Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

54 SCALAR DIFFRACTION THEORY

n

V S

Figure 4.6. The volume and its surface used in Green’s theorem.

with the Helmholtz equation and converting it to an integral equation using Green’s theorem.

Green’s theorem involves two complex-valued functions UðrÞ and GðrÞ (Green’s function). We let S be a closed surface surrounding a volume V, as shown in Figure 4.6.

If U and G, their first and second partial derivatives respectively, are singlevalued and continuous, without any singular points within and on S, Green’s theorem states that

where @=@n indicates a partial derivative in the outward normal direction at each point of S. In our case, U corresponds to the wave field.

Consider the propagation of an arbitrary wave field incident on a screen at an initial plane at z ¼ 0 to the observation plane at some z. The geometry to be used is shown in Figure 4.7. In this figure, P0 is a point in the observation plane, and P1, which can be an arbitrary point in space, is shown as a point at the initial plane. The closed surface is the sum of the surfaces S0 and S1. There are two choices of the Green’s function that lead us to a useful integral representation of diffraction. They are discussed below.

S0

S1

R

Figure 4.7. The geometry used in Kirchoff formulation of diffraction.

4.5.1Kirchoff Theory of Diffraction

The Green function chosen by Kirchhoff is a spherical wave given by

where r is the position vector pointing from P0 to P1, and r01is the corresponding distance, given by

UðrÞ satisfies the Helmholtz equation. As GðrÞ is an expanding spherical wave, it also satisfies the Helmholtz equation:

where is the solid angle subtended by S0 at P0. The last integral goes to zero as

R ! 1 if

This condition is known as the Sommerfeld radiation condition and leads us to results in agreement with experiments.

What remains is the integral over S1, which is an infinite opaque plane except for the open aperture to be denoted by A. Two commonly used approximations to the boundary conditions for diffraction by apertures in plane screens are the Debye approximation and the Kirchhoff approximation [Goodman, 2004]. In the Debye approximation, the angular spectrum of the incident field has an abrupt cutoff such that only those plane waves traveling in certain directions passed the aperture are maintained. For example, the direction of travel from the aperture may be toward a focal point. Then, the field in the focal region is a superposition of plane waves whose propagation directions are inside the geometrical cone whose apex is the focal point and whose base is the aperture.

The Kirchhoff approximation is more commonly used. If an aperture A is on plane z ¼ 0, as shown in Figure 4.7, the Kirchhoff approximation on the plane z ¼ 0þ is given by the following equations:

It is observed that Uðx; y; zÞ and its partial derivative in the z-direction are discontinuous outside the aperture and continuous inside the aperture at z ¼ 0 according to the Kirchhoff approximation. These conditions are also called Kirchoff boundary conditions. They lead us to the following solution for UðP0Þ:

4.5.2Fresnel–Kirchoff Diffraction Formula

Assuming r01 is many optical wavelengths, the following approximation can be made: