Calculation of the Rayleigh-Sommerfeld diffraction integral by exact integration of the fast oscillating factor

We describe a numerical method that can be used to calculate the propagation of light in a medium of constant (possibly complex) index of refraction n. The method integrates the Rayleigh-Sommerfeld diffraction integral numerically. After an appropriate change of integration variables, the integrand of the diffraction integral is split into a slowly varying and an (often fast) oscillating quadratic factor. The slowly varying factor is approximated by a spline fit, and the resulting Fresnel integrals are subsequently integrated exactly. Although the method is not as fast as methods involving a fast Fourier transform, such as plane-wave propagation or Fresnel approximation, it is accurate over a greater range than these methods.     

Mean and variance of energy reflected from a diffuse object illuminated by radiation with partial temporal coherence

Closed-form integral expressions are developed for the mean and variance of power and energy received from a diffusely reflective object upon illumination by laser radiation with partial temporal coherence. Expressions are presented in dimensionless form and analytic approximations to the integrals are given for signal variations at a receiver caused by fully developed laser speckle. Results are presented in terms of three parameters: the mutual Fresnel number of the receiver and object, the number of longitudinal modes of the illuminating source, and the dimensionless mode spacing of the illuminating source. The calculations assume high light levels and free-space geometry.     

Computation of Fresnel Integrals. II

This paper describes an improved method for computing Fresnel integrals with an error of less than 1 × 10⁻⁹. The method is based on a known approximate formula for a different integral which is due to Boersma and referenced by Abramowitz and Stegun.     

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.     

Calculation of diffraction patterns on a spatial surface

An approximated formulation of the Fresnel function is put forward and is used in the approximate evaluation of the Fresnel diffraction integral. By comparing the approximate formulation with the experimental measurements and calculations in the fast Fourier transform (FFT) method of the diffraction integral, we demonstrate that the proposed method is sufficiently accurate for calculating the Fresnel diffraction. For the diffraction field calculation on a spatial surface, the calculation speed of this method is usually higher than that of the FFT method.     

Two innovations in diffraction calculations for cylindrically symmetrical systems

Two mathematical innovations are presented that relate to calculating propagation of radiation through cylindrically symmetrical systems using Kirchhoff diffraction theory. The first innovation leads to an efficient means of computing Lommel functions of two arguments (u and nu), typically denoted by U(n)(u, nu) and V(n)(u, nu). This can accelerate computations involving Fresnel diffraction by circular apertures or lenses. The second innovation facilitates calculations of Kirchhoff diffraction integrals without recourse to the Fresnel approximation, yet with greatly improved efficiency like that characteristic of the latter approximation.     

Radiant fluxes from on- and off-axis point sources irradiating a circular disc through two different homogeneous isotropic media

The multidomain integral equation method is used to calculate fluxes of radiation from various on- and off-axis point sources passing through two different homogeneous isotropic media and striking a surface of a circular disc perpendicular to the optical axes of these sources. This method is dedicated to radiation passing through attenuating or nonattenuating media with a Fresnel interface and is applicable for arbitrary radiation patterns of point sources. The paper presents, firstly, the generalized multidomain integral equation method, expressed by double iterated integrals, for calculating radiant fluxes from arbitrary emitting point sources. This generalized method is simplified then to the form of multidomain single integral equation method applicable for rotationally symmetrical radiation patterns with optical axes perpendicular to the disc. Next, the simplified method is used for computer simulation of radiant fluxes incident on the disc from small Lambertian and Gaussian sources represented by point source models. All numerical results obtained from this simulation have shown high accuracy and efficiency of the presented method. Selected results are illustrated graphically and validated by Optical Software for Layout and Optimization (OSLO) from Lambda Research Corporation. Potential applications of the presented method include optical sensing and metrology, optical coupling, immersion microscopes, light-extraction problems and creative lighting design.     

A time-space decomposition method for calculating the nearfield pressure generated by a pulsed circular piston

A time-space decomposition approach is derived for numerical calculations of the transient nearfield pressure generated by a circular piston. Time-space decomposition analytically separates the temporal and spatial components of a rapidly converging single integral expression, thereby converting transient nearfield pressure calculations into the superposition of a small number of fast-converging spatial integrals that are weighted by time-dependent factors. Results indicate that, for the same peak error value, time-space decomposition is at least one or two orders of magnitude faster than the Rayleigh-Sommerfeld integral, the Schoch integral, the Field II program, and the DREAM program. Time-space decomposition is also faster than methods that directly calculate the impulse response by at least a factor of 3 for a 10% peak error and by a factor of 17 for a 1% peak error. The results show that, for a specified maximum error value, time-space decomposition is significantly faster than the impulse response and other analytical integrals evaluated for computations of transient nearfield pressures generated by circular pistons.     

Computation of Fresnel Integrals

This paper describes a method for spreadsheet computations of Fresnel integrals to six significant figures, based on successive improvements of known rational approximations which are accurate to only three figures. Outside the range of validity of the improved approximations, known series expansions are used to obtain the Fresnel integrals to six figures.     

Diffraction corrections in radiometry: spectral and total power and asymptotic properties

Wolf's result for integrated flux in the case of diffraction by a circular lens or aperture in the scalar, paraxial Fresnel approximation is considered anew. Compact integral formulas for pertinent infinite sums are derived, and the result's generalizations to extended sources and Planckian sources and asymptotic aspects at small wavelength and high temperature are all considered. Simplification of calculations for an actual absolute radiometer is demonstrated.     

