Optical diffraction in close proximity to plane apertures. III. Modified, self-consistent theory

Journal of Research of the National Institute of Standards and Technology, Sept-Oct, 2004 by Klaus D. Mielenz

The classical theory of diffraction at plane apertures illuminated by normally incident light is modified so that diffraction on the source side of the screen is taken into consideration and the energy transport across the aperture plane is described by continuous functions. The modified field expressions involve the sums and differences of the Rayleigh-Sommerfeld diffraction integrals as descriptors of a bidirectional flow of energy in the near zones on either side of the aperture. The theory is valid for unpolarized fields, and a pragmatic argument is presented that it is applicable to metallic as well as black screens. The modified field expressions are used for numerical near-field computations of the diffraction profiles and transmission coefficients of circular apertures and slits. In the mid zone the modified theory is reduced to the Fresnel approximation, and here the latter may be used with confidence.

Key words: bidirectional scalar fields; boundary-value theory; circular apertures; diffraction; Kirchhoff; irradiance; near zone; optics; polarization; Rayleigh; scalar wave functions; slits; Sommerfeld; transmission coefficients.

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1. Introduction

This is a continuation of previous papers [1,2] in which the physical significance of the classical Rayleigh-Sommerfeld and Kirchhoff diffraction integrals was assessed and their suitability for computations in the near zone was analyzed. The need for such computations arises, for example, in the evaluation of radiometric diffraction errors, where it is necessary to know the transmission coefficients of the apertures used for the measurements. The computation of these coefficients is a near-zone task even for large aperture-detector distances.

The specific situation considered is a plane aperture A contained in an infinitesimally thin screen S that occupies the xy-plane of a Cartesian coordinate system and is illuminated from the half space z < 0 by a normally incident monochromatic plane wave with irradiance [E.sub.0] and wavelength [lambda]. The resulting optical field is denoted by a scalar wave function,

U(P) = [square root of ([E.sub.0])]u(P), |u(P)| [less than or equal to] 1, (1)

and is expressed in the Rayleigh-Sommerfeld theory in terms of the surface integrals,

[u.sub.RS.sup.(p)] (P) = -[[ik]/[2[eth]]] [[integral].[A]] dQ [[e.sup.ikQP]/[QP]], z > 0, (2a)

[u.sub.RS.sup.(s)] (P) = [1/[2[eth]]] [[integral].[A]] dQ [[partial derivative]/[[partial derivative]z]] ([e.sup.ikQP]/[QP]) = [1/ik] [[[partial derivative][u.sub.RS.sup.(p)]]/[[partial derivative]z]], z > 0 (2b)

where a metallic screen illuminated by p- or s-polarized light is assumed. (1) The corresponding expression in Kirchhoff's theory, which is usually associated with black screens, is

[u.sub.K] (P) = -[1/[4[eth]]] [[integral].[A]] dQ [ik - [[partial derivative]/[[partial derivative]z]]] [[e.sup.ikQP]/[QP]] [equivalent to] [1/2] [[u.sub.RS.sup.(p)] (P) [u.sub.RS.sup.(s)](P)], z > 0. (2c)

In these equations, A denotes the aperture area, P = (x,y,z) is the point of observation, Q = ([xi],[eta],0) is a point inside the aperture, QP is the distance between these points, dQ is the surface element at Q, k = 2[pi]/[lambda] is the circular wave number, and the time dependence of the field is assumed as [e.sup.-i[omega]t].

Equations (2a,b) were reduced in Ref. [1] to previously unknown single integrals for the respective cases of circular apertures and apertures bounded by straight lines, and these were used for numerical computations of [u.sub.RS.sup.(p)] and [u.sub.RS.sup.(s)] that involved no simplifying assumptions and could be performed for arbitrarily small distances z from these apertures. The numerical results obtained were everywhere finite, free of singularities, and confirmed the well-known prediction that [u.sub.RS.sup.(p)] and [u.sub.RS.sup.(s)] reproduce the boundary values assumed in their derivation ([partial derivative][u.sub.RS.sup.(p)]/[partial derivative]z [right arrow] ik and [u.sub.RS.sup.(s)] [right arrow] 1 as z [right arrow] 0) but not the compatible values ([u.sub.RS.sup.(p)] [right arrow] 1 and [u.sub.RS.sup.(s)]/[partial derivative]z [right arrow] ik) which are implied in the classical postulate that the aperture field is the same as the unperturbed geometrical field incident on the screen. These inconsistencies obscured the differences between the Rayleigh-Sommerfeld and Kirchhoff integrals in the immediate proximity of the screen and made it impossible to assess their physical significance without additional considerations.

This impasse was overcome in Ref. [2] by evaluating Eqs. (2a,b) for the special case of a diffracting half plane and comparing them to the corresponding values, [u.sub.S.sup.(p)] and [u.sub.S.sup.(s)], given by Sommerfeld's rigorous theory of half-plane diffraction [3,4]. The agreement was remarkably good on the positive side of the screen, where the differences ([u.sub.RS.sup.(p,s)] - [u.sub.S.sup.(p,s)]) and their derivatives were negligibly small even at sub-wavelength distances z. Thus, it was decided that the aperture values given by the Rayleigh-Sommerfeld integrals are consistent with Sommerfeld's rigorous theory, so that attempting to improve them would be pointless.

Accordingly, it became apparent that the real problem with the Rayleigh-Sommerfeld and Kirchhoff theories was not their failure to reproduce the assumed boundary values but these boundary conditions themselves. The classical theories involve "inclination factors" which explicitly preclude a backward motion of diffracted light and, thus, any perturbation of the geometrical field on the source side. On the other hand, Sommerfeld's rigorous theory showed that the incident light is modified by diffraction before it reaches the screen, and therefore the notion of an unperturbed incident field is abandoned in this paper by adding a diffraction term to the geometrical field on the source side.