Beyond nonlinear saturation of backward Raman amplifiers

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by  Ido Barth

Beyond nonlinear saturation of backward Raman amplifiers
Ido Barth,1,∗ Zeev Toroker,2 Alexey A. Balakin,3 and Nathaniel J. Fisch4
1Princeton Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA
2Department of Electrical Engineering, Technion Israel Institute of Technology, Haifa 32000, Israel
3Institute of Applied Physics RAS, Nizhnii Novgorod 603950, Russia
4Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08540, USA
(Dated: January 25, 2016)
6 Backward Raman amplification is limited by relativistic nonlinear dephasing resulting in satu-
1 ration of the leading spike of the amplified pulse. Pump detuning is employed to mitigate the
0 relativistic phase mismatch and to overcome the associated saturation. The amplified pulse can
2 then be reshaped into a mono-spike pulse with little precursory power ahead of it, with the maxi-
mumintensityincreasingbyafactoroftwo. Thisdetuningcanbeemployedadvantageouslybothin
n
regimes where the group velocity dispersion is unimportant and where the dispersion is important
a
butsmall.
J

1 PACSnumbers: 52.38.Bv,42.65.Dr,42.65.Re,52.35.Mw
2

] I. INTRODUCTION caused by the relativistic nonlinearity. To this end, we
h
suggesttocompensatetherelativisticphasemismatchby
p
- Laserintensitiesweresignificantlyincreasedduringthe proper detuning of the pump frequency. This will delay
m past decades, mainly by the means of chirped pulse am- the saturation and allow a longer resonant amplification
s plification,uptothedielectricgratinglimitation[1]. The of the leading spike, resulting in a reshaped, single-spike
a next generation of high intensity laser will require a dif- pulse of higher intensity.
l
p ferent medium for amplification, such as plasma, which
. Detuning of the pump frequency, accompanied by a
s can tolerate much higher intensities and fluences. Par-
density gradient, was suggested to suppress unwanted
c ticularly,backwardRamanamplification(BRA)wassug-
i noise[17],forwardRamanscattering[32],superluminous
s gested as such a scheme [2], taking advantage of the res-
y precursors [19], and to get superradiant linear Raman
onant energy transfer between two counterpropagating
h amplification [8–10]. However, these chirps were em-
lasersthatinteractviaanelectrostaticplasmawave. The
p ployed only at early stages, before the pump encounters
possibilityofreachingnearlyrelativisticunfocusedinten-
[ the seed or in the linear regime when the pump is not
sitiesinbackwardRamanamplifiershasbeeninprinciple
yet depleted. In the advanced nonlinear stage, when the
1 demonstrated experimentally as well [3–10].
v pumped signal is highly amplified to relativistic intensi-
The major physical processes that may affect BRA
2 ties, the effect of pump detuning remains to be studied.
include the amplified pulse filamentation [2, 11–13],
3 It is notable that all the aforementionedchirping advan-
detuning due to the relativistic electron nonlinearity
8 tages in the early stages can be included in the chirping
5 [14, 15], parasitic Raman scattering of the pump and
suggested here, which mainly depends on the detuning
0 amplified pulses by plasma noise [2, 13, 16–18], gen-
profileatlatetimes. Thespecificdetuning,however,that
. eration of superluminous precursors of the amplified
1 will cure the relativistic nonlinearity has a more compli-
0 pulse [19], pulse scattering by plasma density inho-
cated functional dependency as derived here. It will be
6 mogeneities [20], pulse depletion and plasma heating
foundherethat,bycompensatingforthephasemismatch
1 through inverse bremsstrahlung [21, 22], the resonant
through pump detuning, a factor of two in the maximal
: Langmuir wave Landau damping [21, 23–25] or breaking
v intensity can be achieved. Moreover,this intensity is at-
i [2,13,16,26,27],andotherprocesses(see,forexamples,
X tained with little leading power ahead of it.
Refs.[28–31]). Mostofthesedeleteriousprocessescanbe
ar mitigatedbyproperpreparationoflaserpulsesandplas- The plan of the paper is as follows: In Sec. II, we in-
mas,choosingparameterrangesandselectivedetuningof troducethephysicalmodel,i.e.,3-waveinteractionequa-
the Raman resonance. Ultimately, the output intensity tions including nonlinearity, dispersion, and detuning.
limit appears to be imposed primarily by the relativistic More details on the model are given in the Appendix.
electron nonlinearity, causing a phase mismatch of the InSec.III weintroduce a theoryforthe phasemismatch
Ramanresonancethatresultsinasaturationofthedom- evolution in the dispersionless regime and compare with
inantleadingspikegrowth[14]aswellasofthesecondary simulations. Sec. IV is devoted to the main results, the
higher spikes in the multi-spike wave train solution [15]. effect of detuning and a roadmapfor chirpoptimization.
