New criteria for efficient Raman and Brillouin amplification of laser beams in plasma

Raman or Brillouin amplification of a laser beam in plasma has long been seen as a way to reach multi-PW powers in compact laser systems. However, no significant plasma-based Raman amplification of a laser pulse beyond 0.1 TW has been achieved in nearly 20 years, and only one report of Brillouin amplification beyond 1 TW. In this paper, we reveal novel non-linear criteria for the initial seed pulse that will finally open the door to efficient Raman and Brillouin amplification to petawatt powers and Joule-level energies. We show that the triple product of the coupling constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}Γ, seed pulse duration \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}τ and seed pulse amplitude a for the Raman seed pulse (or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a^{2/3}$$\end{document}a2/3 for Brillouin) must exceed a specific minimum threshold for efficient amplification. We also analyze the plasma-based Raman and Brillouin amplification experiments to date, and show that the seed pulses used in nearly all experiments are well below our new threshold, which explains the poor efficiency obtained in them. Finally, we analyze a recent Brillouin amplification experiment that used increased seed pulse power to obtain Joule-level amplification, and find excellent agreement with our theory.

The essential data for the Raman amplification experiments displayed in Figure 1 of the main manuscript are given in   Here, ω 0 denotes the pump laser angular frequency, n cr the critical density for that frequency, n 0 the plasma electron density, Γ R = ω 0 (n e /n cr ) 1/4 / √ 4g is the Raman backscattering coupling coefficient, g = 1 (g = 1/2) denotes linear (circular) polarisation, τ 1 and a 1 are the signal pulse's duration and amplitude. In table I, P1-P5 denote experiments at Princeton University, L1-L3 denote experiments at Lawrence Livermore National Laboratory, S1-S3 denote experiments at Strathclyde University (Glasgow, United Kingdom) and M1-M4 denote miscellaneous experiments. R1-R3 denote pulses from the double-pass experiment by Ren et al. [6] at Princeton University: R1 is the signal pulse before interaction, R2 is the pulse after the first pass and R3 is the pulse after the second pass. We note that this is the only Raman amplification experiment that reported enough information on the parameters of the amplified pulses (both intensity and duration) to do our analysis. In particular, duration and envelope characterisation for the output pulse are missing from all other experiments. Pulse parameters were determined as follows. Most papers provide the energy, duration, wave length and spot diameter of the initial signal pulse and the background plasma density in an unambiguous fashion. The polarisation of the laser beams is not always given, but can usually be taken as linear, especially when Ti:sapphire lasers are used [16]. Energy and duration yield power, power and spot diameter yield intensity, intensity and wave length yield a 1 . The pump pulse wave length yields ω 0 and n cr , and together with the plasma density this yields Γ R . Thus, we find all factors of the triple product Γ R τ 1 a 1 . This was fairly straightforward for the initial signal pulses of almost all experiments. However, in the 2011 experiment by Kirkwood et al. [9], a thermal plasma was assumed with a dispersion relation (∆ω/ω 0 ) 2 = n e /n cr + 12k B T e /(m e c 2 ), where ∆ω denotes the difference between pump and signal pulse frequencies. Then (∆ω/ω 0 ) 2 = (1 − 1054/1220) 2 = 0.0185 was found, and k B T e ∼ 250 eV was taken from a simulation to infer n e /n cr ∼ 0.012. Neither n e nor T e were measured directly though.
Regarding pulse energies: E in is the energy of the input pump beam, while E out is the energy of the output (amplified) seed pulse, both as listed in the publication. No output energy is given for pulse R1, since this is a double-pass experiment with two output energies, after one (R2) or two (R3) passes. The highest efficiencies have been obtained by experiments P4 and the Ren experiment, both conducted at Princeton [4,6]. This is mainly due to the somewhat larger spot diameters that could be maintained in these experiments (see below). For the Livermore experiments L1-L3, we find that the pump pulse is 1 nanosecond long, while the plasma column is only 3 mm wide, so only 20 picosecond (2%) of the pump beam can interact meaningfully with the seed pulse. To improve the comparison with other experiments, we have used 2% of the published pump pulse energy in table I for these experiments (marked with †). In spite of this, we find that the efficiencies of the Livermore experiments remain below 1%, below those reported by Princeton or Strathclyde. In fact, the level of Raman backscattering in the Livermore experiments is suspiciously low [7][8][9], as evidenced by the null shot (only pump, no seed) in Figure 5 of Ref. [8]. In this shot, a 220 J pump beam (1 ns duration and 0.2 mm spot diameter, so 0.22 TW and 5.5 × 10 14 W/cm 2 , a 00 = 0.016) is launched at a 3 mm plasma column with ω 0 /ω p = 8 (i.e. quite high density, Γ R = 3.1 × 10 14 s −1 ). Even so, hardly any Raman backscattering is seen here, although Γ R a 00 L/c ∼ 50 for this shot. Thus, it seems that the Raman backscattering instability itself is inhibited in the Livermore experiments. This may have been caused by their use of random phase plates to smooth the pump beam. Random phase plates are best known as devices to smooth laser beam envelopes [19]. However, they are also known to suppress laser-plasma instabilities like stimulated Raman and Brillouin backscattering because they reduce the spatial coherence of the laser light [20][21][22][23][24], which may be bad for Raman amplification. Princeton and Strathclyde do not report the use of random phase plates, while the LULI Brillouin experiments only use them for the ionizing beam, not for the pump and seed beams. The reduction of the growth rates for stimulated Raman scattering due to an increase in wave number bandwidth (and thus a reduction of the spatial coherence) has also been studied by Santos et al. [25].
It is of interest to study the correlation between the value of the triple product a 1 Γ R τ 1 (or a 2 1 Γ 3 sc τ 3 1 for sc-Brillouin) for the initial seed pulses, and the efficiency obtained in the experiments using those seed pulses. For Raman amplification, a weak positive correlation is found between the values for efficiency and the triple product: R = 0.16. The small value for R has two causes: (i) there are no Raman experiments with large values for either the triple product or the efficiency, so it is difficult to spot any trend, and (ii) the Princeton experiments P4 [4] and Ren [6] report much higher efficiencies (∼ 4%) than one would expect from their values for the triple product (∼ 0.1). Without those two experiments, the correlation between triple product and efficiency would be much stronger: R = 0.58. For sc-Brillouin amplification, we find a strong positive correlation: R = 0.62. This is mainly driven by the results of the Marquès experiment [29], which covers wide ranges of values for both the triple product a 2 1 Γ 3 sc τ 3 1 and the efficiency, with a strong correlation between them.
