Scalar Wave Diffraction Research Articles

The Fresnel Approximation and Diffraction of Focused Waves

In this paper, diffraction of scalar waves by a screen with a circular aperture is explored, considering the incidence of either a collimated beam or a focused wave, a historical review of the development of the theory is presented, and the introduction of the Fresnel approximation is described. For diffraction by a focused wave, the general case is considered for both high numerical aperture and for finite values of the Fresnel number. One aim is to develop a theory based on the use of dimensionless optical coordinates that can help to determined the general behaviour and trends of different system parameters. An important phenomenon, the focal shift effect, is discussed as well. Explicit expressions are provided for focal shift and the peak intensity for different numerical apertures and Fresnel numbers. This is one application where the Rayleigh–Sommerfeld diffraction integrals provide inaccurate results.

Tip Diffraction Coefficient of Wide-Angle Cones

We derive a new asymptotic procedure for the analysis of scalar wave diffraction by semi-infinite circular cones and apply it to derive a new closed-form expression for the tip-diffracted field in the wide-angle limit. The wide-angle cone is important, for example, for terrain or urban propagation. The phenomenological complexity of the wide-angle cone case, compared with the narrow-angle cone that has been solved in the past, is due to the profound role of the interplay between the tip diffraction and the geometrical optics reflections. The new formulation significantly simplifies the calculations and provides a cogent physical interpretation for the scattered field. The results are validated against the exact conical-harmonics solution.

Principles of optics for engineers: diffraction and modal analysis

Introduction 1. Optical plane waves in unbounded medium 2. Superposition of plane waves and applications 3. Scalar wave equation and diffraction of optical radiation 4. Optical resonators and Gaussian beams 5. Optical waveguides and fibers 6. Guided wave interactions 7. Passive guided wave devices 8. Active optoelectronic guided wave components Appendix.

Polarization Aberrations in Astronomical Telescopes: The Point Spread Function

Detailed knowledge of the image of the point spread function (PSF) is necessary to optimize astronomical coronagraph masks and to understand potential sources of errors in astrometric measurements. The PSF for astronomical telescopes and instruments depends not only on geometric aberrations and scalar wave diffraction but also on those wavefront errors introduced by the physical optics and the polarization properties of reflecting and transmitting surfaces within the optical system. These vector wave aberrations, called polarization aberrations, result from two sources: (1) the mirror coatings necessary to make the highly reflecting mirror surfaces, and (2) the optical prescription with its inevitable non-normal incidence of rays on reflecting surfaces. The purpose of this article is to characterize the importance of polarization aberrations, to describe the analytical tools to calculate the PSF image, and to provide the background to understand how astronomical image data may be affected. To show the order of magnitude of the effects of polarization aberrations on astronomical images, a generic astronomical telescope configuration is analyzed here by modeling a fast Cassegrain telescope followed by a single 90° deviation fold mirror. All mirrors in this example use bare aluminum reflective coatings and the illumination wavelength is 800 nm. Our findings for this example telescope are: (1) The image plane irradiance distribution is the linear superposition of four PSF images: one for each of the two orthogonal polarizations and one for each of two cross-coupled polarization terms. (2) The PSF image is brighter by 9% for one polarization component compared to its orthogonal state. (3) The PSF images for two orthogonal linearly polarization components are shifted with respect to each other, causing the PSF image for unpolarized point sources to become slightly elongated (elliptical) with a centroid separation of about 0.6 mas. This is important for both astrometry and coronagraph applications. (4) Part of the aberration is a polarization-dependent astigmatism, with a magnitude of 22 milliwaves, which enlarges the PSF image. (5) The orthogonally polarized components of unpolarized sources contain different wavefront aberrations, which differ by approximately 32 milliwaves. This implies that a wavefront correction system cannot optimally correct the aberrations for all polarizations simultaneously. (6) The polarization aberrations couple small parts of each polarization component of the light (∼10-4) into the orthogonal polarization where these components cause highly distorted secondary, or “ghost” PSF images. (7) The radius of the spatial extent of the 90% encircled energy of these two ghost PSF image is twice as large as the radius of the Airy diffraction pattern. Coronagraphs for terrestrial exoplanet science are expected to image objects 10-10, or 6 orders of magnitude less than the intensity of the instrument-induced “ghost” PSF image, which will interfere with exoplanet measurements. A polarization aberration expansion which approximates the Jones pupil of the example telescope in six polarization terms is presented in the appendix. Individual terms can be associated with particular polarization defects. The dependence of these terms on angles of incidence, numerical aperture, and the Taylor series representation of the Fresnel equations lead to algebraic relations between these parameters and the scaling of the polarization aberrations. These “design rules” applicable to the example telescope are collected in § . Currently, exoplanet coronagraph masks are designed and optimized for scalar diffraction in optical systems. Radiation from the “ghost” PSF image leaks around currently designed image plane masks. Here, we show a vector-wave or polarization optimization is recommended. These effects follow from a natural description of the optical system in terms of the Jones matrices associated with each ray path of interest. The importance of these effects varies by orders of magnitude between different optical systems, depending on the optical design and coatings selected. Some of these effects can be calibrated while others are more problematic. Polarization aberration mitigation methods and technologies to minimize these effects are discussed. These effects have important implications for high-contrast imaging, coronagraphy, and astrometry with their stringent PSF image symmetry and scattered light requirements.

