By Keigo Iizuka; John Wiley & Sons, Inc
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Additional resources for Elements of Photonics, Volume I : In Free Space and Special Media
Example text
6 Shah Function (Impulse Train Function) An array of equally spaced delta functions, such as shown in Fig. 13a, is called the shah function (or comb function), and it is denoted by x . 112 n nD 1 Two major applications of x are the following: 1. The generation of a sampled function. 13 Shah function and its applications. (a) Shah function (c) Step and repeat function R x . x . (b) g x sampled by x. SPECIAL FUNCTIONS IN PHOTONICS AND THEIR FOURIER TRANSFORMS 31 Fig. 13b. 113 x A typical use for gs x is in optical signal processing, as, for example, processing a newspaper photograph that is made of closely sampled points.
7. Compared to sinc(f), the side lobes of sinc2 f are significantly lower in height. An input aperture having a x0 distribution is used to reduce the side lobes in its diffraction pattern. This technique is called apodizing to reduce the side lobes. Apodal means a creature without legs, as, for example, eels or whales. Apodization of a lens is performed by darkening the lens toward the edge of the lens. Apodization of a radiation pattern from an antenna array is achieved by reducing the element antenna current toward the edge of the array [9].
B) Diffraction pattern of (a). 38 FOURIER OPTICS: CONCEPTS AND APPLICATIONS The extent of the array is limited to c ð n in the x0 and y0 directions. 130 The step and repeat functions in the square brackets are truncated by the rectangle functions x0 /c and y0 /n . The Fourier transform of the transmittance pattern is obtained from Eqs. 131 The order of the dimensions in the transmittance pattern is a