Itistheobjectiveofthepresentstudytoaddressmeans In Sec. V we address the effect of dispersion on the de-
of overcomingthe saturationdue to the phase mismatch tunedsolutionandsuggestphysicalparametersforfuture
experiments. The deleterious influence of the secondary
Raman scattering on the amplified pulse is estimated in
∗ [email protected] Sec. VI. The conclusions are summarized in Sec. VII.
2
II. MODEL
3.5
3 τ = 10 Q = 0
3
We adopt the dimensionless quasi-static 3-wave inter- b
2
action model including dispersion, detuning, and rela- 2.5
1
tivistic nonlinearity [15, 17, 33] (for detailed definitions
2
andderivation,seeEqs.(A.22)–(A.24)intheAppendix), max 0 10 15 20
b 1.5 ζ
a = bf, (1)
ζ
−∗ 1 Leading spike
fζ =ab i∆f, (2) Second spike
− Third spike
b =af∗ iQb +ib2b. (3) 0.5 Detuned pulse
τ − ζζ | | Theory
Here,a, b,andf areproportionaltotheenvelopesofthe
pump pulse, counterpropagating shorter seed pulse, and 0
Φ
resonant Langmuir wave, respectively; subscripts signify
derivativeswithrespecttotheelapsedamplificationtime, -π/2
τ, and the distance (or delay time), ζ, from the original 0 1 2 3 4 5 6 7 8 9 10
seedmaximum,ζ . Thetwoparametersinthemodelare τ
0
the rescaled dispersion coefficient, Q, and the rescaled
detuning, ∆ δω = ω +ω ω , where ω are the FIG. 1. (color online) The maximal amplitude, bmax, (up-
f b a a,b,f
∼ − per panel) and the phase mismatch, Φ, (lower panel) of the
frequencies of the pump, seed, and Langmuir waves, re-
amplified pulse as a function of the amplification time, τ, in
spectively. For strongly under-critical plasma, ω ω ,
e ≪ a thedispersionless (Q=0) regime. bmax and Φ of theleading
we have Q ω /2ω (see Eq. (A.26) in the Appendix).
≈ e b (dashed-dotted), second (dashed) and third (dotted) spikes
Equations (1)–(3) are solved numerically for initial
of the saturated unchirped (∆=0) pulse are compared with
constant pump a(ζ,τ = 0) = 1, zero Langmuir wave, thoseofthedetunedpulse(solidline)andwiththetheoretical
f(ζ,τ = 0) = 0, and small input Gaussian seed pulse of predictions (solid lines with squares), Eq. (15) for bmax and
the form Eq. (20) for Φ. The final τ =10 profiles of the singled-spike
detuned(solid)andmulti-spikeundetuned(dashed-doted)are
b0 (ζ ζ0)2 compared in the inset.
b(ζ,τ =0)= exp − (4)
√Dπ − D
(cid:20) (cid:21)
with b =0.05, D =1, and ζ =10. Eqs. (1)–(3) as six real equations
0 0
First, let us recall in Fig. 1 the nonlinearly saturated BF
A = BF cosΦ, θ = sinΦ, (5)
(multi-spike) solution for extremely under-critical plas- ζ ζ
− A
masasfoundbyMalkinet al.[14,15]. WesolveEqs.(1– AB
3) with no detuning, ∆ = 0, and neglect group velocity Fζ =ABcosΦ, ϕζ = sinΦ ∆F, (6)
F −
dispersion, Q = 0. Then, we calculate the total phase
AF
mismatch, Φ(ζ,τ) = θ ψ ϕ, where, the real ampli- B =AF cosΦ, ψ = sinΦ+B2. (7)
τ τ
− − B
tudes (A, B, and C) and phases (θ, ψ, and ϕ) are de-
fined via a = Aeiθ, b = Beiψ, and f = Feiϕ. For a It is clear from Eq. (7) that when Φ = π/2 the three-
−
given amplification time, τ, we find the pulse maximum wave coupling term in the right hand side of amplitude
b (τ) = B(τ,ζ = ζ ) = max [B(τ,ζ)], where ζ Eqs. (5)-(7) vanishes, and thus, the amplification of the
max max ζ max
is the location of the maximum. At this location we cal- seed stops. Next we develop a theory for the phase mis-
culate the phase mismatch, Φmax(τ) = Φ(τ,ζ = ζmax). match evolution of the first spike Φmax,1 for small τ.
