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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.

Under two-wave dynamical diffraction conditions, in the nonlinear case, we have two external free parameters: the deviation from the Bragg exact angle, as in the linear case, and the incident beam intensity. In the present work, continuing the themes of the works of Balyan (2015a,b, 2016a,b,c), an exact solution is obtained for the third-order nonlinear symmetrical Laue case dynamical diffraction in crystals, which takes into account both the intensity of the incident wave and the deviation from the Bragg exact angle. Using this exact solution, the third-order nonlinear two-wave X-ray dynamical diffraction is investigated. These investigations can be a basis for experimental investigations of the third-order nonlinear dynamical diffraction in crystals, can be used for manufacturing Bragg diffractive optical elements (focusing elements, monochromators and collimators) for high-intensity X-ray beams. The exact solutions have a definite advantage with respect to numerical solutions, since they allow determining the behaviors of essential physical quantities depending on the input parameters. Such essential physical quantities of Bragg diffraction are reflection and transmission coefficients, extinction length, rocking curve full width at half-maximum, the behavior of the wavefields depending on depth and other quantities. The exact solutions also allow finding new effects and new peculiarities of the phenomena under investigation.

for the diffracted beam. These replacements are non-symmetrical with respect to the diffracted and transmitted beam. Therefore an asymmetry will be seen also in the solutions with respect to the sign of p. Note also that in the equations for phases, in the nonlinear case, . Thus in the nonlinear case the beams propagate in a medium with a local deviation parameter and local susceptibilities, which depend on the amplitudes and phases of the waves.

Nonlinear X-ray phenomena became more relevant with the advent of intense X-ray synchrotron sources and XFELs. In this work the third-order nonlinear X-ray two-wave dynamical diffraction in crystals is considered. A new exact solution for the Laue symmetric case of diffraction is obtained. The exact solution is presented via Jacobi elliptic functions. Two essential input parameters are the deviation parameter from exact Bragg direction and the intensity of the incident wave. The sign of a parameter, being the combination of these two parameters, defines the behavior of the wavefield inside the crystal. The exact solutions of three different types are found, corresponding to the positive and the negative signs of this parameter and to its zero value. For the non-zero values of this combined parameter the solutions are periodic and a nonlinear Pendellösung effect takes place. For the nonlinear Pendellösung distance, the appropriate formulas are obtained. For some zero values of the mentioned parameter the solutions are nonperiodic and are presented via elementary functions. The Pendellösung distance in this case tends to infinity. In comparison with the linear Pendellösung distance, the nonlinear one is greater due to the fact that the third-order nonlinear susceptibility has the opposite sign of the linear one. The obtained exact solution can be a base for experimental investigations of the third-order nonlinear dynamical diffraction, can be used for preparation of intense X-ray beams with given parameters, for manufacturing X-ray Bragg diffractive elements (monochromators, collimators, focusing elements) and for investigation of nonlinear X-ray interaction with matter. 2b1af7f3a8