X-Ray Amplification by Stimulated Brillouin Scattering

TL;DR

This paper explores plasma-based x-ray amplification via stimulated Brillouin scattering, predicting it can produce ultra-intense, ultra-short x-ray pulses surpassing current sources, with potential for significant advancements in laser technology.

Contribution

It introduces a novel approach using stimulated Brillouin scattering for x-ray amplification, demonstrating its advantages over Raman amplification in the high-energy regime.

Findings

Analytic predictions support feasibility of Brillouin x-ray amplification.

Numerical and particle-in-cell simulations show potential for pulse compression.

Predicted intensities exceed current x-ray sources by orders of magnitude.

Abstract

Plasma-based parametric amplification using stimulated Brillouin scattering offers a route to coherent x-ray pulses orders-of-magnitude more intense than those of the brightest available sources. Brillouin amplification permits amplification of shorter wavelengths with lower pump intensities than Raman amplification, which Landau and collisional damping limit in the x-ray regime. Analytic predictions, numerical solutions of the three-wave coupling equations, and particle-in-cell simulations suggest that Brillouin amplification in solid-density plasmas will allow compression of current x-ray free electron laser pulses to sub-femtosecond durations and unprecedented intensities.

Equations

$$\left[{\partial }_{\tilde{t}}+{\tilde{v}}_1{\partial }_{\tilde{x}}+{\tilde{\nu }}_1\right]a_1$$

$$\left[{\partial }_{\tilde{t}}+{\tilde{v}}_2{\partial }_{\tilde{x}}+{\tilde{\nu }}_2\right]a_2$$

$$\left[{\partial }_{\tilde{t}}+{\tilde{v}}_3{\partial }_{\tilde{x}}+{\tilde{\nu }}_3^R\right]n_3=-\frac{1}{4}\frac{1}{\sqrt{N}}\frac{c^2{k}_3^2}{{\omega }_2^2}\left[a_1^{\ast }{a}_2\right]$$

$$\left[\frac{i}{2}\frac{{\omega }_2}{{\omega }_3}{\partial }_{\tilde{t}}^2+{\partial }_{\tilde{t}}+{\tilde{\nu }}_3^B\right]n_3=-\frac{1}{4}\frac{Z{m}_e}{m_i}\frac{{\omega }_2}{{\omega }_3}\frac{c^2{k}_3^2}{{\omega }_2^2}\left[a_1^{\ast }{a}_2\right]$$

$$\left[{\omega }_3^2-c_s^2{k}_3^2\right]\left[{\omega }_3^2-2{\omega }_2{\omega }_3-c^2{k}_3^2+2c^2{k}_2{k}_3\right]=\frac{k_3^2{c}^2{a}_0^2{\omega }_{pe}^2}{4m_i/Z{m}_e}$$

$${\hat{\Gamma }}_{R,B_{wc}}=\frac{1}{2}{\left[4{\tilde{\Gamma }}_{R,B_{wc}}^2+{\left({\tilde{\nu }}_3^{R,B}-{\tilde{\nu }}_1\right)}^2\right]}^{\frac{1}{2}}-\frac{{\tilde{\nu }}_3^{R,B}+{\tilde{\nu }}_1}{2}$$

$$C_{sc}{\kappa }^3+\left(C_{sc}{\tilde{\nu }}_1+C_{wc}\right){\kappa }^2+\left(C_{wc}{\tilde{\nu }}_1+{\tilde{\nu }}_3^B\right)\kappa +K=0$$

$${\tilde{\nu }}_{\text{ei}}=\frac{2\sqrt{2}}{3\sqrt{N\pi }}\frac{r_e{\Lambda }_C{\omega }_2}{c{q}_T^{3/2}}{\tilde{\nu }}_{\text{Lnd}}=\frac{\sqrt{N\pi }}{(2q_T{)}^{3/2}}{e}^{-\frac{1}{q_T}}$$

$${\tilde{\Gamma }}_R=\frac{a_0{k}_{3}c{N}^{\frac{1}{4}}}{4\sqrt{{\omega }_1{\omega }_2}}$$

$${\tilde{\Gamma }}_{B_{wc}}=\frac{a_{0}c}{2\sqrt{2}}{\left[\frac{Z{m}_{e}N{k}_2}{m_i{\omega }_2{c}_s}\right]}^{\frac{1}{2}}{\tilde{\Gamma }}_{B_{sc}}=\frac{\sqrt{3}}{2}{\left[\frac{a_0^2{c}^2{k}_2^{2}N}{2{\omega }_2^2{m}_i/Z{m}_e}\right]}^{\frac{1}{3}}$$

$${\tilde{\nu }}_1{a}_1$$

$$({\tilde{v}}_2-{\tilde{v}}_1){\partial }_{\tilde{x}}{a}_2$$

$$-{\tilde{v}}_1{\partial }_{\tilde{x}}{n}_3+{\tilde{\nu }}_3{n}_3$$

$$a_1(\tilde{x},\tilde{t}\to \infty )=\frac{K_1{a}_0^2}{{\tilde{\nu }}_1\sqrt{2D_1}}\frac{e^{D_2{a}_0^2\tilde{x}}}{1+(1/2D_2)e^{2D_2{a}_0^2\tilde{x}}}$$

$$a_{1,\text{max}}=\frac{K_1{a}_0^2}{2{\tilde{\nu }}_1}{\left[\frac{K_3}{K_2}\left(1+\frac{{\tilde{v}}_2}{{\tilde{v}}_1}\right)\right]}^{\frac{1}{2}}$$

