narrow
laser beams r0  L  the transverse ponderomotive effect becomes significant and affects
the propagation dynamics [74].
1.2(c) Laser pulse compression
In order to analyze the compression dynamics of short laser pulses in
plasma, it becomes important to study its nonparaxial propagation since finite pulse
length and group velocity dispersion effects [75-79] become significant. Relativistic and
33
ponderomotive nonlinearity arising due to laser-plasma interaction provides the source
for laser pulse compression. Theoretical and simulation studies have shown that two
colliding laser beams in plasma lead to laser pulse compression [80-88]. Theoretical and
experimental work has also shown the possibility of obtaining pulse compression in
plasma using a single laser pulse. The interplay of relativistic self-phase modulation
(SPM), group velocity dispersion (GVD) and the presence of high amplitude nonlinear
plasma waves can lead to laser pulse self-compression.
Considering the nonparaxial wave equation (1.11) and assuming ar , z,   to be


slowly varying with propagation distance, the higher order diffraction term  2 z 2 is
neglected in comparison to 2k 0  z  . Also in the limit L  Z R 1  c 2 k 02  02  2 the




mixed derivative term 2 2 z is neglected in comparison to 1   g2  2  2 . With
these approximations, the one-dimensional form of Eq. (1.11) is given by
2
 

a 


2 
2
2
2
2

a .




2
ik

1


a

k
1


k


1
g
P
0
P
 0

z
4 
 2 
 





(1.28)
While deriving Eq. (1.28), density perturbations have been neglected since it is assumed
that r0  L   p . In order to solve Eq. (1.28), a trial function for the laser amplitude is
assumed as

 2 i 2 
a  f z  exp i  2 
 
L

(1.29)
34
where f z  and  are the amplitude and chirp parameter of the laser pulse respectively.
The evolution of the laser pulse length may be analyzed with the help of the heuristic
method. In this context, the trial function (Eq. 1.29) is substituted into the Eq. (1.28) to
give
 1 f z  i 2 2 L i 2  
2ik 0 

 3

a
z
L z  2 z 
 f z  z
 4 2 4 2 8i 2 2 2i 
 1   g2  4  2  2  2   a

L L  
 L


k P2 f 2 z   2 2 
1  2 a  k P2  k 02  P2  1 a .

4
L 




(1.30)
Comparing imaginary and real parts and equating coefficients of  2 and  0 on both
sides of Eq. (1.30), the chirp, amplitude, length and phase of the laser pulse respectively
evolve as
1


k0
L
,
2
2 L 1   g z
(1.31)
L0
,
L
(1.32)

f z   f 0


2
2L 4 1 g

z 2
L3 k 02
  1   k
2
2
g
2
P
f 02 L0
2k 02 L2
,
35
(1.33)
and


1   g2 k P2 f 2 z  1




k P2  k 02  P2  1 .
2
z
8
k
2
k
k0 L
0
0



(1.34)
The first term on the right side of Eq. (1.33) leads to pulse broadening on account of
group velocity dispersion whereas the second term leads to compression due to effect of
relativistic nonlinearity. Figs. 1.3 and 1.4 respectively show the variation of normalized
pulse length L L0  with propagation distance z Z R  in the absence and in the presence
of nonlinearity for f 02  0.05, r0  20m, L0  15m and P  10m . It is seen that in
the absence of nonlinearity the laser pulse broadens slightly (Fig. 1.3) with propagation
distance while in the presence of nonlinearity pulse length compression is observed (Fig.
1.4).
36
1.005
1.004
1.003
L
L0
1.002
1.001
1
0.999
0
1
2
3
z ZR
4
5
6
Fig. 1.3 Normalized pulse length ( L L0 ) against normalized propagation distance
( z Z R ) in the absence of nonlinearity.
37
1.1
1
L
L0
0.9
0.8
0.7
0
1
2
3
4
5
6
z ZR
Fig. 1.4 Normalized pulse length ( L L0 ) against normalized propagation distance
( z Z R ) in the presence of nonlinearity.
1.3 Propagation of a laser beam in a plasma channel
For most laser-plasma applications it is necessary that the laser beam should
propagate several Rayleigh lengths in plasma. Since short laser pulses do not self-focus in
38
homogeneous plasma (Sec.1.2a), pre-formed plasma channels have been proposed [8994] as a means for allowing extended propagation distances for short pulses. There are
several techniques for creating a plasma channel. In the hydrodynamic expansion [95-96]
approach, a laser pulse having energy ~100 mJ and length ~100 ps, is focused with an
axicon to generate a line focus in a gas at an ambient pressure of about 100 mbar. The


intensity at the line focus 1013  1014 Wcm 2 , ionizes the gas and heats the resulting
plasma by inverse bremsstrahlung (IB). The hot plasma column so formed then expands
on a nanosecond time scale, driving a shock wave into the surrounding cold gas and
leaving a plasma channel behind the shock front. The plasma channels formed by this
method are not fully ionized, and so may only be used below the intensity at which
substantial further ionization of the channel would occur.

In the heater-ignitor method, [97-98] a short (<100fs), intense  5  1014 Wcm 2

ignitor pulse produces initial ionization by optical field ionization (OFI). This initial
plasma is then heated and further ionized by a long (~100ps) “heater” pulse of relatively
low intensity. This approach is implemented by overlapping the line foci produced by
focusing the ignitor and heater beams with cylindrical mirrors. Cylindrical channels are
formed if the ignitor and heater beams propagate perpendicular to each other and
common line focus. This technique has been used to guide laser pulses with a peak input
intensity as high as 1019 Wcm 2 over 4mm.
Plasma channels have also been formed by the opposite process i.e.
hydrodynamic compression, [99-100] using fast capillary discharges. In this approach a
capillary of a few millimeters diameter is filled with gas at low pressure and ionized by a
discharge current with a rise time of 10-50 ns, and a peak current of 5-20 kA. Since the
39
current rises rapidly, the skin effect ensures that the ionization occurs close to the
capillary wall. The large magnetic field generated
by the current then compresses the plasma through the J  B force to drive a strong shock
towards the capillary axis. A plasma channel is formed just before the rapidly collapsing
annulus of plasma reaches the axis.
Plasma channels have also been formed in slow capillary discharges [101-102]. In
this technique, a discharge pulse with a rise time of the order of 100 ns and a peak current
of a few hundred amperes is passed through an initially evacuated capillary formed in a
soft material such as polypropylene. This discharge current ablates and ionizes the wall
material to fill the capillary with plasma and forms a plasma channel by radiative and
collisional heat transfer to the capillary wall. This causes the temperature of the plasma to
be greater on axis and since the pressure across the capillary is uniform, an axial
minimum in the plasma density is achieved.
In order to study the evolution of the laser spot in a plasma channel, consider a
laser beam propagating in a pre-formed plasma channel. The paraxial wave equation (Eq.
(1.12)) describing the evolution of the laser beam amplitude in a parabolic plasma density
channel modifies to
 a 2 nr 2 

 2
2
2
2
   2ik 0 z  a  k P 1  4  n r 2  a  k 0  P  1 a

0 ch 


40

(1.35)
where
n and
rch are the channel depth and radius respectively. Using the same
procedure as described in Sec. (1.2a), the evolution of the laser spot in a plasma channel
is described by,
 2 rs
4  k P2 f 02 r02 rs4 

 4
1 
8
z 2 k 02 rs3 
rM 
(1.36)
where rM  rch2 re n  , re  e 2 mc 2 is the classical electron radius. The third term on
14
the right side of Eq. (1.36) represents the channel focusing effect. It may be noted that the
laser spot-size remains matched  2 rs z 2  0, rs  r0  as it propagates in the plasma
channel if the depth of the channel n is kept equal to a certain critical value given by
nc 
4n0 r ch2
k P2 r04
 k P2 f 02 r02 
1 
.
8 

(1.37)
The laser spot evolution equation (1.36) can be analytically solved by assuming an
oscillatory solution of the form
rs2  A1 cos 2
k M
2
z  A2 sin 2
k M
2
z
(1.38)
41
where k M  2 Z RM  is the betatron wave number and Z RM  k 0 rM2 2 . A1 and A2 are
constants which can be obtained with the help of initial conditions. Differentiating Eq.
(1.38) gives
2r
 r 
 A  A2  2
2 s   2rs 2s   1
k M cos k M z .
z
 2 
 z 
2
(1.39)
Substituting Eq. (1.36) into Eq. (1.39) gives
8
 r 
2 s   2 2
k 0 rs
 z 
2
 k P2 f 02 r02 rs4 
 A  A2  2
1 
 4    1
k M cos k M z .

8
rM 
 2 

(1.40)
Assuming that at z  0, rs  r0 and rs z  0 Eq. (1.38) gives
A1  r02 .
(1.41)
Substituting Eq. (1.41) into Eq. (1.39) and applying the same initial conditions, gives
rM4
A2  2
r0
 k P2 f 02 r02 
1 
.

8 

(1.42)
Now substituting Eq. (1.41) and Eq. (1.42) into Eq. (1.38) leads to
42
r02
r 
2
2
s
4

 rM
1  4
 r0

4
 k P2 f 02 r02 
 
 r
1 
  1  M4

8 

 
 r0

 k P2 f 02 r02 

1 
 cos k M z  . (1.43)

8 



The variation of spot-size with propagation distance is shown in Fig. 1.5. The
laser and plasma parameters are f 02  0.15,
  1m,
P  10m,
r0  10m ,
n  2  1018 cm 3 and rch  20m. Curve a represents the linear evolution of the laser
spot (vacuum diffraction) while curve b represents the spot-size variation in
homogeneous plasma in the presence of relativistic effects which reduce the diffraction of
the beam. Curve c shows the spot-size evolution in a plasma channel in the presence of
relativistic effects. It is seen that laser spot focuses and undergoes oscillation and thus
remains confined within the channel.
43
2.5
a
2
b
1.5
rs
r0
c
1
0.5
0
0
2
4
z ZR
6
8
10
Fig.1.5 Evolution of normalized spot-size versus normalized propagation distance
for a) vacuum diffraction, b) homogeneous plasma and c) a plasma channel.
1.4 Variational technique
Interest in nonlinear properties of short pulses propagating in plasma has grown
tremendously during the last few years. The two main reasons for this interest are the (a)
44
possibility of undistorted laser pulse propagation over large distances for applications
based on laser-plasma interaction such as particle acceleration and (b) the possibility of
extreme compression of laser pulses resulting in pulse widths lying well into the
femtosecond domain. Therefore, it is important to obtain differential equations for the
evolution of the macroscopic quantities that characterize the laser profile. The laser
parameters that undergo simultaneous evolution as the laser pulse propagates in plasma
are its spot-size, amplitude, phase, pulse length, radius of curvature and centroid. One of
the most powerful techniques for establishing the simultaneous evolution equations is the
variational method. The variational structure for dissipationless dynamical equation relies
on the existence of a Lagrangian density. The Lagrangian density for the wave equation
(1.10) is written as


 a 
a   a a  a  a 
a a 
  1   g2
l    a  .  a  ik 0  a
 a    

z    z
 z 
 
 z



a  a
a 2 a 2 n  
 k P2  aa  
 aa   k 02  P2  1 aa  .
z z
8
n0


 


(1.44)
The correctness of this Lagrangian density, can be verified by using the Euler-Lagrange
equation given by


l
  l
n
  k 1

x k   
 
  x k




    l
  t   
 
  t

 


0



45
(1.45)

where   a  or a
 is the dependent variable. For independent variables x  r, z,   ,
k
Eq. (1.45) reduces to


l
  l
n
  k 1

xk   
 
  xk


  0.

