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

96 WIDE-ANGLE NEAR AND FAR FIELD APPROXIMATIONS

where

Equations (7.3-8) and (7.3-9) remain the same because the term involving xx0 and yy0 is not modified. When z w as in the Fraunhofer approximation, the term pðwÞ in Eq. (7.4-3) is skipped. With the new approximations, Eq. (7.3-11) is replaced by

For a given z, the maximum error in the Fresnel approximation is proportional to ðx20 þ y20Þ2 whereas, in the NFFA, it is approximately given by

This error corresponds to the neglected terms in the second component u22 of the Taylor series expansion.

We note that changes are made only in the object coordinates with Eqs. (7.3-8) and (7.3-9). Otherwise, the integral in Eq. (7.4-5) is the same as the integral in Eq. (7.2-9) or Eq. (7.2-14). The output points ðx0; y0Þ are determined by Eqs. (7.3-8) and (7.3-9) in terms of ðxs; ysÞ given by Eqs. (7.2-15) and (7.2-16).

An interesting observation is that the Fresnel approximation examples cited in the literature are approximately correct in the near field at the points ðx0; y0Þ given by Eqs. (7.3-8) and (7.3-9), and not at the points ðxs; ysÞ. As such, these results are really the NFFA results.

In order to use the FFT, it is necessary to satisfy Eqs. (7.2-19) and (7.2-20). mx 0x and my 0y are the output sampling positions that are modified to do so. It is possible to consider other sampling strategies in which mx 0x and my 0y are replaced by some other output sampling positions, say, xs and ys. The results obtained above are valid in all such cases.

7.5INVERSE DIFFRACTION AND ITERATIVE OPTIMIZATION

Inverse diffraction involves recovery of uðx; yÞ from Uðx0; y0; zÞ as discussed in Chapter 6. In the case of the Fresnel approximation, Eq. (7.2-9) is a Fourier transform, and its inversion yields

ðð

Figure 7.1. Minimum z distance for the approximations to be valid as a function of x0 at the output plane with x equal to 1 mm.

interval. Hence, the new approximation is sufficiently accurate at all the output sampling points for x=l greater than 2. Figure 7.3 shows the growth of the difference between x0 and xs calculated with Eq. (28) as x0 increases when z ¼ 10 cm.

Figure 7.3. Change of x0 in mm as a function of x0 at the output plane when z ¼ 20 cm and x ¼ 1 mm.

Equations (7.3-8) and (7.3-9) show the relationship with the regularly sampled output points as in the Fresnel or Fraunhofer approximations, and the semiirregularly sampled output points as in the NFFA. Examples of these relationships are given in Tables 7.1 thru 7.3. The (x,y) sampling coordinate values in these tables are shown as complex numbers x þ iy. Table 7.1 shows the example of regularly sampled values. Table 7.2 shows the corresponding example of semi-irregularly sampled values. Table 7.3 is the difference between Tables 7.1 and 7.2.

In the next set of experiments in 2-D, a comparison of the exact, Fresnel and NFFA approximation integrals was carried out for diffraction of a converging spherical wave from a diffracting aperture. The parameters of the experiments are as follows:

Diffracting aperture size ðDI Þ ¼ 30 m; z ðfocusing distanceÞ ¼ 5 mm;

N ¼ 512; l ¼ 0:6328 m

The numerical integration was carried out with the adaptive Simpson quadrature algorithm [Garner]. The output aperture sampling was chosen according to the FFT conditions. Then, the full output aperture size is given by

However, only the central portion of the output aperture corresponding to 64 sampling points and an extent of 6.7 mm was actually computed. As z ¼ 5 mm, this represents a very wide-angle output field.