From Figure 1 in the main manuscript, certain trends can be observed: experiments at Livermore National Laboratory use high plasma densities, long pulse durations and low amplitudes, while experiments at Strathclyde University use low densities and short pulse durations, but higher amplitudes. Experiments at Princeton University occupy the middle ground. We note that none of the input pulses satisfies our new criterion Γ R τ 1 a 1 ≈ 3.4, usually falling far short. However, if Γ R τ 1 from one of the Livermore experiments could be combined with a 1 from one of the Strathclyde experiments, a seed pulse with a good value for Γ R τ 1 a 1 would result.
In many Raman amplification experiments with weak seed pulses, the growing seed pulse may only reach the nonlinear regime towards the end of the interaction. This can be explained as follows. In most experiments, the initial seed pulse has a finite duration in the range of 0.1-1 ps. When its amplitude is too low, the first effect we observe is not seed pulse growth, but seed pulse stretching, see e.g. Section III of the main manuscript. From figure  3 in that section, we find that this process typically takes about 1000 laser cycles, or 1 mm for a pump laser with 1 micron wave length. The experiments carried out at Princeton and Strathclyde use plasma columns of about 1 mm in length, so nearly the entire plasma column is used up for laser pulse stretching rather than amplification. This goes some way towards explaining the poor efficiency of those experiments. A second cause of inefficiency in this experiment is the reduction of the seed pulse spot diameter, see Section I C below.

B. Brillouin amplification
The essential data for the Brillouin amplification experiments displayed in Figure 1(b) of the main manuscript are given in table II: Quantities have the same meaning as before; Γ 3 sc = ω 3 0 (Zm e /m i )(n 0 /n cr )(1−n 0 /n cr )/(2g) is the coupling coefficient for strongly-coupled Brillouin scattering, where Zm e /m i is the electron-ion charge-mass ratio. L1 and L2 denote the  Figure 1(b) of the main manuscript.
input pulses for the first two experiments at LULI (Palaiseau, France) by Lancia et al. [26,27], and G1 denotes the input pulse for the experiment at the Central Laser Facility (RAL, Didcot, United Kingdom) by Guillaume et al. [28]. Mi1-Mi4 denote input pulses from the recent experiment by Marqués et al. [29], having energies of 4, 40 and 200 mJ and 4 J respectively. We note that the best results were obtained for pulses Mi2-Mi3, which match our criteria best (an initial seed energy of 52 mJ would have been optimal for this experiment), while results were not as good for Mi1 and Mi4, which do not match our criteria. We also note that the points in Figure 3a of Ref. [29] form an S-curve, whose centre sits roughly at 40-50 mJ, just where our criteria predict that it should be.
Obtaining the duration of the output pulses Mo1 and Mo2 was more involved, since those durations are not stated explicitly in Ref. [29]. The procedure was as follows. We noted that the duration of the seed pulse in vacuum was given as 0.55 ps, while the FWHM of the autocorrelator trace of this same pulse (black curve in Fig. 5a) is found to be 0.95 ps. From this, a deconvolution factor of 1.73 is estimated. We then take the FWHM of the central peak of the purple curve in Fig. 5b, 0.5 ps, and apply the deconvolution factor to obtain an estimated pulse duration of 0.29 ps. Together with the pulse energy of ∼ 1 J and a spot diameter of 0.1 mm, this yields point Mo1. However, as explained below under "Transverse effects" and clearly shown in Fig. 8 of Ref. [29], the amplified pulse takes on a "horseshoe" shape, which can easily lead to a factor 2 overestimation of its duration in the autocorrelator trace. Applying this correction to τ 1 increases a 2 1 by a factor 2, while decreasing a 2 1 Γ 3 R τ 3 1 by a factor 4. This leads to point Mo2 in table II, which matches our criterion better than Mo1. Obviously, there are many uncertainties in our approach, caused by the paucity of the available data. In order to do a proper analysis of the output pulses, more accurate characterisation of these pulses is needed.
Regarding pulse energies: again, E in and E out denote the energies of the input pump and output (amplified) seed pulse as published. The highest efficiencies have been obtained by pulses Mi2 and Mi3, whose values of a 2 1 Γ 3 sc τ 3 1 are closest to the predictions by our model. Pulse Mo4 is so far above the predicted value initially that energy flows back from the seed pulse into the pump pulse; therefore this pulse has a "negative" efficiency. Experiment G1 has a seed pulse that matches our criteria, but the interaction length is too short for the given pump intensity, so this experiment did not reach its full potential. To compare: Guillaume et al. [28] use I pump = 3 · 10 14 W/cm 2 , τ pump = 15 ps, and 1.65 mm interaction length (which is too short for the given pump duration), while Marquès et al. [29] use I pump = 4 · 10 16 W/cm 2 , τ pump = 1.7 ps, and 1.2 mm interaction length. If the Guillaume experiment would be repeated with a longer interaction length (e.g. 3-4 mm) and a somewhat higher pump intensity (e.g. I pump ∼ 10 15 W/cm 2 ), significantly higher efficiencies could probably be obtained.

C. Output pulse analysis
In contrast to the input pulses, the output pulses of several experiments [3,4,6,17,18,26] show promise. In those experiments where a full characterisation of the output pulse is actually provided, this is usually given in terms of the pulse's energy, duration and spot diameter. The relationships between those parameters and the "triple products" a 1 Γ R τ 1 or a 2/3 1 Γ sc τ 1 given in Eqns. (7) and (13) below are as follows: Here, E 1 is the energy of the amplified seed pulse in Joule, and τ 1 its duration in seconds, while λ 1 and R 1 are the seed pulse's wave length and spot diameter. We apply these equations to the output pulses discussed in various papers. For Ren et al [6], we find a 1 Γ R τ 1 = 3.66 after the first pass, and a 1 Γ R τ 1 = 4.05 after the second pass, close to the theoretical prediction of Γ R τ 1 a 1 ∼ 3.4 above, as already discussed above. For experiment P4 by Turnbull et al. [4] we find an output pulse with 12 mJ energy and τ 1 ∼ 0.44 ps duration for a plasma with ω pe /ω 0 ∼ 0.1. Using the pump pulse diameter of 0.1 mm (the output pulse's spot diameter was not given), we find I 1 ∼ 2.7 × 10 14 W/cm 2 , a 1 ∼ 0.01 and Γ R τ 1 a 1 ∼ 2, (or Γ R τ 1 a 1 ∼ 4 if a 50 µm spot diameter is assumed). For the Brillouin amplification experiment L1 [26], we find E 1 ∼ 60 mJ, τ 1 ∼ 0.2 ps, R 1 ∼ 17 µm, which yields I 1 ∼ 2 × 10 17 W/cm 2 , a 1 = 0.28 and a 2 1 (Γ sc τ 1 ) 3 ∼ 9.2, not too far from the predicted value of 13.8. We find that in all those experiments, the amplified seed pulse is at or near the non-linear regime at least within its central spot, where its intensity is highest.
