Nonlinear interaction of intense laser beams with magnetized plasma

1
Nonlinear interaction of intense laser beams with
magnetized plasma
Rohit Kumar Mishra
Department of Physics, University of Lucknow
Lucknow 226 007
2

• Interaction of intense lasers with plasma
  involves a number of interesting nonlinear
  physical phenomenon including self-focusing,
  wakefield generation and quasi-static magnetic
  field generation.

• Experiments report that quasi-static magnetic
  fields (both axial and azimuthal) of the order of
  MG are generated when intense laser beams
  interact with underdense plasma.

• These fields affect the propagation
  characteristics of the laser pulses and hence
  play vital role in fast ignition schemes of
  inertial confinement fusion, charged particle
  acceleration and harmonic generation.

3

• In the present thesis a theoretical analysis of
  intense laser plasma interaction , in the
  presence of a uniform magnetic field has been
  presented.
• The effect of magnetic field on
• (a) self-focusing property,
• (b) modulation instability
• and
• (c) possible generation of second harmonic
  frequencies
• have been shown.

4
Self-focusing of intense laser beams propagating
in magnetized plasma

• For a laser beam having Gaussian radial profile,
  the intensity is peaked on axis
  causing the plasma electrons to be expelled away
  from the axis. Therefore, the refractive index
  tends to maximize along the axis . Due to this
  refractive index gradient the phase velocity of
  the laser wavefront increases with the radial
  distance, causing the wavefronts to curve inwards
  and the laser beam to converge

5
(a) Linearly polarized laser beam propagating in
transversely magnetized plasma

• Consider a linearly polarized laser pulse
  propagating in a uniform plasma embedded in a
  constant external magnetic field

Amplitude
Wave number
Frequency

• Basic equations describing the evolution of
  laser beam in magnetized plasma are

• Wave equation

Jha et al Phys. Plas. 13, 103102 (2006)
6

• Current density equation

• Lorentz force equation

and

• Continuity equation

is the relativistic factor and is the
magnetic vector of the radiation
field.
Jha et al Phys. Plas. 13, 103102 (2006)
7

• Using perturbative technique all quantities are
  simultaneously expanded in orders of the
  radiation field. Using Eq. (4) first order
  velocities are given by

• is the cyclotron frequency
  and is the normalized
  field amplitude.

• Presence of magnetic field increases the
  transverse quiver velocity of plasma electrons
  and also leads to the generation of a
  longitudinal velocity component due to
  force acting on plasma electrons.

Jha et al Phys. Plas. 13, 103102 (2006)
8

• Second and third order velocities are

• The second order high frequency x-component of
  velocity is generated due to uniform magnetic
  field and reduces to zero in its absence. However
  z- component and third order tranverse velocities
  are modified due to external magnetic field.

Jha et al Phys. Plas. 13, 103102 (2006)
9

• Density perturbations introduced in the plasma
  due to interaction with the laser beam can be
  obtained by expanding the continuity Eq. (5).
  Thus first order density perturbation is given by

• The first order density perturbation arises due
  to the presence of external magnetic field and
  reduces to zero in its absence. The second order
  density perturbation is given by

Jha et al Phys. Plas. 13, 103102 (2006)
10

• Perturbed velocities and densities are used to
  obtain the transverse current density Eq. (3).

Nonlinear current density terms
Linear current density
external magnetic field
Relativistic mass correction

• Using the value of current density obtained with
  the help of Eq.(8) and using it in the wave
  equation and assuming the radiation amplitude to
  be slowly varying function of z, the paraxial
  wave equation is given by

Jha et al Phys. Plas. 13, 103102 (2006)
11

• N includes nonlinear perturbations due to

• Relativistic effects
• Density fluctuations
• Coupling of radiation field with magnetic field

• Using source dependent expansion (SDE) method the
  equation for laser spot-size is obtained as

Jha et al Phys. Plas. 13, 103102 (2006)
12

• Here is the normalized laser
  power and is the
• Rayleigh Length. defines the critical
  laser power for nonlinear
• self-focusing of a laser beam in magnetized
  plasma and its value is

• A graphical analysis of the normalized laser spot
  size variation with propagation distance and
  magnetic field and the variation of critical
  power with magnetic field is presented.