 
 
(1.46)
Replacing  by a  and x k by the independent variables z,  and r in Eq. (1.46) gives




l
  l  


a  z   a    
 
 
  z  




 l     l   0 .


  a   
 a 

 
 
    


(1.47)
The wave equation is recovered when the value of l (Eq. (1.44)) is substituted into Eq.
(1.47). This verifies the correctness of the Lagrangian density given by Eq. (1.44).
Further, a system of partial differential equations is recast in terms of Hamilton’s
principle according to which an action integral S (   d n rldt , where n is the number of
spatial co-ordinates on which the variational field depends), is stationary with respect to
independent, first-order variations of the dependent variables. Substituting the
 
 
Lagrangian density into the action  0 r  
and performing explicit integrals
  ld  dr
across the transverse and axial coordinates yields a reduced action (reduced Lagrangian
density) with only z as the independent variable. Thus with the help of the Euler-
46
Lagrange equation the evolution of the laser parameters with propagation distance is
obtained.
1.5 Approach
The present study deals with the nonlinear, nonparaxial propagation of short
laser pulses in a parabolic plasma channel. The wave dynamics of laser pulses interacting
with plasma has been described with the help of Maxwell’s, Lorentz force and continuity
equations. A wave equation for the laser field is set up.
Considering the mildly
relativistic regime a perturbative approach is used to obtain the source driving the laser
field amplitude. All independent variables describing the laser field amplitude are
transformed into the pulse frame in order to include the role of finite pulse length and
group velocity dispersion effect. Further, in this thesis a regime has been defined in
which the group velocity dispersion effect dominates over the finite pulse length effect.
An appropriate Lagrangian density has been obtained for the wave equation. Suitable trial
function for the laser pulse amplitude is assumed and substituted into the Lagrangian
density. This density is used to obtain the action integral. Integrating over r and  leads
to the reduced
Lagrangian density. In this reduced form of the action integral the
parameters of the trial function represent a set of dependent variables. With the help of
the Euler-Lagrange equation, variation of the action with respect to the dependent
variables yields a set of coupled differential equations for the evolution of the laser
parameters with propagation distance. An analytical solution for the spot-size (pulse
length) by considering the pulse length (spot-size) to be constant is derived. Further,
47
numerical methods are used to graphically analyze the simultaneous evolution of the laser
spot-size and pulse length.
The propagation dynamics of an initially chirped ultrashort laser pulse in a preformed plasma channel is studied. For ultrashort laser pulses, relativistic and
ponderomotive nonlinearities cancel each other. Hence the source driving the laser
amplitude becomes linear. In this limit, the evolution of the laser spot-size and phase shift
has been obtained using the Variational technique. The variation in amplitude and
frequency have been derived for a matched laser beam propagating in a plasma channel.
A graphical analysis is presented for the effective pulse length and chirp parameter of the
laser pulse due to its interaction with plasma.
For short laser pulses, radial and axial ponderomotive forces acting on the plasma
electron become significant. This leads to density perturbations which in turn generate
electric and magnetic wakefields. The density perturbation has been derived with the help
of second order expansion of Lorentz force, continuity and Poisson’s equations. The
plasma response in the nonlinear regime has been examined within the quasi-static
approximation (QSA) according to which the plasma fluid equation is a function of 
alone (Sec.1.2b). Assuming a wide channel the radial dependence of the plasma
frequency has been neglected. Evolution equation for laser spot-size, pulse length and
intensity have been obtained using the variational technique. As the pulse length
approaches the plasma wavelength a singularity occurs. In this regime l’Hospital’s rule is
used to derive the maximum
L  P 
density perturbation. Using this density
perturbation the evolution equation for the laser spot, pulse length and intensity of the
laser have been obtained in this regime. For a matched laser pulse numerical methods are
48
used to graphically analyze the evolution of the pulse length and intensity of the laser for
the cases when the pulse length is greater than less than and equal to the plasma
wavelength.
1.6 Aim
The present study will be significant for applications such as laser plasma
based accelerators where the laser intensity and pulse length have to be optimized with
respect to the plasma wavelength. Since the laser pulse length and spot-size vary with
propagation distance, it becomes important to study the simultaneous variation of both
these parameters. Further, the possibility of compressing the laser pulse length for
obtaining enhanced intensities can also be explored. The present work is aimed at a
detailed theoretical analysis of the simultaneous evolution of the spot-size and length of
a laser pulse propagating in a parabolic plasma channel. The effects of finite pulse length,
group velocity dispersion, chirping and relativistic as well as ponderomotive
nonlinearities have been taken into account.
49
Chapter 2
Simultaneous evolution of spot-size and length and induced chirping of short laser
pulses in a plasma channel
In this chapter, nonlinear, nonparaxial propagation of intense, short, Gaussian
laser pulses in a pre-formed plasma channel having a parabolic radial density profile has
been studied [103]. Taking into account the effect of group velocity dispersion, the wave
equation describing the evolution of the laser field amplitude driven by relativistic
nonlinearity has been set up. Chirping of the laser pulse induced due to propagation in
plasma is analyzed and a comparison of group velocity dispersion and self-phase
modulation induced chirp is presented. Variational technique is used to obtain equations
describing the simultaneous evolution of the laser spot-size and pulse length in the
plasma channel. Numerical methods are used to study the simultaneous evolution of the
laser spot and pulse length.
2.1 Evolution of spot-size and pulse length
50
Consider the propagation of a linearly polarized laser pulse in a pre-formed
plasma channel having a parabolic density profile of the form n(r )  n0  (nr 2 rch2 ) ,
where n0 is the ambient plasma density and n and rch are the channel depth and
radius respectively. The vector potential of the radiation field is given by,

A
A  xˆ (r , z , t ) exp( ik 0 z  i 0 t )  c.c.
2
(2.1)
where A( r , z , t ) , k 0 and 0 are the amplitude, wavenumber and frequency of the laser
field respectively. The wave equation governing the evolution of the vector potential of
the radiation field is given by
 2 1 2  

4
nr ev.
  2 2  A  
c
c t 

(2.2)


 
While deriving Eq. (2.2), Coulomb gauge . A  0 has been used. Also considering the

limits r0  L, k P L  1 , where k P  4n0 e 2 mc 2 
12
 is the plasma wavenumber, the
ponderomotive nonlinearity has been neglected (Sec.1.2(b)). In this limit, only the
relativistic nonlinearity will drive the laser field. In Eq. (2.2), the relativistic plasma


electron velocity v (  eA mc , where  is the relativistic factor) is obtained with the
help of Lorentz force equation (1.7). Substituting the value of plasma electron density
and relativistic electron velocity into Eq. (2.2) the wave equation reduces to
51
 nr 2 a 2 nr 2 a 2  
2
1 2 
2
a ,
(  2  2 2 )a  k P 1 



4
z
c t
n0 rch2
n0 rch2 4 


2


(2.3)



where a  eA mc2  1 is the normalized vector potential. The second, third and fourth
terms on the right side of Eq. (2.3) are due to plasma channel, relativistic nonlinearity and
coupling of relativistic nonlinearity with the plasma channel (channel coupling).
Substituting Eq. (2.1) into Eq. (2.3) and transforming independent variables z, t to z,
 ( z   g ct where  g  v g c the normalized is group velocity of the pulse) gives,
( 2  2ik 0
 2

2
2  2
2
 (1   g2 ) 2   20  k 02  k P2   2 )a(r , z,  )
z
z

c
 z
 a 2 nr 2
 k 

4
n0 rch2

2
P
 a 2 

 1a(r , z,  ).
4



(2.4)
Assuming ar , z,   to be slowly varying with z, the higher order diffraction term  2 z 2
may be neglected in comparison with 2k 0  z . The mixed derivative term 2  2 z is
neglected in comparison with the term describing the group velocity dispersion effect
(1  
2
g

)  2  2 in the limit k 0 L  1  k P2 r02 4. Thus Eq. (2.4) reduces to


 2  02
2
(  2ik 0
 (1   g ) 2   2  k 02  k P2 )a(r , z,  )
z

c

2

 a 2 nr 2
 k 

n0 rch2
 4
2
P
 a 2 

 1a(r , z,  ).
4



52
(2.5)
The Lagrangian density for Eq. (2.5) is written as,
*
 * 
 02

a *
* a
2 a a
l    a .  a  ik 0 (a
a
)  (1   g )
  2  k 02  k p20 aa 
z
z
   c

2
 a 2 a *2 a 2 a *2 nr 2
* nr 
k 

 aa
.
8 n0 rch2
n0 rch2 
 8
2
P
(2.6)
In order solve the wave equation (2.5), a trial function is assumed as,

i z  2
2
i z r 2
r2 
a  f ( z ) exp i z   2
 2  2
 2 ,
L z 
L z 
rs z 
rs z 

where

(2.7)

f , rs , L ,  ,  and   k 0 rs2 2 Rz  represent the laser amplitude , spot-size,
pulse length, phase, chirp and curvature respectively. In Eq. (2.7) the radial beam as well
as axial pulse profiles of the laser are assumed to be Gaussian. Substituting Eq. (2.6) into
 
 
the action integral  0 r  
  ld  dr yields the reduced Lagrangian density,
f
lˆ 

2
z L1   2  
2 2
 k 0 f 2 z Lrs2  
2 2
 f 2 z rs2 1   g2 1   2 
4 2L

1   2 L  1  2 rs  

 


  


z
4

z
L

z
2
r

z

z


 s


 f 2 z rs2 L  02

 2  k 02  k P2 
c



4 2
  f 4 z rs2 L
 f 4 z rs4 Ln
 f 2 z rs4 Ln 
 k P2 


.
128
512n0 rch2
8 2n0 rch2


53
(2.8)
The
Euler-Lagrange
equation
for
various
laser
parameters
 j  j  1, 2,.......6   ,  , rs ,  , L and f 2 is given by


  lˆ
z    j
 
  z


ˆ
  l  0 .
   j
 
 
(2.9)
Substituting the reduced Lagrangian density (Eq. (2.8)) into Eq. (2.9) for each value of
 j leads to equations relating various laser parameters with each other and a set of
coupled differential equations describing the evolution of the laser parameters. Thus
for 1   , Eq. (2.9) gives,



ˆ

l  lˆ


0.
z      
  z  


(2.10)
Substituting Eq. (2.8) into Eq. (2.10) leads to


 2 2
f rs L  0
z
(2.11)
Eq. (2.11) shows the conservation of energy of the laser pulse. Therefore,
54
f 2 rs2 L  f 02 r02 L0  constant  M
(2.12)
and
f z  
f 0 r0 L0
(2.13)
rs L
where f 0 , L0 , r0 are the initial values of amplitude, pulse length and spot-size
respectively. For 2   , Eq. (2.9) gives,




lˆ  lˆ


 0.
z      
  z  


(2.14)
Substituting Eq. (2.8) into Eq. (2.14) yields,


  1   2 k 0 rs 


  0.
  rs2
rs z 
(2.15)
Solving Eq. (2.15) gives the evolution of  with propagation distance as

k 0 rs rs
.
2 z
(2.16)
55
In order to obtain the evolution equation for the laser spot-size, consider  3  rs . Thus Eq.
(2.9) gives


  lˆ
z   rs
  z
 


ˆ
  l  0 .
  rs


(2.17)
Substituting Eq. (2.8) into Eq. (2.17) gives,


 1   rs  2 1   2

 2M
nrs 
  r 

k0 
 2
 k 0  2 s   k P2 

 0.

