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

It is observed that the sin c function above mixes the plane wave components in the original object wave. This means that each plane wave component in the original object wave is truncated by the aperture (-R,R) and therefore ‘‘spreads’’ by diffraction.

EXAMPLE 6.2 When M ¼ Q ¼ 1=l, show that T ðx; zrÞ can be exactly recovered from Uðx0; z0Þ by forward propagation of U ðx0;z0Þ instead of inverse propagation. Solution: Suppose that forward propagation from the plane z ¼ z0 to z ¼ 2z0 zr is carried out. As

2z0 zr z0 ¼ z0r

the forward propagation equation is given by

In other words, the original image can be reconstructed by back space propagation of the recorded wave or forward propagation of the complex conjugate of the recorded wave.

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. Semiirregular 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.

7.2 A REVIEW OF FRESNEL AND FRAUNHOFER APPROXIMATIONS

The first Rayleigh–Sommerfeld diffraction integral can be written as

where Uðx; y; 0Þ is the input field, Uðx0; y0; zÞ is the output field, k is the wave number equal to 2p=l, l is the wavelength, and r is the radius vector length from the point ðx; y; 0Þ to the point ðx0; y0; zÞ. In general, z is often large enough to approximate z=r2 by 1=z. Then, Eq. (7.2-1) becomes

94 WIDE-ANGLE NEAR AND FAR FIELD APPROXIMATIONS

In order to be able to use the Fourier transform, Eq. (7.3-3) is to be approximated by

The simplest radial set of approximations is obtained by approximating the radial distance r as

Equations (7.3-4) and (7.3-5) are also valid when x; y; x0, and y0 are sufficiently smaller than z (sufficiency conditions will be more clear later). Equation (7.3-5) is further improved in the next section. In order to obtain a Fourier transform relationship with the diffraction integral given by Eq. (7.2-9 ), the following is defined:

where ðxs; ysÞ is the regular output sampling point. The simultaneous solution of Eqs. (7.3-6) and (7.3-7) is given by

w in Eq. (7.3-5) is skipped. The resulting approximations will be referred to as the near field approximation (NFA) and the far field approximation (FFA) below. The general case covering both NFA and FFA will be referred to as NFFA.