Spherical-wave dynamical theory: I. Kato’s theory
This chapter is the first of two dealing with the dynamical diffraction of incident spherical waves. It makes use of Kato's theory, which is based on a Fourier expansion of the spherical wave. The transmission and reflection geometries are handled separately. Two methods of integration are given — direct integration and integration by the stationary phase method. The amplitude and intensity distributions of the reflected and refracted waves on the exit surface are calculated. It is shown that equal-intensity fringes are formed within the Borrmann triangle (Pendellösung fringes) that can be interpreted as due to interferences between the waves associated with the two branches of the dispersion surface. The integrated intensity is calculated and the influence of the polarization of the incident wave discussed. The last section describes the diffraction of ultra-short pulses of plane-wave X-rays such as those emitted by a free-electron laser and which can be handled by considering their Fourier expansion in frequency space.
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