3
3
2 

z
rs
32
r
L
2
n
r


r
 rs z rs z 
s
0
ch


 s

(2.18)
The evolution equation for the laser spot-size is obtained by substituting Eq. (2.16) into
Eq. (2.18) as,
 2 rs
4
 2 3
2
z
k 0 rs
where
 P rs4 
1   4 
 Pc rM 


P Pc  2 f 02 r02 L0 k P2 64L represents the laser power normalized by critical

 
Pc  2 2cLe 202 L0 re2P2 GW

(2.19)
power
for
self-focusing,
rM4  rch2 re n
and

re  e 2 mc 2 is the classical electron radius. The first term on the right side of Eq. (2.19)
represents vacuum diffraction, the second term representing relativistic self-focusing
effect is a function of the pulse length and the third term leads to channel focusing. The
matching condition of the laser spot as it propagates in the plasma channel is obtained by
56
assuming that rs  r0 ,  2 rs  z 2  0 and that the pulse length remains constant L  L0 
with propagation distance. With these conditions, Eq. (2.19) gives
n  nc 
4rch2 n0
k P2 r04

2 f 02 r02 k P2
1 

64


.


(2.20)
The critical density nc  [104-105] represents the density gradient for which the laser
spot-size remains matched rs  r0  as it propagates in the plasma channel, when the
evolution of the pulse length is ignored. Similarly substituting
 4   into Eq. (2.9)
gives,

  lˆ

z   
  z


 lˆ

 0.
  


(2.21)
Substitution of the Lagrangian density leads to





 1   g2 1   2
k 0  L 


  0.
L z 
L2

(2.22)
Solving Eq. (2.22) gives the evolution of chirp with propagation distance as
 
k 0 L L
.
2 1   g2 z


(2.23)
57
In order to obtain the evolution of the laser pulse length,  5  L is substituted into
Eq. (2.9). Thus,




lˆ  lˆ


 0,
z   L   L
  z  


(2.24)
and


2
2
 1   L  k 0  L 2 1   g 1  
k0 
 2


2

L3
 L z L z  L z

2k P2 M
2k P2 Mn


32rs2 L2 128n0 rch2 L2
(2.25)
Substituting the value of chirp parameter   from Eq. (2.23) into Eq. (2.25) gives
2 2
2 (1   g2 )k P2 M
nrs2 1
 2 L 4(1   g ) 1


(
1

) .
z 2
k 02
L3
16k 02 rs2
4n0 rch2 L2
58
(2.26)
The first term on the right side of Eq. (2.26) represents the linear contribution due to group
velocity dispersion while the second term consists of the terms due to relativistic
nonlinearity and its coupling with the plasma channel.
The shift of central wave number k   z  of the laser pulse can be obtained
from Euler-Lagrange equation (2.9) by substituting  6  f


  lˆ
z   f 2
 
  z
2
z  , as


ˆ
  l  0 ,
  f 2
 
 
(2.27)
or
1   L  Lr
2

  1   2 L  1  2 rs  
r2
  1   g2 1   2 s
k 
 


  
L z  2  rs z z 
2L
 z 4  z

2
s 0

 f 2 rs2 L f 2 rs4 Ln

rs2 L   02
nrs4 L 
 2  k 02  k P2   2k P2 


  0.

2  c
128n0 rch2
4 2n0 rch2 
 32


(2.28)
Substituting Eqs. (2.16) and (2.23) into Eq. (2.28) gives


2
2
 

k 0 L  2 L k 0 rs  2 rs  1   g rs rs2 L   02

 2  k 02  k P2 
L  Lr k




2 


 z 8 1   2 z 2
4 z 
2L
2 c
g


2
s 0


 f 2 rs2 L f 2 rs4 Ln
nrs4 L 
 2k P2 


  0.
128n0 rch2
4 2n0 rch2 
 32
59
(2.29)
Further, substituting Eqs. (2.19) and (2.26) into Eq. (2.29) gives
2k P2 f 02 r02 L0

2 1
1  1   g 
  2  2  

z k 0  r0 rs 
k 0 L2
128k 0 rs2 L
2

5nrs2 
7 
.

4n0 rch2 

(2.30)
The third and fourth terms on the right side of Eq. (2.30) represent the phase shift induced
by GVD and relativistic nonlinearity respectively.
2.2 Analytical solution for laser spot-size and pulse length
The laser spot evolution equation (2.19) can be analytically solved by assuming the
pulse length to remain constant L  L0  . Multiplying Eq. (2.19) by 2rs z and
integrating with respect to z gives,
r2
4
4 P
 rs 
 s2  A

  2 2  2 2
k 0 rs k 0 rs Pc 0 Z RM
 z 
2
(2.31)
where P Pc 0  2 f 02 r02 k P2 64 and Z RM  k 0 rM2 2. The constant A is determined with the
help of the initial condition rs  r0 , rs z  0 at z  0 as
60
r02
4
4 P
A 2 2  2 2
 2 .
k 0 r0 k 0 r0 Pc 0 Z RM
(2.32)
Substituting Eq. (2.32) into Eq. (2.31) yields
4
 rs 

  2 2
k 0 rs
 z 
2

P 
4
1 
  2 2
 Pc 0  k 0 r0

r2
P  rs2
1 
  2  02 .
 Pc 0  Z RM Z RM
(2.33)
Multiplying both sides of Eq. (2.33) by rs2 gives
2
2
1  4 Z 2 
P  2  2 4Z RM
 rs 
  r0 rs 
r 
  2  2 RM2 1 
Z RM  k 0 r0  Pc 0 
k 02
 z 

2
s

P  4
1 
  rs 
 Pc 0 

(2.34)
or
rs
1

z Z RM rs
2
 4Z RM
 2 2
 k 0 r0
2

P  2  2 4Z RM
1 
  r0 rs  2
k0
 Pc 0 

12

P  4
1 
  rs  .

 Pc 0 
(2.35)
Integrating Eq. (2.35) with respect to z gives

 rM4
 2
 r0
rs rs


P  2 2
P  4
1 
  r0 rs  rM4 1 
  rs 
P
P
c0 
c0 




12
Substituting rs2  t , 2rs rs  t in Eq. (2.36) gives
61

1
Z RM
z .
(2.36)
1
2   r 4
 M2
 r0
t


P  2
P  2
1 
  r0 t  rM4 1 
  t 
 Pc 0 
 Pc 0 


12

1
Z RM
z .
(2.37)
Solving Eq. (2.37) gives
1
2
 4
 1  rM
 4  r02

t

P  2   1  rM4
1 
  r0   t 
 2
 Pc 0 
  2  r0
2

P  2 
1 
  r0 

 Pc 0 

2




12

1
Z RM
z . (2.38)
Solving the integrals in Eq. (2.38) gives

 1  rM4 
P  2  
  r0  

 t   2 1 
2
P
r
1  1 
c0 
 0 
   z  B .
sin 


4
2
 1  rM 1  P   r 2   Z RM
 2  r 2  P  0  

c0 
 0 
 


(2.39)
Since rs  r0 at z  0 the constant of integration is given by

 2 1  rM4 
P  2  

  r0  
r

1


 0

2  r02  Pc 0 
1  1 
     .
B  sin 

4
2
4
 1  rM 1  P   r 2  


 2 r2  P  0


c0 
 0 
 


62
(2.40)
Therefore Eq. (2.39) leads to
 2 1  rM4 
P  2 
  r0  
 rs   2 1 
2  r0  Pc 0 

    cos 2 z


4
Z RM
 1  rM 1  P   r 2  


 2 r2  P  0

c0 
 0 
 

(2.41)
or
rs2 
r02  rM4
1 
2  r04

P   rM4
1 
  1  4
P
c 0 

 r0


P 
1 
 cos k M z 
 Pc 0 

(2.42)
where k M  2 Z RM  is the wavenumber describing the oscillation of the laser spot-size.
Similarly the pulse length evolution equation (2.26) can be analytically solved by
assuming the spot-size to remain constant rs  r0 . Multiplying both sides of
Eq.
(2.26) by 2L z and integrating with respect to z gives
U 2V
 L 
 C1
   2 
L
L
 z 
2


where U  4 1   g2 k 02 and V 
(2.43)
 2 1   k
2
g
2
P


M 16k 02 rs2 1  nrs2 4n0 rch2 . C1 is the
constant of integration which can be determined by applying the initial condition
dL dz  0 , L  L0 at z  0 . Thus,
63
C1 
U 2V

.
L20 L0
(2.44)
Substituting Eq. (2.44) into Eq. (2.43) gives
U  2VL0 L2  2VL20 L  UL20 .
 L 
  
L20 L2
 z 
2
(2.45)
Assuming U  2VL0  Q, 2VL20  D and UL20  E , Eq. (2.45) reduces to
QL2  DL  E
L
.

z
L0 L
(2.46)
Integrating both sides of Eq. (2.46) with respect to z and multiplying by 2Q gives

2QL
QL2  DL  E
L 
1
2Qz .
L0 
(2.47)
Substituting QL2  DL  E  x, 2QL  DL  x into the first term on the left side of
Eq. (2.47) gives

x
x

D
QL  DL  E
2
L 
1
2Qz .
L0 
64
(2.48)
Solving Eq. (2.48) gives
2 QL  2VL L  UL 
2
2
0

2
0

VL20
2VL20 L UL20 
2
log  L 
 L 

.
Q
Q
Q 
Q

2VL20
2Q
z  C2 .
L0
(2.49)
The constant C2 is obtained with the help of the initial condition that at z  0, L  L0 .
Therefore,
C2  
UL  VL0 
log  0
.
Q
 U  2VL0 
2VL20
(2.50)
Substituting Eq. (2.50) into Eq. (2.49) gives
2U  2VL0 z
2VL20
2
2
2
 2 (U  2VL0 ) L  2VL0 L  UL0 
L0
U  2VL0
  UL  VL
0
 log  0
  U  2VL0


VL20
2VL20
UL20
2
  log  L 
 L 


U  2VL0
U  2VL0 U  2VL0



 .