Diffraction of a plane wave by an infinitely long circular cylinder or a sphere: solution from Mie theory

Diffraction of an infinitely long circular cylinder normally illuminated by a plane wave is discussed from the classical Mie theory. A rigorous expression of the diffracted light is obtained, which is simply characterized by a factor (θ/2)/sin(θ/2) and the sinc function sin(αθ)/(αθ). Numerical calculation shows an apparent difference between our results and those from scalar wave diffraction theory, especially in large diffraction angles. The factor (θ/2)/sin(θ/2) is introduced into the diffracted light by a sphere, which leads to an alternative approximation of the diffracted light.

Field sensitivity to Lp variations of a scatterer

For the problem of diffraction of harmonic scalar waves by a lossless periodic slab scatterer, we analyze field sensitivity with respect to the material coefficients of the slab. The governing equation is the Helmholtz equation, which describes acoustic or electromagnetic fields. The main theorem establishes the variational (Fréchet) derivative of the scattered field measured in the H 1 (root-mean-square-gradient) norm as a function of the material coefficients measured in an L p ( p -power integral) norm, with 2 < p < ∞ , as long as these coefficients are bounded above and below by positive constants and do not admit resonance. The derivative is Lipschitz continuous. We also establish the variational derivative of the transmitted energy with respect to the material coefficients in L p .

Diffraction from arbitrarily shaped bodies of revolution: analytical regularization

A mathematically rigorous and numerically efficient approach, based on analytical regularization, for solving the scalar wave diffraction problem with a Dirichlet boundary condition imposed on an arbitrarily shaped body of revolution is described. Seeking the solution in an integral-equation formulation, the singular features of its kernel are determined, and the initial equation transformed so that its kernel can be decomposed into a singular canonical part and a regular remainder. An analytical transformation technique is used to reduce the problem equivalently to an infinite system of linear algebraic equations of the second kind. Such system can be effectively solved with any prescribed accuracy by standard numerical methods. The matrix elements of this algebraic system are expressible in the terms of the Fourier coefficients of the remainder. Due to the smoothness of the remainder a robust and efficient technique is obtained to calculate the matrix elements. Numerical investigations of structures, such as the prolate spheroid and bodies obtained by rotation of “Pascal’s Limacon” and of the “Cassini Oval”, exhibit the high accuracy and wide possibilities of the approach.

Regularization of a Boundary Value Problem of an Arbitrary Scalar Wave Diffraction by the Spherical Segment. The Neumann Problem

Regularization of a Boundary Value Problem of an Arbitrary Scalar Wave Diffraction by the Spherical Segment. The Neumann Problem

E-Polarized Scalar Wave Diffraction by Perfectly Conductive Arbitrary Shaped Cylindrical Obstacles with Finite Thickness

The Analytical Regularization Method is applied to the problem of E-polarized wave diffraction by arbitrary shaped cylindrical obstacle which is perfectly conductive and homogeneous in longitudional direction. The initial electromagnetic boundary value problem is reduced to the infinite algebraic system of the second kind by means of constructed analytical regularization procedure. As it is well-known, an equation of such a kind, in principal, can be solved with any predetermined accuracy by means of truncation procedure. Numerical implementation of corresponding analytical regularization procedure is suggested. Numerical results thus obtained show high quality of the algorithm, including relatively small values and uniform boundness of condition number of truncated algebraic systems when their size tends to infinity. By the qualitative property of infinite algebraic system of the second kind, it guarantees the stability of numerical process of truncated solving as well as the convergence of solution of truncated system to the solution of infinite system. Relevant numerical results, including condition number behaviour, current density, field space distribution and far field pattern for a few different resonant and non-resonant obstacles are presented.

