Variance of intensity for Gaussian statistics and partially developed speckle in complex ABCD optical systems

We consider physical situations where both the real and imaginary parts of the optical field obey Gaussian statistics. Within the limitations of the generalized ray-matrix method, an expression for the variance of intensity is obtained for arbitrary cylindrically symmetric optical systems in the presence of partially developed speckle, where the rms surface roughness is comparable to the optical wavelength. We assume beam illumination of reflective targets with arbitrary values of surface roughness whose surface-height fluctuations are taken as a zero-mean stationary Gaussian random process. In contrast to previous work, the present analysis is valid for an arbitrary complex optical system that can be characterized by an ABCD matrix (e.g., simple and complex imaging systems, free-space propagation in both the near- and far-field, and Fourier transform systems), including those that exhibit correlation between the real and imaginary parts of the optical field. As a direct application, we consider an optical system for probing angular deflections that is sensitive to the speckle dynamics, and derive the relationship between the surface roughness and the corresponding lateral scale length that yields the maximum AC signal strength.