Current limiting effects on laser compression by resonant backward Raman scattering

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by  Nikolai A. Yampolsky

Current limiting effects on laser compression by resonant backward Raman scattering
Nikolai A. Yampolsky
Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
Nathaniel J. Fisch
Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA
Through resonant backward Raman scattering, the plasma wave mediates the energy transfer
between long pump and short seed laser pulses. These mediations can result in pulse compression
1 at extraordinarily high powers. However, both the overall efficiency of the energy transfer and the
1 durationoftheamplifiedpulsedependuponthepersistenceoftheplasmawaveexcitation. Atleast
0 with respect to the recent state-of-the-art experiments, it is possible to deduce that at present the
2 experimentallyrealizedefficiencyoftheamplifierislikelyconstrainedmainlybytwoeffects,namely
thepumpchirp and theplasma wave wavebreaking.
n
a
J PACSnumbers: 52.38.Bv,42.65.Re,42.65.Dr,52.35.Mw

5

I. INTRODUCTION TheBRAcanbeconsideredasaphotondecayinstabil-
]
h ityofthepumpphotonintotheseedphotonandtheplas-
p Highpowerfemtosecondlaserpulsesarecurrentlygen- mon. Resonantconditions(1)canbeinterpretedthenas
m- erated through the chirped pulse amplification (CPA) the energy and momentum conservation laws of the de-
scheme [1]. The damaging of the solid state amplifier by cay. The pump energy transferred into the plasma wave
s
a high power radiationis avoidedby the temporalstretch- energy can be small as long as the plasma-to-laser fre-
l ing of the laser pulse, its amplification to high intensity, quency ratio remains small. However, the plasma den-
p
. and only then its compression. However,the high power sity should be high enough to provide significant laser
s radiationcanstilldamagesolidstateelementsoftheCPA coupling. In the nonlinear stage, in which there is com-
c
i laser at the compression stage. Thus, the peak intensity pletepumpdepletion,theBRAefficiencyapproachesthe
ys andfluenceatthelastcompressinggratingshouldremain theoretical limit of 1−ωp/ωa ≈1.
h belowthecriticalvalues,sothathigheroverallCPAlaser This paper will review the implications of the cur-
p powercanbereachedonlyiflargersizegratingsareused. rentplasma-basedresonantBRA experiments, including
[ ThatsignificantlyincreasesthecostoftheCPAlaserand those at performed at Princeton University [5–9], Insti-
1 CPA technology approaches its practical limit. tute of Applied Physics (IAP) in Russia [10], Lawrence
v Alternatively, plasma can be used as a nonlinear Livermore National Laboratory(LLNL) [11–13], and In-
1 medium to mediate laser amplification, since it cannot stitute ofAtomic andMolecularSciences (IAMS)inTai-
6 be damaged by high power radiation. The resonant wan [14]. Although by some measures still large,the ex-
8 backward Raman amplifier [2, 3] is at present the most perimentallyachievedefficienciesweresignificantlylower
0 promising and the best studied plasma-basedscheme for thantheoreticallimits. Earlieranalytical[15–23]andnu-
.
1 highpowerlaseramplification. Thisschemereliesonthe merical [19, 21, 24–30] studies were focused on identify-
0 resonantcoupling of two counterpropagatinglaserwaves ing mechanisms which may limit the BRA efficiency and
1 mediated by the plasma wave. The counterpropagating
finding regimes where these limitations can be avoided
1
geometry in BRA allows one to convert the long pump [31–35]. Itisofinterest,therefore,toassesswhatarethe
:
v energy into the energy of the short seed pulse . The res- current limitations in experimental achievements.
i onantnature of interactionallows one to use a relatively
X
low intensity pump pulse and reach significantly higher Although these experiments are only partially diag-
ar efficiencies compared to mechanisms based on nonreso- nosed, it is nonetheless possible to rule out some limit-
nant wave coupling [4]. ing mechanisms. For example, none of the experiments
The resonant conditions for the wave coupling are reported significant signal corresponding to the forward
Raman scattering (FRS) of the amplified pulse, unlike
ω =ω +ω , k =k +k , (1) the concern raised in several numerical studies [32, 34].