Diffraction of scalar waves by an impedance cone of arbitrary cross-section

The diffraction of scalar waves by a cone of arbitrary cross-section, with constant impedance conditions on its surface, is studied for fields satisfying the Helmholtz equation. For this purpose, a particular use of integral transformations, with complex variables, is developed, which allows to consider boundary condition with radial dependence. The problem in question is reduced to that for a spectral function satisfying a Helmholtz type equation on the unit sphere with a hole cut out by the conical surface. This has to satisfy an impedance type boundary condition on the boundary of the hole, with a non-local term involving an integral operator on the spectral variable. Then the problem for the spectral function can be transformed to an integral equation of the second kind with non-oscillatory kernel. As an application, a closed form expression for the scattering diagram is deduced in the narrow cone approximation. Illuminations by a point source and by a plane wave are considered.

Scalar wave diffraction by tangent rays

Modern asymptotic methods are used to provide as complete as possible a description of the scattering of a high-frequency time-harmonic acoustic plane wave by a two-dimensional convex obstacle. The cases of infinite and finite scatterers are considered separately, and descriptions of the diffracted fields and the transition solutions valid across shadow boundaries are presented. For a finite scatterer, expressions are given for the directivity function (or angular variation) of the scattered field in the very far-field encompassing all angular directions; in particular this results in the prediction of a narrow range of angles for which the directivity is an order of magnitude lower than in general.

Diffraction of Scalar Waves At Plane Apertures: a Different Approach

We present a different approach of dealing with integral equations developed to tackle diffraction of scalar harmonic waves by a hole in a perfectly reflecting screen with a thickness assumed to be negligibly small. This approach uses a cartesian representation of the Green's functions which satisfy on the screen the same Dirichlet or Neumann boundary conditions as the total field incident, reflected and diffracted while the continuity of the total field is imposed through the aperture. The solution of the corresponding integral equations is given harmonic plane waves with an application to the diffraction at slits and rectangular apertures.

Diffraction of Electromagnetic Waves At Plane Apertures: a Different Approach

An approach based on integral equations, different from the conventional ones, was previously applied [1] to the diffraction of scalar waves at plane apertures. The same technique generalized to vector integral equations with tensor Green's functions solutions of Maxwell's equations, is used to deal with the diffraction of electromagnetic plane waves. Explicit formulae are given for the electric field with an application to the diffraction by a slit.

Deblurring by a local extrapolation scheme

A special deconvolution problem, where a parameter describes the degree of additive blurring, is considered. No a priori knowledge on the unblurred data is assumed. A standard solution based on the output least-squares formulation includes a regularization parameter into a linear, shift-invariant filter. An alternative approach that takes advantage of the analyticity of the smoothing process with respect to the blurring parameter is introduced. Here a simple local extrapolation scheme is employed. The problem is encountered in applications involving potential theories dealing with magnetostatics, electrostatics and gravity data. Inverse diffraction of scalar waves is also discussed. Examples are presented and the two methods compared numerically.

Weak-contrast edge and vertex diffractions in anisotropic elastic media

Diffraction phenomena caused by edges and vertices are important in a number of studies and applications of wave propagation methods. Diffractions play a fundamental role, for example, in the interpretation of seismic data from faults and other complex geological structures in which petroleum reservoirs are located. The geometric theory of diffraction provides a good description of the diffraction of scalar waves from an edge in a homogeneous medium assuming free or rigid boundary conditions. The solution becomes infinite at the shadow boundary between the illuminated zone and the shadow zone, so that, within this region, a boundary layer solution must be used. This can be achieved by analytic continuation of the geometrical ray solution or, equivalently, using a one-dimensional or two-dimensional diffusion equation for modelling the amplitude of the diffracted wave. An alternative, older method for obtaining asymptotic expressions for different parts of the wavefield is by asymptotic expansion of the Kirchhoff integral. The authors have developed a new reciprocal surface-scattering integral by applying the divergence theorem to the Born volume integral. This new integral is called the Born–Kirchhoff (BK) integral. The scattering surface is a finite sum of smooth surfaces separated by smooth curves with a finite number of corners. The BK integral is now a sum of integrals over each smooth surface. These integrals are evaluated by the method of stationary phase, resulting in specularly reflected waves and boundary diffracted waves represented by a line integral along the boundary of each surface. Further asymptotic evaluation of these line integrals results in expressions for edge-and corner-diffracted waves. Smooth approximations of the wavefield are given for the case that a reflection point and an edge-diffraction point are close to each other, and when an edge-diffraction point and a corner-diffraction point are close to each other. These formulas can be cascaded to provide asymptotic expressions for multiple converted, reflected, transmitted and diffracted waves in anisotropic, inhomogeneous, elastic media. In the case that the rays are well separated from all shadow zones, the new expressions satisfy reciprocity.

