Suppression of a Parasitic Pump Side-Scattering inBackward Raman Amplifiers of Laser Pulses in Plasmas

A. A. Solodov, V. M. Malkin, N. J. Fisch

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.

Abstract

Conceptual Scheme of Backward Raman Amplifiers (BRA)

pump beam seed beam

ω2, k2 ω1, k1

kp

plasma wave

pump beam

seed pulse

depleted pump

amplified pulse

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

seed pulseplasma

target

pump pulse

Conceptual Scheme of BRA (Continued)

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

pumpSRS

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]

0 zpump front

δωpump

δω

δωdetuning

δωplasma

-zpulse

location

δω=δωplasma-δωpump

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

pump

SRS

z

y

x

θ

2/ ),2/(sin 000 pa ωωγθγγ ==

The Raman growth rate maximizes for backscattering:

(linearly polarized pump with normalized amplitude a0=eA/mc2, scattering in the plane perpendicular to the pump polarization).

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

where the vector-potential envelopes of the pump and scattered Stokes waves and Langmuir wave electric field are defined:

The pump wave frequency, wavenumber, and the unit polarization vector are

[]**ˆ (),ˆ (),tbtvbfifbγωγ∂+ =⋅∇ ∂−∂=

[][]20020/exp()../2,/exp()../2,ˆ /exp()../2.eaasebbbpepfffemcaikzitccemcbiitccemcfiitccωωωωω⊥=−+=−+⎡⎤=−+⎣⎦AeAekrEkkr

222*in particular, Im0 f , //, aor linear polarization and for circularpoland ()/1|| r(1ization.),aaapaxykcciμωωωωμμμ⊥⊥⊥=−≈=++=⋅==±eeeee

pumpSRS

z

y

x

θ

φ

z1

Main Equations (Continued)

222(/)(sincos,sThe Stokes wave freqinsinuency, wave vector, and unitpolarization vector ,(|,cos) and |/), The frequency and wave vector of the Langmui(w).rbbpbbbababckckkωωθφθφωωωθω⊥⊥≈=≈===−=−⋅kkeekke2202 .ave and The Raman growth| |sin cossin (/2)sin rat1|e.21|pfabpfazbakωωωωδωωθφμφγθμω=−==−=−+−+kek

201122The frequency detuningterms with and correspond to a plasma frequency detuning and a pump chirp() ,4(sincos,sinsin).,,cospdddddddazqqtccqqωωδωθφθφθ⋅⎡⎤⎛⎞=−−⎜⎟⎢⎥⎝⎠⎣⎦=ree

Main Equations (Continued)

1111111Making transformation to a new coordinate frame (,,) with||, (,), ||(,), introdu cing new dimensionless v (/)/2, ariabl espbbbtxyzzxzzcyzζωω=−⊥kkk1*11 (/)/2, ˆ and performing a simple phase transforma (,)( , ) (),tion ' ' expwher2/ (,,)e ne,, oppzcbfbfxyζτωωγωω=Γ==ΦΦ=Φ2012 ,'' ( ) ,' '' [' coscossinsinc ha ()][1 ].os cosswheredddbfiaqfbqqqτζτθθθθφφθ∂=Γ∂−=Γ=+−−−

Green’s Function

011The Green's function, corresponding to a localized source term added to the r.h.s. of the equ()() ation for ', has the for(,)expln1(/;m [32]CdpiiqbpFiqipfqpδζδτζτηπ⎡⎤Γ⎛⎞=+−=Γ−⎢⎥⎜⎟⎝⎠⎣⎦∫22011 where the contour encompasses in the po -sitive direc 0 , tion singularities at and is the co1;) nfluent hypergeometric func,, (/), ' .tioniqqiqCpapqFηηζτ===Γ=Γ

z1=ct/2

b0

c/2γ

0

Green’s Function (Continued)

0-1204/1,||||/2exp If The effect of detuning the initial amplitude of the localizedLangmuir wave perturbation is much l(becomeess then snoticeable at||1,for || saturates:/|)ex|.MsatqbqbqqηηππηΓ==Γ: ??p(/||), the Stokes wave amplitude remains forever smaThe pump depletion similarlyl remainssmalller then 1..qπ−

z1

|b0|

c

0

0||satb

Premature Pump Backscattering and Side-Scattering in Absence of Detuning

A rectangular in the longitudinal and transversal directions laserpump enters at 0 the plasma layer. A uni form Langmuirwave seed is assumed in plasma before the pump coming.The Stokes wave amplitutz==[]()()2/ (/2)sin 0000220ˆ (0,2/) 22(1 )cos1ˆ ( ),exp '22 (/2)sin 21' . . . :de at the l h s plasma boundary /2 2/where and ' , 0:For linear polarization ,sincos (/2), sin 'i1.LpLlcbztlcfdZIZLZfLaLθωωθθθμθφ===ΓΓ−−≈ΓΓ=−Γ==Γ=Γ∫?20n the plane orthogonal to the pump polarization (cos0)' is independent of .For circular polarization, '1cos. :iaφθμθ=Γ=±Γ=+

z

z1

θ

l

0

||0 f one should provide it is desirable to have in the sor all aneed pulse propad ;a minimal |gati|In Order to Suppress Pump Side-Scattering in All Directions:qqθφ≠gg212121/20 oron /20 direction (toPump chirp is simplify a useful amplificatinecessary, 0, as otherwise0 for scattering at /2.If for linearon) po ,..0BRA, ():ddqqqqqqqzθπθπ>≥<≤=≠==↑↓=e222larization: || minimizes for scattering in the plane perpendicular to the pump polarization (/2), for which it is independent of ;for2/1si circular polarizatincos,qqqφπφθθ==−222/(1cos)o:/2.n qqθ=+

