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.     

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.     

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 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.     

Acoustooptic interaction of beams close to forbidden directions

The theory of acoustooptic interaction of beams taking into account the drift and spreading of beams in an anisotropic medium is presented. The problem is solved analytically for a weak interaction of Gaussian beams close to forbidden directions. The possibility of existence of collinear diffraction of wave beams along the direction forbidden for interaction of plane waves is shown theoretically. The interaction appearing close to such a direction was found to result in a certain distortion of the beam shape. The calculation performed for a paratellurite crystal shows that the efficiency of the diffraction along the forbidden direction can reach 10% at the acoustic power usually used.     

双轴晶体中平面波解在主轴坐标系下的表达

在双轴晶体中,除光轴方向外,沿任一方向只存在2个具有不同偏振方向的平面波解,每个解有各自的传播常数。运用拉格朗日乘数法并借助双轴晶体的折射率椭球的概念,在主轴坐标系下求出双轴晶体的平面波解,得到每个解的空间色散关系,并从所得平面波解出发讨论平面波沿光轴方向传输的情况。最后将所得的结果运用到单轴晶体中检验其正确性。     

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.     

单轴晶体Ⅰ类相位匹配时的转换效率分析

 在考虑了二次谐波的吸收对转换效率影响的情况下,由光在介质中的波动方程,推导出负单轴晶体Ⅰ类相位匹配时平面二次谐波的耦合波方程和二次谐波转换效率的理论公式。在此基础上,再考虑走离效应的影响,给出了二次谐波转换效率的理论公式。根据这些公式,以负单轴晶体K₂Al₂B₂O₇（KABO）为例,对二次谐波的转换效率进行了数值计算。结果表明：考虑二次谐波的吸收时,存在倍频晶体长度的最佳值,当倍频晶体取该值时,二次谐波的转换效率最高;走离效应的存在使二次谐波的转换效率降低。     

Phase retrieval and coherent diffraction imaging by a linear scanning pinhole sampling array

We propose a new method for phase retrieval and coherent diffraction imaging by a specially designed pinhole sampling array (PSA) based on a liquid crystal spatial light modulation. We demonstrate that the phase and the amplitude of the wave front passing through a pinhole sampling array plate can be directly extracted from the inverse Fourier transform of the far-field diffraction intensity pattern. Scanning the whole surface of the wave front by such a series of the PSA plates, we can assemble the extracted complex amplitude to a two-dimensional discrete distribution of the sampled wave front covering the entire PSA plate plane in the scanning consequence. We called it linear scanning pinhole sampling array (LSPSA). Thus the wave front can be reconstructed which avoids any iterative algorithms. Furthermore, it provides a feasible approach for lensless coherent diffraction imaging in real-time.     

Eikonal approximation of dynamical diffraction equations for X-ray beams with a two-dimensional curvature of the wave front. II. Focusing of an X-ray beam by a crystal with non-plane entrance and exit surfaces. Symmetrical Laue case

On the basis of eikonal approximation developed in the first part of this work, diffraction of X-ray beam with a two-dimensional curvature of wave front and its focusing by a crystal with nonplane entrance and exit surfaces is studied in symmetrical Laue case. The expressions for eikonal and focusing distances inside and outside the crystal in the vacuum, are obtained. The size of the focal spot and the intensity gain in the spot, are estimated. The condition of point focusing is obtained. The conditions for obtaining practically achievable focusing distances are investigated.     

Lensless imaging of an arbitrary object

A new class of imaging systems that do not require the use of lenses or similar optical devices is introduced and theoretically investigated. In particular, it is demonstrated that, if an arbitrary plane object is illuminated by an appropriate spherical wave front (generated from a monochromatic point source), then a magnified image of the object intensity distribution can be observed in any transversal plane along the light-propagation direction within the far-field (Fraunhofer) diffraction region. The phenomenon is based on the fact that, under certain conditions, the spherical wave front can modify the energy's angular spectrum of the field distribution in the object plane such that this spectrum replicates the spatial intensity distribution of the object.     

Solitary Wave and Wave Front as Viewed From Curvature

The solitary wave and wave front are two important behaviors of nonlinear evolution equations. Geometrically, solitary wave and wave front are all plane curve. In this paper, they can be represented in terms of curvature c(s), which varies with arc length s. For solitary wave when s → ±∞, then its curvature c(s) approaches zero, and when s=0, the curvature c(s) reaches its maximum. For wave front, when s → ±∞, then its curvature c(s) approaches zero, and when s=0, the curvature c(s) is still zero, but c'(s) ≠ 0. That is, s=0 is a turning point. When c(s) is given, the variance at some point (x,y) in stream line with arc length s satisfies a 2-order linear variable-coefficient ordinary differential equation. From this equation, it can be determined qualitatively whether the given curvature is a solitary wave or wave front.     