$$\left[{\partial }_t^2-c^2{\partial }_x^2+{\omega }_{pe}^2+{\nu }_1{\partial }_t\right]{\mathbf{A}}_1=-{\omega }_{pe}^2\frac{n_{e,1}}{n_{e,0}}{\mathbf{A}}_2$$

$$\left[{\partial }_t^2-c^2{\partial }_x^2+{\omega }_{pe}^2+{\nu }_2{\partial }_t\right]{\mathbf{A}}_2=-{\omega }_{pe}^2\frac{n_{e,1}}{n_{e,0}}{\mathbf{A}}_1$$

$$\left[{\partial }_t^2-3v_e^2{\partial }_x^2+{\omega }_{pe}^2+{\nu }_3^R{\partial }_t\right]\frac{n_{e,1}}{n_{e,0}}=\frac{e^2}{m_e^2{c}^2}{\partial }_x^2\left({\mathbf{A}}_1\cdot {\mathbf{A}}_2\right)$$

$$\left[{\partial }_t^2-c_s^2{\partial }_x^2+{\nu }_3^B{\partial }_t\right]\frac{n_{e,1}}{n_{e,0}}=\frac{Z{e}^2}{m_e{m}_i{c}^2}{\partial }_x^2\left({\mathbf{A}}_1\cdot {\mathbf{A}}_2\right)$$

$${\mathbf{A}}_{1,2}\left(x,t\right)=$$

$$+\frac{m_e{c}^2}{2e}{a}_{1,2}^{\ast }\left(x,t\right)e^{-i\left(k_{1,2}x-{\omega }_{1,2}t\right)}\hat{\mathbf{y}}$$

$$\frac{n_{e,1}(x,t)}{n_{e,0}}=\frac{i}{2}{n}_3(x,t)e^{i\left(k_{3}x-{\omega }_{3}t\right)}+\text{c.c.}$$

$$\left[{\partial }_t^2-2i{\omega }_1{\partial }_t-{\omega }_1^2-c^2{\partial }_x^2+2i{c}^2{k}_1{\partial }_x+c^2{k}_1^2+{\omega }_{pe}^2+{\nu }_1{\partial }_t-i{\nu }_1{\omega }_1\right]a_1=\frac{-i{\omega }_{pe}^2}{2}{n}_3{a}_2$$

$$c^2{k}_1^2={\omega }_1^2\left[1-\frac{{\omega }_{pe}^2}{\omega \left(\omega -i{\nu }_{ei}\right)}\right]\approx {\omega }_1^2\left[1-\frac{{\omega }_{pe}^2}{{\omega }_1^2}\left(1+\frac{i{\nu }_{ei}}{{\omega }_1}\right)\right]={\omega }_1^2-{\omega }_{pe}^2-i{\nu }_{ei}{\omega }_1\frac{{\omega }_{pe}^2}{{\omega }_1^2}$$

$$\left[{\omega }_1{\partial }_t+c^2{k}_1{\partial }_x+{\omega }_1{\nu }_1\right]=\frac{1}{4}\frac{{\omega }_{pe}^2}{2}{n}_3^{\ast }{a}_2$$

$$\left[{\partial }_{\tilde{t}}+{\tilde{v}}_1{\partial }_{\tilde{x}}+{\tilde{\nu }}_1\right]a_1=-\frac{1}{4}\frac{{\omega }_2}{{\omega }_1}N\left(n_3^{\ast }{a}_2\right)$$

$$\left[{\partial }_{\tilde{t}}+{\tilde{v}}_2{\partial }_{\tilde{x}}+{\tilde{\nu }}_2\right]a_2=\frac{1}{4}N\left(n_3{a}_1\right)$$

$$\left[{\partial }_{\tilde{t}}+v_3{\partial }_{\tilde{x}}+{\tilde{\nu }}_3^R\right]n_3=-\frac{1}{4}\frac{1}{\sqrt{N}}\frac{c^2{k}_3^2}{{\omega }_2^2}\left(a_1^{\ast }{a}_2\right)$$

$$\left[\frac{i}{2}\frac{{\omega }_2}{{\omega }_3}{\partial }_{\tilde{t}}^2+{\partial }_{\tilde{t}}+{\tilde{\nu }}_3^B\right]n_3=-\frac{1}{4}\frac{Z{m}_e}{m_i}\frac{{\omega }_2}{{\omega }_3}\frac{c^2{k}_3^2}{{\omega }_2^2}\left(a_1^{\ast }{a}_2\right)$$

$$\left[{\partial }_{\tilde{t}}+v_1{\partial }_{\tilde{x}}+{\tilde{\nu }}_1\right]a_1=-\frac{1}{4}\frac{{\omega }_2}{{\omega }_1}N\left(n_3^{\ast }{a}_2\right)$$

$$\left[{\partial }_{\tilde{t}}+v_2{\partial }_{\tilde{x}}+{\tilde{\nu }}_2\right]a_2=\frac{1}{4}N\left(n_3{a}_1\right)$$

$$\left[{\partial }_{\tilde{t}}+v_3{\partial }_{\tilde{x}}+{\tilde{\nu }}_3^R\right]n_3=-\frac{1}{4}\frac{1}{\sqrt{N}}\frac{c^2{k}_3^2}{{\omega }_2^2}\left(a_1^{\ast }{a}_2\right)$$

$$\left[{\partial }_{\tilde{t}}+{\tilde{\nu }}_3^B\right]n_3=-\frac{1}{4}\frac{Z{m}_e}{m_i}\frac{{\omega }_2}{{\omega }_3}\frac{c^2{k}_3^2}{{\omega }_2^2}\left(a_1^{\ast }{a}_2\right)$$

$$\left[{\partial }_{\tilde{t}}^2-2i\frac{{\omega }_3}{{\omega }_2}{\tilde{\nu }}_3^B\right]n_3=\frac{i}{2}\frac{Z{m}_e}{m_i}\frac{c^2{k}_3^2}{{\omega }_2^2}\left(a_1^{\ast }{a}_2\right)$$

