Investigation of third-order nonlinear dynamical X-ray diffraction based on a new exact solution

Third-order nonlinear two-wave dynamical X-ray diffraction in a crystal is considered. For the Laue symmetrical case of diffraction a new exact solution is obtained. The solution is presented via Jacobi elliptic functions. Two input free parameters are essential: the deviation parameter from the Bragg exact angle and the intensity of the incident wave. It is shown that the behavior of the field inside the crystal is determined by the sign of a certain combination of these parameters. For negative and positive signs of this combination, the wavefield is periodic and the nonlinear Pendellösung effect takes place. For the nonlinear Pendellösung distance the appropriate expressions are obtained. When the above-mentioned combination is zero, the behavior of the field can be periodic as well as non-periodic and the solution is presented by elementary functions. In the nonperiodic case, the nonlinear case Pendellösung distance tends to infinity. The wavefield diffracts and propagates in a medium, whose susceptibility is modulated by the amplitudes of the wavefields. The behavior of the wavefield can be described also by an effective deviation from the Bragg exact angle. This deviation is also a function of the wavefields.