X-ray third-order nonlinear asymmetric diffraction in the transmission geometry. (7th September 2021) - Balyan, M K
Abstract: For high intensity x-ray sources, the investigations of the nonlinear x-ray effects become essential. The x-ray third-order nonlinear asymmetric transmission case dynamical diffraction is investigated theoretically in perfect crystals. The three essential input parameters are taken into account: the deviation from the Bragg exact orientation, the asymmetry angle of reflecting atomic planes and the intensity of incident wave. The third-order nonlinear equations for the asymmetrical dynamical diffraction and two integrals of motion are presented. The exact solutions for the wave amplitudes, presented via Jacobi elliptic functions, have been found. The solutions are periodic functions of the depth. It is shown that the behavior of the waves inside the crystal is determined by the sign of a combined parameter, which includes the three input parameters, mentioned above. The regions, where this parameter is positive, negative or zero, are found. For positive values, the energy is concentrated both in the transmitted and diffracted waves while for the negative values, the energy is concentrated mainly in the transmitted beam. It is found that when this combined parameter is zero, the amplitudes are elementary periodic or non-periodic functions. For the periodic solutions, the nonlinear Pendellösung effect takes place, i.e. the transmitted and diffracted waves periodically exchange with their energies. For the period, called an extinction length, an exact expression is found. It is shown that for large values of the asymmetry factor, the observation of the nonlinear diffraction effects can be realized for sufficiently low intensities of the incident wave. The obtained results can be used for experimental investigation of the nonlinear Bragg diffraction and for preparation of high intensity x-ray beams with given parameters. The expression for the nonlinear extinction length allows to estimate the third-order nonlinear susceptibilities of crystals by measuring the extinction length in experiments.