a b p a b p
Also, the efficiency in high power LLNL experiment was
whereω ,ω ,andω =(4πe2n/m)1/2arethefrequencies similartotheefficienciesinothersmallscaleexperiments,
a b p
ofthepump,seed,andplasmawave,respectively;k ,k , whichindicatesthatself-focusingandself-phasemodula-
a b
and k are the wavenumbers of the pump, seed, and the tioninstabilities[2,15,24,34]donotaffectamplification
p
plasma wave, respectively. These conditions are easy to in current experiments. However, the influence of other
satisfy since the plasma wave dispersion relation weakly limiting effects is less obvious. It is the purpose of this
depends on the plasma wavenumber and so one needs paper to identify these limiting effects in current BRA
just to choose appropriate plasma density n for efficient experiments,therebytoimproveupontheseexperiments
laser coupling. by avoiding the current limitations.
2
ThepaperisorganizedasFollows: InSec.II,weintro- In this regime the maximum amplitude of the ampli-
ducethebasicmodeldescribingwavecouplinginBRA.In fied pulse, b (τ), growslinearly with the amplification
max
Sec. III we demonstrate that detuning can significantly lengthwhileitsdurationdecreasesinverselyproportional
reduce the BRA efficiency in current experiments, but with the amplification length; b τ, ∆ζ 1/τ.
max
∝ ∝
thatthedetuningcausedbythepumpchirpcanbecom- Moreover,atsmallvaluesofdetuning,q 1,thepumpis
≪
pensated by the plasma density gradient. In Sec. IV we almostcompletelydepletedafterinteractingwiththeam-
discuss the influence of plasma wavebreaking on pulse plifiedpulse[15]. TheefficiencyofBRAthenapproaches
amplification. In Sec. V we demonstrate that the com- 1 ω /ω. These scalings were observed experimentally
p
−
bined limitation due to detuning and the wavebreaking [8],whereitwasconcludedthattheamplificationreached
suggests a pump intensity window for efficient BRA. In thenonlinearstageaccompaniedbysignificantpumpde-
Sec.VIweestimatethehighestplasmatemperaturethat pletion. Similar features were also observed in recent
may be toleratedin view of the increasedLandaudamp- experiments [12, 13], in which the output pulse energy
ing of the plasma wave. In Sec. VII we discuss the re- scales linearly with the pump energy and is almostinde-
quirements on the seed pulse quality. pendent of the seed pulse energy.
However, these features do not necessarily prove that
significant pump depletion was actually reached. The
II. BASIC MODEL same scalings of the amplified pulse energy can be de-
scribed by Eqs. (5) with significant detuning q 1
∼
The parameters of the above experiments (except for [15, 22]. Thus, measurements of the output pulse energy
the experiment at IAP) were similar to each other: am- scalings can be a useful tool suggesting different regimes
plification was performed by a pump with intensities on of BRA but these scalings cannot determine without
theorderofI 1014W/cm2 inafewmillimeterplasma doubt whether significant pump depletion was experi-
a
having density∼n 1019cm−3. These parameters are mentally reached. Only accurate estimate of the pump
∼
close to those suggested for BRA experiments [15]. The depletion within the temporal and spatial overlapof the
plasma-to-laserfrequency ratio under these conditions is pump and the amplified pulses can be a reliable tool for
onthe orderof1/10,whichenablesstronglasercoupling determiningwhetheramplificationreachedthenonlinear
on one hand, but keeps high maximum laser conversion stage.
efficiency (on the order of 90%), on the other hand. In The latest experiments with tilted laser geometry at
these regimes, neglecting focusing and refraction effects, Princeton [9] can be deduced to reach significant pump
theBRAcanbedescribedadequatelyintermsofasetof depletion, perhaps about 0.4 [36]. The pump depletion
equations for the wave envelopes of circularly polarized in other experiments appears to be well below this level
laser pulses [2, 15], ( 0.1% in IAP experiment, 0.3% in LLNL exper-
∼ ∼
iments, and 0.5% in IAMS experiment). These es-
at+caz = √ωωpbf (2) timates sugge∼st that nonlinearity caused by the pump
−
bt cbz =√ωωpaf∗ (3) depletion is not the leading limiting mechnism in cur-
−
∗ rent experiments and the observed scalings are caused
ft iδωf =√ωωpab /2, (4)
− byotherphysicaleffects. Inthefollowingsectionswede-
where a,b are the amplitudes of the pump and the seed, scribethemechanismswhicharelikelytoplaysignificant
respectively, normalized so that the laser intensity I = role in current experiments limiting the BRA efficiency.
