Transcript Suppression of a Parasitic Pump Side-Scattering in

Suppression of a Parasitic Pump Side-Scattering in
Backward Raman Amplifiers of Laser Pulses in Plasmas
A. A. Solodov, V. M. Malkin, N. J. Fisch
Abstract
In backward Raman amplifiers (BRA), the pump laser pulse can be prematurely
depleted through Raman scattering, seeded by the plasma noise, as the pump
encounters plasma before reaching the counter-propagating seed pulse. It was shown
previously that detuning of the Raman resonance, either by a plasma density gradient
or a pump frequency chirp, can prevent the premature pump backscattering, even
while the desired amplification of the seed pulse persists with a high efficiency.
However, parasitic pump side-scattering is not automatically suppressed together
with the parasitic backscattering, and might be even more dangerous for BRA. What
we show here is that by combining the above two detuning mechanisms one can
suppress parasitic pump side-scattering as well. Apart from the simplest
counterpropagating geometry, we examine BRA for arbitrary angles between the
directions of pump and seed propagation. We show that, by selecting an appropriate
direction of the plasma density gradient, one can favorably minimize the detuning in
the direction of the seed pulse propagation, while strongly suppressing the parasitic
pump side-scattering in all the other directions. This work was supported in part by
DOE and DARPA.
Conceptual Scheme of Backward Raman Amplifiers (BRA)
pump beam
seed beam
pump beam
seed pulse
plasma wave
amplified pulse
ω2, k2
ω1, k1
kp
depleted pump
kp=k1-k2, ωp=ω1-ω2
The plasma wave forms a 3D Bragg
cell grating that scatters power from
the pump into the seed
At the nonlinear amplification stage
the amplified seed can completely
deplete the pump
Conceptual Scheme of BRA (Continued)
seed pulse
plasma
target
pump pulse
Resent studies showed that short pumped
pulses of nearly relativistic non-focused
intensities are expected: I~1017 W/cm2 for
λ=1 μm [1].
This is 5 orders of magnitude higher than
currently available through chirped pulse
amplification [2].
Additional intensity gain is provided by
focusing.
[1] V. M. Malkin, G. Shvets, and N. J.
Fisch, Phys. Rev. Lett. 82, 4448 (1999).
[2] G. A. Mourou, C. P. J Barty, and M. D.
Perry, Phys. Today 51, 22 (1998).
Problem: Pump Can be Prematurely Depleted by Raman
Scattering Seeded by Thermal Langmuir Noise, before it
Reaches the Seed Pulse
pump
SRS
It is the same efficiency of stimulated
Raman scattering, that makes possible
the fast compression, which can
complicate the pump transporting to the
seed.
The problem is aggravated by the fact
that the linear Raman scattering
(responsible for the noise amplification)
has a larger growth rate than its
nonlinear counterpart (responsible for
the useful amplification of the seed).
The Premature Pump Backscattering Can be Suppressed by
an External Detuning of the Raman Resonance [3]
δω=δωplasma-δωpump
δω
δωdetuning
δωplasma
-z
pulse
location
δωpump
0
z
pump
front
It appears to be possible to suppress the
unwanted Raman backscattering of the
pump by noise, while not suppressing
the desirable seed pulse amplification.
The filtering effect occurs because in the
nonlinear regime the pumped pulse
duration decreases inversely
proportional to the pulse amplitude. The
increased frequency bandwidth allows to
tolerate larger and larger external
frequency detuning.
[3] V. M. Malkin, G. Shvets, and N. J. Fisch, Phys. Rev. Lett. 84, 1208 (2000).
The Goal of the Present Paper is to Analyze How a Parasitic
Pump Side-Scattering in BRA Can be Suppressed
The Raman growth rate maximizes for
backscattering:
x
   0 sin ( / 2),  0  a0  p / 2
(linearly polarized pump with normalized
amplitude a0=eA/mc2, scattering in the plane
perpendicular to the pump polarization).
SRS
θ
pump
z
y
However, the side-scattered radiation has
more time and a longer distance to be
amplified by the pump, before it leaves the
plasma.
These two effects practically compensate
each other making suppression of pump
side-scattering an important task.
Main Equations
The linear stage of SRS is described by
t  (vb )b   fˆ * ,
( t  i ) fˆ *   b,
where the vector-potential envelopes of the pump and scattered Stokes
waves and Langmuir wave electric field are defined:
A 0e / mec 2   a0e  exp(ika z  iat )  c.c. / 2,
A s e / mec 2  beb exp(ik br  ibt )  c.c. / 2,
E p e / me  p c  k f 0  fˆ exp(ik f r  i f t )  c.c. / 2.
The pump wave frequency, wavenumber, and the unit polarization vector are
a , ka  a2   2p / c  a / c, and e  (e x  e y ) / 1 |  |2 (e  e*  1),
in particular, Im  0 for linear polarization and   i for circular
polarization.
Main Equations (Continued)
x
The Stokes wave frequency, wave vector, and unit
polarization vector b  a   ,
z1
φ
pump
k b  ( b2   2p / c)(sin  cos  ,sin  sin  ,cos )
SRS
θ
z
y
kb
(k b  e ).
2
k
The frequency and wave vector of the Langmuir
wave  f  a  b   p   and k f  ka e z  k b .
(| k b | ka   / c  k ), and eb  e  
The Raman growth rate
  a0
 p
2
sin 2  | cos    sin  |2
sin( / 2) 1 
.
2
1 |  |
a02 p  (r  ed )
 z 
The frequency detuning  
q

