Analytical and Numerical Approximations in Fresnel Diffraction

Abstract

Procedures for the fast and accurate numerical computation of Fresnel diffraction integrals are developed on the basis of geometrical properties of the Cornu spiral. The methods proposed allow the highly oscillatory integrals in Fresnel diffraction to be approximated by means of three simpler integrals and permit the calculation of these final integrals using analytical formulae.     

On the Validity of the Anomalous Diffraction Approximation

Abstract

The derivation of the anomalous diffraction approximation from Mie theory has been re-examined. The usual requirement ¦m ? 1¦ ? 1, m being the relative refractive index of the scatterer, is shown to be over-restrictive. A more appropriate requirement is found to be ¦m ? 1¦² ? ¦m + 1¦². The relationship between the anomalous diffraction approximation and another m→1 approximation, namely the eikonal approximation, has been investigated.     

A Comparison of Diffraction Theories for Off-bragg Replay

Two first-order analytic solutions and a second-order analytic solution for the case of lossless-dielectric unslanted volume transmission gratings are compared for the case of weak grating index modulation and significantly off-Bragg replay. It is shown that the analytic solution predicted using the Beta-value method, which has previously been found to agree more closely with experimental results for the unslanted case than the first-order K -vector closure method (of Kogelnik), also agrees more closely with the analytic expression produced by the second-order coupled-mode equations of Kong for this case. A numerical comparison of the first order theories and the rigorous coupled wave theory gives a similar result. Thus the Beta-value method offers definite advantages over the Kogelnik K -vector closure method for the unslanted transmission geometry.     

Investigation of Domains of Validity of Corrections to the Eikonal Approximation in Forward Light Scattering from Homogeneous Spheres

Corrections to the eikonal approximation in light scattering from non-absorbing, transparent homogeneous spheres are considered. Numerical comparisons with the exact theory for various values of relative refractive index m and diameter d are presented. Of particular interest are the results in the diameter range 0·2 μm ? d ? 5·0 μm where it has been shown that for m? 1·15, predictions of diameters from the eikonal approximation and Mie theory agree to within 6 per cent.     

Small-angle Light Scattering by Spherulites in the Anomalous Diffraction Approximation

Using the anomalous-diffraction (AD) light-scattering approximation, a relationship is derived between the scattering matrix of an object and its full Jones' matrix. This allows scattering to be calculated for dielectric objects having any kind of optical anisotropy, e.g. birefringent or optically active particles. The theory is used to calculate the small-angle light scattering (SALS) by a spherulite placed between two linear polarizers. Serious disagreements are found between the AD theory and the currently accepted theory of SALS based on the Rayleigh-Gans-Debye approximation.     

A boundary-only meshless method for numerical solution of the Eikonal equation

The radial basis function (RBF) collocation methods for the numerical solution of partial differential equation have been popular in recent years because of their advantage. For instance, they are inherently meshless, integration free and highly accurate. In this article we study the RBF solution of Eikonal equation using boundary knot method and analog equation method. The boundary knot method (BKM) is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution (MFS), the BKM uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to MFS, the RBF is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method (AEM). According to AEM, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Finally numerical results and discussions are presented to show the validity and efficiency of the proposed method.     

Residual Stresses Comparison Determined by Short-Wavelength X-Ray Diffraction and Neutron Diffraction for 7075 Aluminum Alloy

High strength Al alloy 7075 is widely used in aeronautic and space domains. However, when the thick plate of 7075 alloy is quenched improperly, large residual stress (RS) will be produced which leads to the loss of performance in resistance corrosion, fatigue and fracture. It is necessary to get the detailed information about residual stress distribution in the quenched plate. Conventional X-ray diffraction, a viable residual stress test method, can only reveal the RS state near surface layer due to limited penetrated depth. RS in the bulk components now can usually be analyzed by neutron diffraction or hard X-ray diffraction of high energy synchrotron in non-destructive way. Here a novel non-destructive method using short-wavelength characteristic X-ray diffraction meter (SWXRD) with the X-ray tube of tungsten anode target to determine the RS within the materials is developed in China. This paper revealed the RS distribution in a 20 mm thick plate of 7075 Al alloy determined by SWXRD and typical neutron diffraction (in LLB laboratory of France). The RS distribution through the thickness of Al plate shows a good agreement with the result determined by SWXRD compared with neutron diffraction.     