Scalar wave diffraction from zero-range scatterers

The zero‐range approximation for wave scattering is popularized. The general idea of the approximation is outlined and the theory is applied to solve an illustrative example: diffraction on a system of four identical scatterers forming a d-symmetry structure. The group theoretical methods are used to decompose a scattered wave into four partial waves adequate to the symmetry group of the target. Concepts of partial scattering amplitudes, associated phase shifts and partial cross sections for the nonspherical target considered are introduced and utilized. It is shown that under some conditions a phenomenon of resonant scattering may occur.

Scalar Wave Diffraction by a Rigid Cylindrical Rod of Finite Length with Impedance Ends

AbstractA uniform asymptotic high frequency solution is presented for the problem of diffraction of plane harmonic sound waves by an acoustically rigid cylindrical rod of finite length with the ends characterized by the same surface impedance. This problem is described by a modified Wiener‐Hopf equation of the third kind and then solved approximately. The solution involves two infinite sets of constants satisfying two infinite systems of linear algebraic equations. Numerical solutions of these systems are obtained for various values of the parameters of the problem and their effects on the diffraction phenomenon are shown graphically.

Numerical solution of the Huygens–Fresnel–Kirchhoff diffraction of spherical waves by a circular aperture

Using the rigorous wave-front formulation for scalar wave diffraction of Kraus [ J. Opt. Soc. Am. A6, 1196 ( 1989); J. Opt. Soc. Am. A9, 1132 ( 1992)], it is shown that the two-dimensional integral used to calculate the diffraction of spherical waves by a circular aperture may be reduced to a one-dimensional integral by choosing an appropriate coordinate frame. Both the two-dimensional integral and the one-dimensional integral must be evaluated numerically, but because each dimension must be sampled at approximately N locations to calculate accurately the integral (where N is the number of wavelengths across the aperture) the two-dimensional integration will require of the order of N2 evaluations of the integrand, whereas the one-dimensional integration will require of the order of only N evaluations, a substantial decrease in computing time for apertures that are large compared with optical wavelengths.

Relativistic Theory of Diffraction by Plane Apertures at Oblique Incidence

Abstract Presented is a relativistic theory for the diffraction of scalar and electromagnetic waves impinging at any incidence on plane apertures. The diffracted field is obtained by an integral relation with, as diffracting kernal, a socalled focus wave mode. The paraxial approximation of the diffracted field yields the usual Fresnel expressions.

On the solutions of a class of dual integral equations occurring in diffraction problems

It is shown that there exists a category of two-dimensional diffraction problems, which can be put into a ‘standard form‘ of dual integral equations. These diffraction problems include: diffraction of electromagnetic waves by a finite strip, a finite slit, the diffraction of scalar or vector elastic waves by a rigid strip or crack, etc. A general method for solving such dual integral equations is given by the artifice of constructing a set of functions of compact support biorthogonal to another given set of functions. The sufficient conditions for a given dual integral equations to be solvable in this manner are also determined. Hence, the method forms a complement to the Weiner-Hopf method. To illustrate the method solutions are obtained for a bench-mark problem : the diffraction of light by a finite perfectly conducting strip (or equivalently the diffraction of SH waves by a crack). Comparison with results obtained by others for low, intermediate and high frequencies show the utility and accuracy of the method for the entire range of frequencies. Both the near field and the far field are obtained, the latter is shown to correspond to the Fraunhoffer diffraction pattern for high frequency. It is also shown that for the equivalent crack problem the stress intensity factor (SIF) fluctuates rapidly with changes in the angle of incidence for high frequencies, thus making the SIF especially sensitive to angle of incidence at high frequencies.