q
1
2  t   ,

4 
c
 c 
terms with q1 and q2 correspond to a plasma frequency detuning and a pump chirp,
ed  (sin  d cos d ,sin  d sin d ,cos d ).
Main Equations (Continued)
Making transformation to a new coordinate frame ( x1 , y1 , z1 ) with
z1 || k b , x1  ( z , k b ), y1 || ( z, k b ), introducing new dimensionless variables
  (t  z1 / c)  p / 2,
  ( z1 / c)  p / 2,
  2 /  p ,
and performing a simple phase transformation (b, fˆ * )  (b ', f ')exp(),
where    ( x1 , y1 ,  ), one has
 b '  f ',
(  ia02 q ' ) f '  b ',
where
q '  q1[cos cos d  sin  sin  d cos(  d )]  q2 [1  cos ].
Green’s Function
The Green's function, corresponding to a localized source term  ( ) ( )
added to the r.h.s. of the equation for f ', has the form [3]
b0 ( , ) 

2 i

dp
i  iq  
exp  p  ln 1     1F1 (i / q;1; iq ),
C p
q 
p 


where    2 , q  q '(a0 / ) 2 , the contour C encompasses in the positive direction singularities at p  0 and p  iq, 1F1 is the confluent hypergeometric function.
For | q | 
b0
γ
0


1, b0  I 0 2  , as in a uniform
plasma [4], where I 0 is the modified Bessel function,
c/2
z1=ct/2
in original variables    2 p (t  z1 / c) z1 / 4c,
so the maximum, reached at z1  ct / 2, moves with
the speed c / 2 and increases with the growth rate for
the monochromatic wave instability  .
[4] D. L. Bobroff and H. A. Haus, J. Appl. Phys. 38,
390 (1967).
Green’s Function (Continued)
The effect of detuning becomes
|b0|
noticeable at | q | 
1,
for 
1, | b0 | saturates:
M  4 / q 2
| b0sat |  | q | / 2 exp( / | q |).
|b0sat |
0
c
z1
If the initial amplitude of the localized
Langmuir wave perturbation is much
less then  -1 exp( / | q |), the Stokes
wave amplitude remains forever
smaller then 1.
The pump depletion similarly remains
small.
Premature Pump Backscattering and Side-Scattering in
Absence of Detuning
A rectangular in the longitudinal and transversal directions laser
pump enters at t  z  0 the plasma layer. A uniform Langmuir
wave seed is assumed in plasma before the pump coming.
The Stokes wave amplitude at the l.h.s. plasma boundary:
z1
b( z  0, t  2l / c)  fˆ0
θ
 fˆ0
z
0

L / sin 2 ( / 2)
0

dZ I 0 2 Z  2 L  Z (1  cos  ) 
1
exp( L '),
2 2 sin( / 2)
where L  l  p / 2c and  '   2 / sin( / 2), L '

1.

For linear polarization,   0 :  '  a0 2 1  sin 2  cos 2  ,
in the plane orthogonal to the pump polarization (cos   0)
 ' is independent of  .
l
For circular polarization,   i :  '  a0 1  cos 2  .

Suppression of a Parasitic Pump Side-Scattering in a
Backward Raman Amplifier
In Order to Suppress Pump Side-Scattering in All Directions:
one should provide | q | 0 for all  and  ;
it is desirable to have a minimal | q | in the seed pulse propagation
direction (to simplify a useful amplification).
BRA, ed  z ( d   ) :
q2  q1 / 2  0 or q2  q1 / 2  0.
Pump chirp is necessary, q2  0, as otherwise
δω
δωplasma δωpump
q  0 for scattering at    / 2.
If q1  0, for linear polarization:
q  2q2 / 1  sin  cos  , | q | minimizes
2
2
0
for scattering in the plane perpendicular to
the pump polarization (   / 2), for which it is independent of  ;
for circular polarization: q  2q2 / (1  cos 2  ) / 2.
z
Suppression of a Parasitic Pump Side-Scattering in a
Backward Raman Amplifier (Continued)
δω
δωdetuning
δωplasma
-z
Stokes
pulse front
location
δωpump
0
z
pump
front
Plasma density gradient and pump chirp,
q1  0, q2  0 :
detuning (| q |) is minimal for
backscattering (   ),
both for linear and circular polarization.
Suppression of a Parasitic Pump Side-Scattering in a
Backward Raman Amplifier (Continued)
linear polarization,
φ=π/2
linear polarization,
φ=π/6
circular polarization;
also
linear polarization,
φ=π/4
q1 / 2q2  0
q1 / 2q2  0.5
q1 / 2q2  0.25
q1 / 2q2  0.75
Scattering Cross-Section of the Pump Beam in BRA with
Frequency Detuning
A white noise statistics for the stochastic thermal Langmuir wave seed is assumed:
f 0 ( z1 , t )  0,
f 0 ( z1 , t ) f 0* ( z1', t )  n1D ( z1  z1').
The 1-D spectral density of thermal fluctuations n1D is connected with the usual
3-D thermal fluctuations spectral density,
n
2
 e 

