Issues in Optical Diffraction Theory

This paper focuses on unresolved or poorly documented issues pertaining to Fresnel’s scalar diffraction theory and its modifications. In Sec. 2 it is pointed out that all thermal sources used in practice are finite in size and errors can result from insufficient coherence of the optical field. A quarter-wave criterion is applied to show how such errors can be avoided by placing the source at a large distance from the aperture plane, and it is found that in many cases it may be necessary to use collimated light as on the source side of a Fraunhofer experiment. If these precautions are not taken the theory of partial coherence may have to be used for the computations.In Sec. 3 it is recalled that for near-zone computations the Kirchhoff or Rayleigh-Sommerfeld integrals are applicable, but fail to correctly describe the energy flux across the aperture plane because they are not continuously differentiable with respect to the assumed geometrical field on the source side. This is remedied by formulating an improved theory in which the field on either side of a semi-reflecting screen is expressed as the superposition of mutually incoherent components which propagate in the opposite directions of the incident and reflected light.These components are defined as linear combinations of the Rayleigh-Sommerfeld integrals, so that they are rigorous solutions of the wave equation as well as continuously differentiable in the aperture plane. Algorithms for using the new theory for computing the diffraction patterns of circular apertures and slits at arbitrary distances z from either side of the aperture (down to z = ± 0.0003 λ) are presented, and numerical examples of the results are given. These results show that the incident geometrical field is modulated by diffraction before it reaches the aperture plane while the reflected field is spilled into the dark space. At distances from the aperture which are large compared to the wavelength λ these field expressions are reduced to the usual ones specified by Fresnel’s theory. In the specific case of a diffracting half plane the numerical results obtained were practically the same as those given by Sommerfeld’s rigorous theory.The modified theory developed in this paper is based on the explicit assumption that the scalar theory of light cannot explain plolarization effects. This premise is justified in Sec. 4, where it is shown that previous attempts to do so have produced dubious results.     

Optical Diffraction in Close Proximity to Plane Apertures. I. Boundary-Value Solutions for Circular Apertures and Slits

In this paper the classical Rayleigh-Sommerfeld and Kirchhoff boundary-value diffraction integrals are solved in closed form for circular apertures and slits illuminated by normally incident plane waves. The mathematical expressions obtained involve no simplifying approximations and are free of singularities, except in the aperture plane itself. Their use for numerical computations was straightforward and provided new insight into the nature of diffraction in the near zone where the Fresnel approximation does not apply. The Rayleigh-Sommerfeld integrals were found to be very similar to each other, so that polarization effects appear to be negligibly small. On the other hand, they differ substantially at sub-wavelength differences from the aperture plane and do not correctly describe the diffracted field as an analytical continuation of the incident geometrical field.     

Aberrations of diffracted wave fields. II. Diffraction gratings

The Rayleigh-Sommerfeld theory is applied to diffraction of a spherical wave by a grating. The grating equation is obtained from the aberration-free diffraction pattern, and its aberrations are shown to be the same as the conventional aberrations obtained by using Fermat's principle. These aberrations are shown to be not associated with the diffraction process. Moreover, it is shown that the irradiance distribution of a certain diffraction order is the Fraunhofer diffraction pattern of the grating aperture as a whole aberrated by the aberration of that order.     

Numerical Evaluation of Diffraction Integrals

This paper describes a simple numerical integration method for diffraction integrals which is based on elementary geometrical considerations of the manner in which different portions of the incident wavefront contribute to the diffracted field. The method is applicable in a wide range of cases as the assumptions regarding the type of integral are minimal, and the results are accurate even when the wavefront is divided into only a relatively small number of summation elements. Higher accuracies can be achieved by increasing the number of summation elements and/or incorporating Simpson’s rule into the basic integration formula. The use of the method is illustrated by numerical examples based on Fresnel’s diffraction integrals for circular apertures and apertures bounded by infinite straight lines (slits, half planes). In the latter cases, the numerical integration formula is reduced to a simple recursion formula, so that there is no need to perform repetitive summations for every point of the diffraction profile.     