For the multi-spike solution (as in the inset of Fig. 1),
we similarly define the local maxima, b , maxima lo-
max,j
III. PHASE MISMATCH EVOLUTION
cation, ζ , and the phase mismatch, Φ , where
max,j max,j
j = 1,2, and 3, stand for the leading, second, and third
spikes, respectively. Since Eqs.(1–3) for ∆=0satisfies a2+ f 2 =0,
| | | | ζ
In Fig. 1 we plot the evolution of the amplitudes of
the envelopes can be expressed by (cid:16) (cid:17)
these maxima, b (upper panel) and the associated
max,j
phase mismatches, Φ , (lower panel). What is im- A=cos(u/2) (8)
max,j
portant here is that the nonlinear saturation of the am- F = sin(u/2) (9)
−
plified pulse maximum b , (upper panel) is time- u
max,j ζ
B = . (10)
correlated with its relativistic phase mismatch, Φmax,j, −2cosΦ
(lower panel). It is clearly seen that the amplification of
BysubstitutingEqs.(9-10)intoEqs.(5-7)weobtainthat
eachspike stops when the phase mismatch at the spike’s
the seed envelope dynamics is described by
maximum reaches Φ = π/2. The reason for this
max,j
correlation can be seen if we−rewrite the three complex u +u Φ tanΦ=sinu cos2Φ, (11)
ζτ ζ τ
3
while its phase dynamics is so ξ = 5.5. In the upper panel we plot the analytical
M
solution (15) for the maximal amplitude of the leading
sinΦcosΦsinu u 2
ψτ = + ζ . (12) spike, Bmax(τ), (black line with squares). In the lower
u 2cosΦ
ζ panel of the figure, we compare our theoretical predic-
(cid:16) (cid:17)
tion(20) for the phase mismatch, Φ, of the leading spike
In the absence of the relativistic effect (the term B2
(black line with squares) with simulation results (blue
in Eq. (7)) and detuning (∆ = 0) accompanied by
dashed-dottedline). Itis notable,thatthe theory,which
zero phase (i.e., real fields) initial conditions, θ(ζ,0) =
does not contain any adjustable parameter, is in a high
φ(ζ,0) = ψ(ζ,0) = 0, Eqs. (5)-(7) yield Φ = ψ = 0
agreement with the numerical solution.
for all ζ and τ. In such a case, the second term in left
hand side of Eq. (11) vanishes and Eq. (11) becomes the
sine-Gordon equation,
IV. THE EFFECT OF DETUNING
u =sinu. (13)
ζτ
Forinitialconditions(4),oneobtainstheπ-pulsesolution We are now in a position to show how frequency de-
tuning compensates the relativistic phase mismatch, en-
2τ hancesthetotalamplification,andreshapestheamplified
B = . (14)
(ξ+1)cosh(ξ ξ ) pulse. Consider detuning either by pump chirping or by
M
−
density gradient. However, the relevant regime for our
Here, ξ = 2√τζ, and ξ = ln 4√2πξ /ǫ is calculated
M M problem is the strongly under-critical plasma, Q 1,
iteratively,whereforourdefinitionsǫ=2 b(ζ,τ =0)dζ because only in this regime, the dynamics is domin≪ated
(cid:0) (cid:1)
is the seed capacity [2, 14, 16]. By substituting ξ = ξM by the relativistic nonlinear saturation and not by dis-
R
one finds the maximum of the amplified pulse persion[15]. Inthisregime,ω ω ,andthus,theover-
e b
≪
2τ all detuning can be mainly achieved by pump detuning
Bmax = . (15) rather than via density gradient.