a
πc(mc2/e)2 a2/λ2 = 2.736 1018 a2/λ2[µm] W/cm2;
| | × | |
f is appropriately normalized amplitude of the plasma
III. DETUNING
wave; δω is the detuning frequency between the waves
which is caused by failure of exact resonant condition
(1); t is time and z is longitudinal coordinate along the TheBRAmechanismreliesontheresonantinteraction
direction of the pump propagation. betweenwaves. Deviations fromthe resonancecondition
The advanced stage of amplification is characterized (1) change the instability bandwidth and may signifi-
by pump depletion, where Eqs. (2) — (4) reduce to cantlyreducewavecoupling. Thisreductionisthelargest
when the instability bandwidth constantly changes over
a = bf, b =af∗, f ia2qτf =ab∗, (5)
ζ − τ ζ − 0 the amplification length. Often the shift of the ampli-
fied pulse resonant frequency can be approximated with
where ζ = (t + z/c)√ωaωp/2 is the normalized lon-
a linear function and BRA canbe described by Eqs.(5).
gitudinal coordinate behind the seed pulse and τ =
The detuning can be characterized by the dimensionless
z√ωaωp/c is the normalized amplification length, q =
− parameter
2(∂ ω 2∂ ω )c/(a2ω ω ) is the detuning factor [15].
z p− z a 0 a p
This set of equations allows a self-similar solution [15] 2(∂ ω 2∂ ω )c
z p z a
having the following form q = a2−ω ω , (8)
0 a p
b=a20τB(a20ζτ) (6) where a0 is the undepleted amplitude of the pump. It
a=a A(a2ζτ), f =a F(a2ζτ) . (7) describes the change of the detuning frequency in the
0 0 0 0
3
frame moving with the amplified pulse δω q(t + z) not efficient for increasing the BRA efficiency since high
∝
in Eq. (4). Detuning results in incomplete pump de- pump intensity can result in the plasma wavebreaking
pletion, which, in turn, limits the BRA efficiency. For as will be discussed in Sec. IV and increased pump
an infinitely short seed pulse having integrated ampli- duration results in seed amplification by only a fraction
tude ǫ = √ωaωp b0dt/2, the pump depletion rate can of the pump. Alternatively, the detuning caused by the
beapproximatedbyanasymptoticexpressioninthelimit pump chirp can be partially compensated by detuning
R
q,ǫ 1 [15] caused by the plasma density gradient as follows from
≪
Eq. (8) In this regime, the resonant frequency for the
D 1 q2ξ∗2, (9) seedamplificationremainsconstantin the frame moving
a ≈ − 16 with the seed pulse, i.e. (∂ c∂ )(ω ω )=0.
t z a p
− −
There are two primary mechanisms which can drive
the interaction away from the exact resonance. The de-
tuning can be caused either by the plasma density gra-
dient or by the pump chirp. Earlyexperiments atLLNL
[11] suggest that spontaneous Raman spectrum broad-
ening can be explained by plasma density fluctuations
producedbyaspeckledpump. Randomdetuningcaused
bythesedensityfluctuationscanstronglyreduceamplifi-
cation. Thisscenariocanbedeterminedonlyuponavail-
ability of detailed plasma density measurements which
will determine the characteristicgradientfor the density
variations. Alternatively, the detuning can be caused by
the pump chirp. Many current BRA experiments are
pumped with CPA laser systems [6–10, 14], which are
characterizedbyalargebandwidth. WhentheCPAlaser
pulse is stretched, the induced pulse chirp can be large
enough resulting in significant detuning parameter q .
a
613 ∆ω 1019cm−3 1/2 1014W/cm2
a
q , (10)
a ≈ ∆t [ps] ω n I λ2[µm]
a a (cid:18) (cid:19) a
where ∆t is the FWHM pump duration and ∆ω /ω
a a a
is the relativepump bandwidth. In the experiments per-
formedatPrinceton,IAP,andIAMS,thepumpintensity
and bandwidth were similar to each other. However,the
detuning parameter induced by the pump chirp was sig-
nificantly different.
