Spectrum analysis is the most widely used signal-processing technique in the physical sciences for gaining information about unknown signals. It is used in applications such as pattern recognition, cloud-cover analysis, inspection of manufactured items, particle-size analysis, measurements of turbulence, sea state analysis, characterization of the electromagnetic spectrum, determining direction of arrival of emitters, and structural analysis. The Fourier transform, as developed in Chapter 3, plays a central role in optical spectrum analysis. Signal features, such as periodic structures, are more easily detected in the Fourier domain than in the space domain because the energy from each frequency in the signal is concentrated at a particular point in the Fourier plane.

In the optical system shown in Figure 4.1, a signal stored on a spatial light modulator at plane *P*_{1} is illuminated by coherent light. In Chapter 3 we discussed the range of geometrical conditions for which the Fourier transform occurs; for convenience, we use a system in which the signal and Fourier planes are at the front and back focal planes of the lens. The complex-valued light at plane *P*_{2}, the image plane of the primary source, is the Fourier transform of the light at plane *P*_{1}:

where *s*(*x*, *y*) is the amplitude of the signal, *a*(*x*, *y*) is an aperture function, *x* and *y* are the ...

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