It is analytically shown that X‐ray Bragg reflections from periodical multilayer structures can be used for the excitation of guided modes. The X‐ray guided modes in (i) a guiding film embedded in a periodical multilayer and (ii) an air gap between two space‐separated multilayers have been considered. In the former case, analytical expressions have been obtained which describe a resonant X‐ray intensity enhancement taking place inside the guiding film under the guided mode excitation. Optimal parameters of the waveguide have been found, which are needed to achieve a maximum intensity gain. In the latter case, eigen guided modes arising in the air gap have been determined. Properties of the eigen modes have been studied.

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The well known Darwin approach is generalized to study the problem of X-ray dynamical diffraction in distorted crystals with a three-dimensional deformation ®eld of arbitrary form. The recursion equations describing X-ray diffraction in the above crystals have been derived. The analytical solution of these equations has been obtained within the kinematical approximation.At present, there exist at least two different methods that are widely used in theoretical studies of X-ray dynamical diffraction from distorted crystals. One method is based on the Takagi±Taupin differential equations (Taupin, 1964; Takagi, 1969) and the other on the recursion equations for the re¯ection and transmission coef®cients of layered crystals. The recursion equations were ®rst obtained by Darwin (1914) for solving the problem of Bragg-case dynamical diffraction by perfect crystals. Then Borie (1967) demonstrated that Darwin's approach could be extended to Laue-case diffraction. In fact, the method based on recursion formulae is a convenient approach to the problem of X-ray scattering from layered crystals and multilayered materials such as superlattices, heterostructures etc. [see e.g. Belyaev & Kolpakov, 1983; Vardanyan et al., 1985; Bartels et al., 1986; Caticha, 1994;Andreev & Prudnikov, 1998]. Up to now, the recursion formulae have generally been used for the description of X-ray diffraction by layered crystals with one-dimensional variation of the deformation ®eld uz and/or the polarizability 1z along the z axis normal to the crystal surface. The aim of the present communication is to extend Darwin's approach to the case of X-ray dynamical diffraction from distorted crystals which are characterized by a three-dimensional deformation ®eld ur of arbitrary form.In the spirit of Darwin's approach, we consider ®rst the problem of X-ray scattering by a single distorted atomic plane (Fig. 1). Let the atomic plane be irradiated by a plane monochromatic wave where r e is the classical electron radius, F25 0 is the structure factor, N xy is the number of scattering centers per unit area in the XY plane, 5 0 is the incidence angle, P is the polarization factor (P 1 for ' polarization and P cos25 0 for % polarization), and r H xy , r xy are the distances depending on the position of dxdy. In the case of the distorted atomic plane (see Fig. 1), r H xy q uq, r xy r 0 À q uq, where r 0 is the position vector of the observation point A, q xY yY 0 is the two-dimensional position vector in the XY plane, and uq is the displacement ®eld. The vector r 0 lies in the XZ plane and makes the angle 5 0 with the X axis. Note that in (1) we omit the time factor expÀi3t. If the conditions uqar 0 ( 1 and &ar 0 ( 1 are simultaneously ful®lled, then the phase 9 k 0 Á r H xy k 0 r xy in (1) can be written in the following approximate form:Here, Q k s À k 0 and k s k 0 r 0 ar 0 is the wave vector of the scattered wave. For a distorted atomic plane, the phase 9 differs from that in the case of an ideal (undistorted) atomic plane by the term Q Á uq. In the...

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