TheimportanceofthepumpchirpinBRAexperiments
was recognizedfirst in the experimentperformed at IAP
[10] which was identified as one of the main causes of
limited amplification. The large values of detuning pa- FIG. 1: (Color online) The output energy of the amplified
rameter in that experiment, q 1, result mainly from pulse versus the tilt angle of the laser beams. Experiments
low plasma density, n = (1.4≈ 4) 1016 cm−3. On with positive chirp (upper plot) demonstrate increased BRA
the other hand, detuning was−less i×mportant in other efficiencyintiltedgeometryunlikenegativechirpexperiments
experiments. The plasma density in IAMS experiment (lowerplot)whichdemonstratethemaximumBRAefficiency
at axial propagation of the laser pulses. The parameters of
[14] was roughly two orders of magnitude higher than in
theexperiment are described in Ref.[36].
IAP experiments. Under these conditions, the detuning
in IAMS experiments was on the order of q 0.1 and
∼
most likely did not significantly affect pulse amplifica- This was first illustrated [36] following Princeton ex-
tion. However, the detuning factor can be large even in periments [9], in which two laser pulses propagated at
denseplasmaatsmallpumppulsedurationsimilartoex- an angle with respect to the channel axis. In this setup,
periments performed at Princeton University [9, 36]. In the plasma frequency changesalongthe interactionpath
that setup the pump duration was about 20 ps, roughly since the channel has a transverse density gradient. The
equaltotheseedpulseamplificationtimein2mmplasma tilted geometry of laser propagation resulted in about a
resulting in q 0.27. factorof3 largeramplifiedpulseenergycomparedto ax-
≈
The detuning parameter caused by the pump chirp ial experiments (upper plot of Fig. 1). It is critical to
can be reduced either by increasing the pump duration note that the enhanced pump depletion in experiments
or intensity as follows fromEq. (10). These methods are with tilted alignmentof the laserbeams canbe observed
4
only if the detunings have opposite signs and partially plasma wave remains constant after reaching the crit-
cancel each other. ical value [10]. This model assumes that the plasma
The detuning compensation was demonstrated by wavedoes notvanishafter reachingcriticalamplitude so
changing the sign of the pump chirp, as seen in Fig. 1 that the pulse amplification survives in the wavebreak-
[36]. It is noteworthy that this demonstration is con- ing regime. However, the output pulse spectrum in that
clusive even without an accurate estimate of the plasma experiment consists of several peaks [10, 38] similar to
density gradient in the interaction region. In the experi- the observed spectrum in the experiment Ref. [39] per-
ments with negative (reversed) pump chirp (lower plot), formedinthe superradiantregime[4]. Inthis regimethe
the efficiency in tilted experiments is smaller than the bounce frequency of the laser beatwave ponderomotive
efficiency in axial experiments. This tendency is oppo- potential exceeds the plasma frequency. Thus, an alter-
site to the experiments with positive pump chirp (upper native explanation is that the superradiant regime may
plot). This data verifies that the sign of the pump chirp havebeenreachedintheIAPexperimentsbecauseofthe
isveryimportantforBRAefficiency. Flippingthesignof lowplasmadensity, n (1.4 4) 1016 cm−3,compared
∼ − ×
the pump chirp results in the addition of the detunings to other BRA experiments.
rather than their subtraction. The overall detuning fac- Todescribemoreaccuratelytheplasmawavebreaking,
tor q increases so that the efficiency drops significantly. PICcodesimulationshavebeenemployed,althoughcom-
The sensitivity of the overall detuning factor to the prehensive modeling including wavebreaking is difficult
signofthepumpchirpshouldbetakenintoaccountwhile since many plasma wavelengths should be taken into ac-
planning future experiments. Choosing the wrong chirp count [28, 34]. This task becomes even more challenging
sign (positive or negative depending on the sign of the [32]forsimulatingwarmplasmaduetoalargeincreaseof
plasmadensitygradient)canresultinincreasedvaluesof particle number (on the order of exp( 2/k2λ2 )) needed
− p D
the detuning factor and reduce the BRA efficiency simi- to resolve the electron distribution function in the reso-
lar to IAP experiments [10] in which the plasma density nant region.
increasedandthepumpfrequencydecreasedintheframe
moving with the seed pulse (Fig. 5 of Ref. [10]).