ξ +1
M Following the theoretical scaling Φ µ τ3 of
≈ − ∼ −
When the relativistic effect is small (i.e., the weakly Eq.(20)intheweaklynonlinearstage,µ 1(i.e., τ <3
≪
nonlinear regime), the phase mismatch between the en- for the example shown in Fig. 1), we consider detuning
velopes is small, Φ ψ 1. In this limit, cosΦ 1 of the form
≈ − ≪ ≈
and tanΦ sinΦ Φ. Therefore, we can neglect the
second term≈ in the≈left hand side of Eq. (11) and ap- τ γ
∆(τ)=α . (21)
proximatethe seedenvelopeasthe π-pulsesolution(14). τ
(cid:18) f(cid:19)
Then, Eq. (12) becomes
Here, τ is the total amplification time (proportional to
f
sin(u) u 2
Φ = Φ ζ (16) the plasma length) and the values of the parameters α
τ
− uζ − 2 and γ are to be optimized. Optimization over γ is also
(cid:16) (cid:17)
needed, because for larger amplification times, τ > 4,
and Eq. (11) becomes f
different values of γ may yield a better results. For ex-
u +u ΦΦ =sin(u). (17) ample, in Fig. 1 we illustrate the effect of chirping by
ζτ ζ τ
consideringdetuning withparametersα=55andγ =4.
Since at the point of the maximum amplified pulse This value of γ was chosen because, in this case, where
uζτ/uζ = Bτ/B = O(τ−1), Eq. (17) can be approxi- τf = 10, it results in a larger pulse intensity and better
mated as sin(u)/uζ ΦΦτ. Then Eq. (16) reads pulseprofilethanγ =3(seealsoFig.2). Thespikesmax-
≈
ima of the aforementioned unchirped solution (dashed
2
1 2τ
Φ = (Φ3) , (18) and dashed-dotted lines) are compared in Fig. 1, with
τ τ
−3 −(cid:18)ζM +1(cid:19) themaximumofthe chirpedsolution(solidline). Thefi-
nal(τ =10)amplified (chirped andunchirped) profiles,
that, in turn, is integrated over τ to get f
b(ζ,τ =τ ) are compared in the inset.
f
| |
1 We note that the chirped solution is better than the
Φ+ Φ3+µ=0, (19)
3 unchirped solution in two aspects. First, the maximal
amplitude of the amplified pulse increases by about 50%
where µ= 4τ3 . The real solution of Eq. (19) is
3(ζM+1)2 sotheintensityisenhancedbymorethanafactoroftwo.
Moreover, the pump detuning reshapes the multi-spike
1 −1
Φ=η3 η 3, (20) unchirped solution into a single spike pulse, resulting in
−
ahigherintensityintheleadingspikeandlessprecursory
whereη = 3µ+ 1+ 9µ2. Itisnotablethatforagiven power ahead of it. For the example shown in Fig. 1, the
− 2 4
amplificationtimeqτ,thissolutiondependsononeparam- leadingspikemaximumincreasesbyafactorof2.6sothe
eter only, ǫ, which is determined by initial conditions. maximal intensity of the leading spike increases by more
For the example presented in Fig. 1, we have ǫ = 0.1 than a factor of 6 without additional precursory spikes.
4
b b
max max a minimum detuning amplitude, α, is needed for signifi-
1 2 3 1 2 3 4
cant pulse amplification. For the example of Fig. 2 one
100 100 finds the condition α 15. Note that pulse reshaping
≥
is not included in the figure. Actually, although the ex-
80 80
ample of Fig. 1 has only b 3.5 while the maximum
max
≈
b in Fig. 2 is about 4, it has a mono-spike pulse with
60 60 max
α α less precursory power.
40 40
20 20
V. DETUNING IN DISPERSIVE MEDIUM
0 0
0 2 4 6 8 10 5 10 15
γ τ Next we study the case of non-negligible, but small,
f
dispersion i.e., Q 1. We solve Eqs. (1–3) for the
≪
same initial conditions as in the dispersionless (Q = 0)
FIG. 2. (color online) The maximal amplitude amplification,
bmax asafunctionofthedetuningparameters(γ,α)forfixed case, but now with Q = 0.01. Similarly to Fig. 1, we
τ =10(leftpanel)andof(τ ,α)forfixedγ =4(rightpanel) compare in Fig. 3, the unchirped, multi-spike solution
f f
in thedispersionless regime, Q=0. (dashed and dashed-doted lines) with the chirped so-
lution (solid line). The detuning parameters here are
δ = 4, α = 15, and τ = 8. In the detuned solution the
f
This reshaping effect might be essential for experi- pulse amplificationis significantlyhigher thaninthe un-
ments in which contrast ratio plays an important role. detuned solution and the maximum intensity is achieved
As can be seen in Fig. 1 (upper panel), the leading spike at the pulse front without precursory spikes. However,
maximum of the detuned solution agrees with the an- the effect here is modest compared to the dispersionless
alytical expression of Eq. (15) for the π-pulse solution caseduetothedispersivebroadeningofthepulse. Asbe-
also for τ > 4, when the unchirped solution is already fore,weillustrate,inFig.4,anoptimizationroadmapby
saturated. This agreement, accompanied by the pulse plotting b asa functionofthe detuning parametersα
max
reshaping,supportthe understandingthatthe pump de- andγ andthe amplification time τ . A comparisonwith
f
tuning maintains the amplification of the π-pulse solu- Fig. 2 reveals that for finite Q, smaller detuning ampli-
tion beyond the (unchirped) nonlinear saturation limit.