(2.51)
The evolution of the laser spot-size and pulse length as given by Eqs. (2.19) and (2.26)
are graphically depicted in Fig. (2.2) of Section (2.3).
2.3 Numerical solution for simultaneous evolution of laser spot-size and pulse
65
length
The simultaneous evolution of the laser spot and pulse length in homogeneous as
well as inhomogeneous plasma is studied numerically by solving equations (2.19) and
(2.26), using the fourth order Runge Kutta method, with the assumption that at z  0,
rs z  0, rs  r0  33m, L z  0, L  L0  14m. The other laser and plasma
parameters are
f 02  0.03,   1m and p  10m. Fig. 2.1 shows a comparison
between the evolution of the normalized pulse length L L0  with propagation distance
z
Z R  for a laser pulse propagating in homogeneous plasma curve a
and in
plasma channel with n  2  1017 cm 3 and rch  50 m
1.002
1
0.998
0.996
L
L0
0.994
a
0.992
b
0.99
0.988
0
0.2
0.4
z Z R 0.6
66
0.8
1
1.2
a
Fig.2.1 Variation of normalized pulse length L L0  with propagation distance
z
Z R  in the absence of (curve a) and in the presence (curve b) of the
plasma channel for
f 02  0.03 ,   1m, r0  33m, L0  14m
P  10m, n  2  1017 and rch  50m .
(curve b). It is seen that compression of the laser pulse length occurs for both cases.
However, in the presence of the channel compression is enhanced in comparison to the
homogeneous case. This is due to the combined effects of relativistic nonlinearity and
plasma channel.
Fig. 2.2 depicts the simultaneous evolution of the spot-size (curve a ) and pulse
length (curve b ) of a laser pulse propagating in a plasma channel. Curve c (curve d)
shows the variation of the spot-size (pulse length) when the pulse length (spot-size) is
non-evolving. All the parameters are the same as in Fig. 2.1. It is seen that the laser pulse
length compresses for both cases due to the dominance of relativistic nonlinearity over
GVD effects. However, curve b shows more compression in comparison to curve d due
to the evolution of the spot-size. Curve a shows that the focusing of the spot-size
improves since its amplitude of oscillation decreases, while the spot-size is seen to
oscillate with constant amplitude in curve c . In the former case the evolution of the pulse
length leads to a reduction of the spot-size.
67
The simultaneous variation of the normalized spot-size rs r0  and normalized pulse
length L L0  with the normalized propagation distance, keeping the channel density
gradient equal to the critical density ( n  nc  0.238  1018 1  9.4 f 02  cm 3 , Eq.
(2.20)), is plotted in Fig. 2.3. The set of curves a, b and c (d, e and f)
variation
represent
of the laser spot size (pulse length) for
1.1
1
c
a
0.9
rs L
r0 L0
0.8
d
b
0.7
0.6
0
1
2
3
4
5
6
7
z ZR
Fig.2.2 Simultaneous
variation of normalized
spot size rs r0  (curve a)
and the normalized pulse length L L0  (curve b) with
normalized
propagation distance z Z R  . Evolution of the spot-size keeping the pulse
68
the
length constant (curve c). Evolution of the pulse length keeping the
spot
size constant (curve d)
for
f 02  0.03,
  1m,
r0  33m,
L0  14m, P  10m, rch  50m and n  2  1017 .
1.1
a
b
c
1
rs L
r0 L0
0.9
d
e
0.8
f
0.7
0
1
2
z ZR
3
4
5
Fig.2.3 Simultaneous variation of normalized spot-size rs r0  (curves a, b and
c) and the normalized pulse length L L0  (curves d, e and f ) with
normalized
propagation
distance
69
f 02  0.03
n
c

 1.7  1017 cm 3 ,
0.04
n
c

 1.4  1017 cm 3 and 0.05
n
c

 1.2  1017 cm 3 ,   1m ,
r0  33m, L0  14m, P  10m and rch  50m .
.
different laser intensities f 02  0.03, 0.04 and 0.05 respectively. All other parameters
are the same as in Fig. 2.1. The critical density represents the density gradient for
which the laser spot-size remains matched
rs  r0 
as it propagates in the plasma
channel, without considering the evolution of the pulse length. Consideration of the pulse
length evolution in the present study leads to mismatched propagation of the spot-size.
Also when the initial laser intensity is increased, mismatching and compression of the
laser pulse increases with propagation distance. Fig. 2.4 shows a plot of the evolution of
laser intensities corresponding to the parameters used in Fig. 2.3. It is seen that for higher
initial laser intensities, the rate of increase in intensity enhances due to rapid compression
of the laser pulse and decrement in the spot-size with propagation distance.
2.4 Group velocity dispersion and self-phase modulation induced chirping of
short laser pulses in a plasma channel
Consider a matched n  nc  laser beam propagating in the plasma channel.
The variation of frequency across the laser pulse may be obtained by writing the resultant
phase of the laser pulse as,

2 
    2   .
L 

(2.52)
70
0.08
c
0.07
0.06
b
0.05
f 2 z 
0.04
a
0.03
0.02
0.01
0
0
1
Fig. 2.4 Variation
of
2
intensity

z ZR
for
the
3
initial
4
5
values f 02  0.03
n
c
 1.7  1017 cm 3 ,
n
c
 1.2  1017 cm 3 . Other laser plasma parameters are   1m ,
0.04
n
c
 1.4 1017 cm 3

r0  33m, L0  14m , P  10m and rch  50m.
The frequency shift across the laser pulse (chirp) is given by
71

and
0.05
      
 .
   
 t    t 
   
(2.53)
Combining Eqs. (2.52) and (2.53) and substituting  t  v g gives
  
2v g
L2
.
(2.54)
Eq. (2.54) indicates that if   0 ( 0) the laser pulse will be positively (negatively)
chirped i.e. the frequency increases (decreases) from the front to the back of the pulse
[106]. Further, substituting the value of  ( Eq. (2.23)) into Eq. (2.54) leads to
 
v g k 0

L 1 
2
g

L
.
z
(2.55)
Eq. (2.55) shows that the frequency chirp is a function of propagation distance and varies
with the laser pulse length evolution. Since the pulse length evolves (Eq. (2.26))
primarily on account of GVD and SPM [107-108], therefore GVD as well as SPM are
also responsible for inducing chirp of the laser pulse. In Fig. 2.5, solid
72
0.05
0.04
d
0.03
f
0.02
0.01

0
b
0
0
1
2
3
4
5
z ZR
6
e
-0.01
-0.02
c
-0.03
a
-0.04
-0.05
Fig. 2.5 Variation of normalized frequency
 0 
with z Z R . Solid (dotted)
curve a (curve d) , curve b (curve e), curve c (curve f) are due to SPM, GVD
and combined effect of SPM and GVD at front (back)   0.5 (-0.5) of
the pulse for f 02  0.03,
  1m, P  10m , r0  33m, L0  14m,
rch  50m and n  nc  1.7  1017 cm -3 .
curves (dotted curves) a, b and c d , e and f  show the variation of frequency induced
chirp due to SPM, GVD and combined effect of both, respectively at the front (back) of
the laser pulse. All laser and plasma parameters are considered to be the same as in Fig.
73
2.3 for f 02  0.03 . Curve a (d) shows that due self-phase modulation the frequency at the
front (back) of the pulse decreases (increases). Group velocity dispersion leads to a
converse frequency variation as seen in curves b and e. It may be noted that when both
effects are taken into account SPM dominates over GVD since curve c (f) follows curve a
(d) with a lesser magnitude. Therefore GVD effects tend to reduce the chirp generated by
self-phase modulation of the laser pulse.
Frequency variation of the pulse at a given normalized propagation distance
z
Z R  is shown in Fig. 2.6. Curves a and b show frequency variation at
z Z R  3 and z Z R  6 respectively, all other parameters being the same as in Fig. 2.5. It
is seen that the laser frequency varies linearly over the pulse. The variation of the
normalized induced chirp  0  is negative (red shifted) near the leading edge and
becomes positive (blue shifted) near the trailing edge of the pulse. Since the
instantaneous frequency increases linearly from the leading to the trailing edge, the laser
pulse is positively (up) chirped. Fig. 2.7 shows the variation of the intensity
 f z,     f
2
2
0
 
L0 L exp  2 2 L2
 across the matched laser pulse. Curve a shows the
initial intensity distribution at z Z R =0 while curves b and c depict the intensity variation
after the pulse has propagated a distance z Z R  3 and
74
0.06
0.04

0
0.02
0
-1
-0.5
0
0.5
 L0
1
-0.02
a
-0.04
b
-0.06
Fig. 2.6 Variation of normalized chirp  0  across the laser pulse after traversing
distance z Z R  3 (curve a) and z Z R  6 (curve b) for f 02  0.03,   1m,
P  10m,
L0  14m,
r0  33m,
rch  50m .
75
n  nc  1.7  1017 cm -3
and
0.045
c
0.04
0.035
b
0.03
a
0.025
0.02
f
0.015
2
z,  
0.01
0.005
0
-1.5
-1
Fig.2.7 Variation
-0.5
of
intensity f
0
2
z,  

0.5
with  after pulse
1
1.5
has traversed a
distance z Z R  0 (curve a), z Z R  3 (curve b) and z Z R  6 (curve c) for
f 02  0.03,
  1m, P  10m,
L0  14m, n  nc  1.7  1017 cm -3
r0  33m, and rch  50m.
76
z Z R  6 respectively. All other laser-plasma parameters being the same as in Fig. 2.5. It
is seen that the intensity increases with respect to the initial intensity as the propagation
distance increases.
Chapter 3
Propagation of ultrashort chirped laser pulses in a plasma channel
77
In this chapter, propagation of an initially chirped, Gaussian laser pulse in a preformed parabolic plasma channel is analyzed [109]. The pulse is considered to be
ultrashort, so that relativistic and ponderomotive nonlinearities cancel each other and the
source driving the laser amplitude becomes linear. Evolution of the laser spot-size and
phase shift, with propagation distance of the laser pulse is obtained using a variational
technique. The effect of initial chirp on the laser pulse length and intensity of a matched
laser beam propagating in a plasma channel has been analyzed. Also the variation of the
initial chirp parameter of the laser pulse, as it propagates in plasma, has been obtained
and graphically depicted.
3.1 Analysis of wave equation
Consider a linearly polarized, ultrashort laser pulse propagating in a pre-formed
plasma channel of the form nr   n0  nr 2 rch2 where n and rch are the channel depth
and radius respectively. The vector potential of the laser field is given by

Ar , z , t 
A  xˆ
exp ik 0 z  i 0 t   c.c.
2
(3.1)
where Ar, z, t  , k 0 and 0 are the amplitude , wavenumber and frequency of laser field
respectively. For ultrashort ( L   p , r0 ) laser pulses, relativistic and pondermotive
nonlinearities cancel each other (Sec.1.2 b). Therefore, the wave equation governing the
evolution of the normalized vector potential of such laser pulses is given by
78
 nr 2  
 2 1 2 
a
   2 2 a  k P2 1 
c t 
n0 rch2 


(3.2)

12

where a  eA mc 2 and k P  4n0 e 2 mc 2  is the on-axis plasma. Substituting Eq. (3.1)
into Eq. (3.2) and transforming independent variables z, t to z ,  (  z   g ct , where
 g (  v g c) is the normalized group velocity) leads to
 2

 
2
2 
2




2
ik


(
1


)

 
 ar , z,  
g
 0   z
 2 z 2 



  nr 2
= k P2 1 
n0 rch2
 


  k 02  P2  1  ar , z ,  





(3.3)
where  p (  0 ck 0 ) is the phase velocity of the laser pulse. The terms 2 2 z and
1   
2
g
2
 2 on the left side of Eq. (3.3) represent finite pulse length and GVD effects
respectively. Assuming ar , z,   to be slowly varying with z , the higher order diffraction
term  2 z 2 on the left side of Eq. (3.3) is neglected in comparison to 2k 0  z . Thus
Eq. (3.3) reduces to
2
 2

 
2  




2
ik


(
1


)
 
 a(r , z,  )
g
 0   z
 2 



79
  nr 2
 k P2 1 
n0 rch2
 


  k 02 (  P2  1) a(r , z,  ) .