V. PUMP INTENSITY WINDOW FOR BRA
IV. PLASMA WAVEBREAKING
As discussed in Secs. III and IV, detuning and plasma
wavebreaking are important in current BRA experi-
Another mechanism which can prevent full pump de- ments. The plasma wavebreakinglimits amplification at
pletion is plasma wavebreaking which occurs when the high pump intensity while the detuning becomes impor-
plasmawaveistoostrong. Whentheplasmawavegrows, tant at low pump intensity. Therefore, there is a pa-
the electron oscillations in the plasma wave field also rameter regionfor the pump intensity which is favorable
grow. Electrons originating at different spatial positions for BRA. This idea is illustrated in Fig. 2 (upper plot).
then oscillate with different phases, since the phase of Here we plot individual efficiencies due to the detuning
the plasmawavechangeslongitudinally, E exp(ik z). andtheplasmawavebreaking. TheBRAefficiencydueto
p p
∝
The electron flow is not laminar then, damping the thewavebreakingistakenfromtheassumptionthatonly
plasma wave. criticalpumpintensity(11)canbeabsorbed. Furtherab-
The amplifiedpulse energyatthe highestpumpinten- sorption of the pump leads to the plasma wavebreaking
sity is almost independent of the input pulse energy in and cancels the interaction. The pump depletion rate
several BRA experiments [8, 10, 11]. Since the plasma then is Da = Iacr/Ia if Ia > Iacr. The pump depletion
wave at constant pump depletion is proportional to the rate caused by the detuning was calculated in Ref. [15]
pump intensity, this suggests that amplification may be and it is described by Eq. (9). The overall efficiency of
limited in its nonlinear stage by the plasma wavebreak- BRA is assumed to be a product of partial efficiencies
ing. Assuming that the pump can be fully depleted in due to the detuning and the plasma wavebreaking.
BRA, one can find the critical pump intensity Icr which AsillustratedinFig.2,theoverallBRAefficiencyhasa
a
leads to the plasma wavebreaking [2, 15] well-defined maximum at a certain pump intensity. The
experimentally observed efficiency in Princeton experi-
n 3/2 ments [9, 36] has the same kind of intensity dependence
Icr =1.45 1014λ[µm] W/cm2. (11)
a · 1019cm−3 aspredictedtheoretically(lowerplotinFig.2). Although
(cid:16) (cid:17) a detailed comparison of the experimental and the theo-
However, finite plasma temperature can significantly re- reticalefficiency plots is complicatedsince the pump has
duce this threshold [32, 37]. nonuniformintensityprofilealongthechannel,theeffects
The plasma wavebreaking effect is not analytically ofdetuning andwavebreakingarerobustandthe depen-
tractable. Some heuristic models for the plasma wave- denceofthe BRAefficiencyversuspumpenergycanstill
breaking were used to explain results of IAP experi- be observed. The overall efficiency is the largest in the
ment [10]. The experimental data might fit the so-called regime in which these two limiting effects are competing
“soft”wavebreakingmodelinwhichtheamplitudeofthe withneitherdominant. Thus,theexperimentaldatasup-
5
where q = 2(∂ ω 2∂ ω )c/(a2 ω ω ) is the detun-
1.2 ingfactocrratcritizcaplp−umpz inatensit0ycrofathpewavebreaking.
Therefore, the optimum pump intensity for the highest
1 BRAefficiencycanexceedthecriticalpumpintensityfor
the plasma wavebreaking at high detuning parameters
0.8 qcr > 0.25 (assuming ξ∗ 10 as will be discussed in
∼ ∼
y Sec. VII). At the same time, the maximum pump deple-
n
e tionrateremainsbellow2/3inthisregime. BRAexperi-
ci0.6
effi mentsshouldbedesignedintheoppositeregimeinorder
toreachhighamplificationefficiency. Theoptimalpump
0.4
intensity in that regime is equal to the critical pump in-
tensity for the wave breaking described by Eq. (11) and
0.2
the detuning parameter should be small. If detuning is
dominated by the pump chirp, then this regime can be
0 achieved at small pump bandwidth
1013 1014 1015 1016
pump intensity
∆ω ∆t [ps]λ3[µm] n 2
a a
[%]<0.55 . (14)
5.5 ωa ξ∗ 1019cm−3
Experimental data (cid:16) (cid:17)
Fitted curve Notethatthisconditionisbasedontheestimate(11)for
5
the critical pump intensity due to plasma wavebreaking
in cold plasma. The critical pump intensity in experi-
4.5 ments may be significantly smaller due to finite plasma
%
cy temperature. As a result,the criticaldetuning factor qcr
en 4 may be larger and the requirements for the pump band-
ci
Effi width are more severe. This assumption is supported by
3.5 the experimental data at Princeton experiments [9, 36].