Hyper singular boundary element formulation for the Grad-Shafranov equation as an axisymmetric problem

The Grad-Shafranov equation describes the magnetic flux distribution of plasma in an axisymmetric system such as a tokamak-type nuclear fusion device. This paper presents a scheme to solve the hyper singular boundary integral equation (HBIE) corresponding to this Grad-Shafranov equation. All hyper and strong singularities caused by differentials of the complete elliptic integrals have been regularized up to the level of the Cauchy principal value integral. Test calculations commonly using discontinuous boundary elements have been made to compare the HBIE solutions with the solutions of the standard boundary integral equation (SBIE).     

Study on numerical approaches for the evaluation of J-Integral Results in 3D problems

For numerical analyses of fracture mechanics problems often a combination of the finite element method with a post processor program for the evaluation of J-integral values is used. Instead of calculating a line integral as originally proposed by RICE it is reasonable to evaluate surface or volume integrals (for two or three dimensional calculations, respectively) in this case. In this paper 3D-formulations proposed by DE LORENZI and ATLURI are considered and several possibilities of the numerical realisation of these formulations are described. While the De Lorenzi proposal is contained as subroutines in the GRS-version of the standard finite element program ADINA, the Atluri formulation is used in separate post-processing programs developed by GRS. As a first application three-dimensional calculations of a compact-tension-shear specimen are considered.     

Advances in J‐integral estimation of circumferentially surface cracked pipes

This paper describes enhanced J‐integral estimation schemes for pipes with circumferential semi‐elliptical cracks subjected to tensile loading, global bending and internal pressure. These schemes are given in two different forms to cover the wide ranges of geometries and material parameters; the modified GE/EPRI method and the modified reference stress method. In the former method, new plastic influence functions for fully plastic J‐integral estimation are developed based on extensive three‐dimensional finite element calculations. In the latter method, new optimized reference loads are suggested and utilized to predict the J values. To verify the feasibility of these two schemes, J‐integral values obtained from further detailed FE analyses are compared to those from the proposed schemes. Because the estimated J‐integrals agree fairly well with the detailed FE analysis results, the new solutions can be applied for accurate structural integrity assessment of different size pipes with a circumferential surface crack.     

Fresnel diffraction by a circular aperture with off-axis illumination and its use in deconvolution of microscope images

The Fresnel approximation for off-axis illumination of a circular aperture is reexamined. The point-spread function for an aberration-free system can be expressed in terms of redefined optical coordinates. An improved expression is given for contours of constant intensity in the focal plane. The variation in axial width of the focal spot with angle of offset is discussed. The predictions are compared with exact calculations of the Rayleigh-Sommerfeld diffraction integral. Limitations for application in deconvolution of microscope images formed with objectives of finite tube length are discussed.     

A General Method for Evaluation of 2D and 3D Domain Integrals Without Domain Discretization and its Application in BEM

A robust method is presented to evaluate 2D and 3D domain integrals without domain discretization. Each domain integral is transformed into a double integral, a boundary integral and a 1D integral. Both integrals are evaluated by adaptive Simpson quadrature method. The method can be used to evaluate domain integrals over simply or multiply connected regions with any arbitrary form of integrands. As an application of the method, domain integrals produced in boundary element formulation of potential and elastostatic problems are analyzed. Several examples are provided to show the validity and accuracy of the method.     

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.     

The Intensity Distribution of a Gaussian Schell-model Beam Focused by an Aperture Lens

Abstract

Starting from the generalized Huygens-Fresnel diffraction integral, we study the intensity distribution of a Gaussian Schell-model (GSM) beam diffracted at an aperture lens. A great number of numerical calculations have been performed to illustrate the focused field characteristics. Isophote diagrams are given for systems of different Fresnel numbers, which focus GSM beams, and the related analysis is presented.     

Uniform theory for the diffraction of evanescent plane waves

A uniform diffracted field is obtained in terms of Fresnel functions with complex argument by subtracting the unit step function from the Fresnel integral. The method is applied to the problem of diffraction of inhomogeneous plane waves by a perfectly conducting half-plane and wedge. The results are plotted numerically and compared with results reported in the literature.     

Modal analysis of plates using the dual reciprocity boundary element method

This paper presents a new method for determining the natural frequencies and mode shapes for the free vibration of thin elastic plates using the boundary element and dual reciprocity methods. The solution to the plate's equation of motion is assumed to be of separable form. The problem is further simplified by using the fundamental solution of an infinite plate in the reciprocity theorem. Except for the inertia term, all domain integrals are transformed into boundary integrals using the reciprocity theorem. However, the inertia domain integral is evaluated in terms of the boundary nodes by using the dual reciprocity method. In this method, a set of interior points is selected and the deflection at these points is assumed to be a series of approximating functions. The reciprocity theorem is applied to reduce the domain integrals to a boundary integral. To evaluate the boundary integrals, the displacements and rotations are assumed to vary linearly along the boundary. The boundary integrals are discretized and evaluated numerically. The resulting matrix equations are significantly smaller than the finite element formulation for an equivalent problem. Mode shapes for the free vibration of circular and rectangular plates are obtained and compared with analytical and finite element results.     

Transformation of domain integrals to boundary integrals in BEM analysis of shear deformable plate bending problems

In this paper two techniques, dual reciprocity method (DRM) and direct integral method (DIM), are developed to transform domain integrals to boundary integrals for shear deformable plate bending formulation. The force term is approximated by a set of radial basis functions. To transform domain integrals to boundary integrals using the dual reciprocity method, particular solutions are employed for three radial basis functions. Direct integral method is also introduced in this paper to evaluate domain integrals. Three examples are presented to demonstrate the accuracy of the two methods. The numerical results obtained by using different particular solutions are compared with exact solutions. Received 27 January 1999