The impact of thermal effects on Raman amplification in plasma is investigated theoretically. It is shown that damping and the shift in plasma resonance at finite temperature can alter the evolution of an amplified pulse. It is shown that pulse compression can occur which is not predicted by the cold plasma model. Although thermal effects can lead to a reduction in the efficiency of the interaction, this can be avoided by using a chirped pump. In this case these effects can be beneficial, suppressing the development of a train of pulses behind the amplified seed, as observed in the cold plasma model

Ersfeld, Bernhard

Farmer, John Patrick

Jaroszynski, Dino

English

The impact of thermal effects on Raman amplification in plasma is investigated theoretically. It is shown that damping and the shift in plasma resonance at finite temperature can alter the evolution of the amplified pulse and lead to pulse compression which is not predicted by the cold plasma model. Although thermal effects can lead to a reduction in the efficiency of the interaction, this can be ameliorated by using a chirped pump. In this case thermal effects can be beneficial and suppress the development of the train of pulses that develops behind the amplified pulse, as observed in the cold plasma model

Farmer, J.

Ersfeld, B.

Jaroszynski, D.A.

Name not available

Raman amplification in plasma: thermal effects

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryFarmer, John Patrick and Ersfeld, Bernhard and Jaroszynski, Dino (2009) Raman amplification inplasma : thermal effects. [Proceedings Paper]Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/Raman Amplification in Plasma: ThermalEffectsJohn Farmer, Bernhard Ersfeld and Dino JaroszynskiDepartment of Physics, SUPA, University of Strathclyde,107 Rottenrow East, Glasgow, G4 0NG, UKAbstract. The impact of thermal effects on Raman amplification in plasma is investigated theoret-ically. It is shown that damping and the shift in plasma resonance at finite temperature can alter theevolution of an amplified pulse. It is shown that pulse compression can occur which is not predictedby the cold plasma model. Although thermal effects can lead to a reduction in the efficiency of theinteraction, this can be avoided by using a chirped pump. In this case these effects can be beneficial,suppressing the development of a train of pulses behind the amplified seed, as observed in the coldplasma model.Keywords: Raman, laser, amplification, plasma, thermal, Bohm-Gross, heating, damping, Lang-muirPACS: 52.38.BvINTRODUCTIONRaman amplification in plasma is a potential method for the creation of intense, ultra-short laser pulses [1]. Two laser pulses of different frequencies are coupled by the motionof the plasma electrons, allowing energy transfer between the two. A counterpropagat-ing geometry allows a short seed pulse to be amplified by a long pump beam, allowingthe seed to grow to intensities greater than that of the pump [2].Raman amplification is an attractive prospect as it could potentially lead to significantreductions in both the size and cost of high power laser systems. Unlike conventionaloptical media, plasma has no damage threshold, thus removing the need for chirpedpulse amplification and so avoiding the need for compression gratings, which becomelarge and expensive at high powers. Further, plasma has the potential of offering highefficiencies over shorter distances than conventional optical amplifiers.Although such sources will have uses across a wide range of science and technologyapplications, it is of particular interest to the laser-plasma accelerator community, whoalready have much of the skills and equipment necessary to exploit this technique.THEORYWhen two counterpropagating laser pulses of different frequencies collide in plasma, theresulting beat wave will act upon the plasma electrons through the ponderomotive force,driving electrons toward areas of lower intensity. If the lasers are detuned by the plasmafrequency a plasma wave will be excited. The resulting density perturbation will act asa moving Bragg grating which scatters photons from the higher frequency laser pulseto the lower, amplifying the latter. In this way, a short, low intensity seed pulse can beamplified using a long counterpropagating pump beam.In the linear regime [1] the amplification bandwidth is very narrow, due to the har-monic motion of the plasma electrons. Pump depletion is negligible and seed growthexponential. However, the density perturbation that is created continues to scatter thepump after the probe has passed, leading to an increase in the duration of the amplifiedseed proportional to the propagation distance.In the pump-depletion regime [3] depletion of the pump is sufficiently high as tosuppress amplification towards the rear of the seed, leading to pulse compression andspectral broadening. The amplified seed develops as a train of short pulses of decreasingintensity, due to Burnham-Chiao ringing [5], which occurs when energy is transferredfrom the probe back in to the pump.It has been shown [4] that using a chirped pump allows a pulse to be amplified in thelinear regime without the