Furthermore, we note that experiment P3 [3] includes a graph (figure 5 of Ref. [3]) of the duration τ 1 versus the energy E 1 of the amplified seed pulse. The same is true for later papers on the Ren experiment [17,18]. We use equation (1) to analyze this data. For Ref. [3] we use λ 1 = 873 nm and R 1 ∼ 8 µm, while for Refs. [17,18] we use λ 1 = 873 nm and R 1 ∼ 15 µm. The main uncertainties in this approach are related to the measurements of the seed pulse energy and spot diameter. We note that curve fitting with τ 1 = αE −β 1 does not work here (see below). Instead, one should simply use the value of the triple product Γ R τ 1 a 1 to judge how well the output pulse resembles a perfect π-pulse.  in ΓRτ1a1 is about 20% in frame (a), 10% in frame (b). We note that the value of ΓRτ1a1 for experiment P3 is nearly constant, and close to the theoretical value 3.4, while the values of ΓRτ1a1 for the Ren experiment are approaching 3.4 asymptotically. This means that the seed pulses of experiment P3 are more non-linear inside their spot than those of the Ren experiment, in spite of the higher energies obtained in the Ren experiment.
We harvested the data from Refs. [3,17,18] and analyzed it; the results are displayed in Figure 1. While both experiments show clear signatures of proper non-linear Raman amplification, we find that the value of Γ R τ 1 a 1 for experiment P3 is nearly constant (see Figure 1(a)), and close to the theoretical value 3.4, while the values of Γ R τ 1 a 1 for the Ren experiment are approaching 3.4 asymptotically (see Figure 1(b)). This means that the seed pulses of experiment P3 are more non-linear inside their spot than those of the Ren experiment, in spite of the higher energies obtained by Ren et al. Furthermore, we note that all the output pulses for which we could do an energy analysis are mostly or fully non-linear. Thus, the success of a Raman or Brillouin amplification experiment is no longer determined by its degree of non-linearity. Other parameters are becoming more important, in particular the spot diameter of the amplified seed pulse. This is expressed in table III: In table III, R 0 , R 1i and R 1f denote the spot diameters of the pump pulse, initial seed pulse and final seed pulse respectively; τ 1f , a 1f and "tr. prod." denote the duration, amplitude and triple product (Γ R τ 1 a 1 or a 2 1 (Γ sc τ 1 ) 3 as appropriate) for the seed pulse after amplification.
Tr. prod. Eff. (%) P3 [3] 0. From the data in table III, we find that the experimets P3, Ren and L1 [3,6,26] suffer from a significant spot size reduction during amplification. This is the main reason why the output pulses in those experiments carry so little energy, even though they appear to have reached the non-linear regime within their spots. This reduction in spot diameter is likely caused by the initial seed pulse being so weak that it does not have enough intensity in its wings to kick-start non-linear amplification. Non-linear amplification then takes place only within a narrow spot near the axis of propagation, while there is no amplification at all away from the axis. Furthermore, we see that the Ren experiment [6] produced a more energetic output pulse and received wider publicity mainly because of its larger spot diameter, not because of its degree of non-linearity. Experiment P4 reports an output seed energy of 12 mJ, one of the highest ever reported in a plasma-based Raman amplification experiment. This result is mainly due to its fairly large spot diameter (50-100 µm); the long duration and low amplitude of the output seed pulse indicate that the amplification had not yet reached an advanced stage. The Brillouin amplification experiments L1 [26] and Marquès [29] have comparable values for a 1 , τ 1 and a 2 1 (Γ sc τ 1 ) 3 , but the output pulses by Marquès et al. contain much more energy because amplification could be maintained across a much wider spot diameter (100 µm vs. 17 µm). Each time, we see that amplification across a larger spot diameter lies at the base of increased efficiency of the process. To compare: in a recent experiment on crossed-beam energy transfer (CBET, [30]), a large amount of energy could be transferred between laser beams interacting in plasma because the transfer took place across a large spot (300-600 µm).
Further analysis is also possible using Eq. (6) below. We calculate a 2 0 Γ 2 R τ 0 τ 1 for both the most energetic pulse from experiment P3 [3] (a 0 = 0.01, Γ R = 3.313 × 10 14 s −1 , τ 0 = 10 ps, τ 1 = 150 fs, a 2 0 Γ 2 R τ 0 τ 1 ∼ 16) and for the pulse R2 after the first pass of the Ren experiment [6] (a 0 = 0.01, Γ R = 3.425 × 10 14 s −1 , τ 0 = 7 ps, τ 1 = 90 fs, a 2 0 Γ 2 R τ 0 τ 1 ∼ 7). Given that the initial seed pulses are well below the threshold for non-linear amplification, these values seem to be on the low side, indicating that the seed pulse is shorter (and thus more intense) after the given interaction length than could be expected from Raman amplification alone. This points to self-focusing as a contributing factor to the intensity increase, in line with the measured spot diameter decrease. Since self-focusing does not contribute to the total seed pulse power but does lead to a reduction in seed pulse duration, it follows that it has a negative effect on the total seed pulse energy and the efficiency of the amplification process.
Thus, in order to produce seed pulses with high power and energy using Raman or Brillouin amplification, it is vital to ensure that the amplification process takes place across a large spot diameter. To this end, it is vital to use seed pulses that satisfy Eq. (7) or Eq. (13) from the start, and do so across a large spot. Preferrably, the initial seed pulse spot diameter should be larger than the pump pulse diameter, not smaller as is sometimes advised [26,29].