Jha et al Phys. Plas. 13, 103102 (2006)
13
Fig. 1 Variation of rs/r0 with z/ZRfor (a)
unmagnetized plasma
(b) 0.2 and (c) 0.4, with,
, and
0.1.
14
Fig. 2 Variation of with at
0.3 for 0.271,
s-1 and 0.1.
15
Fig. 3 Variation of with
for ,
s-1 and 0.1.
16
(b) Circularly polarized laser beam propagating
in axially magnetized plasma

• A circularly polarized laser beam propagating in
  plasma is embedded in a uniform, axial magnetic
  field . The normalized electric
  field vector of the radiation
  field propagating along the z-direction is
  represented by

where k0 and ?0 are the
normalized amplitude, wave number and
frequency of the radiation field,
respectively. s takes values 1for right or left
circularly polarized radiation, respectively.

• Wave equation governing the propagation of a
  circularly polarized laser beam in presence of
  axial magnetic field is given by

17

• Proceeding in the same manner as in the case of
  linearly polarized laser beam the spot-size of
  the circularly polarized laser beam is given by

where S is given by
18
Fig. 4 Variation of rs/r0 with z/ZR for (a) ?c/?0
0, (b) ?c/?0 0.15, s -1 and (c) ?c/?0
0.15 s 1with a0 0.271 and ?0 1.881015
s-1.
19
Fig. 5 Variation of rs/r0 with ?c/?0 for right
circularly polarized laser beam having z/ZR 0.3,
a0 0.271, ?0 1.881015 s-1 and ?c/?0 0.1.
20
Fig. 6 Variation of rs/r0 with ?c/?0 for left
circularly polarized laser beam having z/ZR 0.3,
a0 0.271, ?0 1.881015 s-1 and ?c/?0 0.1.
21
Modulation instability of laser pulses in axially
magnetized plasma

• Modulation instability is the process in which
  the pump wave amplitude gets modulated in space
  or time. Modulation occurs due to the interplay
  between the nonlinearity and dispersive effects
  Due to this instability the actual wave number
  (k0) of laser beam change into k0K (where K is
  modulation wave number).

• Modulation instability of a circularly polarized
  laser beam propagating through axially
  magnetized, cold and underdense plasma has been
  studied. The governing wave equation is

22

• Considering only linear source term and taking
  the Fourier Transform of wave equation gives

where is the Fourier
Transform of slowly varying amplitude a0( ,t)
and
is the linear part of the total refractive
index, having contributions due to vacuum, finite
spot-size of the laser radiation and presence of
magnetized plasma respectively. Defining mode
propagation constant and
considering the limit that mode propagation
constant is close to the unperturbed wave number
(k0), Eq. (15) may be written as
23

• Using Taylor series expansion the frequency
  dependent function ßm (?) may be expanded
  about ?0 as

where . In
Eq. (17) is related to the group velocity
dispersion (GVD).

• Substituting Eq. (17) in Eq. (16), retaining
  terms up to ß2m (?) and introducing nonlinear
  current source term on the right hand side gives
  the nonlinear non-paraxial wave equation as

24

• In order to study the spatial modulation
  instability, transformations are carried out from
  spatial and temporal coordinates (z, t) in the
  laboratory frame to the spatial coordinates (z,
  ?)in the pulse frame. The transformation is
  achieved by substituting ?z vgt and z z.
  Substituting the nonlinear parameter
  , setting ß0 k0,
  and neglecting in
  comparison to 2
  Eq. (18) may be written in the 1-D limit as

Solution of Eq.(19) may be written as
where a10 (z, ?) is the perturbed beam amplitude
and is the normalized laser power in
presence of axial magnetic field.
25

• The exponentially varying perturbed amplitude may
  be taken to be of the form

where k is the propagation wave number of the
perturbed wave amplitude. Taking to vary
with z as exp(Kz), where K is the modulation
wave number, the dispersion relation for
one-dimensional modulation instability is written
as
where , and
are normalized dimensionless
quantities.
26