Condition (14) was roughly satisfied in that setup, how-
ever, the pump depletion rate in axial experiments was
3
on the order of 10% in the laser coupling region. This
valueiswellbelowthetheoreticalestimate(13)whichin-
2.5
40 50 60 70 80 90 100 dicates that criticalpump intensity due to wavebreaking
Pump energy (mJ)
was significantly reduced in warm plasma. At the same
time, the critical detuning parameter q was reduced in
cr
FIG. 2: (Color online) The upper plot shows theoretically
the tilted laser pulse experiments, which resulted in the
estimated BRAefficiencyasdescribedinthetext. Thelower
pump depletion rate (about 40%) significantly closer to
plot shows experimentally measured dependenceof the BRA
the analytical estimate.
efficiency versus the pump energy in Princeton experiment
[9,36]. Theintensityofthepumpisproportionaltothepump
energysincetheotherparametersofthepumparefixed. The
VI. LANDAU DAMPING
solid line is theleast square parabolic fit of theexperimental
data points.
As discussed in Sec. IV, plasma wavebreakingin BRA
occurswhentheplasmawaveamplitudeexceedsthecrit-
ports the theoretical assumption that both the detuning ical value. However,in warmplasma there might be sig-
andtheplasmawavebreakinglimittheBRAefficiencyin nificant number of electrons interacting with the plasma
the experiment. wave even at small wave amplitudes resulting in Landau
Using the estimates for BRA efficiencies due to de- damping of the plasma wave. As a result, the laser cou-
tuning and wavebreaking described above, the optimum pling is reduced, diminishing the BRA efficiency.
pump intensity andthe pump depletionrate canbe esti- AsimilarLandaudampingeffectoccursininertialcon-
mated as finement fusion (ICF) plasma [40–42]. The plasma wave
Icr√3qcrξ∗ , q > 4 bmoatnh isncaBttReArinagnd(BIRCSF).isSgteundeireasteodfbLyanthdeaubadcakmwapridngRian-
Iopt =  a 4 cr √3ξ∗ (12) ICF plasma are motivated by the suggestion that the
a 4
 Iacr , qcr < √3ξ∗ BARs SopcpaonsebdetsotrothneglcyassuepopfreICssFed, LaatnladraguedvaamlupesinogfiknpλtDhe.
 8 1 , q > 4 case of BRA is unwanted since it reduces laser coupling
cr
Daopt =  13√−3qqc21cr6rξξ∗2∗ , qcr < √√334ξξ∗∗, (13) araenpgdpi,lmicteahtoeirfoensfmsoraietl,litkshpeiλmDapm.opWrtlaihfineettrhtseohroewusatldinmtbeadeteoorpacencroauttr,eadftoerilnybotthhthee

6
Landau damping, which is complicated by the presence for the longest time. In that case, if Landau damping
ofthe pumpwaves. Theinversebremsstrahlunglossesof were strong, the detuning compensation might not be
thepumpcanincreasetheplasmatemperatureleadingto effective.
theregimeofhighk λ . Ontheotherhand,theelectron On the other hand, the estimated plasma tempera-
p D
distribution function might be modified by a large am- ture in the LLNL experiments (T = 200 250 eV [12],
−
plitude plasma wave so that the Landau damping might k λ =0.38 0.42,)suggeststhatLandaudampingprob-
p D
−
saturate[43–46]. Therefore,thesecompetingeffectsmust ably played a more significant role in those experiments,
be considered in describing Landau damping when the whichisconfirmedby2DPICsimulations[13]. Thesame
plasma wave is externally driven by the ponderomotive estimate for the parameters of experiment performed at
force of the laser beatwave. IAMS (T = 150 200 eV [14], k λ = 0.55 0.65)
p D
− −
A fully kinetic description of nonlinear Landau damp- suggests that that Landau damping was the dominant
ing is a challenging task both numerically and analyt- limiting effect in that experiment. It would appear,
ically. A number of reduced analytical models [47–49] however, that the large amplification of the seed pulse
as well as simplified codes [50–52] describing nonlinear (15) suggests either a much lower temperature or that
Landau damping of a driven plasma wave were recently the BRS instability is not suppressed in hot plasma.
