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

5

Fresnel and Fraunhofer Approximations

5.1INTRODUCTION

Fresnel and Fraunhofer approximations of the scalar diffraction integral allow simpler Fourier integral computations to be used for wave propagation. They also allow different input and output plane window sizes. However, they are valid only in certain regions, not very close to the input aperture plane. The valid regions for the Rayleigh–Sommerfeld integral, Fresnel, and Fraunhofer approximations are shown in Figure 5.1.

The Rayleigh–Sommerfeld region is observed to be the entire half-space to the right of the input diffraction plane. The Fresnel and Fraunhofer regions are parts of the Rayleigh–Sommerfeld region. Approximate bounds indicating where they start will be derived below as Eqs. (5.25) and (5.27) for the Fresnel region and Eq. (5.41) for the Fraunhofer region. However, these bounds need to be interpreted with care. See Chapter 7 for further explanation.

The term far field usually refers to the Fraunhofer region. The term near field can be considered to be the region between the input diffraction plane and the Fraunhofer region.

This chapter consists of seven sections. Section 5.2 describes wave propagation in the Fresnel region. The FFT implementation of wave propagation in the Fresnel region is covered in Section 5.3. The fact that the Fresnel approximation is actually the solution of the paraxial wave equation is shown in Section 5.4. Wave propagation in the Fraunhofer region is discussed in Section 5.5.

Diffraction gratings are periodic optical devices that have many significant uses in applications. They also provide excellent examples of how to analyze waves emanating from such devices in the Fresnel and Fraunhofer regions. Section 5.6 discusses the fundamentals of diffraction gratings. Sections 5.7, 5.8, and 5.9 highlight Fraunhofer diffraction with a sinusoidal amplitude grating, Fresnel diffraction with a sinusoidal amplitude grating, and Fraunhofer diffraction with a sinusoidal phase grating, respectively.

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

Copyright # 2007 John Wiley & Sons, Inc.

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It is observed that the second condition given by Eq. (5.2-7) yields an excessive value of z. However, this condition does not have to be true for the Fresnel approximation to be valid [Goodman]. In general, kr01 is very large, causing the quadratic phase factor ejkr10 to oscillate rapidly. Then, the major contribution to the diffraction integral comes from points near x x0 and y y0, called points of stationary phase [Stamnes, 1986]. Consequently, the Fresnel approximation can be expected to be valid in a region whose minimum z value is somewhere between the two values given by Eqs. (5.2-4) and (5.2-7). As a matter of fact, the Fresnel approximation has been used in the near field to analyze many optical phenomena. The major reasons for this apparent validity are the subject matter of Chapter 7.

EXAMPLE 5.4 Determine the Fresnel diffraction pattern if a square aperture of width D is illuminated by a plane wave with amplitude 1.

Solution: The input wave can be written as

The Fresnel diffraction is written in the convolution form as