On spherical X-ray wave focusing upon backscattering from a parabolic cylinder crystal

The focusing of a spherical X-ray wave upon the Bragg backscattering from a crystal uniaxially bent to form a parabolic cylinder is theoretically studied. Based on geometrical optics, an analytical formula for the size of the crystal surface area involved in the diffraction reflection in the backscattering mode is derived. The analytical expression for the wave intensity distribution at the focus is analyzed and compared to that obtained previously.     

The spherical X-ray wave focusing by a bent crystal in the Johann scheme

The focusing of a spherical X-ray beam during the Bragg reflection from a bent crystal in the Johann scheme is considered. It is demonstrated that, with an allowance for the spherical aberration of the beam, the wave intensity in the focus is determined by the square modulus of Airy’s function.     

A kinematic approximation for the coefficient of X-ray wave backscattering from a strongly bent crystal

An analytical expression for the coefficient of Bragg backscattering (θ_B≅π/2) of an X-ray wave from a strongly bent thin crystal is obtained. It is shown that the amplitude reflection coefficient for backscattering from a strongly bent crystal differs from the value for a flat crystal of the same thickness by a small phase-dependent factor.     

Analytical solution of the Takagi equations for X-ray backscattering from a cylindrically bent crystal

An approximate analytical solution of the system of Takagi equations for the amplitudes of diffracted and transmitted waves is obtained for the backscattering of a plane X-ray wave from an elastically bent crystal. In the case of diffraction in the reverse direction (strict backscattering), the obtained solution is exact.     

The Bragg law modified for X-ray diffraction from a bent crystal in the backscattering mode

It is demonstrated for X-ray diffraction from a bent crystal in the backscattering mode that the Bragg angle in the crystal can differ from the value in vacuum by ∼10⁻². This difference exceeds the angular width of the region of total reflection for the CuKα radiation backscattered from the (220) plane of a silicon single crystal. The Bragg law modified with allowance for the refraction and crystal bending is presented.     

Approximate analytical solution of the Takagi equations for X-ray diffraction in the backscattering mode from a cylindrically bent crystal

The dynamic diffraction of a spherical X-ray wave from a cylindrically bent crystal in the backscattering mode has been theoretically studied. An approximate analytical solution of the system of Takagi differential equations is obtained under certain conditions imposed on the coordinates of atoms inside the crystal.     

The diffraction focusing of X-rays upon backscattering from a bent crystal covered with epitaxial film: Sensitivity with respect to the interplanar spacing mismatch between film and substrate

The diffraction focusing of an X-ray wave reflected from a thick bent crystal covered with an epitaxial film is theoretically studied in the case when the Bragg angle θ_B is close to π/2. It is shown that the X-ray wave focusing upon backscattering from this system is sensitive with respect to the relative interplanar spacing mismatch Δd/d between the epitaxial film and the thick crystal substrate. It is suggested that this effect can be employed for evaluating Δd/d with an accuracy on the order of 10⁻¹⁰.     

Formulation of light focusing through a plano-convex spherical lens in wave optics

Focusing of an incident light wave through a plano-convex spherical lens is discussed by calculating the light intensity distribution on the lens's optical axis after the incident wave is multiply scattered inside the lens. It is found that the size and location of the region into which the incident wave is focused are determined by two conditions. It is also found that it is possible for the wave to be focused into two such regions.     

Theory of X-ray diffraction on a lateral crystal with elastically bent lattice planes

A theory of X-ray diffraction on a lateral crystal with elastically bent lattice planes is developed within the framework of the kinematic approximation. Analytical solutions for coherent and diffuse scattering are obtained in application to high-resolution triple-axis diffractometry. The angular distribution of the intensity of coherent and diffuse scattering near a node of the reciprocal lattice are numerically simulated. It is shown that the pattern of X-ray diffraction from a crystal with a trapezoidal cross section and elastically bent lattice planes has a “butterfly” shape that was previously observed in experiment.     

On the coherence of X-ray beams focused by bent crystals

The spatial and temporal coherence of X-ray beams focused by bent crystals is considered for some practical schemes (backscattering and Johann schemes) widely employed in X-ray optics. It is shown that a high beam intensity achieved with crystal optics employing the Johann and backscattering schemes leads to incoherent focusing.     

A focusing X-ray diffractor: Effect of the crystal bending parameters on the spectral resolution

The dependence of the spectral resolution of a focusing X-ray diffractor on the shape and curvature of the reflecting crystallographic faces in the plane perpendicular to the focusing circle plane is theoretically studied in the point source approximation. The curvature parameters ensuring the best spectral resolution are determined for ellipsoidal and toroidal types of the crystal bending.     

Quasi-stationary distribution of relativistic particles in a bent crystal

It is shown that a quasi-stationary distribution of the transverse energies of relativistic particles (protons) moving in a bent crystal has a characteristic width of less than a few thousandths of an electron-volt.     

Diffuse X-ray scattering from a crystal with spheroidal pores

A theory of diffuse X-ray scattering from a crystal with spheroidal pores has been developed. Expressions for the Debye-Waller factor, intrinsic correlation function, and correlation volume have been derived. Reciprocal-space maps of the diffuse scattering have been numerically simulated for model crystals with pores having the shapes of spheroids and rectangular parallelepipeds. It is established that the pore size fluctuations influence the angular distribution of X-ray scattering intensity in the reciprocal space.     

Vertically bent single crystal silicon wafers as a neutron monochromator

A simple technique to construct monochromators for neutron beams is presented. The monochromator consists of a set of Si wafers stacked together and elastically bent to focus in the vertical geometry. The apparatus is described. It is shown that the peak reflectivity of the set is mainly governed by the number of wafers, while symmetry and smoothness of the rocking curve is achieved by the bending process. A prototype monochromator based on the (220) reflection of Si wafers has been tested and has successfully performed as a fixed monochromator selecting a flux of 3 × 10⁶ n/(s cm²) (at 45 meV) at the sample position (with 20′ collimation in the white beam and 10′ after the reflection from the monochromator). We conjecture that such a technique could be applied to any kind of material provided the wafers are cut from a single crystal with the surface parallel to the reflecting planes.     

Design of bent photonic crystal fiber supporting a single polarization

In this work, it is shown that the differential loss between the TE- and TM-polarized fundamental modes in a highly birefringent photonic crystal fiber (PCF) can be enhanced by bending the fiber. As a result, a design approach for single-mode single-polarization operation has been developed and is discussed. A rigorous full-vectorial H-field-based finite element approach, which includes the conformal transformation and the perfectly matched layer, is used to determine the single-polarization properties of such a highly birefringent PCF by exploiting its differential bending losses.     

Satellite peak splitting in the X-ray diffraction rocking curves of a crystal modulated by surface acoustic wave

A theory of the X-ray diffraction modulated by surface acoustic wave in a crystal has been developed in a kinematical approximation in application to double-and triple-crystal diffractometry. The proposed theory is applied to interpretation of the experimentally observed splitting of satellite peaks in the rocking curves.