Optical Diffraction in Close Proximity to Plane Apertures. IV. Test of a Pseudo-Vectorial Theory

Rayleigh’s pseudo-vectorial theory of the diffraction of polarized light by apertures which are small compared to the wavelength of light is analyzed with respect to its mathematical rigor and physical significance. It is found that the results published by Rayleigh and Bouwkamp for s-polarized incident do not obey the conditions assumed in their derivation and must therefore be dismissed. It is also found that the theory leads to paradoxical predictions concerning the polarization of the diffracted field, so that the pseudo-vectorial approach is intrinsically incapable of describing polarization effects.     

Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number

Analytic expressions are given for the on-axis intensity predicted by the Rayleigh-Sommerfeld and Kirchhoff diffraction integrals for a scalar optical system of high numerical aperture and finite value of Fresnel number. A definition of the axial optical coordinate is introduced that is valid for finite values of Fresnel number, for high-aperture systems, and for observation points distant from the focus. The focal shift effect is reexamined. For the case when the focal shift is small, explicit expressions are given for the focal shift and the axial peak in intensity.     

Young-Kirchhoff-Rubinowicz theory of diffraction in the light of Sommerfeld's solution

Kirchhoff's theory of diffraction is derived by transforming the exact solution of Sommerfeld into surface integrals for the half-plane problem. It is shown that the exact solution directly yields the integral theorem of Kirchhoff in the context of the modified diffraction theory of Kirchhoff. The line integrals of Young-Rubinowicz are also derived by considering the rigorous solution of the reflected scattered fields for grazing incidence.     

Diffraction of Gaussian beams by periodic and aperiodic rulings

The diffraction of Gaussian beams by periodic and aperiodic rulings is considered. The theory of diffraction is based on the Rayleigh-Sommerfeld integral equation with Dirichlet conditions. The transmitted power and the normally diffracted energy are analyzed as a function of the beam radius. Two methods to determine the Gaussian beam radius by means of periodic and aperiodic lamellar gratings are proposed. One is based on the maximum and the minimum transmitted power, and the other one considers the normally diffracted energy. Small and large Gaussian beam radii can be treated with these two methods.     

An Analytical Solution of Vector Diffraction for Focusing Optical Systems

Abstract

Vector diffraction theory for optical systems has been of interest for a long time. Ignatovsky and Wolf have formulated these problems in terms of diffraction integrals and Wolf has presented very interesting results. Usually, the quadrature of diffraction integrals is numerically intensive, therefore these problems have remained of interest and many authors have worked on the Ignatovsky-Wolf formulation or some variation thereof. This paper presents yet another method of solving diffraction integrals. Since a certain part of the kernel of these integrals is Riemann integrable in the interval [0, π], the Weierstrass theorem says that it can be approximated by a uniformly convergent series of orthogonal functions. Thus it is possible to expand these functions into a series of Gegenbauer polynomials of the first kind. Once these expansions are substituted in the diffraction integrals, the resulting integrals are readily evaluated, over the surface of unit sphere, in terms of the spherical Bessel functions and Gegenbauer polynomials. The results are particularly simple if the image plane is the focal plane. In this paper, we evaluate the diffraction integrals for several optical systems of arbitrary numerical aperture with or without obscuration, and for a parabolic reflector. The results presented here are in agreement with previously published results. The numerical computations are easy since all the functions are evaluated by adding a finite series. The calculations which for the basis of results presented in this paper were performed on a personal computer.     

Scalar and electromagnetic diffraction point-spread functions

We describe an accurate technique for computing the diffraction point-spread function for optical systems. The approach is based on the combined method of ray tracing and diffraction, which implies that the computation is accomplished in a two-step procedure. First, ray tracing is employed to compute the wave-front error in a reference plane on the image side of the system and to determine the shape of the vignetted pupil. Next the Rayleigh-Sommerfeld diffraction theory, combined with the Kirchhoff approximation and the Stamnes-Spjelkavik-Pedersen method for numerical integration, is applied to compute the field in the region of the image. The method does not rely on small-angle approximations and works well for a pupil of general shape. Both scalar and electromagnetic computations are discussed and numerical results are presented.     

