Nonlinear refraction of hard x rays

We study the nonlinear refraction of x rays in highly ionized condensed matter by using a classical model of a cold electron plasma in a lattice of still ions coupled with Maxwell equations. By employing a group-theoretical technique, we reduce the governing equations of the system to an integrable set of nonlinear ordinary differential equations, discussing the existence and stability of nonlinear waves. This allows us to define the effective Kerr coefficient $n_2$ at x rays. With reference to real-world crystalline materials (B, C, Li, and Na), we consider beam self-defocusing and predict that nonlinear processes become comparable to the linear ones for focused beams with powers on the order of $m{c}^3/r_0$ $\left(\approx 10\text{GW}\right)$, the classical electron power. As a consequence, nonlinear phenomena are expected to largely affect imaging experiments in currently exploited x-ray free-electron lasers and in their future developments.

Figure 1

(Color online) (a) First Brillouin zone and (b) x-ray dispersion relation of a SC lattice in the case of an infinitesimal periodic modulation. High-symmetry points are indicated with their reduced wave vector $\mathbf{k}\pi /a$.

Figure 2

(Color online) (a) First Brillouin zone and (b) x-ray electromagnetic band structure for a fcc lattice.

Figure 3

(Color online) Solution of nonlinear dispersion relation (17): The continuous (dashed-dotted) line is the forward (backward) propagating wave; the horizontal dashed line corresponds to the asymptotic value $k_{nl}=-k$.

Figure 4

Divergence vs input peak intensity $I_0$ in dimensionless units.

Figure 5

(Color online) (a) Atomic-loss-limited intensity for nonlinear defocusing and (b) material absorption length versus wavelength.

Figure 6

(Color online) (a) Electronic-loss-limited intensity for nonlinear defocusing and (b) absorption length due to electron-ion collisions versus wavelength.