Full text

Ultrafast sources of high-intensity coherent x rays have dramatically advanced our understanding of physics at atomic spatial and temporal scales Ackermann et al. (2007); Emma et al. (2010); Fuchs et al. (2015); Krausz and Ivanov (2009), permitted high-resolution study of protein, ceramic, and semiconductor structure Neutze et al. (2000); Miao et al. (2015); Bostedt et al. (2016), and, due to the small size of an x-ray diffraction-limited spot, may allow Schwinger-limit intensities Schwinger (1951); Brezin and Itzykson (1970); Popov (1971) to be reached with moderate pulse energies Ringwald (2001); Alkofer et al. (2001); Popov (2001); Malkin et al. (2007). Further advances require amplifying and compressing x-ray pulses beyond the current capabilities of x-ray free-electron lasers (FELs) Ackermann et al. (2007); Emma et al. (2010); Fuchs et al. (2015); Bostedt et al. (2016), x-ray lasers Sebban et al. (2000); Zeitoun et al. (2004); Suckewer and Jaegle (2009); Alessi et al. (2011); Rohringer et al. (2012), or high-order-harmonic-generation based sources Burnett et al. (1977); McPherson et al. (1987); Ferray et al. (1988); Gibbon (1996); Lichters et al. (1996); Chang et al. (1997); Hentschel et al. (2001); Quéré et al. (2006); Dromey et al. (2007); Popmintchev et al. (2015); Edwards and Mikhailova (2016). The parametric amplification of x rays in high-density plasmas using stimulated Raman scattering (SRS) has been suggested for compressing the output of FELs at nanometer wavelengths, potentially reaching attoscond durations Malkin and Fisch (2007); Malkin et al. (2007); Malkin and Fisch (2009, 2010); Sadler et al. (2015). However, Raman amplification is constrained by Landau and collisional damping at low and high densities and high and low temperatures, respectively. Without the application of extraordinarily large magnetic fields Shi et al. (2017), the window between these damping mechanisms closes at x-ray wavelengths, rendering SRS amplification impractical for reasonable pump intensities.

We demonstrate that stimulated Brillouin scattering (SBS) is a more promising approach for the amplification of x-ray beams. Even though SBS is less practical than SRS at optical wavelengths, in the x-ray regime the weaker damping of the ion wave is critical. We show that in solid density plasmas with current pump intensities SBS amplification may provide sub-femtosecond x-ray pulses orders-of-magnitude more intense than FEL output.

Consider parametric plasma amplification based on three-wave coupling between a seed beam (frequency $\omega_{1}$, wavenumber $k_{1}$), pump beam ($\omega_{2}$, $k_{2}$), and Langmuir (SRS) or ion-acoustic (SBS) plasma wave ($\omega_{3}$, $k_{3}$). When the resonance conditions for counterpropagating beams are approximately met ($\omega_{2}=\omega_{3}+\omega_{1}$, $k_{2}=k_{3}-k_{1}$), energy is efficiently transferred from the pump to the seed beam, resulting in amplification and compression of the seed pulse to up to relativistic intensities and plasma-wave-period durations Malkin et al. (1999). Since the Langmuir wave has a shorter period than the ion-acoustic wave, Raman amplification tends to give higher growth rates, more intense amplified pulses, and shorter pulse durations Edwards et al. (2016), and SRS has been studied in greater depth Malkin et al. (2000); Ping et al. (2004); Cheng et al. (2005); Ren et al. (2008); Yampolsky et al. (2008); Ping et al. (2009); Vieux et al. (2011); Trines et al. (2011); Turnbull et al. (2012); Depierreux et al. (2014); Toroker et al. (2014); Lehmann and Spatschek (2014); Edwards et al. (2015). However, because SBS allows the pump and seed to be at almost the same wavelength, it has been considered as an alternative to SRS in both the weakly-coupled regime (WC-SBS), where the plasma mode is the ion acoustic wave Milroy et al. (1977); Andreev and Sutyagin (1989), and the strongly-coupled regime (SC-SBS), where the plasma response is a driven ion quasi-mode Andreev et al. (2006); Lancia et al. (2010); Lehmann et al. (2012); Lehmann and Spatschek (2013); Weber et al. (2013); Riconda et al. (2013); Guillaume et al. (2014); Lehmann and Spatschek (2016); Lancia et al. (2016); Chiaramello et al. (2016). For Brillouin amplification it is often necessary to consider both the strongly and weakly-coupled contributions Edwards et al. (2016); Schluck et al. (2016), as well as the distinction between SRS and SBS Jia et al. (2016). Particle-in-cell (PIC) simulations have also suggested that SBS performance improves in a slightly collisional plasma Humphrey et al. (2013). Here we consider the x-ray regime, where SBS may be more practical than SRS because its robustness to damping allows shorter wavelengths, lower pump intensities, and a broader range of plasma densities and temperatures.