(3.4)
Considering a chirped Gaussian laser pulse, the solution of Eq. (3.4) may be written in
the form of a trial function given by

i z,  r 2
r2 
ar , z,    bz,   exp i z,    2
 2

rs z,  
rs z,  

where b( z ,  ) (  f ( z,  ) exp(   2 (1  i ) L20 ), f ( z,  ),
L0 and 
(3.5)
being the pulse
amplitude, constant length and constant chirp parameter respectively),  ( z ,  ) ,  ( z,  )
and rs ( z,  ) represent the phase, radius of curvature and spot-size of the laser beam
respectively. Taking the Fourier transform of Eq. (3.4) with respect to  gives
 2  nr 2 


 2
2
2
2
2


ˆ
ˆ


2
ik
a

k
1


k
(


1
)


k
(
1


)

k
p
0
p
g ak ,
 
z 
n0 rch2 

 

(3.6)
where k  k 0  k , k is the spread in laser wavenumber and the Fourier transform of
a( ) is given by
aˆ k 
1
2
  d exp( ik )a( ).
(3.7)
80
In order to obtain the Fourier transform of the trial function, Eq. (3.5) is
substituted into Eq. (3.7). Thus

i z, k r 2
r2 
f z, k  exp i z, k   2
 2

rs z, k 
rs z, k 
2

1
aˆ k 




 2

exp   2 1  i   ik d .
 L0

(3.8)
2
2
4 p where p  1  i  L20 and q  ik , Eq.
Since  
  exp  px  qx dx   p exp q
(3.8) gives
aˆ k 
f z, k L0
2 1  i 
12
 k 2L20
i z, k r 2
r2 
,
exp  
 i z, k   2
 2



4
1

i





r
z
,
k
r
z
,
k
s
s


(3.9)
or
aˆ k 
f z, k L0 1  i 1 2
1   
2 12
2
 k 2L20
ir 2
r2 
.
 exp  




1

i


i

z
,
k


2
2
2





4
1


r
z
,
k
r
z
,
k
s
s




(3.10)
Assuming 1  i  r cos1  i sin 1   r exp  i1  where r  1   2 and 1  tan 1  , Eq.
(3.10) leads to,

i z, k r 2
r2 

aˆ k  bk z, k  exp  i z, k   2
 2
rs z, k 
rs z, k  

81
(3.11)
where the beam amplitude in k - space is given by
bk z , k  

f z , k L0 1   2

1 4
2
 k 2 L20

1  i   i tan 1   .
exp  
2
2
 4 1 



(3.12)
Applying a variational technique, the Lagrangian density of Eq. (3.6) is given by


 aˆ 
a 
l    aˆ k .  aˆ k  ik  aˆ k k  aˆ k k 
z
z 

  nr 2
+ k P2 1 
n0 rch2
 


  k 02 (  p2  1)  k 2 (1   g2 ) aˆ k aˆ k  0.



(3.13)


Substituting Eq. (3.11) into Eq. (3.13) and taking action integral  
0 rdrl yields the
reduced Lagrangian density

f 2 rs2 L20 1   2
lˆ 
4
k P2

2

r 2 n
1  s 2
 2n r
0 ch


1 2
 k 2 L20
exp  
2
 2 1 




  1 2
  1   rs 
 

 k 


2

 z 2 z rs z 
  rs

 1
  k 2 1   g2  k 02  P2  1  .
 2


 

82


(3.14)
Proceeding


  lˆ
z    j
 
  z
as
in


ˆ
  l  0
   j
 

Chapter
2,
the
Euler–Lagrange
equation,
where  j  j  1, 2,.......4   ,  , rs and f 2 define various laser
parameters, is considered. Now substituting 1   and using the Lagrangian density
(Eq. (3.14), the Euler–Lagrange equation gives
 2 2
bk rs  0 , bk2 rs2  bk20 r02
z

where

(3.15)



L0
k 2 L20
i
2 1 4
1

bk 0  f 0
(1   ) exp  
(
1

i

)

tan

2

2

2
 4(1   )

is
the
expression for bk (Eq. 3.12) at z  0. Eq. (3.15) shows the power conservation relation.
For  2   the Euler-Lagrange equation gives

  f 2 rs2 L20 1   2

z 
8

1 2
k
 k 2 L20
exp  
2
 2 1 








 f 2 L2 2 1   2
0

4




 f 2 L2 r  1   2
0 s

4

1 2

 k 2 L20
exp  
2
 2 1 

1 2



k




 k 2 L20
exp  
2
 2 1 


 rs 

  0.
 z 
(3.16)
On account of the power conservation relation, the first term on the left side of Eq (3.16)
vanishes. Thus
83

krs rs
.
2 z
(3.17)
For  3  rs , the Euler-Lagrange equation gives


  k    1   2 k rs k P2 rs2 n 



  0.
 
z  rs  rs  rs2
rs z
4n0 rch2 
(3.18)
With the help of Eq. (3.17) the evolution of the laser spot-size is described by
 2 rs
4  k P2 nrs4 
1 
.

z 2 k 2 rs3 
4n0 rch2 
(3.19)
Assuming rs  r0 and  2 rs z 2  0 in Eq. (3.19) the critical channel depth is given by
nc (  rch2 re r04 where re  e 2 mc 2 is the classical electron radius). If the channel depth is
equal to the critical channel depth the laser beam propagates with a constant spot-size
which is equal to its beam waist (matched beam). For a matched laser beam, Eq.(3.19)
reduces
to
 2 rs
nrs4 
4 



1
z 2 k 2 rs3  nc r04 
(3.20)
84
Eq. (3.20) shows the evolution of the laser spot-size in a parabolic plasma channel and is
the same as that obtained in Ref. [110]. The Euler–Lagrange equation for amplitude of


the laser pulse  4  f 2 z  is given by

f 2
k P2
2

 f 2 r 2 L2 1   2
s
0

4


r 2 n
1  s 2
 2n r
0 ch


1 2
 k 2 L20
exp  
2
 2 1 




 1   2
  1   rs 


 k 


2

z
2

z
r

z
r
s


 s
 1 2

  k 1   g2   k 02  P2  1  0 .
 2


(3.21)
Solving Eq. (3.21) gives,
  krs  2 rs  k P2 rs2 
rs2 n

1 
1  kr 

4 z 2 
2  2n0 rch2
 z
2
s

 







rs2
k 2 1   g2  k02  p2  1  0 .
2
(3.22)
Substituting Eq. (3.19) into Eq. (3.22) and using the linear dispersion relation

2
0

c 2  k 02  k P2  4 r02 gives the equation governing the evolution of the phase of the
laser amplitude as

2 1
1  (k ) 2
   2  2  
(1   g2 ) .
z
k  rs r0 
2k
3.2 Effect of chirp on matched laser pulse propagation
85
(3.23)
In order to study the effect of chirp on the length and intensity of the laser
pulse, consider a matched ( rs  r0 , n  nc ) laser pulse propagating in a plasma
channel. For a matched laser beam, Eqs.(3.15), (3.17) and (3.23) yield bk  bk 0 ,   0
 k 2 
(1   g2 ) z . Hence the Fourier transform of the trial function as given by
and    
 2k 
Eq. (3.11) reduces to
 r 2  k 2

ˆak  bk 0 exp  2  i
(1   g2 ) z  .
 2k

 r0
(3.24)
The second term in the exponent (proportional to k 2 ) on the right side of Eq. (3.24)
arises due to group velocity dispersion (GVD) effects. Taking the inverse Fourier
transform of Eq. (3.24) gives the laser field amplitude as
a

f 0 L0 1   2

1 4
2 



 i
r2 
exp   tan 1   2 
r0 
 2


 L20 1  i  i 1   g2 z  2
k exp ikdk .
exp  

 4 1  2

2
k
0




(3.25)
While deriving Eq. (3.25) it is assumed that k ~ 1 L << k 0 , so that k 2 k  k 2 k 0 . The
integral on the right side of Eq. (3.25) may be solved as before (Eq. 3.8) to give
86


12


1   2 z 

1  i  

Z D 0 
 i
r2  

2 1 4
1
a  f0 1  
exp   tan   2 
12
r0  
 2
1   2 z 2 2z 

1 

Z D0 
Z D2 0







 2 

1   2 z  

 

1

i



2 

Z D 0  

 L0 
exp 

1   2 z 2 2z
 1


Z D2 0
Z D0





where



 


Z D0  k 0 L20 2 1   g2  k 03 L20 r02 8 1  k P2 r02 4



1  2 z 
  r exp i 2 
Assuming 1  i  
Z D 0 


(3.26)

12

where
is the dispersion length..



2 1   2 z 
1  2 z 
 ,  2  tan 1   
 , Eq.(3.26) reduces to,

Z D0
Z D 0 



z2
2 z 
a  f 0 1  (1   2 ) 2 

Z D0 Z D0 

1 4




z
2 
  
 
1

i
(
1


)
 2

Z D0
r
i
z
2 

  . (3.27)
 exp  2  tan 1 (1   2 )


2 
2
 r
2
Z D 0 L0 
z
2  z 
2
0
1  (1   ) 2 




Z
Z

D
0
D
0


Comparing Eq. (3.5) and (3.27) shows that the effective pulse length is modified to
12

z2
2z 
 .
Le 0  L0 1  (1   2 ) 2 

Z
Z
D
0
D
0


87

2 2 2

1


z
2
r  1   
2

Z D0

(3.28)
The second term on the right hand side of Eq. (3.28) tends to broaden the pulse length
while the third term leads to compression (broadening) for a positively (negatively)
chirped laser pulse. Variation of the effective normalized pulse length Le 0 L0  with
propagation distance z Z D 0  is presented in Fig. 3.1. Curves a and c show the variation
in pulse length for positively
  0.3
and negatively
  0.3
chirped pulses
respectively, while curve b has been plotted for an unchirped pulse. For a positively
chirped laser pulse, the pulse length initially compresses with propagation distance.
Maximum compression occurs at a critical distance z Z D 0   1   2 where the effective
pulse length becomes L0 1   2 1   2  . When z Z D 0 exceeds the critical distance,
12
the pulse length starts to broaden. The effective pulse length becomes equal to the initial
pulse length after the laser beam has traversed a distance equal to twice the critical
distance. Beyond this distance, broadening of the laser pulse length occurs. Unchirped as
well as negatively chirped pulses are seen to broaden with propagation distance.
Apart from changes in pulse length, comparison of Eq. (3.5) with Eq. (3.27)
shows that the effective chirp parameter becomes
88
1.8
b
1.6
c
1.4
a
Le 0 1.2
L0
1
0.8
0.6
0
0.2
0.4
z
0.6
0.8
1
Z D0
Fig.3.1 Variation of effective pulse length Le 0 L0  with propagation distance
z
Z D 0  for a positively chirped pulse ((curve a),   0.3 ), unchirped pulse
(curve b) and negatively chirped pulse ((curve c),   0.3 ).
(1   2 ) z
Z D0

.
2
(1   ) z 2 2 z
1

Z D0
Z D2 0

 e0
(3.29)
89
Eq. (3.29) shows that if the laser pulse is positively chirped, the effective chirp becomes
zero at the critical distance z Z D 0   (1   2 ) . This implies that the
initially (positively) chirped pulse becomes unchirped. When z Z D 0 exceeds the critical
distance, the positively chirped laser pulse becomes negatively chirped (chirp reversal).
This reversal occurs because the group velocity induced (negative) chirp and its coupling
with initial chirp becomes more effective in comparison to the initial (positive) chirp. If
z Z D 0 is less than the critical distance, reversal of chirp does not occur. Also, negatively
chirped pulses do not undergo chirp reversal.
2
The matched pulse intensity ( a ) can be obtained with the help of Eq. (3.27) and
is given by
a  f 02
2
 2r 2 2 2 
L0
exp   2  2 .
Le 0
Le 0 
 r0
(3.30)
Eq. (3.30) shows that the on-axis intensity of the laser pulse increases as the effective
pulse length ( Le0 ) decreases. For a positively chirped pulse, maximum intensity will be
obtained at the critical distance where compression of the pulse length is maximum. Fig.

