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Imaging of periodic objects in free space brings us to the Talbot effect in optics, for x-rays and for matter waves. X-ray dynamical diffraction imaging of periodic objects inside a crystal has scientific and applied interest. Thus, we arrive at the concept of the dynamical diffraction Talbot effect in a crystal. This work is the first attempt to investigate the Talbot effect inside a medium. Using the Green function formalism, an exact formula for the diffracted wave amplitude is found. By means of 'paraxial' approximation, which is an analogue of the paraxial approximation in optics, it is shown that the dynamical diffraction Talbot effect takes place. Expressions for polarization sensitive Talbot and corrected Talbot distances are obtained. We analyse the influences of absorption, Bragg filtration of harmonics and polarization on the dynamical diffraction Talbot effect. For the first time, simulated Talbot carpets inside the crystal are obtained, which show that the predictions, obtained by 'paraxial' approximation, are true. We present the dynamical diffraction Talbot carpets observation method by means of a wedgeshaped crystal. The dynamical diffraction Talbot effect can be used for investigation of objects and crystal defects and deformations. Dynamical diffraction Talbot effect investigations in optics, for electrons and neutrons, are possible.

Two-wave symmetric Bragg-case dynamical diffraction of a plane X-ray wave in a crystal with third-order nonlinear response to the electric field is considered theoretically. For certain diffraction conditions for a non-absorbing perfect semi-infinite crystal in the total reflection region an analytical solution is found. For the width and for the center of the total reflection region expressions on the intensity of the incidence wave are established. It is shown that in the nonlinear case the total reflection region exists below a maximal intensity of the incidence wave. With increasing intensity of the incidence wave the total reflection region's center moves to low angles and the width decreases. Using numerical calculations for an absorbing semi-infinite crystal, the behavior of the reflected wave as a function of the intensity of the incidence wave and of the deviation parameter from the Bragg condition is analyzed. The results of numerical calculations are compared with the obtained analytical solution.

The theoretical investigation of double-slit dynamical X-ray diffraction in ideal crystals shows that, on the exit surface of crystals, interference fringes similar to Young's fringes are formed. An expression for the period of the fringes was obtained. The visibility of the fringes depending on temporal and spatial coherent properties of the incident beam is studied. The polarization state of the incident beam also affects the visibility of the fringes, which in turn depends on the size of the slits. The deviation from Bragg's exact angle causes a shift of the fringes and can also affect the amplitude of the intensity. One of the parameters on which the visibility of the fringes depends is the source-crystal distance. The proposed scheme can be used as a Rayleigh X-ray interferometer. Use of the scheme as a Michelson X-ray stellar interferometer is also possible.

An X-ray dynamical diffraction Fraunhofer holographic scheme is proposed. Theoretically it is shown that the reconstruction of the object image by visible light is possible. The spatial and temporal coherence requirements of the incident X-ray beam are considered. As an example, the hologram recording as well as the reconstruction by visible light of an absolutely absorbing wire are discussed.

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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. RECEIVED

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