D. Effect of the intensity of the input seed pulse on the amplifier output
In support of our findings, we have conducted 2-D particle-in-cell simulations using the code XOOPIC [55] where a 10 14 W/cm 2 pump pulse interacted with a seed pulse in a 4 mm long plasma column, with ω 0 /ω p = 20 or 10. We decreased the seed pulse intensity from 10 15 W/cm 2 to 10 12 W/cm 2 , leaving all other parameters constant, and also included "null" simulations with a pump beam but no seed pulse at all. Simulation results are displayed in Figure  2. We found the following. (i) For every factor 10 decrease in the input seed intensity, we found a factor ∼ √ 10 decrease in the output seed intensity. If we assume that the amplified seed pulse is mostly non-linear on-axis, so Γa 1 τ 1 does not vary much between simulations, the energy flux is proportional to a 2 1 τ 1 ∝ a 1 , and will thus be reduced by a factor 4 √ 10 ≈ 1.78. This confirms the need to keep the input seed intensity up to achieve a high output seed intensity and power over a fixed finite interaction length. (ii) The transverse pulse width decreases with decreasing 10 12 10 13 10 14 10 15 . Blue dots and line: initial seed pulse duration is 50 fs, pulse spot diameter is 600 micron, plasma electron density is ncr/400. Orange: initial seed pulse duration is 200 fs, pulse spot diameter is 60 micron, plasma electron density is ncr/400. Green: initial seed pulse duration is 50 fs, pulse spot diameter is 60 micron, plasma electron density is ncr/100. (a): Final seed intensity versus initial seed intensity. The dashed line represents the level of pump Raman backscatter induced by noise in a simulation with no seed pulse. Each decrease of the initial seed pulse intensity by a factor 10 leads to a decrease in output intensity by a factor ∼ √ 10. In the simulations with the lowest initial seed intensity, 10 12 W/cm 2 , the contribution of RBS from noise becomes comparable to the amplified seed. (b) Final seed spot FWHM versus initial seed intensity. The dashed line represents the FWHM of pump Raman backscatter induced by noise in a simulation with no seed pulse. For I10 ≥ 10 13 W/cm 2 , each decrease of the initial seed pulse intensity by a factor 10 leads to a decrease in final seed FWHM by 15-20%. In the simulations with the lowest initial seed intensity, 10 12 W/cm 2 , the contribution of RBS from noise causes the final spot FWHM to increase again. initial seed intensity, by 15-20% for a factor 10 decrease of the input seed intensity, reaching a minimum for 10 13 W/cm 2 initial seed intensity. (iii) Between the decrease in intensity and in spot diameter, we find that the efficiency of the process decreases by a factor ∼ 2.5 for a factor 10 decrease of the input seed intensity. (iv) Every time we decrease the input seed intensity, the relative importance of noise (and Raman-amplified noise) and instabilities like filamentation increases. For the lowest input seed intensity, 10 12 W/cm 2 , noise-driven RBS makes up about a third of the output signal. (v) Filamentation leads to the emergence of high intensity "speckles" on top of the laser envelope where the intensity gain is driven in part by the filamentation instability rather than Raman amplification (like in the experiment by Ren et al.). This does not lead to an overall increase amplified power though. (vi) For high initial seed pulse powers and intensities, the noise and noise-related instabilities are drowned out by the "regular" Raman amplification process.
We conclude that both the efficiency of the amplification and the quality of the pulse envelope decrease with decreasing pulse intensity, and speckles develop; while the speckles reach a relatively high intensity, the average output intensity and power decrease; efficient amplification to high powers across a wide spot with a good envelope is only seen for high input seed pulse intensities.
We repeated our simulations with a 10 15 W/cm 2 pump pulse, leaving other parameters the same, but this led to a significant increase in noise and noise-driven instabilities, so we did not pursue this further.
Since our simulations without any seed pulse still showed significant Raman scattering from noise, we have investigated the importance of noise-driven Raman backward scattering (RBS) in experiments, wherever information was published on a "pump only" shot with no seed pulse. We looked at the energy from "thermal RBS" as a percentage of the energy of the amplified seed. Results were as follows. Princeton [1][2][3][4][5][6] [29]: up to 25%, probably because of the large pump intensity (4 × 10 16 W/cm 2 ). Clearly, the energy of the output pulse in all relevant experiments contains a sizeable contribution of RBS from thermal noise, similar to what we see in our simulations. This cannot be fully eliminated, but its relative importance can be reduced by using a more powerful seed pulse and (for the LULI 2019 experiment) a reduction of the pump pulse intensity.

II. DERIVATION OF EQUATIONS (3), (4), (6) AND (7) IN THE MAIN MANUSCRIPT, AND THEIR PROPERTIES
We define a 0 and a 1 to be the scaled envelopes of pump and seed pulse respectively, a 0,1 ≡ 8.55 × 10 −10 g 1/2 (I 0,1 λ 2 0,1 [Wcm −2 µm 2 ]) 1/2 , where g = 1 (g = 1/2) denotes linear (circular) polarisation. Let ω 0 , k 0 and n cr denote the pump laser frequency, wave number and critical density, and n e and ω pe the background electron density and corresponding plasma frequency. The group velocity of the pump pulse is v g = c 2 k 0 /ω 0 = c(1 − n e /n cr ) 1/2 and the electron thermal velocity is v e = (k B T e /m e ) 1/2 . We note that we use a 0,1 = ||a 0,1 || cos(θ 0,1 ) for the laser pulses, and δn e = ||δn e || cos(θ L ) for the Langmuir wave (where θ denotes a fast phase), same as in Forslund et al. [31] (and contrary to Drake et al., who use E = 2E 0 cos(θ)), so the relationship between the amplitude of a real-valued physical quantity and the amplitude of its (sometimes complex) envelope function is obvious.
We will see below that the growth rates Γ R,B a 0 and Γ sc a 2/3 0 , the wave-breaking conditions a 0 /α R,B > 1/ √ 2 and the threshold for a 2 0 /g for strongly coupled Brillouin scattering all depend on a 0 and g only via the combination a 0 / √ g = 8.55 × 10 −10 (I 0 λ 2 0 [Wcm −2 µm 2 ]) 1/2 , which is independent of the laser polarisation g. The reason we include the polarisation explicitly is, that many theoretical works write these expressions in terms of the amplitude a 0 without being too clear about the effects of polarisation: Forslund et al. [31] and Malkin et al. [33] use circular polarisation, Andreev et al. [52] "probably" use circular polarisation, while Drake et al. [32], Kruer [34] and Lehmann and Spatschek [53] use linear polarisation. As a result, all works claim to provide "the" growth rate or "the" threshold, but they never seem to agree on the numerical coefficients in their expressions. Thus, we supply expressions with explicit polarization dependence to link theory papers with both linear and circular polarization to each other and to experiments. Our readers can then decide for themselves.