中子衍射应力测量试验中赝偏移现象的影响

主要介绍了中子衍射应力谱仪试验过程中,由非应力原因引起的衍射偏移的各种因素的影响。讨论了样品的定位与样品测量体积的选择对衍射峰赝偏移的影响,最后给出了赝偏移校正方法与试验过程中减小赝偏移的有效技术手段,使试验数据得到有效的处理,从而得到样品内部应力的真实信息。     

阶梯形浮雕和折射率位相光栅的衍射性质比较

位相全息图一般既有浮雕调制又有折射率调制,它们都会对衍射特性产生影响。利用以严格电磁波理论为基础的模式匹配法（ＭＭＴ）,从理论上比较了阶梯形浮雕位相调制光栅和折射率位相调制光栅对ＴＥ波的各级衍射效率。尽管它们同一周期内的相应部分所引起的位相变化相同,但随着光栅层厚度的加大两种位相调制光栅的各级衍射效率会出现差别。这种差别与周期有关。并将矢量理论的结果与标量理论的结果进行了比较,研究结果对研究全息术     

Optical Diffraction in Close Proximity to Plane Apertures. II. Comparison of Half-Plane Diffraction Theories

The accuracy and physical significance of the classical Rayleigh-Sommerfeld and Kirchhoff diffraction integrals are assessed in the context of Sommerfeld’s rigorous theory of half-plane diffraction and Maxwell’s equations. It is shown that the Rayleigh-Sommerfeld integrals are in satisfactory agreement with Sommerfeld’s theory in most of the positive near zone, except at sub-wavelength distances from the screen. On account of the bidirectional nature of diffraction by metallic screens the Rayleigh-Sommerfeld integrals themselves cannot be used for irradiance calculations, but must first be resolved into their forward and reverse components and it is found that Kirchhoff’s integral is the appropriate measure of the forward irradiance. Because of the inadequate boundary conditions assumed in their derivation the Rayleigh-Sommerfeld and Kirchhoff integrals do not correctly describe the flow of energy through the aperture.     

A Method of Representing Nonstationary Temperature Fields in the Best Approximations

Depending on the body geometry and on the types of internal source and external boundary heat loadings, a space is found for the heat equation, in which a high accuracy of representing a multidimensional temperature field by a single component is attained. An application of the developed analytical-numerical algorithm to solutions of hyperbolic heat equations is given, for which the synthesis of the sought temperature leads to the sum of terms of responses of simple inertial-wave or wave links.     

Comparison of the Two-dimensional Fraunhofer and the Two-dimensional Fresnel Approximations in the Analysis of Surface Roughness by Angle Speckle Correlation

Abstract

In this paper we discuss some new theoretical results which we reached in the analysis of a random surface roughness in the framework of the two-dimensional (2D) Fresnel approximation. A comparison with 2D Fraunhofer approximation results is made. It is shown for experimentally reasonable values of the geometrical arrangement parameters of the method that the surface roughness measurement can be interpreted in the 2D Fraunhofer approximation only for surfaces with high slopes of surface irregularities. Surfaces with small slopes of irregularities require interpretation of the surface roughness measurement in the 2D Fresnel approximation. In this case, not only the standard deviation of surface height fluctuations but also the correlation length of the surface can be determined.     

A Modification of the Enskog Equation for a Dense Gas of Soft Spheres

A modification of the Enskog kinetic equation for dense gases and gas mixtures of hard spheres is given for the case where the particle diameter depends on the relative velocity in a collision. Analytical expressions are obtained for transport coefficients of a simple dense gas, and the results of comparison of the viscosity coefficient with smoothed experimental data are given.     