3.2 shows the variation of normalized on axis r  0 intensity a
2

f 02 across the laser
pulse. The dotted curve represents the initial z Z D 0  0 intensity spread for chirped as
well as unchirped laser pulses. Curve a has been plotted for a positively chirped   0.3
laser pulse after it has traversed a distance z Z D0  0.275 (critical distance). It is seen that
90
after traversing the critical distance the peak intensity (at the pulse centroid) of the
positively chirped laser pulse increases by about 4% in comparison to the initial peak
intensity. Curves b and c respectively show the intensity variation of an unchirped
  0 and negatively   0.3 chirped laser pulse after traversing the same distance.
The maximum intensity of a negatively chirped laser pulse is damped by about 10.5% as
compared to its initial value. The intensity of the unchirped pulse also shows a slight
decrease due to GVD effects.
3.3 Variation of laser frequency across chirped laser pulses
In order to study the variation of frequency across the laser pulse, the resultant
phase of the laser pulse is obtained by comparing Eqs. (3.5) and (3.27) as,
2

2 
1
2
1 (1   ) z


    2     tan
 2  e0 .
2
Z
L
L0
D0
0


91
(3.31)
1.2
a
b
c
0.8
a
2
f 02
0.4
0
-1.5
-1
-0.5
Fig 3.2 Variation of normalized
 L0
0
intensity
a
0.5
2
f 02
1

1.5
with  L 0 for the
initial laser pulse z Z D 0  0, dotted curve  and after the laser pulse
has traversed a distance z Z D0  0.275 (critical distance) for chirp
parameter   0.3 (curve a),   0 (curve b) and   0.3 (curve c).
The   z  v g t  dependence of the phase ( ) implies that the instantaneous frequency
varies across the pulse from the central value  0 . The frequency shift ( ) is given by
the time derivative of the phase   t  as
92
  
2
 e0 v g
L20
   2
4c 2
where v g  c1  P2  2 2
   0  0 r0
(3.32)



12

 is the group velocity of the pulse. It may be noted that

the frequency at the pulse centroid (   0 ) remains unaffected by propagation of the
pulse in the plasma channel, while the frequency shift across the pulse tends to zero after
a positively chirped pulse has traversed a distance equal to the critical distance where the
pulse becomes unchirped  e 0  0  .
Figs. 3.3-3.5 show the variation of the normalized frequency shift  0  with  L 0
after the pulse has propagated a normalized distance z Z D 0  0.1, 0.275 and 0.45
respectively for   0.3 (line a)   0 , (line b)   -0.3 (line c). The laser and plasma
parameters are r0  25m, L0  10m,   1m and p  40m . Fig. 3.3 shows that the
frequency increases (decreases) linearly from the front to the back of a positively
(negatively) chirped laser pulse. It may be noted that an
93
Fig.3.3 Variation of normalized frequency  0  with  L 0 at z Z D 0  0.1 for
chirp parameters   0.3 line a ,
  0 line b and   -0.3 (line c ),
for r0  25m , L0  10m ,   1m and p  40m.
94
Fig. 3.4 Variation of normalized frequency  0  with  L 0 at z Z D 0  0.275
for chirp parameters   0.3 line a ,   0 line b and   -0.3 (line c)
for r0  25m , L0  10m ,   1m and p  40m.
95
Fig. 3.5Variation of normalized frequency  0  with  L 0 at z Z D 0  0.45 for
chirp parameters   0.3 line a,   0 line b and   -0.3 (line c) , for
r0  25m , L0  10m ,   1m and p  40m.
unchirped laser pulse attains a slight negative chirp due to GVD effect. This
induced negative chirping will superimpose upon initial chirping and will result in
decrease (increase) in effective chirp of a positively (negatively) chirped pulse. After
96
propagating a distance z Z D0  0.275 , the frequency variation of the laser pulse is shown
in Fig. 3.4. The negative chirping of the unchirped and negatively chirped laser pulse
increases in comparison to Fig. 3.3 due to increase in propagation distance. However, the
frequency variation of the positively chirped pulse reduces to zero, since at this critical
distance a positively chirped pulse becomes unchirped. Fig.3.5 shows that with increase
in propagation distance beyond the critical value, the positively chirped pulse becomes
negatively chirped (chirp reversal). This reversal occurs because the group velocity
induced chirp and its coupling with the initial chirp becomes more effective in
comparison to the initial (positive) chirp. However for unchirped and negatively chirped
pulses, the effective chirp becomes more negative in comparison to Figs. 3.3 and 3.4.
The study shows that the length as well as the chirp of an ultrashort laser pulse
changes due to propagation in a plasma channel. A positively chirped pulse may be
compressed after propagating a certain critical distance, thus leading to a shorter pulse
length laser source. It is important to note that there is a possibility of unchirping a
positively chirped laser pulse by allowing it to propagate a given distance in plasma.
CHAPTER 4
Wakefield effects on the evolution of symmetric laser pulses in a plasma channel
The present chapter deals with the study of the combined effects of wakefields
(inside the laser pulse) and relativistic nonlinearity on the propagation of a short laser
pulse in a parabolic plasma channel [111]. The density variation of plasma electrons
97
produced by the combined effect of transverse and axial ponderomotive nonlinearity have
been obtained. The wave equation describing the evolution of the laser amplitude driven
by relativistic and ponderomotive nonlinearities has been set up. Further, the reduced
Lagrangian density is obtained and variational technique is used to derive the coupled
equations governing the evolution of the pulse length and spot-size of the laser.
Numerical methods are used to graphically analyze the evolution of the laser pulse length
and intensity.
4.1 Wave dynamics
Consider a linearly polarized, short laser pulse propagating in a pre-formed
plasma channel of the form nr   n0  nr 2 rch2 where n and rch are the channel depth
and radius respectively. The vector potential of the radiation field is given by

Ar , z , t 
A  xˆ
exp ik 0 z  i 0 t   c.c.
2
(4.1)
where Ar, z, t  ,  0 and k 0 are the field amplitude, frequency and wavenumber
respectively. The wave equation describing the evolution of the vector potential driven by
relativistic and ponderomotive nonlinearities is given by
98
 nr 2 a 2 n nr 2 a 2  
 2 1 2 
2
a
   2 2 a  k P 1 

 
2
2


4
n
4
c

t
n
r
n
r


0
0 ch
0 ch




a (r , z , t ) ( eA(r , z , t ) mc 2 )
where

k P  4n0 e 2 mc 2 
12
is
the
normalized
(4.2)
vector
field
and
 is the plasma wavenumber. The second, third, fourth and fifth
terms on the right side of Eq. (4.2) represent the contributions due to the plasma channel,
relativistic nonlinearity, ponderomotive effect and coupling of the relativistic nonlinearity
with the channel (channel coupling).
Substituting Eq. (4.1) into Eq. (4.2) and transforming independent variables z, t
to z ,  (  z   g ct , where  g  ck 0  0 is the normalized group velocity of the laser
pulse) gives,
2
 0 2

 2

2
2
2
2
2 



k

k


2
ik

2

1




0
P
0
g
 
2
2
2
 c

z
z

z




2

n nr 2
a
k 
 
4
n0 n0 rch2


2
P


 a2




 1  ar , z,    0 .
 4



(4.3)
Neglecting the higher order diffraction effect  2 z 2 in comparison to the terms
2  2 z
2k 0  z ,
and

1   
L2 r02  1  k P2 r02 4  k P2 r02 f 02 16
2
g

2
 2
in the limit k 0 r0 >1, L  2Z R and
respectively, the Lagrangian density of the wave
equation (4.3) is given by
99


a   a a  a  a 
 a *

l    a.  a * ik 0  a
 a *   

z    z
 z 
 z
 1   g2 

a a *  02
  2  k02  k P2 aa *
   c

 a 2 a 2 n  nr 2 2 2 nr 2  
 k P2 
 aa 
a a 
aa  .
n0
8n0 rch2
n0 rch2
 8

(4.4)
Considering a sinusoidal laser pulse the trial function for its amplitude may be
represented by

r 2    
a  f exp  i  1  i  2  sin  
rs   L 

(4.5)
where f ( z ),  ( z ),  ( z ), rs ( z ) and Lz  are the amplitude, phase shift, curvature, spotsize and length of the laser pulse respectively.
4.2 Density perturbation
In order to determine the density perturbation (wakefield) inside the laser pulse,
its evolution equation (Sec.1.2b) is written as
 2
 2  c 2 k P2 1  nr 2 n0 rch2
 t
 
 n c 2 2 2
   ( a ).
4
 n0

100
(4.6)
Transforming independent variables z, t  to z,   z   g ct  in Eq. (4.6) gives
 2
k P2


1  nr 2 n0 rch2
  2  2
g

 

 nn 

0
1
4 g2
 2 ( a ) 
2
2
2
( a ). (4.7)
2
2
4 g 
1
While deriving Eq. (4.7) the quasi-static approximation (Sec.1.2b) has been used. In this
approximation the laser does not evolve significantly as it transits a plasma electron. Thus
plasma electrons experience a laser field which is a function of  and r variables only
(independent of z ).
The solution of Eq. (4.7) is obtained by first writing the corresponding
homogeneous equation as
 2
k P2

1  nr 2 n0 rch2
 2
2
g
 
 

 nn  0 .

(4.8)
0
Eq. (4.8) has a solution of the form
n
n0
 C1 cos k P   C 2 sin k P 
(4.8a)
where C1 and C2 are constants. For obtaining the particular solution, C1 and C2 are
replaced by u  and v  respectively. Thus
101
n
n0
 u   cos k P   v sin k P  .
(4.8b)
Differentiating Eq. (4.8b) with respect to  gives
n n0 
u  
v 
 cos k P 
 sin k P 



 k P v cos k P  u sin k P  .
(4.9)
In order to obtain the value of u  and v  , it is assumed that
cos k P 
u  
v 
 sin k P 
 0.


(4.10)
Therefore, Eq. (4.9) reduces to
 n n0 
 k P v cos k P  u sin k P  .

Second differentiation of Eq. (4.11) with respect to  gives
 2 n n0

v
u 
 k P cos k P
 sin k P

2

 


102
(4.11)
 k P2 u cos k P  v sin k P .
(4.12)
Substituting Eqs. (4.8b) and (4.12) into Eq. (4.7), and assuming a wide channel
k P rch
 1 and  g2  1 , gives

v
u  1 2 2
1 2
2
k P cos k P
 sin k P 
a   .
    a   
2

  4
4 

(4.13)
Multiplying Eq. (4.10) by sin kP and Eq. (4.13) by cos kP and adding gives
v
1

 4k P
 2 2

2
2



a


a   cos k P .
 
2



(4.14)
Similarly multiplying Eq. (4.10) by cos k P and Eq. (4.13) by sin kP and subtracting
gives

u
1  2 2
2
2

  a    2 a   sin k P .

4k P 


(4.15)
Integrating Eqs (4.14) and (4.15) yields
v  
1   2 2
  a   
4k P L

2
2
2
L   2 a   cos k P d 

and
103
(4.16)
u   
2
 

1   2 2
2




a


a   sin k P d  2

2


L  
4k P  L

(4.17)
respectively. Substituting the values of u  and v  into Eq. (4.8b) gives the density
perturbation as,
n
n0

1
4k P


L

2

a  sin k P    d 
2
2
2
L   2 a  sin k P    d  .

