Rocking Curves of X-ray Asymmetrical Laue Diffraction with Two-Dimensional Curvature of Wave Front

X-ray asymmetrical Laue diffraction in a perfect crystal with plane front surface was considered taking into account the two-dimensional curvature of a wave incident on the crystal. Using an appropriate Green function the dependences of rocking curves on deviation angles from selected exact Bragg direction were examined in the diffraction plane and in the direction perpendicular to the diffraction plane in locally plane wave approximation. The dependence of rocking curves on the degree of asymmetry of diffraction geometry was investigated. Analytical expressions for half widths of rocking curves in the diffraction plane and in the direction perpendicular to the diffraction plane are obtained. Some requirements to spatial and temporal coherence are estimated.     

X-ray Laue diffraction with allowance for second derivatives of amplitudes in dynamical diffraction equations

Asymmetrical Laue diffraction in a perfect crystal with a plane input surface is considered. The second derivatives of amplitudes in the direction, perpendicular to diffraction plane in the dynamical diffraction equations are taken into account. Using the corresponding Green function a general form of the amplitude of diffracted wave in the crystal is derived. The sizes of the source in both directions as well as the source-crystal distance and non-monochromaticity of the radiation incident on the crystal are considered. On the base of obtained expression the coherent properties of the field depending on sizes of source and on the width of the spectrum of the incident radiation are analyzed. Taking into account the second derivatives of amplitudes with respect to the direction, perpendicular to the diffraction plane, the time-dependent propagation equations for an X-ray pulse in a perfect crystal, are given.     

X-ray Bragg diffraction with allowance for the second derivatives of amplitudes in dynamical diffraction equations

Asymmetrical Bragg diffraction in a perfect crystal with plane entrance surface is considered theoretically. In the dynamical diffraction equations the second derivatives of amplitudes in the direction, perpendicular to the diffraction plane are taken into account. Using the corresponding Green function, a general form for the amplitudes of waves in the crystal is presented. The sizes of the source in both directions as well as the source-crystal distance and non-monochromaticity of the radiation incident on the crystal are taken into account. On the basis of obtained formulas, the coherent properties of the field depending on the sizes of the source and on the width of spectrum of the incident radiation, are analized.     

Rocking Curves of X-Ray Asymmetric Bragg Diffraction with Two-Dimensional Curvature of Wave Front

The X-ray asymmetric Bragg diffraction in a perfect semi-infinite crystal with plane entrance surface is considered taking into account the two-dimensional curvature of the wave front of an incident wave. An expression for reflection coefficient using the Green function is obtained in the approximation of locally plane wave and the rocking curves are investigated as functions of angular departure from the selected exact Bragg direction in the diffraction plane and in the direction perpendicular to the diffraction plane. The shape, position and dimensions of total reflection region of rocking curves are studied depending on the degree of asymmetry of diffraction geometry. The requirements to the spatial and temporal coherence for obtaining the rocking curves are estimated.     

X-ray Laue diffraction with allowance for two-dimensional curvature of the wavefront: The concept a locally plane wave

Symmetrical Laue diffraction in a perfect crystal with a plane entance surface is considered. The two dimensional curvature of the wave front of the incident beam is taken into account. Using the corresponding Green function, a general expression for the amplitude of diffracted wave in the crystal is presented. Based on this expression, the concept of a locally plane wave is analyzed taking into account two-dimensional curvature of the wave front. This concept is used for obtaining the rocking curves depending on the angles of deviation from the chosen exact Bragg direction in both the diffraction plane and in perpendicular direction. The obtained result is compared with the result of the standard dynamical diffraction theory.     

Eikonal approximation of dynamical diffraction equations for X-ray beams with two-dimensional curvature of the wave front. I. Basic formulas

Taking into account the second derivatives of amplitudes with respect to the coordinate perpendicular to the diffraction plane, the eikonal approximation of dynamical diffraction equations for X-ray beams with two-dimensional curvature is presented. The diffracted field outside the crystal, in vacuum is described by the corresponding eikonal approximation of the parabolic equation of diffraction. The corresponding eikonal equations and complete integrals are obtained in case of a perfect crystal. A method is described, by which the eikonals and trajectories inside the crystal as well as in vacuum, outside the crystal, satisfying the given boundary conditions, can be obtained. This method allows to describe diffraction using sufficiently general assumptions on the incident wave front and for the non-plane entrance and exit surfaces of the crystal.     

X‐ray third‐order nonlinear plane‐wave Bragg‐case dynamical diffraction effects in a perfect crystal

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.     