An adequate model for x-ray plasma amplification must include treatment of wave damping. Previous works Malkin and Fisch (2007); Malkin et al. (2007); Malkin and Fisch (2009, 2010); Balakin et al. (2011) have assumed damping of the plasma wave to be dominant, neglecting electromagnetic-wave damping. Although this assumption produces only moderate errors for SRS, the reduced damping of the ion-acoustic mode means that electromagnetic wave damping substantially affects SBS calculations. To find the linear growth rates and estimate the limits of amplification, we use the three-wave coupling equations Malkin et al. (1999), extending the treatment of Malkin and Fisch for quasitransient backward Raman amplification (QBRA) Malkin and Fisch (2009, 2010) to include SBS and collisional damping of the seed. Assuming that an envelope approximation is valid and the phase-matching conditions are met, the fluid model reduces to three-wave coupling equations for the seed ($a_{1}$), pump ($a_{2}$), and electron density fluctuation ($n_{3}$) envelopes (Appendix A). In normalized units (time $\tilde{t}=t\omega_{2}$, position $\tilde{x}=x\omega_{2}/c$), the seed and pump equations are:

[TABLE]

where $a_{1,2}=eA_{1,2}/m_{e}c$ is the normalized vector potential, $\nu_{1,2}=\tilde{\nu}_{1,2}\omega_{2}$ is the damping rate, $\tilde{v}_{1,2}=k_{1,2}c/\omega_{1,2}=(1-\omega_{pe}^{2}/\omega_{1,2}^{2})^{1/2}$ is the normalized group velocity for light in terms of the plasma frequency $\omega_{pe}=(4\pi e^{2}n_{e}/m_{e})^{1/2}$, and $N=n_{e}/n_{c}=\omega_{pe}^{2}/\omega_{2}^{2}$ is the plasma number density ($n_{e}$) normalized by the pump-frequency critical density $n_{c}=m_{e}\omega_{2}^{2}/4\pi e^{2}$ defined in terms of the electron charge ($e$) and mass $(m_{e})$.

The system is closed with an equation for the electron density, which, for SRS, is

[TABLE]

where $\tilde{v}_{3}=k_{3}c/\omega_{3}$ and $(\omega_{3},k_{3})$ are found by solving the phase-matching conditions with the dispersion relation $\omega_{3}=\omega_{pe}$. For SBS,

[TABLE]

where $Z$ gives the ion charge state, $m_{i}$ is the ion mass, and the plasma response ($\omega_{3},k_{3}$) satisfies the dispersion relation Edwards et al. (2016):

[TABLE]

with $c_{s}=\sqrt{ZT_{e}/m_{i}}$ for electron temperature $T_{e}\gg T_{i}$ Nicholson (1983). In Eq. (4), the group velocity term for SBS is small and has been neglected. The first derivative in time is dominant for WC-SBS, where $\omega_{3}=c_{s}k_{3}$, and the second derivative dominates for SC-SBS.

Assuming negligible pump depletion, the asymptotic linear growth rate of the seed may be calculated from the impulse response of the seed-plasma wave system Bobroff and Haus (1967), as previously applied for QBRA Malkin and Fisch (2009, 2010), or by calculating a self-similar solution for the system as $\tilde{t}\to\infty$ (derived in Appendix B). Including collisional damping of the seed, the growth rate becomes

[TABLE]

for SRS and WC-SBS, where $\tilde{\nu}_{1}=N\tilde{\nu}_{ei}\omega_{2}/\omega_{1}$ Ginzburg (1970), and $\tilde{\nu}_{3}^{R}=\tilde{\nu}_{\textrm{Lnd}}+\tilde{\nu}_{\textrm{ei}}/4$ Malkin and Fisch (2010). For SBS with $T_{i}\ll T_{e}$, previous work Epperlein et al. (1992) suggests a damping coefficient $\tilde{\nu}_{3}^{B}=0.01\times 2c_{s}/c$ for conditions considered here; this rate is small, and even substantial variations do not affect the final results. For SC-SBS and SBS in the intermediate regime, the effective growth rate $\hat{\Gamma}_{B}=\kappa$ is the largest real root of the cubic equation:

[TABLE]

where $C_{sc}=i\omega_{2}/2\omega_{3}$, $C_{wc}=1$ (assumed 0 for SC-SBS), and $K=\tilde{\nu}_{1}\tilde{\nu}_{3}^{B}-NZm_{e}c^{2}k_{3}^{2}/16\omega_{1}\omega_{3}m_{i}$. The normalized collisional and Landau damping rates are Malkin and Fisch (2010):

[TABLE]

where $q_{T}=4T_{e}/Nm_{e}c^{2}$, $\Lambda_{C}=\ln 12\pi\lambda_{D}^{3}n_{e}/Z$ is the Coulomb logarithm, $\lambda_{D}=(T_{e}/4\pi n_{e}e^{2})^{1/2}$ is the Debye length, and $r_{e}=e^{2}/m_{e}c^{2}$. The undamped Raman growth rate, with $\omega_{3}=\omega_{pe}$, is Kruer (2003):

[TABLE]

where $a_{0}$ is the pump field strength ($a_{2}$) before interaction. The Brillouin growth rate may be evaluated numerically in the intermediate regime Edwards et al. (2016), with the weak and strong coupling limits Kruer (2003):

[TABLE]

The growth rate ($\hat{\Gamma}_{R,B}$) of the seed pulse in the linear (constant pump amplitude) regime is a simple metric for comparing SRS and SBS. Apart from dictating the plasma length required to amplify an initially small seed, the linear growth rate determines whether the amplification process out-competes deleterious effects like filamentation, dispersion, and forward scattering. Using Eqs. (6) and (7), the maximum SRS and SBS growth rates achievable over all densities and electron temperatures at a particular pump wavelength are plotted in Fig. 1. At optical frequencies SRS has a higher growth rate than SBS, but collisional and Landau damping create a pump-amplitude-dependent cutoff in the x-ray regime, where $\hat{\Gamma}_{R}$ rapidly drops; for example, below $\lambda=4$ at $a_{0}=0.01$, $\hat{\Gamma}_{B}>\hat{\Gamma}_{R}$. Since $a_{0}$ scales with wavelength, the intensities required to achieve a sufficient SRS growth rate below $\lambda=1$ nm are currently impractical, e.g. at $\lambda=1$ Å, $a_{0}=0.05$ corresponds to $3\times 10^{23}$ W/cm2. Note that all SBS results presented in this work assume a full-ionized proton plasma ($Z=1$, $m_{i}=1836m_{e}$) with negligible ion temperature; amplifier performance will decrease as $Z/m_{i}$ decreases.

