Laser Speckle and Related Phenomena
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Topics in Applied Physics
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Laser Speckle and Related Phenomena Editor: J. C. Dainty
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Laser Speckle and Related Phenomena
Edited by J. C. Dainty With Contributions by J. C. Dainty A. E. Ennos M. Frank.
(2.4)
Figure 2.3 illustrates the complex addition of the many elementary phasor contributions to produce the resultant A. We wish to know the statistics (e. g., probability density functions) of the complex field, the intensity, and the phase of the speckle pattern at point (x,y,z). With reference to Fig. 2.3, the problem before us is identical with the classical statistical problem of a random walk in the plane, which has been studied for nearly 100 years [2.4, 15, 16]. We shall derive the necessary results here, being careful to delineate the underlying assumptions and their physical significance. Let the elementary phasors have the following statistical properties: (i) The amplitude ak!VN and the phase cpk of the kth elementary phasor are statistically independent of each other and of the ampli-
lm
Re
Fig. 2.3. Random walk in the complex plane
14
1. W.
GooDMAN
tudes and phases of all other elementary phasors (i.e., the elementary scattering areas are unrelated and the strength of a given scattered component bears no relation to its phase); (ii) The phases cPk are uniformly distributed on the primary interval (- n, n) (i.e., the surface is rough compared to a wavelength, with the result that phase excursions of many times 2n radians produce a uniform distribution on the primary interval). With these assumptions, we shall investigate the statistical properties of the resultant complex field. 2.2.2 Statistics of the Complex Amplitude Attention is now focused on the real and imaginary parts of the resultant field, A (~R-A.R u, ~R-A.Rv)~V.~~~_s;:::-~J="""=:H~ยท~{x2-a~2R~.'!.:_Y-:.!:~~R~)~ X
Fig. 4.14. The domain of integration (shaded) for the Wiener spectrum, N(u.r). when the optical system has a circular pupil. For an aberration free system with a uniformly transmitting pupil, N(u,v) is given by the shaded area, suitably normalised
where the normalising constant has been ignored (Fig. 4.14). If we now carry out a two-dimensional Fourier transform on the autocorrelation function, (4.34), a result identical to (4.37) will be obtained. We must therefore conclude that DAINTY's result is equivalent to (4.34), and conseq