increase in duration observed when using a monochromaticpump. The amplification bandwidth is increased due to the varying frequency of thepump. The different frequency components will be in phase close to the head of the seed,but further behind they will destructively interfere, preventing pulse lengthening. Thepeak intensity grows linear with distance: while each frequency component is amplifiedexponentially, they will only interact for a finite time before falling out of resonance.The same effect can be achieved using a plasma density gradient.Impact of thermal effectsDamping and the plasma response at finite temperature have several implications forRaman amplification in plasma.At finite temperature, contributions from the electron pressure lead to an increase inthe resonant frequency of the plasma. This Bohm-Gross shift will alter the resonanceconditions of the Raman process.In addition, the plasma wave will be subject to Landau damping. Electrons withthermal velocities close to the phase velocity of the plasma wave will exchange energywith the wave. Those with velocities just below the phase velocity of the plasma wavewill be accelerated, while those just above will be decelerated. Analogues can be drawnwith plasma wakefield acceleration. For a Maxwellian distribution, the net effect is forthe plasma wave to lose energy.Collisional damping of the laser pulses will not have a significant direct impact ontheir intensities for the parameters used in current experiments (e.g. [2]). However, itwill lead to an increase in the plasma temperature. The plasma wave, on the other hand,may be significantly damped, due to a combination of collisional damping and Landaudamping, which will also contribute to an increase in temperature.SIMULATION MODELSimulations have been carried out using a 1-dimensional fluid model, based on theslowly varying envelope approximation for Raman amplification [1], modified to takeinto account thermal effects.The resonant frequency of the plasma at finite temperatures is given byω ′p =√ω2p +3θc2k2‖ (1)where ωp =√nee2/ε0m is the plasma frequency, corresponding to the cold plasmaresonance, k‖ the wavenumber of the Langmuir wave and θ = 3kBTmc2 the normalisedtemperature. e and m are, respectively, the electron charge and mass, ε0 the permittivityof free space, c the vacuum speed of light, kB the Boltzmann constant and T thetemperature.The plasma will be heated according to∂θ∂ t = ν⊥(|a0|2 + |a1|2)/3+ (ν‖+νL) | f |2/3 (2)where a0,1 =E0,1emω0,1care the reduced vector potentials of the pump and probe, respectively,E0,1 and ω0,1 are the associated transverse electric field and frequency. f = Ezemωpc isa normalised form of the longitudinal electric field Ez. ν⊥,‖ are the transverse andlongitudinal collisional damping rates, and νL the Landau damping rate.The transverse and longitudinal collisional damping rates are given by substitutingthe laser and plasma frequencies, respectively, into the expressionν =Ze2ω2p6(2pi)3/2ε0mc3[ln(4θmc2h¯ω)− γ](3)where h¯ is Planck’s constant divided by 2pi and γ is Euler’s constant. The Landaudamping rate is approximated byνL ≈√pi8ω4pk3‖c3θ−3/2exp(−ω2p2k2‖c2θ−32)(4)The modified envelope equations for the Raman process are then(∂∂ t − c∂∂ z +(ω0ωp)2 ν⊥2)a0 =−ωp f ∗a12(5)(∂∂ t + c∂∂ z +(ω0ωp)2 ν⊥2)a1 =ωp f a02(6)( ∂∂ t +ν‖+νL2+ i(ω ′p− (ω0−ω1))) f = ω0a∗0a12(7)These modified equations include damping of the plasma wave and the laser fields, whileEq. (7) also incorporates the shift in plasma resonance due to temperature.RESULTSThe simulation parameters were chosen to be similar to those used in recent experimen-tal works [2]: Gaussian profile seed of FWHM 0.5ps, peak intensity 1×1012 W cm−2,ω1 = 10 ωp; flat top pump pulse, length 10ps, intensity 1×1014 W cm−2, ω0 = 11 ωp;plasma length 2mm, of density ne = 1×1019 cm−3 and initial temperature 5 eV.Figs. 1.a,b show the simulation results for these parameters if thermal effects are notincluded. The pump depletion regime is accessed and the expected pulse compressionand pi-pulse [3] profile are observed, with a peak intensity ∼ 2000 times that of the inputseed.The inclusion of thermal effects, as shown in Figs. 1.c,d shows a reduction in thepeak intensity, achieving an amplification of ∼ 500 times. However, compression of theamplified pulse is still observed, and the profile is qualitatively similar to the pi-pulse.However, pump depletion reaches ∼ 30%, and is seen to decrease after this point. Thissuggests that pump depletion is not the primary source of pulse compression.This was verified by arbitrarily excluding pump depletion from the simulation model.Although this is clearly nonphysical, it serves as a useful way of examining the processesthat contribute to the observed behaviour. Fig. 1.e shows the result of this modification:there is an increase in seed amplification, which is to be expected, but pulse compressionstill occurs. This behaviour cannot be accounted for using the cold plasma model.However, it can be explained simply by considering the Bohm-Gross frequency shift,as shown in Fig. 1.f. The plasma resonance varies with time, and so will have a similareffect to using a chirped pump or a plasma