The plasma conditions optimal for SRS and SBS differ, with SBS generally more flexible and favoring higher temperatures. Fig. 2 shows the damped SRS and SBS growth rates [from Eqs. (6) and (7)] at $a_{0}=0.01$. Increased collisional damping reduces the growth rate at high densities, and Landau damping of the Langmuir wave sets the low-density cutoff for SRS. To numerically solve the wave-coupling equations [Eqs. (1)-(4)], we employ a step-shifting algorithm for the spatial derivatives Clark (2003); Clark and Fisch (2003a, b) and a fourth-order Runge-Kutta scheme for the time derivatives; numerical solutions are marked with points.

For optical wavelengths SC-SBS is usually considered more promising than WC-SBS, due to a higher growth rate and shorter permitted pulse durations Andreev et al. (2006). However, in extending SBS to the x-ray regime, we find that heavier collisional damping at the high densities and low temperatures associated with SC-SBS means the highest growth rates may be in the weakly-coupled regime for reachable pump intensities, (Fig. 3). Filamentation, the modulational instability, forward Raman scattering, dispersion, and collisional damping of the pump are more significant at higher plasma densities, so the highest output seed intensities will be reached for lower plasma densities than those corresponding to the highest growth rates.

Without seed damping, the wave-coupling equations predict infinite amplification (until the underlying assumptions fail); with $\tilde{\nu}_{3}\neq 0$ the solution approaches a limit at infinite time. This asymptote constrains the maximum possible amplification, since any seed pulse will converge towards it, i.e. a larger seed pulse will decrease in amplitude. In this limit Eqs. (1)-(4) reduce (Appendix C) to:

[TABLE]

where $K_{1}=-\omega_{2}N/4\omega_{1}$, $K_{2}=N/4$, $K_{3}^{R}=-c^{2}k_{3}^{2}/4\omega_{2}^{2}\sqrt{N}$, and $K_{3}^{B}=Zm_{e}c^{2}k_{3}^{2}/4m_{i}\omega_{3}\omega_{2}$ for SBS. An analytic solution of these equations is possible if $\tilde{\nu}_{3}\ll\tilde{\nu}_{1}$, as is the case for SBS. This yields

[TABLE]

where $D_{1}=K_{1}K_{2}/\tilde{\nu}_{1}(\tilde{v}_{2}-\tilde{v}_{1})$ and $D_{2}=-K_{1}K_{3}/\tilde{v}_{1}\tilde{\nu}_{1}$, with maximum value:

[TABLE]

The maximum amplification factor $(a_{1,\textrm{max}}/a_{0})$ at $a_{0}=0.01$ and $\lambda_{0}=3$ nm as calculated from these equations is shown in Fig. 4. For SRS, where $\tilde{\nu}_{3}$ cannot be neglected, the solution is found numerically. For these parameters the amplification factors achievable with SBS are far higher than with SRS and the amplified pulse in Fig. 4b has an intensity full-width-half-maximum of 0.5 fs, suggesting that at its limit SBS may produce intense sub-femtosecond x-ray pulses. Higher amplification factors are associated with lower densities, which give lower growth rates, so a practical device might use a high-to-low density gradient to combine high initial growth rates with large final amplitude limits. Relativistic and dispersive effects will stop growth before the indicated asymptotes. We can estimate the effect of the modulational instability by calculating the energy transferred from pump to seed in the pump-depletion regime during a normalized modulational growth time $2/Na_{1}^{2}$ for the final seed intensity Malkin et al. (1999). Using a compressed duration $200$ $\lambda/c$ at $N=0.01$ and assuming that half the pump energy is transferred to the seed, we find that $a_{0}=0.01$ can produce $a_{1}=0.1$ in a modulational time, or a two order of magnitude increase in intensity, which is similar to the limits suggested by the collisional asymptote.

The above analysis is limited by the assumptions that the fluid model is valid (i.e. kinetic effects are neglected), that damping is adequately described by a linear model, and that density fluctuations may be considered small. These assumptions can be checked by PIC simulations, which include kinetic, dispersive, and relativistic effects. Using the code EPOCH Arber et al. (2015), we demonstrate in Fig. 5 that under these conditions the three-wave model makes reasonable predictions for solutions of the full system; the $\lambda=3$ nm pump amplifies the seed pulse to 20 times the pump intensity in 19 fs, producing a 0.9-fs-duration output pulse. The density fluctuations (Fig. 5b) and amplified spectra (Fig. 5d) are characteristic of Brillouin amplification, and the seed envelope (Fig. 5c) evolves in agreement with the three-wave model. The slightly lower growth in the PIC calculation and the appearance of envelope modulations result from forward Raman scattering, which is overestimated due to the particle discreteness in PIC simulations.