• Modulation instability is excited provided
  is sufficiently negative ,
  , so that can be complex.
  Consequently the range of unstable wave numbers
  for which the instability exists is given by

• The growth rate of modulation instability for the
  laser beam propagating through transversely
  magnetized plasma is given by

27
Fig. 7 Variation of modulation instability growth
rate for right (curve a), and left (curve c)
circularly polarized laser beam propagating in
magnetized plasma and for laser beam propagating
in unmagnetized (curve b) plasma, with normalized
wave number with r015µm, a00.271 ,
?01.881015s-1 , ?p/?00.1 and ?c/?00.05
(curves a and b).
28
Fig. 8 Stability boundry curves showing the
variation of normalized laser power with
for right (curve a), left (curve c) circularly
polarized laser beam propagating in magnetized
plasma and unmagnetized case (curve b). The
parameters used a00.271 , ?01.881015s-1 ,
?p/?00.1 and ?c/?00.05.
29
Second harmonic generation in laser magnetized
plasma interaction

• It has been shown that when an intense laser beam
  interacts with homogeneous plasma embedded in a
  transverse magnetic field, second order
  transverse plasma electron velocity oscillating
  with frequency twice that of the laser field is
  set up

• This plasma electron velocity couples with the
  ambient plasma density leading to a transverse
  plasma current density oscillating at the second
  harmonic frequency. Also first order density
  perturbation oscillating at the laser frequency
  arises due to the presence of the magnetic field.
  This density perturbation couples with the
  fundamental transverse quiver velocity to give
  transverse plasma current density oscillating at
  twice the laser frequency.

30

• Consider a linearly polarized laser beam
  propagating along the z-direction as
• As the beam propagates through transversely
  magnetized plasma, transverse current density at
  twice the laser frequency arises and
  acts as a source of second harmonic generation.
• Corresponding to the frequencies and
  the electric fields are assumed to be
  given by

Laser frequency
Amplitude
Propagation constant
31

• Here and
  . and are wave
  refractive indices corresponding to the
  frequencies and .
• The equation governing the propagation of the
  laser pulse through plasma is given by
• where .
• The plasma electron density is given by

Plasma electron velocity
Plasma electron density
32

• Relativistic interaction between the
  electromagnetic field and plasma electron is
  governed by
• Lorentz force equation
• Continuity equation

Transverse magnetic field
Magnetic vector of radiation field
33

• Using perturbative technique all quantities can
  be expanded in orders of the radiation field.
  Using Lorentz force equation the first and second
  order longitudinal velocities and second order
  transverse velocity of the plasma electrons is
  given by
• where is the cyclotron frequency of
  plasma electrons and
  and are normalized
  amplitudes.

34

• The first order plasma electron density is
  obtained by using continuity equation as
• The transverse current density can now be
  written as
• From the current density equation it is observed
  that current density at second harmonic arises
  via
• Transverse plasma electron velocity oscillations
  at second harmonic frequency.
• Coupling of electron density oscillations at
  fundamental frequency and electron quiver
  velocity also oscillating at fundamental
  frequency. This contribution is attributed to
  the external magnetic field and provides source
  for the generation of second harmonic radiation.

35

• Linear fundamental and second harmonic
  dispersion relations are given by
• For obtaining second harmonic amplitude the
  value of the current density is
  substituted in the wave equation and it is
  assumed that the distance over which
  changes appreciably is large compared with the
  wave length and that depletes (with z)
  very slightly so that quantity can be
  assumed to be independent of z.
• The evolution of the second harmonic amplitude
  is given by
• where .

36

• The second harmonic conversion efficiency is
  defined as
• For a given conversion efficiency is
  periodic in z.
• The minimum value of z for which is
  maximum is given by
• The length represents the maximum plasma
  length up to which the second harmonic power
  increases. For z gt the second harmonic power
  reduces again.