It is also notable that a negative chirp (α < 0) has the
opposite effect, i.e., increasing the phase mismatch, Φ,
reducing the pulse amplification (not shown in the fig- 2 2 τ = 8 Q = 0.01
ures). 1.5
Obviously, one would want to optimize the detuning b1
with respect to a certain objective function; some ex- 1.5 0.5
amples are the maximum output intensity, the amplified
pulse profile, and the energy transfer efficiency. To this max 0 10 15 20
b 1 ζ
end, one should define a detuning profile and optimize
(e.g.,viaageneticalgorithm)itsparameterswithrespect
to an objective function. For simplicity, we choose here 0.5
Leading spike
tolookattheamplifiedpulsemaximum,bmax,andstudy Second spike
its dependency on the detuning parameters α and γ of Detuned pulse
thedetuningprofile(21)andthetotalamplificationtime,
τf. Of course, different objective functions or detuning 0
profiles would be suitable for specific experimental goals Φ
and restrictions. However, these further calculations are -π/2
outside the scope of this paper.
0 1 2 3 4 5 6 7 8
In Fig. 2, we present the maximum amplitude, b
max τ
in the (γ,α) parameter space (left panel) for constant
τ = 10, and in the (α,τ ) plane for constant γ = 4
f f FIG. 3. (color online) The maximal amplitude, bmax, (up-
(right panel). The sharp transitions between very low
per panel) and the phase mismatch, Φ, (lower panel) of the
(bmax < 1) and large (bmax > 3) amplifications in both amplified pulse as a function of the amplification time, τ in
panels indicate that for detuning that is too abrupt (too the finite dispersion (Q = 0.01) regime. The saturation of
small γ or too small τ for a given α), the Raman res- the leading (dashed-dotted) and second (dashed) spikes of
f
onance condition is lost at early times, resulting in an theunchirped(∆=0)pulsearecompared with themaximal
insignificant amplification. This effect is important to amplitude of the chirped pulse (solid line). The final τ = 8
mitigate the premature noise amplification as suggested profiles of the singled-spike chirped (solid) and multi-spike
unchirped(dashed-doted) are compared in theinset.
in Ref. [17]. In addition to the slownessof the detuning,
5
bmax bmax and equals to 3.1 1018W/cm2 while the first spike in-
0.5 1 1.5 0.5 1 1.5 tensity is 1.8 10×18W/cm2. The detuning in this ex-
100 100 ample improve×s the maximal intensity of the first (and
only)spikeupto5.2 1018W/cm2 withoutputduration
80 80 ×
(fullwidthhalfmaximumintensity)of12fs. Notably,al-
60 60 thoughproperdetuningsignificantlyincreasestheoutput
α α intensity,thefluenceandefficiencyremainapproximately
40 40 the same. For no detuning (∆ = 0), the fluence, that is
defined inEq.(A.28), is w =8.5 104J/cm2, where the
20 20 b
×
integrationwastakenbetweenζ =5andζ =15,andthe
0 0 efficiency (see Eq. (A.30)) is η = 0.75. Employing de-
0 2 4 6 8 10 5 10 15 w
γ τ tuning in this example changes these values by less then
f
3%. For the same pump intensity, but with linear polar-
ization (p = 1), shorter plasma but larger detuning are
FIG. 4. (color online) The maximal amplitude amplification,
bmax,asafunctionofthedetuningparameters(γ,α)forfixed required to get the same amplification.
τ =8(leftpanel) andof(τ ,α) forfixedγ =4(rightpanel)
f f
in thefinitedispersion regime, Q=0.01.
