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453.701 Linear Systems, S.M. Tan, The University of Auckland 7-1 Chapter 7 Wave propagation and Fourier optics Fourier optics describes propagation of light in optical systems using Fourier transform techniques. These techniques are useful since many operations are linear and spatially shift-invariant. They form the basis for analyzing and designing optical imaging and computation systems. 7.1 Propagation of light in the paraxial approximation Although the classical wave description of light is as a transverse electromagnetic wave, many effects can be studied using a scalar rather than the full vector wave equation. In free space, we have 2 2 2 2 @ @ @ 1 @ 2 + 2 + 2 = 2 2 (7.1) @x @y @z c @t In this equation represents a component of the electric or magnetic field. For monochromatic, coherent light, we can write (x; y; z; t) = (x; y; z; 0)e−jωt : (7.2) Substituting this into the wave equation, we obtain Helmholtzís equation @2 @2 @2 2 2 + 2 + 2 = °k ; (7.3) @x @y @z where !=k = c. Consider propagation which is nearly parallel to the z axis, so that (x; y; z; 0) = fz (x; y)ej kz ; (7.4) where fz (x; y) varies slowly with z. (Note, for example that for a plane wave travelling parallel to the z axis, fz (x; y) is constant). Substituting (7.