1
4k P
(4.18)
Substituting the trial function (Eq. (4.5)) into Eq. (4.18) gives
 f 2 z 
  2 2 2
n0  k P r0 rˆs

n

2 ˆ2





2r 2  
1
ˆ    L cos 2 

1  2 2 1 
cos
2



L

2 ˆ2

 1   2 Lˆ2
L0 Lˆ 
 r0 rˆs 
 p

 1  L

z 




 2r 2


2 


ˆ
 2 2


cos
2



L

cos
exp




 r rˆ
ˆ

8 1   2 Lˆ2 
L
L
P



 0 s
0 


f
2






(4.19)
where Lˆ  L L0 and rˆs  rs r0 are the normalized laser pulse length and spot-size ( L0 ,
r0 represent the initial values of the pulse length and spot-size respectively), P is the
plasma wavelength and   L0  P . The first and second terms on the right side of Eq.
(4.19) respectively arise due to transverse and longitudinal ponderomotive effects. Since
104
L̂ varies with propagation distance, Eq. (4.19) becomes invalid if Lˆ  1 . Hence in this
limit the l’Hospital’s rule is applied to give the maximum density perturbation as
 f 2 z 
 
n0  k P r02 rˆs2
n


2r 2
1  2 2
 r rˆ
0 s

 1 cos k P     P 

 

sin k P  
 k
2
kP

 P
 2 f 2 z 

4k P L0 L


2r 2 
.
2 2 
 r0 rˆs 
   P sin k P  exp  
2 ˆ2
(4.20)
4.3 Evolution of laser spot-size and pulse length
Inserting Eqs. (4.5) and (4.19) into the Lagrangian density (Eq. (4.4)) and using

L

lˆ   r   ld  dr yields the reduced Lagrangian density as
0 0

f
lˆ 
2
z L0 Lˆ
4

 1 

f
4
2
g



rˆs

   2 rˆs2 Lˆ

ˆ
ˆ
2
r

2


r
 s

s
zˆ
zˆ
zˆ  k 0 L0 Lˆ2 zˆ

2 2
 02
k P2 r04 rˆs4 n 
2
2  r0 rˆs



 k0  k P 


4n0 rch2 
2 L20 Lˆ2  c 2
 2
 2 r02 rˆs2
z L0 Lˆ
32

2
 1    rˆs

 3k P2 r02 rˆs2 3k P2 r04 rˆs 4 n
1

 2 2
1 
2
64n0 rch
16
 Lˆ  1






k P2 r02 rˆs2 Lˆ sin 2Lˆ r02 rˆs2 k P sin 2Lˆ



8
2  2 Lˆ2  1
4 L0 Lˆ  2 Lˆ2  1


 .
 



 sin 2Lˆ  2 Lˆ2

 2Lˆ  2

(4.21)

L

ˆ

l

r
ld

Similarly, substituting Eq. (4.5) and (4.20) into Eq.(4.4) and using
0  0 dr
gives the reduced Lagrangian density in the limit L  P as,
105
f 2 z L0 Lˆ 
2
lˆ 
1     rˆs
4


 1 

3f
4
2
g

 2 r02 rˆs2
2 L20 Lˆ2
z L0 Lˆ 
256
5 


rˆs

   2 rˆs2 Lˆ

ˆ
ˆ
2
r

2


r
 s

s
zˆ
zˆ
zˆ  k 0 L0 Lˆ2 zˆ

2 2
  02
ˆs k P2 r04 rˆs4 n 
2
2  r0 r


  2  k0  k P 


4n0 rch2 
c
 2
k p2 r02 rˆs2
2

k p2 r04 rˆs4 n
8n0 rch2

 2 r02 rˆs2 
.
L20 Lˆ2 

(4.22)
The evolution of various laser parameters may be described with the help of the
Euler-Lagrange
  ˆ   j 
  lˆ  j  0
l 
z 

z


equation,
where  j  j  1, 2,.......4   , L  , and rs . Using the reduced Lagrangian density given by
Eq.(4.21) and substituting 1   and  2  L gives,


 2 2ˆ
f rˆs L  0
zˆ
(4.23)
and
1   
2
g
4 L30 Lˆ3
2
f 02
3 2 f 02 2 3 2 nf 02 r02 2



128L30 rˆs2 Lˆ2 512n0 rch2 L30 Lˆ2 32r02 L0 rˆs4 Lˆ2
106
f 02 L20

16r02rˆs4








f 02 2
64r02 L0 rˆs4

f 02  2 2 3 2 Lˆ2  1
2
64 L3  2 Lˆ2  1 rˆ 2 Lˆ2


 







 









s
  

3 2 Lˆ2  1 sin 2Lˆ 
ˆ

 cos 2L 
 0
2
 2 Lˆ2  1 Lˆ
32  2 Lˆ2  1 L30 Lˆ2 rˆs2 


f 02



 cos 2Lˆ Lˆ sin 2Lˆ 
f 02 2
  2 Lˆ2  1 



 2 2


2
2
2
4
2 ˆ2
2
ˆ
  2 Lˆ2  1 


ˆ

L

1
16
r
L
r

L

1



0 0 s

0


2 2 Lˆ2  1 sin 2Lˆ
  cos 2Lˆ


2 3 3
3 ˆ2
2 ˆ2
2
2 ˆ2
ˆ

 2L0 L  L  1 2  L  1 L0 L

(4.24)
respectively. Differentiating Eq. (4.24) with respect to propagation distance  z  gives the
equation describing the evolution of the laser pulse length as,


 3 2 1   g2
3 2 f 02 2 3 2 nf 02 r02 2
f 02




4 L40 Lˆ4
64 L40 Lˆ3 rˆs2 256n0 rch2 L40 Lˆ3 16r02 L20 rˆs4 Lˆ3

 
 


f 02
2 2 Lˆ2  1 cos 2Lˆ 
ˆ

 sin 2L 

16r02 L20  2 Lˆ2  1 rˆs4 Lˆ2 
Lˆ  2 Lˆ2  1







 
f 02 2 2 Lˆ2  1 cos 2Lˆ
2
16  2 Lˆ2  1 r 2 L2 rˆ 4 Lˆ3

0

0 s



f 02 4 Lˆ
2
32r 2 L2  2 Lˆ2  1 rˆ 4
0
0



f 02 sin 2Lˆ

2
16r02 L20  2 Lˆ2  1 rˆs4 Lˆ4



s

 



2

 6 2 Lˆ2  1 2 2 2 Lˆ2  1 




2 ˆ2
2
 L  1 






 2  2 Lˆ2  1 
1  2 ˆ2

 L 1 

107






 sin 2Lˆ Lˆ cos 2Lˆ 
f 02 3



2
2 2
2 ˆ2
4
2
  2 Lˆ2  1 
ˆ
8r0 L0  L  1 rs 

 sin 2Lˆ Lˆ cos 2Lˆ 3 2 Lˆ2 sin 2Lˆ 
f 02 3



3 
8
8  2 Lˆ2  1 
r02 L20 rˆs4  2 Lˆ2  1  16





f 02 2 2

2
32 L20  2 Lˆ2  1 rˆs2












 

 

 


2 2

3 sin 2Lˆ
  3 Lˆ  1
cos 2Lˆ 


Lˆ







3 2 Lˆ2  1 cos 2Lˆ 
ˆ

sin
2

L



Lˆ  2 Lˆ2  1




  1
 rˆ Lˆ

f 02  1 
f 02 cos 2Lˆ


 
 3
3 5 2 
2 2
 8L0 r0  rˆs Lˆ  8r0 L0  Lˆ  1




f 02  2

3
16 L40  2 Lˆ2  1 Lˆ2 rˆs2

2

3 2 Lˆ2  1 
 3 2

 2 ˆ  2 ˆ3 2 ˆ2


 L0 L L0 L  L  1 

f 02 2 3
2
16 L40  2 Lˆ2  1 rˆs2 Lˆ2





 


2
3
 
 f 02 2 2 Lˆ2  1 sin 2Lˆ

2
2 
8r03 L0  2 Lˆ2  1

5
s


2
Lˆ2  1
 2 2 2
2 Lˆ  Lˆ  1


2


1 
 rˆ 5 Lˆ3 
 s 




f 02 2  2 Lˆ2  1
f 02 2 cos 2Lˆ
3 f 02 2 2
f 02 3 Lˆ sin 2Lˆ



2
3
2
3 3 2
8r03 L0  2 Lˆ2  1 rˆs5 64r0 L0 rˆs Lˆ 4L0 r03 rˆs5  2 Lˆ2  1
16L0 r03 rˆs5  2 Lˆ2  1

f 02 2 2 3 2 Lˆ2  1
f 02 2 2 cos 2Lˆ
f 02 3 2 Lˆ2  1 sin 2Lˆ


2
2
3
32r0 L30 rˆs3 Lˆ2  2 Lˆ2  1
16r0 L30  2 Lˆ2  1 rˆs3 Lˆ2
16r0 L30  2 Lˆ2  1 rˆs3 Lˆ3

r0 rˆs
L0 zˆ


where












 



 Lˆ




 zˆ




(4.25)
ẑ
propagation

 z Z R , Z R  k 0 r02 2 is the Rayleigh length
distance.
Eq.
(4.23)
representing
108
the

is the normalized
conservation
of
energy
f
2

rs2 L  f 02 r02 L0  constant w here f 0 is the initial laser field amplitude , has been used
while deriving the evolution equation for the pulse length. The first, second and third
terms on the left side of Eq. (4.25) arise due to group velocity dispersion (GVD),
relativistic nonlinearity and its coupling with the plasma channel respectively. The
remaining terms are due to the presence of wakefields. When the laser pulse length
approaches the plasma wavelength, Eq. (4.22) is used to obtain the evolution of the laser
pulse length with the help of the Euler-Lagrange equation as


 3 2 1   g2 3 2 2 f 02 r02 n 3 2 f 02 2
15 f 02
9 2 f 02  Lˆ






4
4
256n0 rch2 L40 Lˆ3 64 L40 rˆs2 Lˆ3 128r02 L20 rˆs4 Lˆ3 64 L40 rˆs2 Lˆ5  zˆ
 4 L 0 Lˆ
 3 2 f 02  2
15 f 02
9 2 f 02  r0 rˆs



.

3 3 2
64r03 L0 rˆs5 Lˆ2 128r0 L30 rˆs3 Lˆ4  L0 zˆ
 64r0 L0 rˆs Lˆ
(4.26)
Substituting  3   and the reduced Lagrangian density (Eq.4.21) into the EulerLagrange equation gives
  rˆs
rˆs
.
zˆ
(4.27)
The evolution of the laser spot-size is described by substituting  4  rs into the EulerLagrange equation as
109
2 4
2 2 2
 2 rˆs
1 k p r0 rˆs n 3k p f 0 r0
 

zˆ 2 rˆs3
4n0 rch2
128Lˆ rˆs3

sin 2Lˆ
 2 Lˆ
 sin 2Lˆ 
1


ˆ
2 ˆ2
ˆ2 2  2 Lˆ2  1 2  2 Lˆ2  1 2 
L
2


L

1
L




2 2
2
ˆ
kP
f 0 r0
 sin 2L 


 2
.