offered. The quasilinearmodeldescribedinRef.[49]was The latter possibility wouldhaveimportantimplications
chosento study this effectsince it is formulatedin terms for the NIF program. Accurate conclusions about Lan-
of simple equations which can be analyzed analytically. dau damping in LLNL and IAMS experiments, however,
This model is in a reasonable agreement with more pre- can only be drawn with more accurate measurements of
cise analytical kinetic model developed in Refs. [47] and the plasma temperature. Measuring the plasma tem-
numerical simulations [53]. The quasilinear model, how- perature, however, can be challenging since the plasma
ever, does not account for a number of important effects temperature changes during amplification due to inverse
such as particle detrapping due to their ballistic motion bremsstrahlung losses of the pump, so that the plasma
across the speckle or collisions. temperatureduringlaseramplificationmaybe unrelated
The analysis of nonlinear Landau damping in BRA is to the plasma temperature before interaction.
presented in Ref. [54]. A regime was found in which the Notethatplasmaheatingcanbeavoidedinyetunpur-
initially large linear Landau damping rate can become sued “ionization front” regime wich can be achieved if a
saturated The main scaling in this estimate comes from short intense seed pulse ionizes plasma and seeds ampli-
the exponential dependence of the linear Landau damp- fication at the same time [26, 55]. As a result, the pump
ing rate on the plasma temperature. Therefore, the es- does not heat plasma for a long time prior to laser cou-
timate for the maximum plasma temperature in BRA is plingandLandaudampingisexpectedtobesmallinthis
roughlythesameforvariousexperimentsdescribedabove regime.
[9, 12, 14].
kλ <0.4, T <190 n λ2[µm]eV, (15) VII. EFFECTIVE SEEDING POWER
D 1019cm−3
(cid:16) (cid:17)
for I 1014W/cm2, n 1019cm−3, and few milimeter Estimates for the pump depletion rate in the presence
a
∼ ∼ of detuning (Eqs. (9), (12)—(14) ) as well as estimates
plasma length.
forsaturationofLandaudampinginBRA[54]dependon
Estimate (15) roughly agrees with findings presented
the self-similar position of the amplified pulse maximum
in Ref. [32], where no significant amplification was ob-
served at high plasma temperature, k λ > 0.4. How- ξ∗. This coordinate weakly depends on the integrated
p D
amplitude of the seed pulse ǫ= b(τ =0,ζ)dζ
ever, parameter region for efficient BRA presented in
Ref. [32] was strongly affected by the forward Raman
R
scattering(FRS)oftheamplifiedpulse. Thegrowthrate ξ∗ =ln(4 2πξ∗/ǫ), (16)
of FRS is about (ω /ω)2 times smaller than the growth
p p
rate ofBRS.Therefore,FRS instability is more sensitive
toplasmadensityfluctuationsthanBRS.Currentexper- Ib λ[µm] n 1/4
ǫ=0.18 ∆t [ps] (17)
imental data indicate noticeable plasma density fluctua- s1012 W/cm2 1019 cm−3 b
tions[9,11]whicharelikelytosuppressFRS[16]inmod- (cid:16) (cid:17)
ern BRA experiments unlike analytical and numerical whereI and∆t aretheseedpulseintensityandFWHM
b b
predictions which assume uniform plasma [2, 15, 32, 34] duration, respectively.