A. Raman amplification
For Raman amplification, the envelope equations for pump, seed and plasma wave can be written in the following generic form [31,33]: Here, a 0 and a 1 denote the scaled envelopes of pump and seed pulse respectively, a 0,1 ≡ 8.55 × 10 −10 × g 1/2 (I 0,1 λ 2 0,1 [Wcm −2 µm 2 ]) 1/2 , where g = 1 (g = 1/2) denotes linear (circular) polarisation. Let ω 0 , k 0 and n cr denote the pump laser frequency, wave number and critical density, v e = (k B T e /m e ) 1/2 the electron thermal velocity, and n e and ω pe the background electron density and corresponding plasma frequency. We define b = α R δn e /n e where δn e is the plasma wave density fluctuation and α R is to be determined, and Γ R is the Raman coupling constant, also to be determined (so the Raman backscattering growth rate in homogeneous plasma is given by Γ R a 0 ). The group velocity of the pump pulse is then v g = c 2 k 0 /ω 0 = c(1 − n e /n cr ) 1/2 .
Curiously, if the same reasoning is used to match the equations by Malkin et al. [33] to those by Forslund et al. [31], one finds f = (ω pe /4ω 0 )(δn e /n e ) =Ē L /2 for the Langmuir wave (Ref. [33] uses f in lieu of our b), wherē E L = eE L /(m e ω pe c) is that wave's scaled electrostatic field. This in spite of the fact that Malkin et al. state that "f is the electrostatic field of the Langmuir wave, normalized to m e ω pe c/e", i.e. f =Ē L . Similarly, in papers describing the MBRS simulation code to solve the equations (3)-(5) [36,37], it is claimed that "f is the complex amplitude of the Langmuir wave scalar potential in units (mc 2 /2e) ω p /2ω 0 ", which would correspond to f = 2α R δn e /n e , where f =Ē L = α R δn e /n e is needed for equations (3)- (5). The solution to this apparent paradox is provided in Refs. [38,39]: here, the electrostatic field is explicitly given in the formĒ L = f exp(iθ) + c.c. = 2f cos(θ), and similarly δn e /n e = 2f cos(θ), orf = α R δn e /n e after scaling. So when it is stated that "f denotes the scaled electrostatic potential/field/density fluctuation", it may well mean that f denotes half that quantity. Further support for this is found in Refs. [40,41], where the Manley-Rowe relation ||a 0 || 2 ω 0 = 2g(||Ē L ||/2) 2 ω pe has been derived using an energy-conservation argument rather than a set of three-wave equations. Comparing this to the Manley-Rowe relation ||a 0 || 2 ω 0 = 2g||f || 2 ω pe by Malkin et al. [33], we find that this model must satisfy f =Ē L /2 in order to conserve energy. Compatibility with the model by Forslund et al. then follows by default, and the posted wave breaking limit for circular polarisation, a wb = (ω pe /ω 0 ) 3/2 /4, is obtained for δn e /n e = 1, as it should. While this does not affect the current discussion, it will become important when systems of three-wave equations are used that contain higher-order terms in f . Following Malkin, Shvets and Fisch [33], or Menyuk, Levi and Winternitz [42], we define ζ = Γ R a 00 (x/c + t), t = Γ R a 00 t and ξ = 2 √ ζt where a 00 denotes the pump pulse amplitude. Attractor solutions to the system (3)-(5) can then be obtained in terms of ξ alone. In particular, the first peak of the growing seed pulse can be approximated by a 1 (ζ, t ) ≈ (2a 00 t /ξ)∂u(ξ)/∂ξ where u(ξ) = 2 √ 2 arctan[ exp(ξ)/(4 √ 2πξ)], with < 0.1 depending on the initial seed pulse B-integral a 1 (ζ, 0)dζ. The function ∂u(ξ)/∂ξ has an amplitude A ≈ 1.29 and a width ∆ξ ≈ 2.65, mostly independent of , while the position of its maximum, ξ M , obeys 5 < ξ M < 7 for practical values of [33,43]. Let ∆ζ denote the width of the first peak of a 1 (ζ, t ) for fixed t and let ξ 1,2 = ξ M ± ∆ξ/2. Then ξ M ∆ξ = (ξ 2 2 − ξ 2 1 )/2 = 2t ∆ζ. For ξ = ξ M , we find that ||a 1 || = 2Aa 00 t /ξ M , ∆ζ = ξ M ∆ξ/(2t ). We consider pump and seed pulses with durations τ 0 and τ 1 (after amplification), and setting ∆ζ = Γ R a 00 τ 1 and t = τ 0 /2 (for an interaction time t, the seed pulse sees a length ∼ ct of plasma column, but since the pump pulse is advancing at speed −c, the seed effectively interacts with 2t of pump pulse), we find: The scalings for the seed pulse amplitude and duration are ||a . The asymptotic energy transfer efficiency for the first peak is given by η = ||a 1 || 2 τ 1 (t)/(2a 2 00 t) = A 2 ∆ξ/ξ M ≈ 4.4/ξ M . Thus, η is constant for a given configuration, and decreases with increasing ξ M .
The strength of these new relations is also shown in their potential to obtain a deeper understanding of earlier work. For example, in a paper by Malkin and Fisch on Raman amplification in the presence of plasma wave damping [44], it is suggested that the damping can be overcome by increasing the initial seed pulse amplitude. However, no explicit expression for the minimum seed pulse amplitude necessary is given in Ref. [44]. With the aid of Eqns. (6) and (7), we can now derive that threshold. From Eqns. (18)-(20) of Ref. [44], we obtain (using our own definitions of ζ and Γ R ): a 00 Γ R ∂ ζ b + νb = Γ R a 0 a 1 . Clearly, for nonlinear amplification to dominate over damping, we must have |a 00 Γ R ∂ ζ b| > |νb|. From Eq. (7) above, we obtain ||a 00 Γ R ∂ ζ || ∝ 1/τ 1 ∝ Γ R ||a 1 ||; we take |a 00 Γ R ∂ ζ b| ∼ Γ R ||a 1 ||b as a first approximation. Immediately we find Γ R ||a 1 || > ν as the criterion for non-linear Raman amplification to dominate over Langmuir wave damping. For a constant damping coefficient, e.g. ν = qΓ R a 00 , amplification will overcome damping if ||a 1 ||/a 00 > q. This is directly borne out by figures 2-4 of Ref. [44], where non-linear amplification truly takes off as soon as a 2 1 /a 2 00 exceeds q 2 (25, 100 and 400 respectively). For a linearly growing damping coefficient, e.g. ν = q ν Γ 2 R a 2 00 t, we apply the seed pulse scaling ||a 1 ||(t) = (2A/ξ M )a 2 00 Γ R t. We find that damping dominates over amplification if Γ R ||a 1 ||(t) = (2A/ξ M )a 2 00 Γ 2 R t < q ν Γ 2 R a 2 00 t, or q ν > 0.45. This is directly borne out by Figure 1 of Ref. [44], where true nonlinear amplification (including secondary peaks in the seed) can only be seen for those cases where q ν < 0.5. The ease with which we can derive these thresholds and extend the theory of Ref. [44] is a clear demonstration of the power of our equations (6) and (7).