Suppression of a Parasitic Pump Side-Scattering in aBackward Raman Amplifier

δωpumpδωplasma

z0

δω

Suppression of a Parasitic Pump Side-Scattering in aBackward Raman Amplifier (Continued)

0 zpump front

δωpump

δω

δωdetuning

δωplasma

-zStokes

pulse front location

12Plasma density gradient and pump chirp, 0, 0:detuning (||) is minimal for backscattering (), both for linear and circular polarization.qqqθπ≠≠=

Suppression of a Parasitic Pump Side-Scattering in aBackward Raman Amplifier (Continued)

linear polarization,φ=π/2

linear polarization,φ=π/6

circular polarization;also

linear polarization,φ=π/4

12/20qq=

12/20.25qq=

12/20.5qq=

12/20.75qq=

Scattering Cross-Section of the Pump Beam in BRA with Frequency Detuning

01*0101 A white noise statistics for the stochastic thermal (,)0, Langmuir wave seed is (,)( assumed:',fztfztfz=()133111 1- The D spectral density of thermal fluc tuations is connected with the usual3- D t)( ).' 1 hermal fluct uations spectral density ,/2(2)DDDeeptnzzennTmcδωπ=−⎛⎞=⎜⎝01113202201 The averaged Stokes wave field intensity for scattering fr , by where is the solid angle in the wa ve vecto16, r om a r .spaceegi2 on isbpisatzDDfznnbbfdzckeφπωΔ=ΔΩΔΔΩ⎟⎠=122210120.4zpsatDbnzcωΔ=Δ∫

Scattering Cross-Section of the Pump Beam in BRA with Frequency Detuning (Continued)

10221 The pump energy scattered by a unity p Dividing it by the incident pump intensitylasma volume in a .1unity of time ina solid ang:6le zebdIcmcVzeIωπΔ⎛⎞=⎜⎟Δ⎝Ω⎠Δ=()()222/|2202|2120022033 (/8)(/), a - differential scattering cross section of :the pump pulse in plasma is obtained () (/2cossin (8/3)/ where is t),8 he Thomson sqdeeedIqqqeknIcamceemcVπθθλπωπσπσ=+−=ΔΩ()1/22 cattering cross section on , /4 a single electron is the Debye lengt.hdeTneλπ=

Suppression of a Parasitic Pump Scattering in a Raman Amplifier with Arbitrary Angles between Pump and Seed Pulses

Only the most desirable for Raman amplification case wilAssume that the seed pulse propagates in the direction determined by and . Linear Polarization (0):ssθφμ={} 1222/2.cols be consid It is desirable to use a plasma()sins ,ered when the seed pulse propagates pe[1 ()]in cos 21cos1rfrequesin pendicular to the pump polarizatiocosn ncy g. r,/ sdddqqqφπθθθθφφθθφ=−−−−=−−−1212121 adient in the direction and should satisfy[0/2 ()].cos The minimum detuning gradient is in thedirection , ||of the seed propagation/2 ()/2,and 0/2s|/s (/in () /i 2nddssdsqqqqqqqqπφπθθπθθθ=<+==<−<<22)2|.q−

seed pulse

density gradient

zθs

θd

y

7/8, 3/4, 5/8.dddθπθπθπ===

Suppression of a Parasitic Pump Scattering in a Raman Amplifier with Arbitrary Angles between Pump and Seed Pulses(Continued)

2Linear polarization. ()/2 vs. (a) for /2 and vs. (b)for the minimal values of in the plots (a).dqqθφφπφθ−==

Suppression of a Parasitic Pump Scattering in a Raman Amplifier with Arbitrary Angles between Pump and Seed Pulses(Continued){} ()11222cos()sinsin[1For minimal || in the dir0 for all and if 0/2cos().cos()] 21cos/2,1cos Circular Polarization ():/ddddqqqqqqqiθθθθφφθθθφπθμ−−−−=−+−≠<<−=2221111cos Color in the figure shows the micos()cos(/2)2(1cos)1.cos()sin(/2)sin(ection , one should take: ,, , and should sa2t)isfyssssddsdssdsdssqqqqθθθθθθθθθφφφφθθθθ−−=×−⎡⎤−+−=⎥⎦=−⎣=⎢2snimalvalues of ()/2 for (,) permitted.dqqθθ−

Suppression of a Parasitic Pump Scattering in a Raman Amplifier with Arbitrary Angles between Pump and Seed Pulses(Continued)

2Circular polarization. ()/2 vs. for 5/6 (a), 2/3 (b),and /4 (c). Several plots for each correspond to different .ssssdqqθθπθπθπθθ−===

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