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.

1. Introduction

In a previous paper [1] this author derived mathematically rigorous expressions for the classical Rayleigh-Sommerfeld and Kirchhoff boundary-value diffraction integrals pertaining to circular apertures and slits illuminated by normally incident plane waves. In spite of their functional differences, these diffraction integrals were found to be surprisingly similar and nearly indistinguishable in most of the near zone. They exhibited significant differences only in the immediate proximity of the aperture, but in this region their physical properties were obscured by the fact that they or their normal derivatives, or both, do not reproduce the assumed incident field. In these circumstances it was not possible to assess their physical significance by merely comparing them to one another. In the present paper, they will be re-examined by applying them to the specific case of diffraction by a reflecting half plane and their physical properties will be interpreted in the context of Sommerfeld’s [2] rigorous theory of half-plane diffraction and Maxwell’s equations.

2. Comparison of Scalar Wave Functions

The scalar wave functions U discussed in this paper all denote the complex disturbance at a point of observation P(x, y, z) in the diffraction pattern of a perfectly conducting, infinitesimally thin, semi-infinite screen that occupies the half plane x > 0, z = 0 of a cartesian coordinate system, as depicted in Fig. A1 of Appendix A. The primary field is assumed to be a monochromatic plane wave with irradiance E0, wavelength λ, and circular wave number k = 2π/λ that is normally incident from the half space z < 0 and is plane polarized so that, in accordance with Maxwell’s equations, ∂U/∂z or U are continuous and equal to zero on crossing the screen. The resulting diffraction pattern is independent of y and will be denoted by , so that we have |u| ≤ 1.

Fig. A1

Basic geometry and notation for Sommerfeld’s theory.

Sommerfeld’s half-plane theory dates back to the late 1800s and used to be discussed at length in textbooks [3-5]. However, it appears to be no longer included in modern curricula of theoretical optics, and therefore its main features are summarized and supplemented by new expressions for the diffracted irradiance in Appendix A, below. On combining Eqs. (A3a–c) and (A8a,b) of Appendix A it follows that, for normally incident light, Sommerfeld’s solution is reduced to where and V(ρ) is the complex Fresnel-type integral defined by Eq. (A3d). These expressions are rigorously valid everywhere in the xz-plane of Fig. A1, except that along the x-axis ρ and must be evaluated as where z = ±0 refers to the positive and negative sides of the screen, respectively. This distinction is necessary because and are discontinuous on crossing the screen, and is taken into account in Sommerfeld’s theory by “wrapping” the diffracting half plane in a semi-infinite, two-sided Riemann surface so that its positive and negative sides are distinguished by the values 2π and 0 of the polar angle ϕ in Fig. A1.

The corresponding results given by the Rayleigh-Sommerfeld theory are obtained from Eqs. (10a,b) of Ref. [1] by suitably modifying the limits of integration, leading to where are Hankel functions of the first kind and nth order. These expressions are valid for z > 0, only, and will be supplemented in this paper by the assumptions made in their derivation for z ≤ 0; namely, and respectively.

Kirchhoff’s diffraction integral, which will be required for the discussion in Sec. 3 is equal to the arithmetic mean of the Rayleigh-Sommerfeld integrals (2a,b), and can therefore be easily deduced from the above expressions.

In the paraxial Fresnel approximation where z is positive and large compared to λ and x/z is small all of the above-mentioned solutions converge to the familiar Fresnel limit uF. That is, where the right-hand expression follows from Eqs. (1a–c) by letting , so that , , and .1 The same result is obtained from Eqs. (2a,b) on replacing , and and β by the leading terms of their asymptotic and Taylor expansions. The Fresnel approximation, Eq. (3a), is estimated to be accurate within 1 % for z >> 100λ.

For numerical applications it is also useful to know that the above solutions all predict the same value, in the positive shadow boundary (x = 0, z > 0). In the case of the Rayleigh-Sommerfeld integrals, Eqs. (2a,b), this result follows from the identity and was used in this work as the starting value for recursive numerical integrations as described in Ref. [6].

The above expressions for and were used to compute the squared magnitudes of these functions in the immediate proximity of the positive and negative sides of the aperture plane, as shown in Figs. 1 and 2. For these computations, Eqs. (2a,b) were evaluated as noted above and the Fresnel sine and cosine integrals required for the computation of V(ρ) and were evaluated using the algorithms of Ref. [7]. The main conclusions drawn from these results are as follows.

Fig. 1

(-----) and (–—) vs x/λ at the distance z = +0.1λ from the aperture plane.

Fig. 2

(-----) and (–—) vs x/λ at the distance z = –0.1λ from the aperture plane.