B. Brillouin amplification
Brillouin amplification in plasma is based on principles similar to Raman scattering, only the Langmuir plasma wave is replaced by an ion-acoustic wave. Because the frequency of the ion-acoustic wave will be much smaller than the frequency of the pump laser, pump and seed laser beams can have (nearly) the same frequency. Brillouin scattering in the so-called weak-coupling regime [31,32,34,[45][46][47][48][49] is very similar to Raman scattering and can be treated in the same way. We introduce ω pi = ω pe Zm e /m i and c s = v e Zm e /m i . For a 2 00 < 8g(ω 0 /ω pe ) 2 Zm e /m i v 3 e /c 3 , the electron pressure is the dominant restoring force and the plasma wave dispersion is not significantly affected by the beating between pump and seed pulses. In that case one can reuse equations (3)-(5) and only needs to replace 3v 2 e (k/ω pe ) by c s in Eq. (5). For backward Brillouin scattering, the ion-acoustic wave has wave number k s ≈ 2k 0 and frequency ω s = c s k s ≈ 2c s k 0 . Then we find Γ B α B = ω 2 pe /(4ω 0 ) and Γ B /α B = c 2 c 2 s k 2 s /(4gω s v 2 e ), leading to Γ B = cω pe ω s /(4v e √ gω 0 ω s ) and α B = √ gv e ω pe /(c √ ω 0 ω s ). After substituting Γ B , α B for Γ R , α R , all the above results for Raman amplification also apply to the weak-coupling Brillouin case, including Eqns. (6) and (7), the wave breaking threshold a wb = α B / √ 2, the numerical constants 5 < ξ M < 7, ∆ξ ≈ 2.65 and A ≈ 1.29, and the seed pulse scalings.
For a 2 00 > 8g(ω 0 /ω pe ) 2 Zm e /m i v 3 e /c 3 , or Γ B a 00 /ω 0 > c s /c, the ponderomotive pressure from the beating between pump and seed pulses will take over from the thermal pressure as the primary restoring force for the ion-acoustic wave. In this regime, called strong-coupling (sc) Brillouin scattering, the equation for the plasma wave becomes [31,50,51]: Here, Γ sc denotes the coupling constant for sc-Brillouin backward scattering. From (3) and (8) and using k = 2k 0 ≈ 2ω 0 v g /c 2 as before, we find: Γ sc α sc = ω 2 pe /(4ω 0 ) and Γ 2 sc /α sc = c 2 k 2 ω 2 pi /(2gω 2 pe ). This yields [31,50,51]: where ω sc and γ sc denote the frequency and growth rate of the driven ion wave. The numerical factor for Ω sc must be a (complex) cube root of −1, and must have a positive imaginary part for the plasma wave to grow rather than damp, and the only suitable value is then (1 + i √ 3)/2. To make the connection with earlier work: if one uses a 0,1 · δn e = a 0,1 δn e /2 and ∇ 2 A 2 = −(2k 0 ) 2 a 0 a 1 /g in Hüller et al. [50], then this work agrees with Forslund et al. [31]. In Refs. [52,53], all quantities are ∝ exp(iθ) rather than ∝ cos(θ), so both a 0,1 · δn e and a 0 · a 1 are twice what they should have been. Furthermore, Refs. [52,53] use ∇ 2 a 0 · a 1 = −k 2 0 a 0 · a 1 rather than ∇ 2 a 0 · a 1 = −(2k 0 ) 2 a 0 · a 1 for unknown reasons. As a result, Refs. [52,53] cite the correct value for Γ sc (for linear polarisation), but their value for α sc is twice what it should have been. This appears to have been corrected by Schluck et al. [51] though, whose equations agree with Forslund et al. [31].
C. Properties of the equations equations (7) and (13) As discussed in the main manuscript, the equations (7) and (13) are fully valid only when the seed pulse amplitude exceeds certain thresholds, ruling out the case of a very long initial seed pulse with a very low amplitude. The first threshold follows from the fact that the various models for Raman and Brillouin amplification require fully nonlinear interaction and full pump depletion. The "self-similar" model for Raman amplification by Malkin, Shvets and Fisch [33] has been derived under the condition that ∂a 1 /∂ζ > ∂a 1 /∂t. We use the scalings ∂a 1 /∂t ∝ Γa 2 0 and ∂a 1 /∂ζ ∝ Γa 2 1 to arrive at the threshold condition for the seed amplitude a 10 > a 00 . For sc-Brillouin amplification [52], we use the scalings ∂a 1 /∂t ∝ a 2 00 Γ/(a 2 00 Γt) 1/4 ∝ a 2 00 Γ/a to again arrive at a 10 > a 00 . All this means that our criterion works best if the initial seed pulse intensity is at least that of the pump beam.
Phenomena like plasma wave damping or chirping of the frequency of either the pump beam or the plasma wave can impose further thresholds. To derive the seed amplitude threshold in case of damping, we start from an initial seed pulse with amplitude a 1 and duration τ 1 satisfying Eq. (7). From Eq. (18) and (20) of Ref. [44], we find the equation for the damped plasma wave ∂ ζ b − νb. = Γ R a 0 a 1 . We insert the initial seed pulse into this equation and require that growth dominates over damping, i.e. |Γ R a 0 a 1 | > |νb|. Requiring full pump depletion, i.e. ||a 0 || ∼ ||b||, we then obtain the condition Γ R |a 1 | > ν for a viable seed pulse. This means that the "damping distance" 1/ν is larger than the "pump depletion distance" 1/(Γ R a 1 ). For the case of chirp, we use Eq. (1) of Ref. [54] to obtain the equation for the plasma wave ∂ ζ b − i(δω)b = Γ R a 0 a 1 . At the start of the interaction, we have δω = qτ 1 ∼ q/(Γ R a 1 ).
We require that growth dominates over detuning, i.e. |Γ R a 0 a 1 | > q|b|/(Γ R a 1 ). Again requiring ||a 0 || ∼ ||b||, we obtain the condition Γ 2 R a 2 1 > q for a viable seed pulse. This means that the detuning δω = qτ 1 ∼ q/(Γ R a 1 ) due to chirp over the duration of the seed pulse is smaller than the bandwidth 1/τ 1 ∼ Γ R a 1 of the seed pulse. The seed amplitude thresholds for sc-Brillouin amplification are obtained using similar arguments. We note that we could not have derived these thresholds without using Eq. (7), which underlines its importance.