Anomalous Depression of the Debye-Waller Factor and Observations of the Laue Spots Only at Forward X-ray Diffraction in the Light Atom Crystals as Quantum Solid

The ratio of 6·(1/2m)·(E ₀/c)² of 12.9 meV at E ₀=4 keV to k _{B D} of 2.27 meV for hcp-⁴He in the exponent 2M=[6·(1/2m)·(E ₀/c)²/k _{B D} ]·(t)sin² _B of the Debye-Waller factor D=exp(–2M) becomes 5.7, because of the small atomic mass m and the low Debye temperature _D . Conversely that of the heavy atom crystals becomes smaller than 1. An experiment on hcp-⁴He as an example of a light atom crystal at low temperature limit of (t)1 reveals that not only the maximum value of D is significantly reduced below 0.249 (=e ^–1.395) but also the values of D for all other Laue spots except the following observed six spots are crowded into a range of D<0.05. Therefore, only the six Laue spots of at 11.2 keV, (0002) at 20.3 keV, and at 14.4 keV and and at 23.9 keV in a low angular range of _B10.22° were observed over low values of e ^–1.395>De ^–3. At high angle of _B 45°, the values of E ₀ for the above six Laue spots are assigned to the region below 4.07 keV from the Bragg condition under constant values of D. Therefore, detection of them over the higher angular range of _B 45° is not impossible but quite difficult at low temperature X-ray diffraction, considering X-ray absorption by multi-thermal shields of the refrigerator, the thick sample holder as a pressure vessel and massive air in space. Generally this means that the large reduction of the Debye-Waller factor of hydrogen and helium drastically encloses the diffraction intensity and the number of observable Laue spots within lower bounds. This is the reason why definite Laue spots in the light atom crystals of hcp-⁴He could be observed only by forward X-ray diffraction. Conversely the recoil fraction of the diffuse scattering expressed by 1–D concomitantly increases as a major part of the total scattering. A proposal to measure thermal diffuse scattering as well as pressure experiments will provide insight into the wave-like character of constituent atom in solid helium.     

A Coordinate Transformation Method for Diffraction Gratings in Conical Mountings

Abstract

We present here a differential method to calculate the fields diffracted by corrugated gratings characterized by a dielectric constant. We consider the general case in which the wave-vector of the incident plane wave is not contained in the main section of the grating. To simplify the boundary conditions the regions above and below the grating surface are transformed into half planes by means of conformal and non-conformal mappings. Numerical examples validating the effectiveness of the developed algorithm are provided for the case of cycloidal and triangular profile gratings.     

Comparison of Electronic Speckle Laser Interferometry Hole-Drilling and X-ray Diffraction Techniques for Determination of Residual Stresses in the Heat Treated Steels

Carburizing is widely used to improve wear resistance and fatigue life of high duty machine parts. Fatigue performance of the carburized components is greatly dependent on the residual stress state in the surface layer. The aim of this paper is to measure the depth profiles of residual stresses in the carburized steels by electronic speckle laser interferometry (ESPI) assisted hole-drilling, and to compare the results with those measured by X-ray diffraction technique. To comprehend the differences in the residual stress state, the low-C steel components were carburized, and then, tempered in the range of 180–600 \(^{\circ }\)C. Microstructural investigations and hardness measurements were also conducted. The results obtained from both techniques gave identical results, and showed that the beneficial compressive residual stresses exist at the surface after carburizing, and their magnitudes decrease with increasing tempering temperature. It was concluded that ESPI assisted hole-drilling, with optimized drilling and stress calculation parameters, is suitable for determining the residual stress state of the carburized and tempered steels.     

一种X射线衍射线形分析中位错衬度因子的计算方法

基于Teodosiu各向异性弹性理论计算位错产生的位移场,建立适用于各滑移系中位错衬度因子的计算公式。以立方晶体为实例,对计算过程进行了化简。给出了f.c.c晶体Cu和b.c.c晶体Fe中分布于12个滑移系中位错在8种衍射面指数下的衬度因子,将计算结果同文献进行了对比。对历经不同应变准静态压缩加载的无氧铜开展了X射线衍射实验,阐述了衬度因子在消除衍射谱各向异性展宽中的作用。结合衍射线形分析,获取了样品中的亚晶粒尺寸、螺位错比例和位错密度。