VI. SECONDARY RAMAN BACKSCATTERING
tudes, α, are need for optimal amplification. Note that Theamplificationalsotendstosaturateduetothesec-
these maps show only part of the whole picture, because ondary Raman backscattering (SRBS) of the amplified
neither pulse reshaping nor efficiency considerationwere pulse, b, into downshifted, counter propagating noise of
taken into account. For example, although for τf > 10 frequency ωb ωe. It was shown in Ref. [2] that, for the
(in the right panel) one can find a solution with higher π pulse solut−ion, this effect is minor (about 5 exponen-
bmaxthanshowninFig.3whereτf =8,itwillbeamulti- ti−ationsonly)anddoesnotdepletemuchoftheamplified
spike pulse (not shown) so the amplified pulse profile is pulse. Nevertheless, in the relativistic regime the ampli-
not as good as in Fig. 3. We also found that the pump fiedpulse is longerso thereis moretime for SRBS.Also,
detuning is more effective for small values of Q. This is the group velocity dispersion broadens the pulse, so it
because larger dispersion broadens the amplified pulse, becomes more susceptible to SRBS as Q increases. We
so bmax saturates due to dispersion rather than by rel- define the SRBStotalnoiseamplification, eΓ,where(see
ativistic nonlinearity. Therefore, this saturation cannot Eq. (A.31))
be overcome by pump detuning.
Onemightimaginethatwemaycombineseedchirping
Γ=√σ bdζ, (22)
[31] with pump detuning in order to reduce the amplifi- | |
Z
cationtime andthereforetoimprovethe efficiencyofthe
detuned BRA. However, these two methods operate in and for strongly under-critical plasma, σ 2. For the
≈
different regimes. Seed chirping is useful for high densi- dispersionless profiles in the inset of Fig. 1, one finds
ties (ω > 0.25ω ), where the group velocity dispersion Γ = 7.8 for the unchirped (dashed line), which reduces
e b
oftheseedpulsebecomesadominanteffect(seeFig.1in to Γ = 6 for the chirped solution (solid line). Similarly,
Ref.[31]). Ontheotherhand,pumpdetuningiseffective but less significantly, for the examples of Fig. 3 in the
forsmalldensities(ω <0.1ω ),wherethegroupvelocity dispersive regime, one finds Γ = 7.5 for the unchirped
e b
dispersion does not shadow the relativistic nonlinearity solution (dashed line in the inset of Fig. 3) and Γ = 7
(see Fig. 3 in Ref. [15]). for the chirped solution (solid line). In both cases, the
Finally, let us consider what could be the parame- integration range in Eq. (22) was 5<ζ <15.
ters for a future experiment. For example, for seed Next we estimate the initial (thermal) noise inten-
wave length λ λ = 0.1µm, and Q = 0.01 (cor- sity in order to determine the significance of this ef-
b b
responding to ω≈/ω = 0.02), the plasma density is fect in experiment. To this end, we consider electron
e b
ne = 4.4 1019cm−3. In this case, for pump intensity temperature Te, with associated electric field Enoise =
I0 =Ibr/2×,whereIbr is the wavebreakingthreshold(see Te/eλD, where −e is the electron charge and λD[cm] =
Eq.(A.15)),wehaveI0 =6.7 1013W/cm2. Forcircular 743 Te[eV]/ne[cm−3] is the Debye length. Hence,
polarized field (i.e., p=2), th×e dimensionless pump am- the electric field can be written as Enoise[V/cm] =
10−p3 T [eV]n [cm−3] and the corresponding noise in-
plitude is given by a = 6 10−10λ [µm] I[W/cm2] = e e
0 × a tensity is
5 10−4. Therefore,fortheexampleshownqinFig.3,the p
pu×mp duration is 0.85ns, which corresponds to plasma I [Wcm−2]=10−9T [eV]n [cm−3]. (23)
noise e e
length of 12.75cm, while the totaldetuning ofthe pump
frequency is 7.5%. Without pump detuning, the max- Hereweassumeδ-correlatednoisewithflatspectrumand
imal amplified intensity is achieved in the second spike a cut-off at λ /c. The Raman instability width can be
D
6
estimated as the SRBS increment,˜b V , where V is the Appendix: Derivation of the model
m 3 3
three wave coupling that is defined in Eq. (A.7) and ˜b
m
is a typical seed amplitude (see Eq. (A.21)). Therefore, ThisappendixpresentaderivationofEqs.(1)–(3). We
the effective noise intensity reads beginwiththeusualthreewaveinteractionequationsin-
cludingnonlinearityanddetuningwithinthefluidmodel
˜b V λ [15, 17, 33]
m 3 D
I =I . (24)
eff noise
c a˜ +c a˜ =V f˜˜b (A.1)
t a z 3
˜b c ˜b = V a˜f˜∗+iR˜b2˜b iκ˜b (A.2)
Finally, for λ = 0.1µm, n = 1019cm−3, T = 100eV, t− b z − 3 | | − tt
and typical abmplified pulsee amplitude bm =e 1.5 (i.e., f˜t+iδωf˜=−V3a˜˜b∗. (A.3)