T / 2 
 16 ,
3 e
(2 )
 me p c 
1
3D
by n1D  n3 D k 2f , where  is the solid angle in the wave vector space.
The averaged Stokes wave field intensity for scattering from a region z1 is
bz1
2


z1
0
b0sat e
ib 0  p
2c
2
f 0 dz1

sat 2
b0
 2p
4c
1D
n
z1.
2
Scattering Cross-Section of the Pump Beam in BRA with
Frequency Detuning (Continued)
The pump energy scattered by a unity plasma volume in a unity of time in
a solid angle  :
2
2
bz1
dI
c  mec 


 .
V
z1 16  e 
Dividing it by the incident pump intensity I 0  (c / 8 )a02 (mec / e) 2 , a
differential scattering cross-section of the pump pulse in plasma is obtained:
dI
3
2
 3 q2  (q1  q2 )cos sin 2 ( / 2)e 2 /|q|  k d  ne 0 ,
I 0V 8

where  0  (8 / 3) e / me c
2


2 2
is the Thomson scattering cross section on
a single electron, d  T / 4 ne e

2 1/ 2
is the Debye length.
Suppression of a Parasitic Pump Scattering in a Raman
Amplifier with Arbitrary Angles between Pump and Seed Pulses
Assume that the seed pulse propagates in the direction determined by  s and s .
Linear Polarization (  0):
Only the most desirable for Raman amplification case will be considered,
when the seed pulse propagates perpendicular to the pump polarization, s   / 2.

q  2 q1
cos(   d )  sin  sin  d [1  cos(  d )]
 q2
1  cos
1  sin 2  cos 2  .
It is desirable to use a plasma frequency gradient
in the direction d   / 2 and  d  ( s   ) / 2,
q1 and q2 should satisfy 0  q1 / 2q2  sin( s / 2)
[0  q1 / 2q2  cos(   d )].
The minimum detuning gradient is in the direction
of the seed propagation, | q || q1 / sin( s / 2)  2q2 | .
/
seed
pulse
density
gradient
y
θd
θs
z
Suppression of a Parasitic Pump Scattering in a Raman
Amplifier with Arbitrary Angles between Pump and Seed Pulses
(Continued)
Linear polarization. (q ) / 2q2 vs.  (a) for   d   / 2 and vs.  (b)
for the minimal values of  in the plots (a).
d  7 /8,
d  3 / 4,
d  5 /8.
Suppression of a Parasitic Pump Scattering in a Raman
Amplifier with Arbitrary Angles between Pump and Seed Pulses
(Continued)
Circular Polarization (  i):
cos(   d )  sin  sin  d [1  cos(  d )]
q  2 q1
 q2
1  cos 2   / 2,

1  cos
q  0 for all  and  if 0  q1 / 2q2  cos(   d ).
For minimal | q | in the direction    s ,   s one should take: d  s ,
q1 , q2 , and  d should satisfy
q1
1  cos s


q2 cos( s   d )

/
1
 cos( s / 2   d ) 2(1  cos  s ) 
1  cos(   ) sin( / 2)sin( 2 )  .
s
d
s
s 

Color in the figure shows the minimal
values of ( q) / 2q2 for ( d ,s ) permitted.
2
Suppression of a Parasitic Pump Scattering in a Raman
Amplifier with Arbitrary Angles between Pump and Seed Pulses
(Continued)
Circular polarization. (q) / 2q2 vs.  for  s  5 / 6 (a),  s  2 / 3 (b),
and  s   / 4 (c). Several plots for each  s correspond to different  d .
Conclusions
• Detuning of the Raman resonance by a plasma density gradient
and a pump chirp can suppress parasitic pump side-scattering in
Backward Raman Amplifiers
• By selecting appropriate values of the pump chirp and plasma
density gradient as well as direction of the density gradient, one
can favorably minimize the detuning in the direction of the seed
pulse propagation, while strongly suppressing the parasitic pump
side-scattering in all the other directions