Critical diffraction of irregular structure detonations and their predictability from experimentally obtained D−κ data

The present work reports new experiments of detonation diffraction in a 2D channel configuration in stoichiometric mixtures of ethylene, ethane, and methane with oxygen as oxidizer. The flow field details are obtained using high-speed schlieren near the critical conditions of diffraction. The critical initial pressure for successful diffraction is reported for the ethylene, ethane and methane mixtures. The flow field details revealed that the lateral portion of the wave results in a zone of quenched ignition. The dynamics of the laterally diffracting shock front are found in good agreement with the recent model developed by Radulescu et al. (Physics of Fluids 2021). The model provides noticeable improvement over the local models using Whitham’s characteristic rule and Wescott, Bdzil and Stewart’s model for weakly curved reactive shocks. These models provide a link between the critical channel height and the critical wave curvature. The critical channel heights and global curvatures are found in very good agreement with the critical curvatures measured independently by Xiao and Radulescu (Combust. Flame 2020) in quasi-steady experiments in exponential horns for three mixtures tested. Furthermore, critical curvature data obtained by others in the literature was found to provide a good prediction of critical diffraction in 2D. These findings suggest that the critical diffraction of unstable detonations may be well predicted by a model based on the maximum curvature of the detonation front, where the latter is to be measured experimentally and account for the role of the cellular structure in the burning mechanism. This finding provides support to the view that models for unstable detonations at a meso-scale larger than the cell size, i.e., hydrodynamic average models, are meaningful.     

The influence of progressive light diffraction in Debye-Sears phenomena

Two previous theories of Debye-Sears diffraction of light by plane ultrasonic waves are summarized. The theories assume that significant modulation of the light wave occurs in both phase and amplitude. In our own approach account is taken of both curvature of the light rays inside the ultrasonic beam and diffraction of the system alone at the exit plane, neglecting any diffraction effect in the liquid itself. In the Hargrove's method progressive diffraction in the medium is considered. Results from these theories are compared with experimental data obtained at a frequency of 5 MHz with ultrasonic waves having large amplitude and beamwidth (50 mm).     

Reducing the line curvature error of mechanically ruled gratings by interferometric control

Line curvature error greatly influences the quality of the diffraction wave fronts of machine-ruling gratings. To reduce the line curvature error, we propose a correction method that uses interferometric control. This method uses diffraction wave fronts of symmetrical orders to compute the mean line curvature error of the ruled grating, taking the mean line curvature error as the system line curvature error. To minimize the line curvature error of the grating, a dual-frequency laser interferometer is used as a real-time position feedback for the grating ruling stage, along with using a piezoelectric actuator to adjust the stage positioning to compensate the line curvature error. Our experiments show that the proposed method effectively reduced the peak-to-valley value of the line curvature error, improving the quality of the grating diffraction wave front.     

Optimal conditions of the acousto-optic interaction in crystals with gyrotropy

The conditions of the acousto-optic interaction in the crystals with gyrotropy are considered. The optimal length of the acousto-optic interaction and the diffraction efficiency in the gyrotropic direction are calculated. The diffraction efficiency is determined by the coupled wave method which allows calculation of the field interaction in the gyrotropic crystal [1]. It is theoretically shown and experimentally confirmed that the diffraction efficiency in the gyrotropic direction is independent of incident light polarization, when the polarization plane rotates by 45° along the interaction length.     

Pencil method in elastodynamics: application to ultrasonic field computation

The principles of pencil elastodynamics and, in more detail, some selected applications of pencil techniques to elastodynamics are described. It is shown how a systematic use of a matrix representation for the wave front curvature and for its transformations simplifies the handling of arbitrary pencils and, consequently, the field computations. Pencil matrix representations for the propagation into homogeneous solids made of isotropic or anisotropic media are derived. The use of matrix representations for pencil reflections on, or refractions through, arbitrarily curved interfaces, together with matrix representations for propagation into homogeneous media, allow us to derive an overall matrix formulation for elastodynamic propagation into complex heterogeneous structures. Combined with the classical Rayleigh integral to account for transducer diffraction effects, the proposed theory is applied to the prediction of ultrasonic fields radiated into complex structures by arbitrary transducers. Examples of interest for application to ultrasonic non-destructive testing are given.     

粗糙表面激光散射特性的理论研究

用矢量微扰动方法分析激光束在粗糙表面上散射的分布特性。详细讨论高斯光束的角谱和适用于准直激光束散射问题的波前分割概念认为，当用入射总功率作归一化因子时，准直粗激光束散射可近似地作为平面波处理，而细激光束散射则与平面波存在很大的差异，利用所得结果讨论了一维粗糙表面的激光隐身作用及椭圆偏振光入射时镜反射方向和后向散射场的偏振态。最后介绍两种线偏振光在粗糙表面上散射分布的差异。实验验证了主要结论。