The salient properties of equations (7) and (13), and the equivalent relations (1) and (2) for Γ 2 R E 1 τ 1 and Γ 3 sc E 1 τ 2 1 , are as follows. (i) These relations have been derived for asymptotic attractor solutions for the seed pulse, and Γ R τ 1 a 1 or Γ sc τ 1 a 2/3 1 will therefore only settle on their asymptotic values after a significant interaction distance. The specific distance needed varies significantly with the initial seed pulse size and shape, and can be an object of further study. (ii) The triple products Γ R τ 1 a 1 and Γ sc τ 1 a 2/3 1 are not constants of motion and will vary in particular during the early stages of the interaction. Their values can be used to assess how close the seed pulse is to the asymptotic π-pulse solution. For this reason, curve fitting experimental data pairs (E 1 , τ 1 ) with a curve of the form τ 1 = αE −β 1 does not work, since it assumes that all output seed pulses are perfect π-pulses already, while the seed pulses usually do not fully reach that stage within the limited interaction length available. Instead, one should simply use the value of the triple product Γ R τ 1 a 1 to judge how well a given output pulse resembles a perfect π-pulse. (iii) It is often stated that the π-pulse solution is an attractor [33], even if no complete mathematical proof is ever given. However, the asymptotic values for Γ R τ 1 a 1 and Γ sc τ 1 a 2/3 1 definitely have attractor properties, as borne out by the simulation results in the main manuscript. (iv) Since the equations (7) and (13) are completely independent of the pump pulse, they will continue to hold under circumstances not covered by the original "self-similar" models, e.g. slowly varying pump amplitude or plasma density, two or three dimensions (see below), higher order laser modes such as Laguerre-Gaussian (see below), full numerical simulations or experiments that know nothing about the approximations needed to arrive at the π-pulse solutions of Refs. [33,52]. (v) Following Ref. [33], we define = Γ R dζg(t = 0, ζ), the integrated amplitude of the initial seed pulse. For weak seed pulses, 1. We then find that the position of the first seed pulse maximum is given by ξ M ≈ 2.31 + ln(1/ ). Even though the dependence of ξ M on is "only logarithmic", we find that the intensity of seed pulses used in experiments, and thus , can vary by orders of magnitude, leading to significant shifts in ξ M , as well as in the first-peak efficiency 4.4/ξ M . Since for a finite interaction length the first peak may well be the only one that develops fully (the long tail of trailing peaks needs "infinite" time to develop, see below), this explains why a poor choice for the initial seed pulse can have a lasting negative impact on the efficiency, as shown in Figure 3 of the main manuscript.
It is important to remember that Malkin et al. and Andreev et al. provide asymptotic solutions for the growing seed pulse that are valid in the limit t → ∞, while experiments and most numerical simulations are carried out during a limited, finite interaction time. When a realistic seed pulse evolves towards the asymptotic solution, the head of the pulse will form first, and the tail of secondary pulses only after that. This can be quantified as follows. From the definitions of the "self-similar" coordinates ξ above, we find for Raman amplification: ∂ ζ = t /ζ∂ ξ , ∂ t = ζ/t ∂ ξ = (ζ/t )∂ ζ . Since the π-pulse solution has been derived under the assumption ∂ t ∂ ζ , e.g. ∂ t 0.1∂ ζ this means that for a given distance ζ behind the head, the pulse will only begin to resemble the π-pulse for e.g. t 10ζ. Thus, in realistic situations the leading spike of the growing seed pulse will form first, and the "tail" of subsequent spikes will form much later, if at all. (See e.g. Figure 3 of Ref. [6], where the head of the seed pulse is fully formed but the tail much less so.) This means that when analysing seed pulses from experiments, one cannot offhand assume that a tail of trailing spikes exists and contains a fixed fraction of the total pulse energy. After the limited interaction distance available to many experiments, the amplified seed pulse may consist of the leading spike only. Similarly, we find for strongly coupled Brillouin amplification that ∂ ζ = √ t ∂ ξ , ∂ t = (ζ/2 √ t )∂ ξ = (ζ/2t )∂ ζ , so ∂ t 0.1∂ ζ implies t 5ζ for a given distance ζ behind the head of the seed pulse. In general, if one requires |∂ t | < ε|∂ ζ | for some ε 1, then this only holds for 0 < ζ < εt or 0 < ξ < 2t √ ε for Raman amplification, and for 0 < ζ < 2εt or 0 < ξ < 2ε(t ) 3/2 for sc-Brillouin amplification. The difference between "transient" and "asymptotic" solutions for the seed pulse is also important in the discussion surrounding the duration, amplitude and shape of the "optimal" seed pulse, especially when a seed pulse with a duration shorter than "optimal" still leads to the highest amplification efficiency (see figure 3 in the main manuscript). We note that a truly "optimal" seed pulse does not only have the optimal relationship between amplitude and duration, but also the ideal π-pulse shape with a tail of secondary peaks. This is impossible to generate in an experiment, so we did not use the ideal π-pulse shape in our simulations either; we used simple Gaussian pulse envelopes instead. When using Gaussian pulse shapes with varying values for Γ R τ 1 a 1 , it may be that the pulse that reshapes to the optimal π-pulse fastest has an initial value for Γ R τ 1 a 1 that differs somewhat from 3.4, even if the long-term value for Γ R τ 1 a 1 will always settle on 3.4. Even so, the "optimal" value for Γ R τ 1 a 1 in our simulations is still close enough to 3.4 to make our criteria very valuable for designing experiments, especially since the values for Γ R τ 1 a 1 of previous experiments are always so far below 3.4 (see discussion in the Supplemental Material).
In summary, the main gap between the models by Malkin et al. [33] and Andreev et al. [52] on one hand, and real-life experimental situations on the other, is that the experimental results are mainly determined by "transient" solutions to the three-wave equations while such "transient" solutions are not studied in the above two models at all. Our work aims to (i) bridge this gap by looking at transient behaviour of the seed pulse, and (ii) determine the necessary conditions to get the initial seed pulse as close to the asymptotic solution as possible, in order to make the most of the limited interaction distance and boost the overal efficiency of the amplification. Furthermore, the above-mentioned models are all one-dimensional. While direct Raman or Brillouin backscattering are technically 1-D phenomena, an 1-D model is not enough to capture and explain all the dynamics of Raman/Brillouin amplification using finite-width pulses. As argued above, filamentation and self-focusing play a vital role in the success or failure of any Raman/Brillouin amplification experiment. As shown above, configurations with weak seed pulses suffer more from filamentation and self-focusing than configurations with strong seed pulses, underlining how important the power of the initial seed pulse is to the success of an experiment. None of this is covered by existing 1-D models.

