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 the
wavelength used and in the near field, the ASM has been found to give satisfactory
results [Mellin and Nordin, 2001].With the ASM, one disadvantage is that the input
and output plane sizes are the same. The output plane size is usually desired to be
considerably larger than the input plane size in most applications, and this can be
done with the ASM only by repeatedly using the ASM in short distances and
additionally using filtering schemes to make the output size progressively larger.
The Fresnel, Fraunhofer approximations, and the ASM can be considered to be
valid in practice at small angles of diffraction. It is desirable to have approximate
methods that are valid at wide angles of diffraction in the near field as well as the far
field and also based on the Fourier integral to be implemented with the FFT so that
large scale computations can be carried out in a reasonable amount of time and
storage. Even if the spatial frequencies are sampled such that the FFT cannot be
used, the Fourier integral representation is still desirable as it is separable in the two
variables of integration, reducing 4-D tensor operations to 2-D matrix operations.
In this chapter, a new set of approximations having such features is discussed.
The approximations are based on the Taylor expansion around the radial distance of
a point on the output plane from the origin. With these approximations, what makes
possible to utilize the FFT is semi-irregular sampling at the output plane. Semi-
irregular sampling means sampling is first done in a regular array, and the array
points are subsequently perturbed to satisfy the FFT conditions.
Some interesting results will be related to the similarity of the proposed approximation
to the Fresnel approximation even though the new approximation is valid in
the near field and at wide angles whereas the Fresnel approximation is not. This is
believed to be one reason why the Fresnel approximation has often been used in the
near field in many applications. The results obtained in such studies are actually
valid at the output sampling points perturbed in position, as discussed in this paper.
This chapter consists of eight sections. Section 7.2 is a review of Fresnel and
Fraunhofer approximations previously discussed in Chapter 5. Section 7.3 introduces
the new methods with the radial set of approximations. Section 7.4 provides further
higher order improvements and error analysis. Section 7.5 shows how the method can
be used in inverse diffraction and iterative optimization applications. Section 7.6
provides numerical simulation examples in 2-D and 3-D geometries. Section 7.7 is
about how to increase accuracy by centering input and output plane apertures around
some center coordinates and possibly also using smaller subareas in large scale
simulations. Section 7.8 covers conclusions.
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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