On the positive side of the aperture plane the Sommerfeld and Rayleigh-Sommerfeld solutions are surprisingly similar, even at very small distances z. The real and imaginary parts of and contributing to the results plotted in Fig. 1 agree within ± 1 % or better for z = 0.1λ, and additional computations showed that this agreement improves rapidly for larger values of z. It follows that for all practical purposes the Rayleigh-Sommerfeld integrals are adequate for computations throughout the positive near zone, and hence it may be inferred that this will also be the case for the corresponding solutions for circular apertures and slits derived in Ref. [1].

The agreement for negative values of z is unsatisfactory. In Sommerfeld’s theory diffraction manifests itself as a field phenomenon that occurs on both sides of the aperture plane, so that the incident geometrical field is modified before it reaches the screen. On the other hand, in the Rayleigh-Sommerfeld theory diffraction on the source side is explicitly ruled out, and here the results obtained from Sommerfeld’s theory show that the assumed geometrical field (2d) is only a crude approximation of the true field. Thus, the main problem with the Rayleigh-Sommerfeld and Kirchhoff integrals appears to be not so much that they fail to reproduce the assumed geometrical field values, but that the latter are themselves objectionable.

The residual differences between and for z > 0 can be attributed to the imperfect boundary conditions assumed in the Rayleigh-Sommerfeld theory. These boundary values are step functions that violate the wave equation and are the probable cause of the fact, shown in Appendix B, that the Rayleigh-Sommerfeld integrals also do not obey the wave equation in the immediate proximity of the aperture plane. Although this wave-equation failure is small in most of the near zone, and thus unimportant for practical purposes, it is worthwhile to mention that it might be remedied by replacing the boundary values Eq. (2c) with the corresponding values given by Sommerfeld’s theory for z = +0; namely,

The real and imaginary parts of these functions are plotted in Fig. 3, where it should be noted that is discontinuous and singular, and is not continuously differentiable, for x = 0. Nonetheless, they constitute improved boundary values because Sommerfeld’s theory obeys the wave equation even at the diffracting edge itself (see Appendix B).

Fig. 3

Real (–—) and imaginary (-----) parts of the boundary values Eqs. (4a,b) predicted by Sommerfeld’s theory for p- and s-polarized lincident light.

When Eqs. (4a,b) are substituted into the derivation of the Rayleigh-Sommerfeld integrals for the half plane one finds where the integration now extends from –∞ to +∞. Because the boundary values Eq. (2c) and Eqs. (4a,b) are the same for x < 0 and the former are zero for x ≥ 0, this can be rewritten as where are correction terms that can be added to the Rayleigh-Sommerfeld integrals to convert them to the exact values given by Sommerfeld’s theory. These expressions should be free of errors because Eqs. (4c) and (4d) are rigorous expressions of the Helmholtz’ theorem in which and are the same on both sides of the equal sign.

This method was originally proposed by Braunbek [8-10], who envisioned its use for constructing improved solutions for large apertures of finite width and are bounded by straight or even curved edges. Braunbek’s work involved the assumption that and rapidly become negligibly small on the dark side of the screen, so that the effective ranges of integration in Eqs. (4f,g) are only a few wavelengths wide and approximative methods can be used. According to Fig. 3 this is a valid assumption for but not for , so that computational difficulties could be encountered in the case of .

3. Irradiance and Energy Flow

Although the squared magnitudes of scalar wave functions are commonly identified with the irradiance of the field, the data plotted in Figs. 1 and 2 must not be interpreted in this manner. The diffracted field specified by Sommerfeld’s solution is a bidirectional field composed of two plane waves, uS and ±ûS which propagate in the opposite directions of the incident primary field and its reflection from the screen. When Maxwell’s equations are invoked, as in Eqs. (A5) through (A7) of Appendix A, it is found that in accordance with the principle of interference these waves cannot interfere with one another2 so that the effective energy flow is composed of mutually incoherent components in the forward and reverse directions. For normally incident light, these respective directions are parallel and anti-parallel to the unit vector = [0,0,1] in the direction of the positive z-axis, and the final expression for the time-averaged Poynting vector (A7c) is where ES and ÊS are the forward and reverse irradiances incident on the opposite sides of any given area element dx dy containing the point of observation P.3 These irradiances are given by the squared magnitudes of the basic Sommerfeld functions uS and ûS themselves, and thus the quantities |uS − ûS|2 or |uS + ûS|2 do not represent the irradiances of the field for p- and s-polarized light. Accordingly, the forward and reverse irradiances of the field are independent of the state of polarization of the incident light, and in this connection it should also be noted that in practice the reverse irradiance ÊS is not easily observable as it may be obscured by a detector placed in the path of the forward field.