2 ˆ2
3
ˆ  2 Lˆ2  1 
4

16  L  1 Lˆ rˆs 
L
L
0


f 02
 5
4rˆs










(4.28)
In deriving the Eq. (4.28), Eqs. (4.23), (4.27) and (4.21) have been used. The first term on
the right side of Eq. (4.28) shows vacuum diffraction, the second term represents channel
focusing, the third term leads to relativistic self-focusing and is a function of pulse length.
The fourth and fifth terms arise due to the transverse and axial ponderomotive effects,
respectively. When the laser pulse length approaches the plasma wavelength, singularity
arises in Eq. (4.21). In this limit Eq. (4.22) is used along with the Euler-Lagrange
equation to obtain the evolution of the laser spot-size as,
 2 rˆs
1 k P2 r04 rˆs n 3k P2 f 02 r02 15 f 02 3 2 f 02 r02





.
zˆ 2 rˆs3
4n0 rch2
128Lˆ rˆs3
32rˆs5 Lˆ 64 L20 rˆs3 Lˆ3
(4.29)
In order to obtain the usual matching condition (Sec. 1.3) of the laser spot as it propagates
in the plasma channel with the effect of wakefields neglected and the pulse length kept
constant L  L0  , the right hand side of Eq. (4.28) or Eq. (4.29) is considered to be zero.
Thus the laser beam will propagate with a constant spot-size (matched beam) if the
channel density gradient is equal to a certain critical value given by
110
nc 
4rch2 n0
k P2 r04
 3 f 02 r02 k P2 
1 
.
128 

(4.30)
The intensity of the laser pulse is dependent upon both the spot-size and pulse
length of the laser. Simultaneous evolution of the spot-size and pulse length enhances the
laser intensity with propagation distance if the laser compresses and focuses. Thus, the
intensity of the laser pulse, after it has propagated a given distance, is represented by
f
2

f 02
.
rˆs2 Lˆ
(4.31)
4.4 Graphical description
Maximum wakefields are obtained when the laser pulse length tends to the
plasma wavelength
  1 .
Therefore, numerical studies for three cases   1 ,
  1 and   1 have been presented. Figs. 4.1, 4.2 and 4.3 respectively depict the
evolution of the laser pulse length, spot-size (considering matching condition) and
intensity. Curves a, b and c are plotted for   1,   1 and   1 respectively. Curves a
and b are plotted with the help of Eqs. (4.25) and (4.28) while curve c   1 has been
plotted by simultaneously solving Eqs. (4.26) and (4.29). The channel depth nc  is
considered to be equal to a certain critical value (Eq. (4.30)) for which a laser beam
(assuming a constant pulse length and neglecting wakefields) propagates with a constant
spot-size. The laser and plasma parameters for all these figures are L0  16m (curve a),
111
9 m (curve b) and 11m (curve c), r0  25m, f 02  0.015 P  11m , rch  40m
and n  nc  4.3  1017 cm -3 . Fig 4.1 shows that due to the presence of wakefields, the
laser pulse length tends to compress when   1 (curve a) while the pulse length broadens
when   1 (curve b). It is interesting to note that the results obtained for both cases are
contrary to each other. For the case   1 (maximum wakefield), compression of pulse
length is slow. Fig 4.2 shows the evolution of the laser spot-size with propagation
distance. It is seen that for all three cases the laser spot becomes mismatched (even
though n  nc ) and tends to focus due to the presence of wakefields. For   1 (curve
a) focusing is greater than the other two cases. Consequently, in Fig.4.3 the peak intensity
of the laser pulse (Eq. 4.31) increases with propagation distance for all the three cases
(curves a, b and c). Since both compression and focusing of the laser pulse occurs
when   1 , the enhancement in intensity is more rapid (curve a) in comparison to the
cases when   1 and   1. When   1, the increase in laser intensity is small due to
broadening of the laser pulse. These results will be extremely useful for optimization of
laser and plasma parameters for applications such as laser-plasma based accelerators and
intense radiation generation.
112
1.005
b
1
0.995
L
L0
0.99
c
0.985
a
0.98
0.975
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z ZR
Fig.4.1 Variation of normalized pulse length with propagation distance
when   1 curve a ,   1 curve b
and   1 curve c for
L0  16m , 9m and 11m r0  25m,   1m , f 02  0.015
P  11m , rch  40m and n  nc  4.3  1017 cm 3 .
113
1.005
1
0.995
0.99
rs
r0
b
0.985
c
0.98
0.975
a
0.97
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z ZR
Fig.4.2 Variation of normalized spot-size with propagation distance when
  1 curve a,   1 curve b and   1 curve c for L0  16m ,
9 m and 11m , r0  25m, f 02  0.015 ,   1m P  11m ,
rch  40m and n  nc  4.3 1017 cm 3 .
114
0.0164
a
0.0162
c
0.016
0.0158
f
2
z 
0.0156
0.0154
b
0.0152
0.015
0.0148
0
0.2
0.4
0.6
z Z R 0.8
1
1.2
1.4
1.6
Fig.4.3 Variation of laser pulse intensity with propagation distance when
  1 curve a,   1 curve b and   1 curve c for L0  16m ,
9 m and 11m , r0  25m, f 02  0.015 ,   1m , P  11m ,
rch  40m and n  nc  4.3 1017 cm 3 .
CHAPTER 5
Conclusions
5.1 Conclusions
115
The present thesis is a significant contribution towards understanding the
evolution dynamics of short laser pulses in a pre-formed plasma channel. The study
shows that the evolution characteristics of laser pulses are affected by group velocity
dispersion as well as relativistic and ponderomotive nonlinearities. The possible
important applications arising from this study are in the fields of laser wakefield
acceleration and ultrashort laser pulse generation.
Considering the mildly relativistic regime, the wave equation describing the
evolution of the laser field amplitude in a plasma channel has been set up, taking into
account relativistic nonlinearity and group velocity dispersion effects. The Lagrangian
density for the wave equation is obtained and the variational technique is used to derive
equations describing the evolution of various laser parameters. Numerical techniques are
used to study the spot-size (pulse length) evolution, considering the pulse length (spotsize) to be constant. Further, simultaneous evolution of the laser spot-size and pulse
length has been studied. A higher rate of compression of the pulse length occurs when the
spot-size and pulse length evolve simultaneously as compared to the constant spot-size
case. Also, the focusing effect increases when the spot-size evolves simultaneously with
the pulse length in comparison to the case when the pulse length is considered to be
constant. Thus the enhancement in intensity of the laser pulse is more when the spot-size
and pulse length evolve simultaneously as compared to the case when either the spot-size
or the pulse length is considered to be constant. In order to study the effect of the
presence of the plasma channel, the simultaneous evolution of the laser spot-size and
pulse length in the channel is compared with the evolution in homogeneous plasma. It is
116
seen that the presence of the plasma channel enhances the compression of the pulse
length. This is due to the combined effect of relativistic nonlinearity and its coupling
with the plasma channel.
An important advantage of using a plasma channel for applications based on
laser-plasma interaction is that the channel allows the propagation of the laser beam with
constant spot-size (matched beam). This can be achieved by choosing a certain critical
value of the channel density gradient nc  . The evaluation of the critical channel depth is
independent of the laser pulse length. The present study shows that when a laser spot
satisfying the beam matching condition propagates in a plasma channel it does not remain
matched when the pulse length evolution is taken into account. Since the laser pulse
length undergoes compression the laser spot-size reduces (focuses), instead of remaining
constant. It is also shown that if the initial laser intensity is increased, compression as
well as focusing of the laser spot increases.
It is shown that, an initially unchirped laser pulse becomes chirped as it
propagates in a plasma channel, due to group velocity dispersion (GVD) as well as selfphase modulation (SPM). For a matched rs  r0 , n  nc  laser pulse the frequency at
the front (back) of the pulse decreases (increases) due to SPM effect whereas it increases
(decreases) due to GVD effect. However, when both effects are taken into account SPM
dominates over GVD leading to an increase in frequency from front to back of the pulse
(positive chirping). Since SPM depends upon the intensity of the laser pulse, a reduction
in intensity causes the GVD and SPM induced chirp to approximately cancel each other
so that the initially unchirped laser pulse remains nearly unchirped.
117
The propagation of an ultrashort chirped laser pulse in a parabolic plasma channel
has been analyzed. For such pulses, relativistic and ponderomotive nonlinearities nearly
cancel each other. The effect of initial chirp on the laser pulse length and intensity of a
matched laser beam propagating in a plasma channel is studied. The pulse length and
chirp parameter of the laser pulse are modified due to its interaction with plasma. The
effective values of these parameters have been obtained. It is shown that the pulse length
of a positively chirped laser pulse length, compresses with propagation distance.
Maximum compression occurs at a certain critical distance which depends on the chirp
parameter. When the propagation distance exceeds the critical value, the pulse length
starts to broaden. The effective pulse length becomes equal to the initial pulse length after
the laser beam traverses a distance equal to twice the critical distance. However, the pulse
length of a negatively chirped pulse broadens continuously. Consequently, it is seen that
at the critical distance, the peak intensity (at the pulse centroid) of a positively
(negatively) chirped laser pulse maximizes (decreases) in comparison to its initial peak
intensity.
It is seen that the chirp parameter of an initially chirped laser pulse changes due to
propagation in plasma. Also, after the pulse has traversed a certain critical distance, the
positively chirped laser pulse becomes unchirped. Beyond this critical distance, chirp
reversal occurs and the laser pulse becomes negatively chirped. This reversal occurs
because the group velocity induced (negative) chirp and its coupling with the initial chirp
becomes more effective in comparison to the initial (positive) chirp. If the distance
traversed by the laser pulse is less than the critical distance, reversal of chirp does not
occur. Also chirp reversal does not occur for negatively chirped pulses.
118
The effect of wakefields, generated inside the pulse, on the propagation
characteristics of the laser beam has been studied. The density variation of plasma
electrons produced by the combined effect of transverse and axial ponderomotive
nonlinearity is obtained. Since the density perturbation exhibits a singularity when the
laser pulse length approaches the plasma wavelength (maximum wakefield condition),
source term driving the laser amplitude for this case is obtained using l’Hospital rule.
Using the variational technique, coupled equations describing the evolution of the laser
pulse length and spot-size are obtained for the case when (i) the plasma wavelength P 
is not equal to the pulse length L  and when (ii) L  P . It is seen that for a matched
laser beam the pulse length tends to compress (broaden) when it is initially greater (less)
than the plasma wavelength. When the plasma wavelength is equal to the laser pulse
length the rate of compression of the pulse length is slow. It is important to note that in
the regime where maximum wakefields are obtained the propagation characteristics of
the laser pulse are extremely sensitive to the ratio of the pulse length to the plasma
wavelength and show contrary behaviour when the laser pulse length is slightly increased
or decreased. These results will be extremely useful for optimization of laser and plasma
parameters for various applications, for example, in laser wakefield accelerators,
generation of maximum wakefield amplitude is essential but at the same time, matched
beam propagation is equally important. Therefore the regime   1 may be more
appropriate as compared to   1. On the other hand for generating high intensity laser
beams (via compression), the regime   1 will be suitable.
5.2 Recommendations for future work
119
The thesis presents a comprehensive study of the simultaneous evolution of the
spot-size and pulse length of a laser, as it propagates in a pre-formed plasma channel.
With appropriate choice of laser and plasma parameters it has been shown that focusing
of the laser spot-size and compression of its pulse length can be achieved. The possibility
of further pulse compression could be explored by tapering the plasma density along the
direction of propagation of the laser pulse. Therefore, the study of simultaneous evolution
of the laser spot-size and pulse length in a tapered plasma channel can be an interesting
proposal for future work.
In the presence of a plasma channel, the study of propagation characteristics of
initially chirped ultrashort laser pulses in the linear regime is presented. The study of
propagation of initially chirped laser pulses, for which relativistic and
ponderomotive
nonlinearity effects do not cancel (ultrashort) needs to be carried out in the nonlinear
regime using appropriate techniques. The effect of wakefields generated inside a
sinusoidal laser pulse, on the propagation characteristics of the pulse has been analyzed in
the present thesis. It would be interesting to study the wakefields generated by an initially
chirped laser pulse propagating in a plasma channel and the effect of such wakefields on
the laser pulse dynamics. Simulation studies of the work presented in this thesis could be
undertaken for values of the laser strength parameter equal to or greater than unity. In this
regime higher order nonlinearities will affect the laser plasma interaction.
120
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