37

• The maximum second harmonic efficiency, after
  traversing a distance is given by
• The maximum conversion efficiency is zero in the
  absence of magnetic field and increases in with
  increase in magnetic field. However, near the
  electron cyclotron resonance the theory breaks
  down. The conversion efficiency also increases
  with the increase in the intensity of the laser
  beam.

38
Fig. 9 Variation of conversion efficiency (?)
with the propagation distance z for?c/?0?p/
?00.1, a120.09 and ?01.881015s-1.
39
Fig. 10 Variation of maximum conversion
efficiency (?max) with ?c/?0 for ?p/?00.1,
a120.09 and ?01.881015s-1.
40
Conclusions

• Transverse magnetization of plasma enhances the
  self-focusing property of the laser beam and the
  critical power required to self-focus the
  linearly polarized laser beam propagating in
  transversely magnetized plasma is reduced. This
  above explanation is also valid for a left
  circularly polarized laser beam propagating in
  axially magnetized plasma

• If the laser beam is right circularly polarized,
  the beam will be defocused. Focusing of the right
  circularly polarized beam can be brought about by
  reversing the direction of the external magnetic
  field.

• Magnetic fields alter the growth rate of
  modulation instability. The peak growth rate of
  modulation instability in the presence of the
  magnetic field for a left circularly polarized
  laser beam is found to increase while for right
  circularly polarized beam the spatial growth rate
  reduces as compared to the absence of magnetic
  field.

• The stability boundary curve shows that for left
  circularly polarized beam, the area representing
  the unstable interaction is increased while that
  for left circularly polarized laser beam it
  reduces.

41

• It is seen that second harmonic conversion
  efficiency oscillates as the wave propagates
  along the z-direction.

• It is found that maximum conversion efficiency
  is zero in the absence of magnetic field and
  increases as the magnetic field is increased.

• The conversion efficiency also increases with
  increase in intensity of the laser beam.

• observation of second harmonics in homogeneous
  plasma could point towards the possibility of
  presence of a magnetic field, since second
  harmonics have so far been generated by the
  passage of linearly polarized laser beams through
  inhomogeneous plasma.

42
Journal Publications

• Self focusing of intense laser beam in magnetized
  plasma
• Pallavi Jha, Rohit K. Mishra, Ajay K. Upadhyay
  and Gaurav Raj, Physics of Plasmas, 13, 103102
  (2006).
• Also published in Virtual Journal of Ultrafast
  Science, 5, Issue 10 (2006)

• Second harmonic generation in laser
  magnetized-plasma interaction
• Pallavi Jha, Rohit K. Mishra, Gaurav Raj and
  Ajay K. Upadhyay, Physics of Plasmas, 14, 053107
  (2007).
• Also published in Virtual Journal of Ultrfast
  Science, 6, Issue 5, (2007)

• Spot-size evolution of laser beam propagating in
  plasma embedded in axially magnetic field.
• Pallavi Jha, Rohit K. Mishra, Ajay K. Upadhyay
  and Gaurav Raj, Physics of Plasmas, 13, 103102
  (2006).

43
Conference Proceedings

1. Interaction of laser pulses with magnetized
   plasma
   Rohit K. Mishra, Ajay K. Upadhyay, Gaurav Raj
   and Pallavi Jha
   Presented at 20th National Symposium on Plasma
   Science and Technology Cochin (2005).
2. Modulation instability of a laser beam in a
   transversely magnetized plasma Rohit K.
   Mishra, Ajay K. Upadhyay, Gaurav Raj and Pallavi
   Jha Presented at 21st
   National Symposium on Plasma Science and
   Technology Jaipur (2006).
3. Spot-size evolution in axially magnetized plasma
   
   Rohit K. Mishra, Ajay K. Upadhyay, Gaurav Raj and
   Pallavi Jha Presented at
   6th National Laser Symposium Indore (2007).
4. Magnetic field detection via second harmonic
   generation
   Rohit K. Mishra, Ram G. Singh and Pallavi Jha
   
   Presented at 22nd National Symposium on Plasma
   Science and Technology Ahmedabad (2007).

Thank you