˜bm = 0.1), one finds V3 = 1.9 1015sec−1 and λD = Here, a˜ and ˜b are the vector-potential envelopes of the
1.1 10−6cm. For this exampl×e, the total background
pumppulse andthe counterpropagatingshorterpumped
nois×ecanbeestimatedasInoise =4.4 1012W/cm2while pulse, respectively, measured in units of m c2/e 5
× e
the effective noise intensity available to SRBS would be 105V; f˜is the rescaled envelope of the Langmuir≈wav×e
I =3.3 1010W/cm2. This means that 18 exponenti-
aetffions sep×arate the effective noise intensity and the am- electrostaticfieldinunits of(mec/e)√pωeωa. Here, ωa,b
are the frequencies of the pump and the seedpulses. c is
plified pulse intensity thatwasestimated inthe previous
thevacuumspeedoflight;ω =4πe2n /m istheplasma
section to be of the order of 3 1018W/cm2. For the e e e
aforementioned example the nu×mber of SRBS intensity frequency; −e,me, and ne are the electron charge,mass,
anddensity, respectively; Subscripts t and z signify time
exponentiations is only 2Γ 14, so the SRBS is pre-
≈ and space derivatives and
dicted to be harmless in this case.
c =c 1 (ω /ω )2 (A.4)
a,b e a,b
−
q
are the group velocities of the pump and the seed. The
VII. SUMMARY parameter p determines the polarizations of the pulses,
where
In summary, we found that pump detuning can mit- p=1 Linear polarization (A.5)
igate the relativistic nonlinear saturation of the leading
p=2 Circular polarization. (A.6)
spike for strongly under-critical plasmas. This occurs
when detuning compensates the relativistic phase mis- The 3-wavecoupling constant, V3 (realfor appropriately
match that causes the saturation of the leading spike. defined wave envelopes), can be written as [28, 34]
The benefits of this compensation are twofold. First,
pω
the amplification of the maximal intensity can be en- V =k c e , (A.7)
3 f
16ω
hanced by as much as a factor of two compared to the r b
achievable amplification without pump detuning. Sec-
where k is the wave number of the resonant Langmuir
f
ond, the amplified pulse is reshaped into a single-spike
wave, i.e.,
pulse with significantly less precursory power ahead of
it. Also, the reshaping of the leading spike reduces the kf =ka+kb, (A.8)
effect ofthe secondaryRaman backscatteringof the am-
ω ω2
plified pulse. The precise pulse reshaping and maximum k = a,b 1 e . (A.9)
a,b c − ω2
intensity were shown to depend upon the precise detun- s a,b
ing parameters. It is worth noting that the technique
The corresponding frequency resonance condition is
proposedhere for overcomingthe nonlinearsaturationis
not limited to BRA but is, in fact, a universal solution ωb+ωf =ωa, (A.10)
ofthe threewaveinteractionprobleminvariousphysical
where ω ω is the Langmuir wavefrequency in a cold
systems. f ≈ e
plasma. For under-critical plasma, ω ω , we can
e a
≪
approximateωa ωb and kf 2ka, so V3 √pωeωa/2.
≈ ≈ ≈
Then,thenonlinearfrequencyshiftcoefficientduetothe
relativistic electron nonlinearity, R, reads [35–37]
ACKNOWLEDGMENTS
pω2ω pω2
R= e a e. (A.11)
8ω2 ≈ 8ω
This work was supported by NNSA Grant b a
No. de-na0002948, AFOSR Grant No. FA9550-15-
The group velocity dispersion coefficient is
1-0391, DOE Contract No. DE-AC02-09CH11466,
DTRA Grant No. HDTRA1-11-1-0037, RFBR Grant κ= 1 dcb = ωe2c2 (A.12)
No. 15-32-20641,and by the Dynasty Foundation. 2c dω ω3c2
b b b
7
andthetotaldetuningfromtheperfectRamanfrequency Here,
resonance condition (A.10) is given by
1
δω =ω +ω ω . (A.13) σR 3
f b− a ∆= δω (A.25)
V4a2
Note that, in this normalization, the average square of (cid:18) 3 0(cid:19)
the electronquivervelocity inthe pump laserfield, mea-
is the rescaled detuning and the parameter
suredinunitsofc2,is a˜2,issuchthatv2 =c2 a˜2. Also,
| | ea | |
the average square of the electron quiver velocity in the
seed laser field and in the Langmuir wave field are given (ka+kb)2c2ωb dcb
Q= (A.26)
by v2 = c2˜b2ωa and v2 = c2 f˜2pωa, respectively. 4ω ω (c +c ) dω
eb | | ωb ef | | ωf e a a b
This hydrodynamic model is applicable for pump pulse
intensities Ia smaller than the Langmuir wave breaking characterizes the group velocity dispersion of amplified
threshold Ibr, pulseanddependsonlyontheratiooftheplasmatolaser
frequencyq =ω /ω . Instronglyunder-criticalplasmas,
πm2c5p 5.48 1018 a˜2p W e b
I = e a˜2 × | | . (A.14) whereq 1,onefindsQ=q/2andσ =2;innearlycriti-
a e2λ2a 2| | ≈ (λa[µm])2 2 cm2 calplasm≪as,whereq 1,onehasQ=0.5/ 1 q2 1.
