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

where

Substituting Eq. (4.2-6) into the wave equation (3.5-7) again results in the Helmholtz equation for Uf ðr; f Þ.

e jkr=r is spherically symmetric. In this case, r2 in polar coordinates is simplified to

The Helmholtz equation with a spherically symmetric wave function U(r) becomes

Let U(r) be of the form f ðrÞ=r. It is not difficult to show by partial differentiation that

The solution of this equation is given by

f ðrÞ ¼ e jkr

Thus, UðrÞ ¼ e jkr=r:

Thus, if Uðx; y; 0Þ is known, Að fx; fy; 0Þ can be computed, followed by the computation of Uðx; y; zÞ. The limits of integration in Eq. (4.3-12 ) can be limited to a circular region given by

provided that the distance z is at least several wavelengths long so that the evanescent waves may be neglected.

Under these conditions, Eq. (4.3-8) shows that wave propagation in a homogeneous medium is equivalent to a linear 2-D spatial filter with the transfer function given by

(p

This is schematically shown in Figure 4.1. The impulse response, which is the inverse Fourier transform of this transfer function, is discussed in Example 4.6.

Propagation of a wavefield in the z-direction in a source-free space is correctly described by the propagation of the angular spectrum in the near field as well as the far field. Two other ways to characterize such propagation are in terms of the Fresnel and Fraunhofer approximations, discussed in Chapter 5. However, they are valid only under certain constraints.

Let F[ ] and F 1[ ] denote the forward and inverse Fourier transform operators, respectively. In terms of these operators, Eq. (4.3-12) can be written as

hpi

amplitude of a plane wave traveling in the direction specified by the direction cosines ax ¼ 2pfx, ay ¼ 2pfy, and az ¼ 2pfz, where fz ¼ 21p ½1 a2x a2y &1=2. The

EXAMPLE 4.3 Find the angular spectrum of a wave field at plane z ¼ a (constant)

if the wave field results from a plane wave Be jkx0 xe jkz0 z passing through a circular aperture of diameter d at z ¼ 0.

Solution: The wave at z ¼ 0 can be written as

Uðx; y; 0Þ ¼ Bejkx0 xcyl r

It is observed that the single plane wave component before the circular aperture is changed to a spectrum with an infinite number of such components after the aperture. They are mostly propagating in the direction of the incident wave field, with a spread given by the sombrero function. This spread is inversely proportional to d, the diameter of the circular aperture.

At a distance z, the angular spectrum becomes

4.4 FAST FOURIER TRANSFORM (FFT) IMPLEMENTATION OF THE ANGULAR SPECTRUM OF PLANE WAVES

The angular spectrum of plane waves relating Uðx; y; zÞ to Uðx; y; 0Þ can be implemented by the fast Fourier transform (FFT) algorithm [Brigham, 1974] after discretizing and truncating the space variables and the spacial frequency variables. For a discussion of the discrete Fourier transform and the FFT, see Appendix B. The discretized and truncated variables in the space domain and the spatial frequency domain are given by

x ¼ x n1

y ¼ y n2

ð4:4-1Þ

fx ¼ fx m1 fy ¼ fy m2