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In previous chapters, wave propagation and imaging with coherent and monochromatic wave elds were discussed. In this chapter, this is extended to wave propagation and imaging with quasi-monochromatic coherent or incoherent wave elds. Monochromatic wave elds have a single temporal frequency f. Nonmonochromatic wave elds have many temporal frequencies. Quasi-monochromatic wave elds have a temporal frequency spread f , which is much less than the average temporal frequency fc. In practical imaging applications, wave elds can usually be assumed to be quasi-monochromatic. This chapter consists of 10 sections. The rst few sections lay the groundwork for the theory that is pertinent for analyzing quasi-monochromatic waves. Section 10.2 introduces the Hilbert transform that is closely related to the Fourier transform. Its main property is swapping the cosine and sine frequency components. It is a tool that is needed to de ne the analytic signal described in Section 10.3. The analytic signal is complex, with its real part equal to a real signal and its imaginary part equal to the Hilbert transform of the same real signal. In addition to its use in analyzing quasi-monochromatic waves, it is commonly used in analyzing single-sideband modulation in communications. Section 10.4 shows how to represent a quasi-monochromatic wave in terms of an analytic signal. The meanings of quasi-monochromatic, coherent, and incoherent waves are more closely examined in Section 10.5 with spatial coherence and time coherence concepts. The theory developed up to this point for simple optical systems is generalized to more complex imaging systems in Section 10.6. Imaging with quasi-monochromatic waves is the subject of Section 10.7. The difference between coherent imaging and incoherent imaging becomes clear in this section. A diffraction-limited imaging system is considered as a linear system in Section 10.8, and its linear system properties are derived for coherent and incoherent imaging. One of these properties is the optical transfer function. How it can be computed with a computer is the topic
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(iii) Expression (4.17) follows from the second expression of J00 . The numerical computation of 2 has some dif culties related to the behavior T of the integrand near the diagonal. Since 2 = 0 A+ (u, u) ds dt, one can use s,t the next proposition to describe the function A+ (u, u) when |t s| is small. For s,t the proof, which we are not going to give here, one can use Maple to compute its Taylor expansion as a function of t s in a neighborhood of 0. Proposition 4.5 (Aza s et al., 1999). Let the Gaussian stationary process {X(t) : t R} satisfy the hypotheses presented earlier in this section. Assume, moreover, that 8 is nite. (a) As t 0: A+ (u, u) = 0,t 1 1 4 ( 2 6 4 )3/2 exp u2 t 4 + O(t 5 ). 1/2 2 2 1296 4 2 2 4 2 2 2 2
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(5.2)
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