density gradient, allowing pulse compression.It is important to note that thermal effects do not guarantee pulse compression. Forexample, at high initial temperatures, the shift in plasma resonance is smaller, allowingpulse lengthening as observed in the cold plasma model.While the reduction in pulse length observed in Fig. 1.c is an interesting effect, the lowefficiency probably makes this scenario unsuitable for practical applications. However,it is possible to compensate for the shift in plasma resonance through the use of a chirpedpump pulse. Considering Eq. (7), it is clear that if ω0 increases with time to match theBohm-Gross shift due to heating by the pump (as shown in Fig. 1.f), the two effects willcancel.Fig. 2 shows the effect of using a pump with a linear chirpvarying from 11 ωp to11.075 ωp over its 10ps duration. Although this is a rather rough match to the shift inresonance shown in Fig. 1.f, it can be seen that it is sufficient to significantly increasethe efficiency of the interaction, giving an amplified pulse of peak intensity similar tothat observed in the cold plasma model (Fig. 1.a).While the peak efficiency is matched, the train of pulses behind the seed head issuppressed. This is due to imperfect matching of the shift in plasma resonance: althoughthe shift due to heating by the unperturbed pump could be exactly matched, additionalcontributions due to damping of the plasma wave and seed cannot. This will lead to anincreased shift in resonance behind the head of the seed, which will suppress Burnham-Chiao ringing, both directly, through the shift in resonance, and indirectly, through thereduction in pump depletion.a 05e141.0e151.5e152.0e15 0  2  4  6  8  10Intensity (W cm-2 )Time (ps)Input0.5mm1.0mm1.5mmOutput 05e111.0e12-1 -0.5  0  0.5  1b 02.5e135.0e137.5e131.00e141.25e14 0  2  4  6  8  10  12  14  16  18  20Intensity (W cm-2 )Time (ps)c 01.0e142.0e143.0e144.0e145.0e14 0  2  4  6  8  10Intensity (W cm-2 )Time (ps)Input0.5mm1.0mm1.5mmOutput 05e121.0e131.5e13-1  0  1  2  3  4d 02.5e135.0e137.5e131.00e141.25e14 0  2  4  6  8  10  12  14  16  18  20Intensity (W cm-2 )Time (ps)e 05e141.0e151.5e152.0e15 0  2  4  6  8  10Intensity (W cm-2 )Time (ps)Input0.5mm1.0mm1.5mmOutput 01.0e132.0e13-1  0  1  2  3  4f 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 0  2  4  6  8  10Frequency (ω/ωp)Time (ps)FIGURE 1. Simulation results illustrating the impact of thermal effects on Raman amplification.(a) Seed evolution a and (b) pump depletion b using the cold plasma model.(c) Seed evolution and (d) pump depletion using the thermal effects model.(e) Seed evolution using the thermal effects model modified to remove the effects of pump depletion.(f) Shift in plasma resonance due to heating of the plasma by the pump beam.a The seed intensity observed at different positions in the plasma is plotted against time. For a givenposition, the head of the pulse will appear leftmost.b Temporal profile of the pump intensity as it leaves the plasma.a 05e141.0e151.5e152.0e15 0  2  4  6  8  10Intensity (W cm-2 )Time (ps)Input0.5mm1.0mm1.5mmOutput 05e111.0e12-1 -0.5  0  0.5  1b 02.5e135.0e137.5e131.00e141.25e14 0  2  4  6  8  10  12  14  16  18  20Intensity (W cm-2 )Time (ps)FIGURE 2. Effect of using a chirped pump to match the shift in resonance due to pump heating on(a) seed evolution and (b) pump depletion. Other parameters are as in Fig. 1.c.CONCLUSIONThe impact of thermal effects on Raman amplification has been investigated. The shiftin plasma resonance over time due to heating by the pump acts as a chirp. For amonochromatic pump, this chirp leads to a reduction in the efficiency of the interactionand, in some cases, allows pulse compression without the need for pump depletion, amarked departure from the behaviour observed in the cold plasma model.In order to avoid these effects, an up-chirp may be added to the pump beam tocompensate for the shift in plasma resonance. The two chirps cancel, increasing theinteraction efficiency. It is shown that this allows for an amplified pulse with peakintensity similar to that observed in the cold plasma model. However, additional heatingdue to damping of the plasma wave leads to a further shift in resonance behind thehead of the seed. This limits pump depletion and suppresses Burnham-Chiao ringing,allowing a single intense pulse, rather than the series of pulses observed in the coldplasma model.These results show that thermal effects can have a significant impact on Ramanamplification, affecting not only the efficiency of the interaction, but altering the lengthand profile of the amplified pulse. For this reason thermal effects should be taken in toaccount both when planning experiments and analysing results.REFERENCES1. G. Shvets et al, Phys. Rev. Lett. 81 4879 (1998)2. W. Cheng et al, Phys. Rev. Lett. 94 045003 (2005)J. Ren et al, Nat. Phys. 3 732 (2007)3. V. M. Malkin, G. Shvets and N. J. Fisch, Phys. Rev. Lett. 82 4448 (1999)4. B. Ersfeld and D. A. Jaroszynski, Phys. Rev. Lett. 95 165002 (2005)5. D. C. Burnham and R. Y. Chiao, Phys. Rev. Lett. 188 667 (1969)

Raman amplification in plasma : thermal effects

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