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In previous chapters, the important role of the Fourier transformation in modern physical optics has been demonstrated in diffraction and interference phenomena and in the description of imaging systems, with emphasis on spatial spectral analysis. By solving the wave equation in Chapter 1, it was shown that the optical wavefront in the far field of the diffracting aperture (Fraunhofer zone) or in the focal plane of a lens is the Fourier transform of the optical wavefront in the diffracting aperture. In Chapter 3, the Fourier transform was used to develop the array theorem and the convolution theorem was derived mathematically using an integral representation of the delta function. Chapters 4 through 8 applied Fourier theory to imaging systems in order to treat them as linear. These systems are described equally well by either an optical impulse response or its Fourier transform, the optical transfer function. Fourier analysis was also used in Chapters 11, 22, 23, 25, and 26 to develop the principles of interferometry, coherence, and holography. In this chapter and the next four chapters, we will develop the application of Fourier transforms to the analysis and synthesis of optical signals using both optical analog methods as well as hybrid methods. In particular, we show how an analog optical system can be used to perform optical harmonic analysis resulting in many useful mathematical operations. As has been the case in these chapters, no attempt is made to include references to all the original work; rather, useful supplemental reference material is cited to support the particular approach being taken.
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