Consider a focused intensity of $10^{19}$ W/cm2 at $\lambda=3$ nm, which lies near the capabilities of current and proposed advanced FEL sources Ackermann et al. (2007); Emma et al. (2010); Fuchs et al. (2015). In a plasma with density near $5\times 10^{23}$ cm*-3* and temperature 500 eV, slightly higher than compressed solid hydrogen Weir et al. (1996) and lower than fully-ionized aluminum, the SBS growth rate (in hydrogen) will be around 100 ps*-1*. Metal targets will allow higher electron densities than hydrogen, but the lower $Z/m_{i}$ ratio will reduce the amplification growth rate, and the higher collision frequency will cause additional damping. The modulational and collisional limits, as described previously, give for these conditions a maximum output pulse intensity 100-1000 times greater than the pump intensity, up to $10^{21}-10^{22}$ W/cm2. With a seed provided by laser-driven high-order harmonic generation, SBS could be used to amplify and compress the output of large-scale x-ray FEL by several orders of magnitude in a millimeter scale plasma. For comparison, at this wavelength and pump intensity, SRS is too heavily damped to produce any significant amplification for any plasma density or temperature.

Although FEL beams are only quasi-coherent, previous studies have suggested that parametric plasma amplification is relatively resilient to moderate amplitude and phase fluctuations Balakin et al. (2003); Solodov et al. (2003); Fraiman et al. (2002); Kirkwood et al. (2011). Current x-ray FELs have bandwidths on the order of 0.1-1% of $\omega_{0}$ Geloni et al. (2010); Vartanyants et al. (2011). In Fig. 6, a PIC simulation of amplification of x-ray Brillouin amplification using a pump bandwidth $\Delta\omega=0.4\%\times\omega_{0}$ (FWHM) with corresponding coherence time around 0.5 fs shows that the inclusion of finite coherence does not impede the interaction, provided that the coherence time is not short compared to the final amplified pulse. The pump is constructed by randomly choosing the phase and frequency of 1000 components from a Gaussian frequency distribution, producing the modulated intensity envelope shown as an inset in Fig. 6. The amplified pulse shown in Fig. 6 has an intensity and duration similar to that found using a coherent pump under the same conditions (Fig. 5). It should be noted that the seed pulse maintains its coherence, so SBS may be useful for cleaning FEL output to produce high intensity coherent pulses from a lower-intensity quasi-coherent source.

To conclude, this work predicts an amplification regime where stimulated Brillouin scattering is significantly more useful than Raman scattering, allowing coherent x-ray pulses with unprecedented intensities. In this regime, plasma-based x-ray amplification may be possible with available FEL beamlines.

Appendix A Derivation of the Three-Wave Coupling Equations

The three-wave coupling equations play a crucial role in the analysis of Raman and Brillouin amplification. Since the work presented here relies on a more general treatment of the governing equations than is usually necessary, we provide a derivation for reference below.

We start with the Maxwell-fluid equations, which describe the coupling between light and plasma based on a fluid model of the electron and ion motion Kruer (2003). This model does not include kinetic and non-continuum effects from first principles, though Landau and collisional damping are incorporated empirically through added terms, so the model is only applicable where other kinetic effects are unimportant and the expressions for the damping terms are valid. Considering a seed beam (vector potential $\mathbf{A}_{1}$), a pump beam (vector potential $\mathbf{A}_{2}$) and a plasma of number density $n_{e,0}$, the governing equations are:

[TABLE]

[TABLE]

where $\nu_{1,2}$ are damping coefficients for the light wave, $\omega_{pe}=\sqrt{4\pi e^{2}n_{e,0}/m_{e}}$ is the plasma frequency, and $n_{e,1}$ is the amplitude of small electron number density fluctuations. To close the system, we write an equation for the coupling of an electron plasma wave (SRS),

[TABLE]

or an ion-driven electron density perturbation (SBS),

[TABLE]

to the laser fields Kruer (2003). Here, one ion species is assumed, with $Z$ as the ion charge and $m_{i}$ as the ion mass. The plasma sound speed is $c_{s}=\sqrt{ZT_{e}/m_{i}}$, assuming that $T_{i}\ll T_{e}$ and that the fluctuations are quasi-neutral Nicholson (1983).

Assuming that the laser fields may be described as an envelope over a high frequency carrier wave, for linear polarization, the vector potential is written:

[TABLE]

where $k_{1}$ and $k_{2}$ contain the directionality of the wave propagation, i.e. $k_{1}<0$ corresponds to propagation in the negative $x$ direction. The second term is the complex conjugate (c.c.) of the first. For this article, we consider only linear polarization.

Similarly, the density fluctuations may be written:

[TABLE]

where we have notated $n_{3}$ with the subscript $3$ as a reminder that it represents the third fluctuating quantity. We assume that the phase matching conditions are met: $k_{3}=k_{2}-k_{1}$ and $\omega_{3}=\omega_{2}-\omega_{1}$. Substituting these definitions (for linear polarization) into Eq. (16), evaluating derivatives of the high frequency components, and dropping non-resonant terms, we have:

[TABLE]

with a similar expression for $a_{2}$. We assume that the light propagation satisfies the dispersion relation Ginzburg (1970):

[TABLE]

Substituting this dispersion relation into Eq. (22), with $\nu_{1}=\nu_{ei}\omega_{pe}^{2}/\omega_{1}^{2}$ and noting that $\partial_{t}\ll\omega_{1}$ and $\partial_{x}\ll k_{1}$, we have:

[TABLE]

To simplify this equation, we use the normalized variables $\tilde{t}=t\omega_{2}$, $\tilde{x}=x\omega_{2}/c$, and $\tilde{\nu}_{1}=\nu_{1}/\omega_{2}$, and write the normalized group velocity as $\tilde{v}_{1}=ck_{1}/\omega_{1}$. We then have:

[TABLE]

where $N=\omega_{pe}^{2}/\omega_{2}^{2}=n_{e}/n_{c}$. The pump equation similarly simplifies to:

[TABLE]

Using the same approach, the density equations for SRS and SBS can be written in wave-coupling form. For the Langmuir wave (SRS):

[TABLE]

For SBS, we cannot in general make the assumption $\partial_{t}\ll\omega_{3}$ and are therefore left with both first and second order time derivatives on the left-hand side. However, the group velocity $\tilde{v}_{3}$ is in this case negligible, so all spatial derivatives may be dropped. This yields the following expression for the SBS plasma response:

[TABLE]

which reduces to the strongly-coupled and weakly-coupled limits, as described in Ref. Edwards et al. (2016), if the second-order or first-order time derivatives are neglected, respectively.