Estimate (15) suggeststhat the Landaudamping does The integrated seed pulse amplitude ǫ is equal to the
notaffect stronglythe amplificationinthe Princetonex- pump depletion rate behind the seed pulse in the lin-
periments [9] (k λ 0.23). This conclusion is also ear regime. This parameter fully describes the evolu-
p D
≈
supported by experiments with tilted geometry of laser tion of the amplified pulse within the three-wave cou-
pulses in which the detuning was compensated on the pling model (5) under assumption that the seed pulse is
side of the pump entrance into the channel. The plasma infinitely short. The integrated amplitude of the seed
is likely the hottest in that domain since it was heated pulse in all the experiments is large enough (ǫ 0.1,
∼
7
ξ∗ 5.5 inPrincetonexperiments)to providesignificant Gaussianprofilecauseslargediscrepancywithanalytical
≈
amplification. However,the integratedseed pulse ampli- model. Poor pulse profile can even result in appearance
tude cannot be used to characterize the amplified pulse of precursors [17] which prevent pulse compression and
growth if the seed pulse is long since only a small frac- growth of its peak intensity.
tionofthe seedpulse contributes to BRAseeding inthis Currently,the seedpulses aregeneratedby downshift-
regime. Qualitatively, the effect of the finite seed pulse ing the frequency of the chirped pulse in a Raman cell
duration plays role when the duration of the amplified [9, 11, 12] or through generation of supercontinuum [14]
pulse becomes on the order or smaller then the duration with the following temporal compression. Such pulses
of the seed pulse. In this regime the laser coupling re- areexpectedtohavepureshot-to-shotstabilityandpoor
gionislocatedatthefrontoftheseedpulseandtheback front quality and, thus, are subject to generate precur-
part of the seed pulse remains behind the interactionre- sors. This problem can be avoided upon increasing the
gion. Autocorrelation measurements of the original seed seedpulsecontrastusingwell-developedtechniques. Itis
and the amplified pulse durations in Princeton experi- also worth noting that poor contrast of the seed pulses
ments [9] show that the amplified pulse becomes much in current experiments does not result in a decrease of
shorter than the original seed pulse. The effective seed- theeffectiveseedingpowerbymanyordersofmagnitude.
ing powerwhichcontributes to the pulse amplificationis Therefore,it is reasonableto considerξ 10 in allthe
eff
significantly reduced in this regime. estimates presented in previous sections. ∼
The quantitative estimate for the effective seeding
powerwaspresentedinRef.[56]assumingGaussiantem-
poral profile of the seed. It was shown that the domain
which contributes the most to the pulse amplification VIII. DISCUSSION
slips to the front of the seed pulse and its duration de-
creases during amplification. Eventually, only a small
domain with exponentially small amplitude effectively It is possible to deduce in recent state-of-the-art ex-
drives amplification. The quantitative estimate for ef- periments that the experimentally realized efficiency is
fective seeding power reads as [56] likely constrained mainly by the pump chirp and wave-
breaking, which can be put as a bracket on pump inten-
304 I L[mm]∆t [ps] 2 nλ2[µm] sity. The plasmadensitygradientwasshowntocompen-
a b
ξeff ≈ξ∗+ ξ2 1014W/cm2 1019cm−3, (18) sate for the pump chirp. When these explanations are
eff (cid:18) (cid:19) taken into account, together with an accounting of the
where L is the plasma length. spatial and temporal overlap of the counterpropagating
Theregimeofstronglyreducedeffectiveseedingpower beams, it becomes clear that the experimentally realized
islikelytotakeplaceinexperimentsperformedatPrince- efficiency is within the theoretical expectations. In ad-
ton [9]. The analytical estimate (18) for the effective dition, the plasma temperature should be small enough,
seeding power predicts the increase of ξeff by more than kpλD < 0.4, to avoid Landau damping of the plasma
a factor of 2. Note that the effective seeding power de- wave which mediates the laser coupling.
scribed by Eq. 16 depends exponentially on the effective Moreover, although the experiments were carried out
self-similar coordinate of the amplified pulse maximum in the opticalregime for ps pulses,it is clearthat the es-
ξ . Therefore, the effective seeding power should de- tablishment ofthe physicalmechanisms atplay here can
eff
crease by several orders of magnitude. In this regime, beextrapolatedtootherregimesofresonantRamancom-
the effective seeding domain is located on the front tail pression, such as through ionization-front coupling [55],
of the original seed pulse and the estimate (18) cannot X-ray or UV compression [57–61] , or the quasitransient
be used since small deviation of the seed pulse from the regime [23, 33].
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