X-ray amplification by stimulated Brillouin scattering

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 subfemtosecond durations and unprecedented intensities.

Figure 1

The maximum linear growth rate (${\mathrm{\Gamma }}_{R,B}$), normalized by the pump frequency (${\omega }_L={\omega }_2$), for SRS (dashed lines) or SBS (solid lines) against the pump wavelength ($\lambda =2\pi c/{\omega }_L$) at varied pump strength ($a_0$). Density and temperature are chosen at each wavelength to maximize the growth rate. The water window between the K-absorption edges of oxygen (2.34 nm) and carbon (4.4 nm) is important for biological imaging. Growth rates corresponding to approximate pump strengths and wavelengths demonstrated at the Linac Coherent Light Source (LCLS) [2, 3] are marked.

Figure 2

The effective linear growth rate of SRS and SBS for $a_0=0.01$ and varied $\lambda$, $T_e$, and $N$. (a) $\lambda =1000$ nm, (b) $\lambda =100$ nm, (c) $\lambda =10$ nm, and (d) $\lambda =3$ nm. The gray vertical line indicates the upper limit of valid densities for SRS ($N=0.25$). The orange points are from numerical solution of the wave-coupling equations.

Figure 3

The growth rate of SBS against density ($N$) and temperature ($T_e$) for pump wavelength $\lambda =3$ nm and $a_0=0.01$. The threshold $\mathrm{\Lambda }=0.25=(Z{m}_e/m_i){\left({\omega }_{pe}{a}_{0}c\right)}^2/\left(16{\omega }_2{k}_2{c}_s^3\right)$ [37] between WC-SBS and SC-SBS is marked by a dashed line. The black region at bottom excludes parameters where the Coulomb-logarithm collisional model is substantially invalid (${\mathrm{\Lambda }}_C<1$).

Figure 4

(a) Maximum seed-to-pump amplification factor (asymptotic limit) produced by SRS or SBS at $\lambda =3$ nm and $a_0=0.01$. The orange dashed lines mark the analytically calculated values for SBS, neglecting ${\tilde{\nu }}_3$. Thin lines are logarithmically spaced in density, with $N=0.001,0.01,0.1$ marked by bold lines. (b), (c) The wave envelope limits from solution of the three-wave equations for selected SBS (b) and SRS (c) pulses at $N=0.01$, $T_e=1$ keV and $N=0.01$, $T_e=100$, respectively. In (b), (c) the orange dashed lines mark solutions to Eqs. (11, 12, 13).

Figure 5

PIC simulation (epoch [75]) of Brillouin amplification with $\lambda =3$ nm, $a_0=0.02$, $N=0.04$, $T_e=2$ keV, $T_i=10$ eV, and $m_i=1836m_e$. (a) Intensity ($I$) of seed (red) and pump (blue) beams at $\tilde{t}=1.2\times {10}^4$ (19 fs) from PIC calculations. Inset: Pump intensity only, showing 85% depletion. (b) Electron ($N_e$, green) and ion ($N_i$, black) density. Inset: a $10\lambda$ interval of (b), showing $0.5\lambda$ period of plasma density fluctuations. (c) Evolution of seed envelope from initial time ($\tilde{t}=0$) to $\tilde{t}=4.4\times {10}^3$ for PIC (solid gray) and three-wave (dashed orange) calculations. (d) Spectra of pump and seed at $\tilde{t}=8.2\times {10}^3$. $\mathrm{\Delta }k$ is the wave number downshift required for the SBS phase matching condition and agrees with the observed shift. Here ${\nu }_{ei}^{-1}\approx 1$ fs, and collisions have a small effect on the interaction. The PIC calculations use 80 cells/$\lambda$, 1000 particles per cell, and a simulation window moving at the seed group velocity. No amplification is observed in an equivalent simulation with immobile ions.

Figure 6

PIC simulation (epoch [75]) of Brillouin amplification in an incoherent pump (bandwidth FWHM 0.4% of ${\omega }_0$) with $\lambda =3$ nm, $a_0=0.02$, $N=0.04$, $T_e=2$ keV, $T_i=10$ eV, and $m_i=1836m_e$ at $\tilde{t}=6.3\times {10}^3$. Apart from pump coherence, conditions and simulation parameters are the same as those in Fig. 5. Inset: Pump intensity envelope (before interaction) showing amplitude modulations due to incoherence.

Figure 7

Undamped linear growth rate of SRS and SBS as a function of (a) density (with $T_e=200$ eV) and (b) temperature (at $N=0.05$) at fixed pump strength ($a_0=0.01$) and wavelength ($\lambda =1\mu \mathrm{m}$). The analytic linear growth rates (lines) are compared to those extracted from the rate of growth of the maximum of the seed in wave-coupling calculations.

Figure 8

Analytic estimates of the linear growth rate of a Raman amplified seed in the presence of no damping (solid thin lines), Landau and collisional damping of only the plasma wave (dashed lines), and both damping of the plasma wave and collisional damping of the seed (thick solid lines), compared to the growth rate extracted three-wave simulations (points) with both seed and plasma wave damping. Simulations are shown for $T_e=50$, 200, and 1000 eV, and at (a) $\lambda =1000$ nm and (b) 10 nm. All results are for $a_0=0.01$.

Figure 9

Analytic estimates (lines) compared to values found from solving the three-wave equations (points) for the Brillouin linear growth rate in the WC-SBS, SC-SBS, and SBS models with and without damping (which arises almost entirely due to collisional damping of the seed) at $\lambda =1000$ nm, $a_0=0.01$, and $T_e=50$ eV. The ion temperature is assumed negligible, $m_i=1836m_e$, and $Z=1$.