Diffraction of sound by an elastic cylinder near the surface of an elastic halfspace

Diffraction of an acoustic wave by an elastic cylinder near the surface of an elastic halfspace is considered. The solution relies on a Helmholtz-type integral equation and uses the Green function of an elastic halfspace. The latter function is represented in the form of an integral over the Sommerfeld contour on the plane of a complex variable that has the meaning of the angle of the wave incidence on the halfspace boundary. An integral equation for the sound pressure distribution over the cylinder surface is derived. This equation is reduced to an infinite system of equations for the Fourier-series expansion coefficients of this distribution. The results obtained are valid for the diffraction of a cylindrical wave and a plane wave. They also describe the diffraction of a spherical wave when the transmitter and receiver are far from the cylinder and lie in one plane that is orthogonal to the cylinder axis.     

Theoretical studies of atom-solid elastic scattering by iterative integration using the Green function

The elastic scattering of low-energy light atoms from a perfect crystalline surface is studied by an iterative integration scheme using the Green function. The atom-solid interaction is represented by the often used Morse type surface potential. A varying number of closed and open channels is included in the calculation, according to necessicity. For a beam incident along the cyrstal symmetry directions, a scheme to utilize the symmetry condition for efficient computation is proposed. The diffraction intensities at a bound state resonance (selective adsorption) are calculated by properly selecting a reference potential for the calculation of the Green function. The calculations yield the resonant diffraction intensity patterns in agreement with previous calculations using a different numerical technique and with the experimental observations for the HeLiF and HeNaF systems. A calculation including 69 allowed diffracted beams (open channels) for the HeLiF system at normal incidence is also presented and comparison with experimental results is made to estimate the periodic potential parameter.     

Diffraction of sound by an elastic or impedance sphere located near an impedance or elastic boundary of a halfspace

An exact solution is obtained to the problem of sound diffraction by an elastic or impedance sphere located near an impedance or elastic boundary of a halfspace. The problem is solved using the Helmholtz integral equation in which the field of a point source in the halfspace with an elastic boundary is used as the Green function. The diffracted field is represented as a series expansion in spherical harmonics. The expansion coefficients are determined from a set of independent algebraic systems of equations. The matrix coefficients of these systems are determined as integrals of the products of the associated Legendre polynomials on the complex plane with respect to the real and complex angles of the sound incidence on the halfspace boundary. To decrease the number of such integrals, expansions using the Klebsh-Gordon coefficients are applied. As a result, algorithms for calculating the scattered field in the halfspace are obtained.     

Diffraction of stochastic electromagnetic fields by a hole in a thin film with real optical properties

The classical Kirchhoff theory of diffraction is extended to the case of real optical properties of a screen and its finite thickness. A spectral power density of diffracted electromagnetic fields by a hole in a thin film with real optical properties was calculated. The problem was solved by use of the vector Green theorems and related Green function of the boundary value problem. A spectral and spatial selectivity of the considered system was demonstrated. Diffracted patterns were calculated for the coherent and incoherent incident fields in case of holes array in a screen of perfect conductivity.     

Light diffraction features in an ordered monolayer of spheres

The structure and optical diffraction properties of monolayers of monodisperse spheres crystallized on transparent dielectric substrates are studied. Two types of diffraction phenomena are considered: surface light diffraction on the lattice of spheres and waveguide resonances in the monolayer plane. For experimental study of these phenomena, optical retroreflection and transmission spectra are measured as functions of the light incidence angle and azimuthal orientation of the incidence plane. The monolayer structures determined by scanning electron microscopy and light diffraction methods are in quantitative agreement. It is concluded that one-dimensional Fraunhofer diffraction is applicable to describe surface diffraction in the hexagonal lattice of spheres. In the case of oblique light incidence, anisotropy of diffraction and transmission spectra depending on the light incidence plane orientation with respect to the sphere lattice and linear polarization of incident light is detected. Waveguide resonances of the planar two-dimensional photonic crystal are approximated within the light diffraction model in the “empty” hexagonal lattice. The best approximation of the waveguide resonance dispersion is achieved using the effective refractive index, depending on the wavelength. Surface diffraction suppression by waveguide resonances of the photonic crystal is demonstrated. Surface diffraction orders are identified as diffraction at singular points of the Brillouin zone of the planar twodimensional photonic crystal.     