→ − ≫
With Eqs. (A.22)–(A.24) we arrive at Eqs. (1)–(3) that
If we assume nearly complete pump depletion, then I
br p
comprise the physical model of this paper. In strongly
can be written as [27]
under-criticalplasmas(i.e.,Q ω /2ω 1andσ 2),
e b
m n c3ω whichisofmajorinteresthere,≈theampli≪fiedpulsein≈ten-
e e e
I = . (A.15)
br 16ωa sity, Ib, can be expressed in these variables as
Therefore, the wave breaking vector potential for both
1 1
linear (p=1) and circular (p=2) polarizations is I = I0|b|2 4 3 = Gωe|b|2 2I02 3 ,(A.27)
b Q pa2 4λ p2I2
2e2λ2I 1 ω 23 Q3/2 (cid:18) 0(cid:19) b (cid:18) br(cid:19)
a = a br = e , (A.16)
br spπm2ec5 √p(cid:18)2ωa(cid:19) ≈ √p where G = m2ec4/e2 = 0.3J/cm as in Refs. [14, 15]. In
this limit, the final amplified pulse fluence can be calcu-
where, λ = 2π/k is the pump wavelength, and Q
q/2 = ωa/2ω (seeabelow). Next, we adopt a univers≈al lated at time τ =τf via
e b
model[15]bytransformingtothedimensionlessvariables
1
I 16 3
τ = σRV32a40 13 Lc−b z (A.17) wb =Z Ibdt= Qω0a (cid:18)p2a40(cid:19) Z |b|2dζ. (A.28)
(cid:0) (cid:1)1
V4a2 3 L z
ζ = 3 0 t − (A.18) For constantpump, Ia =I0, the totalpump fluence that
σR − c
(cid:18) (cid:19) (cid:18) b (cid:19) was invested in the system, wa = I0∆ta, where ∆ta =
where σ = 1+ ca 2. Here, τ measures the elapsed 2L/c is the pump duration, can be written as
cb ≈
amplificationtime(orthedistancetraversedbytheorig-
inal seed front); ζ measures the distance (or delay time) 1 1
2τ I 2 3 Gτ I 3
f 0 f 0
from the original seed front; L is the plasma length and wa = Qω p2a4 = λ 4I . (A.29)
a is the input pump amplitude; the seedis injected into a (cid:18) 0(cid:19) a (cid:18) br(cid:19)
0
the plasma at z = L, t = 0 and meets immediately the
pumpfrontinjectedintotheplasmaatz =0,t= L/c . Then, the efficiency can be defined as [38]
a
−
Then, we define new wave amplitudes, a,b, and f, via
w 1
a˜=a0a, (A.19) ηw = b = b2dζ, (A.30)
w τ | |
f˜= a √σf, (A.20) a f Z
0
−
1
V a2√σ 3 where we assumed that the initial energy in the seed is
˜b= 3 0 b. (A.21)
R smallcomparedtothefinalenergyintheamplifiedpulse.
(cid:18) (cid:19)
Finally,wenotethatthenumberofexponentiationsof
By neglecting the “slow” time derivative of the pump
the SBRS (see Sec. VI) can be written in these rescaled
amplitude compared to the “fast” time derivative of the
variables as
pump amplitude, Eqs. (A.1–A.3) read
a = bf, (A.22)
ζ − Γ=V ˜bdt=√σ bdζ. (A.31)
f =ab∗ i∆f, (A.23) 3 | | | |
ζ − Z Z
b =af∗ iQb +ib2b. (A.24)
τ ζζ
− | |
8
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