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##### From Diffraction, Fourier Optics and Imaging

7.1   INTRODUCTION

Approximations for computing forward and inverse diffraction integrals are of vital
significance in many areas involving wave propagation. As discussed in Chapters 4
and 5, approximations such as the Fresnel approximation, the Fraunhofer
approximation, and the more rigorous angular spectrum method (ASM) all involve
the Fourier transform, its discrete counterpart, the discrete Fourier transform, and its
fast computational routine, the fast Fourier transform (FFT).

The Fresnel approximation is valid at reasonable distances from the input plane
whereas the Fraunhofer approximation is valid in the far field. The ASM is a
rigorous solution of the Helmholtz equation; its numerical implementation is usually
done with the FFT, and possibly other digital signal processing algorithms, with their
related approximations [Mellin and Nordin, 2001; Shen and Wang, 2006].

When the sizes of the diffracting apertures are less than the wavelength, scalar
diffraction theory yields nonnegligible errors, and other numerical methods such as
the finite difference time domain (FDTD) method and the finite element method
(FEM) may become necessary to use [Kunz, 1993; Taflove and Hagness, 2005].
However, these methods are not practical with large scale simulations as compared
with methods utilizing the FFT. With diffracting aperture sizes of the order of...

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##### Topics of Interest

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