A simplified method for treating the dynamic response of embedded structures to SH-waves

A simplified method is proposed to treat the dynamic response of rigid embedded structures subjected to plane SH-waves. The soil-structure interface is discretized into a finite number of strips, resulting in an analysis which involves only plane surfaces and plane waves. The interaction force on any strip is determined easily and the contributions from all the strips are superposed to obtain the total force on the structure. The equation of motion then yields the steady state response of the structure. The Green function for the structure is obtained by Fourier synthesis. The transient response of the structure to any other excitation is then obtained by using the Green function and the Duhamel integral. Numerical examples are presented both to validate and illustrate the method.     

X‐ray focusing by the system of refractive lens(es) placed inside asymmetric channel‐cut crystals

An X‐ray one‐dimensionally focusing system, a refracting–diffracting lens (RDL), composed of Bragg double‐asymmetric‐reflecting two‐crystal plane parallel plates and a double‐concave cylindrical parabolic lens placed in the gap between the plates is described. It is shown that the focal length of the RDL is equal to the focal distance of the separate lens multiplied by the square of the asymmetry factor. One can obtain RDLs with different focal lengths for certain applications. Using the point‐source function of dynamic diffraction, as well as the Green function in a vacuum with parabolic approximation, an expression for the double‐diffracted beam amplitude for an arbitrary incident wave is presented. Focusing of the plane incident wave and imaging of a point source are studied. The cases of non‐absorptive and absorptive lenses are discussed. The intensity distribution in the focusing plane and on the focusing line, and its dependence on wavelength, deviation from the Bragg angle and magnification is studied. Geometrical optical considerations are also given. RDLs can be applied to focus radiation from both laboratory and synchrotron X‐ray sources, for X‐ray imaging of objects, and for obtaining high‐intensity beams. RDLs can also be applied in X‐ray astronomy.     

晶面偏角对利用Voigt函数法计算硅单晶本征晶格应变的影响

研究并制备了不同晶面偏角的Si(111)单晶,经过研磨和抛光使表面粗糙度低至3.4?达到超光滑水平,消除了表面和亚表面损伤层以及其所产生的应力变化.利用高精度X射线衍射仪分别测定了在不同晶面偏角条件下衍射曲线的半高全宽和积分宽度.应用Voigt函数法分析计算了微观应变,通过理论计算和实验对比可知,Si(111)单晶在晶面偏角达到0.749o时,偏角本身所带来的衍射峰半高全宽变化使计算出的应变值误差大于5%.研究结果为其他晶体类似研究提供了重要参考.     

Spatial structure of a focused X‐ray beam diffracted from crystals

The spatial structure of a beam focused by a planar refractive lens and Bragg diffracted from perfect silicon crystals was experimentally studied at the focal plane using a knife‐edge scan and a high‐resolution CCD camera. The use of refractive lenses allowed for a detailed comparison with theory. It was shown that diffraction leads to broadening of the focused beam owing to the extinction effect and, for a sufficiently thin crystal, to the appearance of a second peak owing to reflection from the back surface. It was found that the spatial structure of the diffracted beam depends on whether the crystal diffracts strongly (dynamically) or weakly (kinematically). The results help to understand the physical origin of the diffracted intensity recorded in a typical microbeam diffraction experiment.     

吸收限附近非对称反射条件下X射线的异常透射

研究了原子吸收限附近非对称布拉格条件下完整平板晶体的X射线异常透射．当衍射主要由原子散射因子的虚部引起时，在严格的布拉格角处，晶体内部驻波的波节位于衍射原子面上，从而导致异常透射的发生．透射波主要来源于晶体内部坡印廷矢量指向晶体下表面（入射面为上表面）的波场．该波场的有效吸收系数随非对称因子a的增大而减小，所以整个晶体的透射系数随a的增大而增大．当原子散射因子的实虚部对衍射的贡献之比一定时，晶体内坡印廷矢量偏离色散面实部法向的程度随反射的非对称程度的增大而增大． 关键词：     

Plane wave topography on crystals with step-like impurity distributions

In this paper a qualitative and quantitative study is presented of the contrast formation in the plane wave topography of quartz plates with step-like impurity atoms distributions which are connected with different lattice parameters. Contrast distributions are calculated by means of a simple local application of the dynamical theory of the X-ray diffraction from the perfect crystal and using a simple mathematical model of the lattice deformation according to the isotropic theory of elasticity. A surprisingly good agreement between calculations and experimental results can be stated.The authors thank Dr. V. Holý for helpful discussions.     

Ray picture of diffraction gratings

We elaborate on a geometrical picture of diffraction gratings. Exact paraxial propagation including coherence effects is obtained by allowing the geometrical rays to transport Wigner function instead of simply specific intensity. We apply this formalism to perfect and less than perfect gratings, illuminated by plane and Gaussian waves.