Focal shifts in diffracted converging electromagnetic waves. II. Rayleigh theory

Part II of this study is an application of the Rayleigh vector diffraction integrals to an investigation of the effect of focal shifts in converging spherical waves diffracted in systems of arbitrary relative aperture. The results are compared numerically with those obtained in Part I [J. Opt. Soc. Am. A 22, 68 (2005)] from the Kirchhoff vector diffraction theory. The effect of the numerical aperture (NA) on focal shifts can be considered in two regions: When NA < or = 0.5 the system behaves like an paraxial system, and the Fresnel number is the dominant factor. When 0.5 < NA < or = 0.9 the absolute value of the relative focal shift decreases with increasing value of NA.     

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.     

Predictions of Rayleigh's diffraction theory for the effect of focal shift in high-aperture systems

In their work on diffraction [J. Opt. Soc. Am.51, 1050 1961], Osterberg and Smith have computed in an exact manner from the Rayleigh-Sommerfeld diffraction integral of the first kind the irradiance distribution along the axis of a converging spherical wave, and they found that in a scalar optical system of high relative aperture and finite value of Fresnel number, the central peak value of the axial irradiance may occur inside, at, or outside the geometrical focal point as the angular semiaperture of the system is less than, equal to, or greater than, respectively, a particular angle that falls near 70 degrees . These findings are now reexamined using a different assumption that takes into account diffraction at the edge of the aperture. Different results are obtained that agree well with the predictions of other theories of diffraction of light and give confidence to the common conclusions drawn by investigators of the effect of focal shift, that the point of the principal maximum of axial irradiance is not at the geometrical focus but shifted toward the aperture in systems of different relative aperture and finite value of Fresnel number.     

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.     

Focal shift in small-Fresnel-number focusing systems of different relative aperture

Axial irradiance distribution arising from the diffraction of a uniform, converging, spherical wave at a circular aperture is studied on the basis of scalar boundary-diffraction wave theory. The combined effects of Fresnel number and angular aperture on the focal shift are evaluated, and the validity of the results is checked against the Kirchhoff boundary conditions.     

Axial irradiance distribution throughout the whole space behind an annular aperture: comments

We comment on the recent paper by Harvey and Krywonos [Appl. Opt. 41, 3790-3795 (2002)], in which approximate irradiance calculations along the axis of a circular aperture illuminated by a plane wave are performed. As the starting point of their calculations, an approximated version (valid for z > lambda) of the Rayleigh-Sommerfeld diffraction integral is used. They based their numerical conclusions on a misleading "near field criterion," which guides the readers to the wrong idea that their calculations are valid even for the very near field behind the aperture. Their ideas are not original; the exact calculations of diffracted fields behind a circular aperture have been known for 40 years.     

Potential and limits of holographic reconstruction algorithms

For the purpose of ultrasonic nondestructive testing of materials, holography in connection with digital reconstruction algorithms has been proposed as a modern tool to extract crack sizes from ultrasonic scattering data. Defining the typical holographic reconstruction algorithm as the application of the scalar Kirchhoff diffraction theory to backward wave propagation, we demonstrate its general incapability of reconstructing equivalent sources, and hence, geometries of scattering bodies. Only the special case of a planar measurement recording surface, that is to say, a hologram plane, and a planar crack with perfectly rigid boundary conditions parallel to the hologram plane and perpendicular to the incident field yields a nearly perfect correlation between crack size and reconstructed image; the reconstruction algorithm is then referred to as the Rayleigh-Sommerfeld formula; it therefore represents the optimal case matched to that special geometrical situation and, hence, may be interpreted as a quasi-matched spatial filter. Using integral equation theory and physical optics, we compute synthetic holographic data for a linear cracklike scatterer for both plane and spherical wave incidence, the latter case simulating a synthetic aperture impulse echo situation, thus illustrating how the Rayleigh-Sommerfeld algorithm or its Fresnel approximation increasingly fail for cracks inclined to the hologram plane and excited nonperpendicularly. Furthermore, we point out how the physical data recording process may additionally influence the reconstruction accuracy, and, finally, guidelines for a careful and serious application of these holographic reconstruction algorithms are given. The theoretical results are supported by measurements.     

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.