The full set of equations describing wave coupling for Raman and Brillouin scattering is given below, where Eqs. (29) and (30) are used for all models, Eq. (31) for SRS, and Eqs. (32)-(34) for the different forms of SBS.

[TABLE]

These equations can numerically, or, in some limits, analytically, be solved to describe amplification by SRS or SBS subject to the assumptions discussed above.

For this model to be useful, particularly in the x-ray regime, the damping coefficients $\tilde{\nu}_{1,2,3}$, in the above equations need to be evaluated. We include collisional damping of light waves, Langmuir waves, and the ion response, as well as Landau damping of Langmuir waves. The collisional damping of electromagnetic waves is Ginzburg (1970):

[TABLE]

where

[TABLE]

Here, $q_{T}=4T_{e}/Nm_{e}c^{2}$, $\Lambda_{C}=\ln 12\pi\lambda_{D}^{3}n_{e}/Z$ is the Coulomb logarithm, $\lambda_{D}=(T_{e}/4\pi n_{e}e^{2})^{1/2}$ is the Debye length and $r_{e}=e^{2}/m_{e}c^{2}$.

The Langmuir wave is both Landau damped and collisionally damped, so, $\tilde{\nu}_{3}=\tilde{\nu}_{\textrm{Lnd}}+\tilde{\nu}_{\textrm{ei}}/4$ Malkin and Fisch (2009), where:

[TABLE]

Collisional damping of the ion-acoustic wave has a smaller effect, which has been approximated as $\tilde{\nu}_{3}=0.01\times 2c_{s}/c$ Epperlein et al. (1992). Since this damping rate is small compared to the direct collisional damping rate for light waves, its exact value does not substantially affect observed behavior. We neglect Landau damping of the ion-acoustic wave by staying in the regime $T_{e}\gg T_{i}$.

Appendix B The Asymptotic Linear Growth Rate

The growth rate of the seed in the linear regime, where the pump is not significantly depleted, is important for determining the conditions under which plasma amplification is possible. An exact solution for the impulse response of the seed-plasma wave system, which is applicable to SRS and SBS amplification for negligible pump damping, has been derived by Bobroff and Haus Bobroff and Haus (1967), albeit with the wave equations cast in slightly different form than that usually applied for plasma amplification. Using this method, Malkin and Fisch Malkin and Fisch (2009) found the effective linear growth rate of the seed in the quasi-transient backward Raman amplification (QBRA) regime, adjusted for damping of the plasma wave. This growth rate is the asymptotic solution at infinite time, and represents the rate at which the seed maximum will grow.

Here, we use the self-similarity of the asymptotic solution to find the effective growth of the seed pulse, a method which is simple and more readily extended to include direct damping of the seed and strongly-coupled SBS. Starting with the equations applicable to SRS and WC-SBS, we assume that the solution in the limit $t\to\infty$ of the coupled equations:

[TABLE]

is of the form

[TABLE]

where

[TABLE]

Note that this method includes damping of the seed ($\tilde{\nu}_{1}$) and plasma response ($\tilde{\nu}_{3}$), but does not include any depletion or damping of the pump. The constant $v$ represents the unknown rate at which the seed is stretching in time. Substituting the above forms of the solution,

[TABLE]

which, eliminating the density, may be rewritten as a single equation:

[TABLE]

Evaluating the derivatives, we have an equation with which we want to solve for $\kappa$, the long-time growth rate.

[TABLE]

In the infinite time limit, we may drop terms with inverse powers of $\tilde{t}$, which removes all of the $\xi$-derivatives from the equation and leaves only a quadratic equation in $\kappa$.

[TABLE]

This may be solved in straightforward fashion to give the modified growth rate:

[TABLE]

which for SRS is:

[TABLE]

where $\tilde{\Gamma}_{R}=\Gamma_{R}/\omega_{2}$ is the undamped Raman growth rate. The above expression will also be valid for WC-SBS, substituting $\Gamma_{B_{wc}}$ for $\Gamma_{R}$ and using the appropriate damping factors. Although this form is different from that presented by Malkin and Fisch Malkin and Fisch (2009), if $\tilde{\nu}_{1}=0$, it is strictly equivalent.

To treat SBS more generally, including the strongly-coupled and intermediate cases, we return to the wave-coupling equations which include the second derivative in time associated with strongly-coupled stimulated Brillouin scattering:

[TABLE]

where for SBS

[TABLE]

and $C_{wc}=1$. For WC-SBS, we can assume $C_{sc}=0$ and for SC-SBS, we assume $C_{wc}=0$. Similarly to the previous derivation, the derivatives with respect to $\xi$ vanish as $\tilde{t}\to\infty$, leaving the following cubic equation for $\kappa$:

[TABLE]

Since this is a cubic equation in $\kappa$, a solution exists, though not one in as simple a form as that available for SRS. We give the full analytic solution below, though note that, in practice, numerical evaluation of the above equation is likely more readily implemented:

[TABLE]

where

[TABLE]

When $C_{sc}=0$, Eq. (47) should be used directly. When $\tilde{\nu}_{1}$ and $\tilde{\nu}_{3}$ are zero, this equation recovers the plane wave growth rate Edwards et al. (2016).