III. THERMAL EFFECTS
Plasma distinguishes itself from other non-linear media by the importance of thermal effects. While thermal effects can often be neglected in Raman amplification in e.g. non-linear fibers, thermal effects must be considered in plasma because all plasma is necessarily thermal, or it would recombine to neutral gas.
Thermal effects are automatically included in the study of Brillouin amplification, since the ion-acoustic wave involved requires thermal pressure to even exist. The plasma temperature controls both the frequency of the ionacoustic wave and the growth rate of the process in weakly coupled Brillouin amplification. This is already discussed in the main manuscript. In strongly coupled Brillouin amplification, the beating between the EM waves dominates over the thermal plasma pressure, so thermal effects only play a minor role in this process. Thermal effects need to be studied in Raman amplification in particular, since Raman amplification can happen in both cold and warm plasma, but it is more complicated in warm plasma and the behaviour of Raman scattering in cold and warm plasma is not completely the same.
Thermal effects can influence Raman amplification in plasma in three main ways: (i) reduction of the wave breaking limit of the Langmuir wave, (ii) increase of the Langmuir wave frequency via the Bohm-Gross dispersion relation, and (iii) Landau damping of the Langmuir wave.
Wave-breaking limit For a non-relativistic Langmuir wave in cold plasma, the wave breaking limit is given by eE L /(m e ω pe v ϕ ) = 1, where E L is the electric field amplitude of the Langmuir wave, v ϕ its phase speed, and other symbols have their usual meaning [35]. For a non-relativistic Langmuir wave in warm plasma, the wave breaking limit is given by eE L /(m e ω pe v ϕ ) = (1 − 8 3 β 1/4 + 2β 1/2 − 1 3 β) 1/2 , where β = 3k B T e /(m e v 2 ϕ ) and T e is the plasma electron temperature [56]. In Raman backscattering, the Langmuir wave has a phase speed v ϕ /c = ω pe /(2ω 0 ), so β = 12(k B T e /m e c 2 )(ω 2 0 /ω 2 pe ). This likewise reduces the pump pulse amplitude limit for wave breaking: a wb = (α R / √ 2)(1 − 8 3 β 1/4 + 2β 1/2 − 1 3 β) 1/2 . For a pump amplitude above this limit, the Langmuir wave induced by Raman scattering will break and the scattering process will evolve into Stimulated Compton Scattering [57], which obeys entirely different scaling laws. Thus, one has to take care that one's pump amplitude is below the warm-plasma limit when considering Raman amplification in warm plasma, not just below the cold-plasma limit.
Landau damping Landau damping [59,60] is the collisionless damping of longitudinal plasma waves, involving the conversion of plasma wave energy into plasma thermal energy. Raman amplification will be affected by Landau damping via the damping of the Langmuir wave, although this damping will saturate if the Raman process is sufficiently intense [61][62][63]. For Raman amplification, Landau damping of the electron plasma wave becomes significant when its phase speed becomes comparable to the electron thermal speed, i.e. when ω pe /(2ω 0 ) 3v T e /c, or T e 0.025(n 0 /n cr )[511keV]. For (n 0 /n cr ) = 1/400, we find significant Landau damping for T e 100 eV. Such temperatures have been reported in past experiments with long pump beams [7,8]. For sc-Brillouin amplification, assuming that T i ∼ T e , we find that the phase speed of the driven ion mode is larger than the ion thermal speed but smaller than the electron thermal speed, so neither ions nor electrons can cause significant Landau damping of the plasma wave.
Third, we observe that the model used in Eqns. (30)-(34) of Ref. [64] is very similar to the cold-plasma model used by Malkin, Shvets and Fisch [33], except that t = Γ Rã 2 00 t is used instead of t = Γ R a 2 00 t. Eq. (4) of the main manuscript then becomes: Note that the influence of Landau damping on Eq. (3) of the main manuscript is harder to determine, since Landau damping increases the value of the constant ξ M too, and no analytical expression for this increase is supplied. . Frame (a) shows the evolution of the seed pulse in Raman amplification with fixed initial pulse duration and varying pulse intensities, while frame (b) shows the evolution of the seed pulse for fixed intensity but varying seed pulse duration. Simulation parameters are as given in the text. The dashed lines correspond to Eqns. (14). The evolution of each curve starts at the circle and ends at the star. These results show clearly that for a warm plasma, just like for a cold plasma, the triple product Γr||a1||τ1 tends toward a constant for the growing signal pulse, irrespective of pulse intensity or initial pulse duration. While thermal corrections provide a minor correction (up to 10%) to the numerical value of this constant, there is no qualitative change to the concept at all.
We calculate the factor a 00 /ã 00 for various combinations of a 00 , ω 0 /ω pe and k B T e : a 00 ω 0 /ω pe k B T e (eV) a 00 /ã 00 0. We find that the effect of Landau damping decreases with a 00 , as long as there is no Langmuir wave breaking, since Landau damping saturates sooner when the Raman backscattering process is driven harder. The effect increases with increasing ω 0 /ω pe , because this lowers the phase speed of the Langmuir wave and makes it more susceptible to damping.
To verify the above model, we have conducted a number of particle-in-cell simulations of Raman amplification in warm plasma. The results are displayed in Figure 3. Parameters are as follows: Figure 3(a): k B T e = 100 eV, n 0 /n cr = 0.01 and a 0 = 0.01 (black) or 0.015 (blue); a 0 = 0.032, n 0 /n cr = 0.0044 and k B T e = 12 eV (green) or 50 eV (red). Figure 3(b): k B T e = 100 eV, n 0 /n cr = 0.01 and a 0 = 0.015, with τ 1 /τ R the same as in Figure 2(b) in the main manuscript. Parameters not listed are as given in the main manuscript. The dashed lines correspond to Eqns. (14) for Raman, (which is like Eq. (4) in the main manuscript with a 1.1 thermal correction factor).
Once again, we find that the evolving seed pulses closely follow the analytical predictions, irrespective of the pump intensity chosen in the simulations, proving that the non-linear matching conditions for Raman amplification with thermal correction, given by (14), still represents an "attractor" solution and remains valid over a wide range of pulse intensities, and that thermal effects do not make any qualitative change to our findings.

IV. TRANSVERSE EFFECTS
In a multi-dimensional setting, the amplitude a 1 and duration τ 1 of the growing seed laser pulse will of course depend on the transverse coordinate x 2 . The same holds true for the location of the seed pulse maximum ζ M in a frame comoving with the pulse. This leaves an imprint on the full shape of the envelope when parametric amplification is studied in more than one dimension.