It now seems reasonable to interpret the Rayleigh-Sommerfeld theory in a like manner, so that the quantities and defined by Eqs. (2a,b) are also regarded as bidirectional wave functions that can be resolved into mutually incoherent forward and reverse components, uK and ûK. Thus we define, in analogy to Eq. (1a), and hence it follows that the corresponding forward and reverse irradiances, EK and ÊK, will be given by an expression analogous to Eq. (5),

It will be noted that the forward wave function uK defined by Eq. (6a) and Kirchhoff’s integral (2e) are identically the same, and therefore the subscript “K” was retained in the above equations. The Kirchhoff and Rayleigh-Sommerfeld solutions were originally derived on the mutually exclusive assumptions of black and metallic screens, and it is generally agreed that Eq. (2c) has no definable physical meaning as it would somehow imply the coherent superposition of two orthogonal states of polarization. However, in the present context, the Rayleigh-Sommerfeld integrals are interpreted as composite quantities and taking their sum and difference is tantamount to resolving them into their basic components. Accordingly, Kirchhoff’s integral uK now appears as an integral part of the Rayleigh-Sommerfeld theory for metallic screens so that and provide the framework for the evaluation of all field parameters while uK and its counterpart ûK define the flow of field energy. This new interpretation of Kirchhoff’s integral has a precise, physically realizable meaning.

A numerical comparison of the forward irradiances ES and EK defined by Eqs. (5) and (6b) is presented in Figs. 4 and 5. As expected, these quantities are essentially the same on the positive side of the aperture plane, the agreement being on the order of a few percent for z = +0.1λ and increasingly better for larger values of z. This confirms that the identification of |uK|2 with the forward irradiance EK is a valid assumption. As also expected, the agreement is poor on the negative side because in this region EK represents only the undiffracted geometrical field. The even symmetry of the irradiance Es shown in Fig. 5 suggests that the modification of the geometrical field due to diffraction is isotropic in the immediate vicinity of the edge.

Fig. 4

Forward irradiances ES(x,z) (-----) and EK(x,z) (–—) vs x/λ at the distance z = +0.1λ from the aperture plane.

Fig. 5

Forward irradiances ES(x,z) (-----) and EK(x,z) (–—) vs x/λ at the distance z = –0.1λ from the aperture plane.

4. Conclusions

The above comparison of the classical Rayleigh-Sommerfeld boundary-value theories with Sommerfeld’s rigorous theory for diffraction by a perfectly reflecting half plane has added substantially to the understanding of the physical significance of these theories.

It was found that the mathematical expressions and algorithms presented in Ref. [1] for the Rayleigh-Sommerfeld integrals are in very satisfactory agreement with Sommerfeld’s half-plane theory. Thus, they are well suited for computations in most of the positive near zone, and it is inferred that this will also be the case for the corresponding Rayleigh-Sommerfeld integrals and slits derived in Ref. [1]. Sommerfeld’s theory also confirms that, on the whole, the differences between these respective solutions for p- and s-polarized incident light are small so that polarization effects are small, as might be expected for normally incident light. All in all, it appears that the use of Helmholtz’ theorem has proved remarkably effective in compensating for the inadequate boundary conditions assumed in deriving the classical boundary-value integrals. The residual differences between the Rayleigh-Sommerfeld and Sommerfeld solutions are confined to sub-wavelength differences from the screen, and it is shown in Appendix B that in this region the former do not obey the wave equation.

The comparison with Sommerfeld’s theory and its interpretation in terms of Maxwell’s equations has also revealed a previously overlooked aspect of diffraction by a reflecting screen; namely, that the optical field is bidirectional and comprises light traveling in opposite directions even on the positive side of the screen. According to the principle of interference, the observable Poynting vector is given by the incoherent vector sum of its components in the forward and reverse components, and thus it is impermissible to express the near-zone irradiance of the field as the squared magnitudes of scalar wave functions. Rather, the latter must be resolved into their forward and reverse component and it turns out that Kirchhoff’s integral is the appropriate expression for the forward irradiance of the field even in the Rayleigh-Sommerfeld theory. The forward and reverse irradiances were found to be independent of the state of polarization of the incident field.

It was noted that the residual deficiencies of the Rayleigh-Sommerfeld and Kirchhoff solutions in the proximity of the positive aperture plane can be removed by replacing the originally assumed boundary values with those predicted by Sommerfeld’s theory. This was not be pursued further as it would produce only marginal improvements on the positive side of the screen, without removing the problem that the classical boundary-value integrals all exhibit discontinuities with respect to the incident geometrical field. A more effective approach would be the derivation of improved approximations for the entire field by constructing analytical continuations of the existing boundary-value solutions into the half space z ≤ 0. This will be attempted in a subsequent publication.