Appendix C Steady-State Solution for Non-Linear Amplification

With the inclusion of collisional damping of the seed laser pulse ($\tilde{\nu}_{1}$), the solution to the three-wave equations in the pump-depletion (non-linear) regime asymptotically approaches a steady state at infinite time, rather than growing infinitely short and intense (until violation of the envelope approximation invalidates the SRS and SBS models). Although the steady-state solution is of limited utility in finding actual amplified pulse shapes because secondary effects, including relativistic self-modulation, group velocity dispersion, and forward Raman scattering, become increasingly important as the growth decreases, calculation of this asymptote allows us to set a limit on the maximum possible factor of amplification which can be achieved with a particular set of parameters. As we show, this limit is useful for identifying regimes for which plasma amplification is unlikely to be viable, but by itself, does not guarantee that a particular choice of parameters is practical. Since the time taken to reach the steady state solution as important as its final value, for some applications it may be desirable to use a varying plasma density, combining the high growth rates of higher density plasmas for early times with the high steady-state amplification factors of lower plasma densities for the final stages of interaction. With appropriate choice of parameters, the following analysis is valid for both SRS and all forms of SBS.

Starting with the three-wave equations written in the form:

[TABLE]

where the first term on the left hand side of Eq. (57) is immediately negligible for SRS and

[TABLE]

we find the asymptotic solution by choosing the reference frame moving with the seed and by setting the time derivatives to zero. We will also neglect pump damping ($\tilde{\nu}_{2}$) because its inclusion causes the system to be propagation-length dependent (i.e. will not reach a steady state), and pump-depletion can be usefully included after this calculation using a pre-compensated variable-intensity pump boundary condition. These solutions therefore lose validity if the pump is substantially depleted on the timescale of the interaction, a limitation mitigated by the fact that a high degree of pump depletion will prevent pump-seed interaction in a plasma of any reasonable length, making such a regime unlikely to be useful. The steady-state equations, assuming $|\tilde{v}_{1}|\gg|\tilde{v}_{3}|$, are:

[TABLE]

In the case of SBS, damping of the ion-wave ($\tilde{\nu}_{3}$) is small and can be safely neglected, substantially simplifying the system of equations. Substituting for $a_{1}$, the above equations may be rewritten as:

[TABLE]

where

[TABLE]

Note that $v_{1}<0$. This set of equations may be solved to give:

[TABLE]

We are, however, primarily interested in $a_{1}$, which may be written in terms of $a_{2}$ as:

[TABLE]

This simplifies to:

[TABLE]

To get the maximum value of $a_{1}$ in the steady state, we find where the derivative with respect to $\tilde{x}$ is zero to get:

[TABLE]

Note that since $K_{1}$, $K_{3}$ and $\tilde{v}_{1}$ are less than 0, the above envelope is real and greater than 0. It is clear that if $\tilde{\nu}_{1}=0$, the maximum intensity determined from this set of equations is infinite, and they cannot be used to determine maximum possible amplification. If $\tilde{\nu}_{3}$ is not small, for example for SRS, the system of equations is less tractable, and we resort to numerical evaluation of the steady state equations. The results of evaluating these equations are shown in Fig. 4.

Appendix D Numerical Solutions to the Coupling Equations

To numerically solve the wave-coupling equations, we employ a step-shifting algorithm similar to that previously applied to the problem of Raman amplification by Clark Clark (2003); Clark and Fisch (2003a, b). To implement the propagators on the left side of Eqs. (1)-(4), all simulation values are shifted by one grid space at intervals of $\Delta x/\tilde{v}_{(1,2,3)}$. The time derivatives are evaluated with a fourth-order Runge-Kutta scheme. Numerical evaluation of the coupling equations is much faster than fully kinetic or fluid based simulations, and provides solutions for conditions which are not analytically tractable. As code validation, and to justify comments made in the manuscript, we present a series of simple comparisons between simulations and analytic results.

The linear growth rate of the undamped system represents the early-time behavior of the interaction and is important for quantifying the highest degree of amplification that may be expected. It can be shown that the undamped asymptotic seed growth rate in the linear regime is simply the instability plane-wave growth rate, for which expressions are well known for SRS, WC-SBS, and SC-SBS and which may be readily calculated for intermediate SBS conditions Edwards et al. (2016). In Fig. 7, the analytic growth rate for the undamped SRS and SBS instabilities are compared to the asymptotic linear-regime growth rates found from numerical solution of the three wave-coupling equations. Note that, for example, a WC-SBS solution may be found in the strongly-coupled regime, but this solution will disagree with the solution to the full equation and does not represent a useful solution to the full system. Furthermore, as evaluated here, the equations assume that the regime of interaction is set only by the initial boundary field strength, and do not fully include the transition between regimes that would be observed in practice Schluck et al. (2016).

The expressions for the damped growth rate in the linear regime [Eqs. (6) and (7)] may also be compared to the reduced growth rate found using numerical evaluation of the three-wave model. In Fig. 7, the analytic solutions are shown for SRS with no damping, damping of only the Langmuir wave, and damping of both the Langmuir and seed electromagnetic wave. These are compared to three-wave calculations which include both Langmuir wave and seed damping. As should be expected, the numerical solutions agree with the analytic model. Similarly, Fig. 7, compares analytic solutions and numerical solutions for the damped growth rate of SBS, using the weakly-coupled, strongly-coupled, and full models. The agreement suggests the validity of the analytic expressions and the numerical implementation.