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

where fx ¼ x0=lz and fy ¼ y0=lz.

5.8 FRESNEL DIFFRACTION BY A SINUSOIDAL AMPLITUDE GRATING

In order to compute Fresnel diffraction by a sinusoidal amplitude grating, we can use the convolution form of the diffraction integral given by Eq. (5.2-8). The transfer function of the system is rewritten here as

where ejkz can be suppressed as it is a constant phase factor.

The Fourier transform of the transmittance function is given by Eq. (5.7-9). In order to simplify the computations, we will assume that the diffraction grating aperture is large so that the Fourier transform of the transmittance function of the grating can be approximated by Eq. (5.7-7). Then, the frequencies that contribute are given by

ðfx; fyÞ ¼ ð0; 0Þ; ð f0; 0Þ; ðf0; 0Þ

The propagated wave has the following Fourier transform:

82 FRESNEL AND FRAUNHOFER APPROXIMATIONS

It is observed that sinusoidal amplitude grating in Section 5.7 has three orders of energy concentration due to three sin c functions, which do not significantly overlap if the grating frequency is much greater than 2/D. In contrast, the sinusoidal phase grating has many orders of energy concentration. Whereas the central order is dominant in the amplitude grating, the central order may vanish in the phase grating when J0ðm=2Þ equals 0.

5.10DIFFRACTION GRATINGS MADE OF SLITS

Sometimes diffraction gratings are made of slits. A two-slit example is shown in Figure 5.9, where b is the slit width, and f is the diffraction angle as shown in this figure.

The intensity of the wave field coming from the grating in the Fraunhofer region can be shown to be

d

b b

φ

Figure 5.9. A two-slit diffraction grating.

6

Inverse Diffraction

6.1INTRODUCTION

Inverse diffraction involves recovery of the image of an object whose diffraction pattern is measured, for example, on a plane. In the case of the Fresnel and Fraunhofer approximations, the inversion is straightforward. In the very near field, the angular spectrum representation can be used, but then there are some technical issues that need to be addressed.

The geometry to be used in the following sections is shown in Figure 6.1. The observation plane and the measurement plane are assumed to be at z ¼ z0 and z ¼ zr, respectively. Previously, zr was chosen equal to 0. The distance between the two planes is denoted by z0r. The medium is assumed to be homogeneous. The problem is to determine the field at z ¼ z0, assuming it is known at z ¼ zr.

Like all inverse problems, the inverse diffraction problem is actually difficult if evanescent waves are to be incorporated into the solution [Vesperinas, 1991]. Then, the inverse diffraction problem involves a singular kernel [Shewell, Wolf, 1968]. When the evanescent waves are avoided as discussed in the succeeding sections, the problem is well behaved.

This chapter consists of four sections. Section 2 is on inversion of the Fresnel and Fraunhofer approximations. Section 6.3 describes the inversion of the angular spectrum representation. Section 6.4 discusses further analysis of the results of Section 6.3.

6.2 INVERSION OF THE FRESNEL AND FRAUNHOFER REPRESENTATIONS

The Fresnel diffraction is governed by Eq. (5.2-13). As the integral is a Fourier transform, its inversion is given by

ð6:2-1Þ

Diffraction, Fourier Optics and Imaging, by Okan K. Ersoy

Copyright # 2007 John Wiley & Sons, Inc.

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