Richard Barakat

Sums of independent lognormally distributed random variables

The probability-density function of the sum of lognormally distributed random variables is studied by a method that involves the calculation of the Fourier transform of the characteristic function; this method is exact. When the number of terms in the sum is large, we employ an asymptotic series in N−1, where N is the number of terms, developed by Cramer. This method is employed in order to show that the permanence of the lognormal probability-density function is a consequence of the fact that the skewness coefficient of the lognormal variables is nonzero. Finally, a simplified proof, by use of the Carleman criterion, is presented to show that the lognormal is not uniquely determined by its moments.

First-order Probability Densities of Laser Speckle Patterns Observed through Finite-size Scanning Apertures

The first-order probability density function of laser speckle patterns observed through finite size apertures is theoretically studied. An exact solution is obtained through the use of a two-dimensional Kac-Siegert analysis in terms of the spatial correlation function of the input. An approximate solution of Goodman is shown to be a limiting case of the exact theory. The probability density of the logarithm of the intensity is also derived. Representative numerical calculations are presented for a slit aperture.

Direct derivation of intensity and phase statistics of speckle produced by a weak scatterer from the random sinusoid model

The complex amplitude at a point in a speckle pattern that is due to a weak scatterer is modeled as the superposition of N sinusoidal waves of random phase, with the probability density of these phases given by the nonuniform von Mises rather than by the uniform one that characterizes a strong scatterer. Explicit formulas are obtained for both intensity and total phase statistics in terms of a single parameter directly related to the density function of the constituent phasors. The case in which, in addition, N itself is random (governed by a Poisson distribution with mean value 〈N〉) is also studied.

The Influence of Random Wavefront Errors on the Imaging Characteristics of an Optical System

The influence of random wavefront errors on the transfer function and point spread function of an optical system is studied theoretically. The stochastic part of the aberration function is assumed to be Gaussian and spatially stationary (although the requirement of stationarity is relaxed in § 5). The stochastic transformation from wavefront to transfer function is non-linear. The consequences of this non-linear transformation are two-fold: first, the statistics of the transfer function are non-Gaussian, second, the transfer function is non-stationary. (The same statements hold for the point spread function.) Therefore the characterization of these processes requires an infinite number of averaged products (moments), not just the first two if the processes were Gaussian. These averaged products are obtained in the form of multiple integrals involving the characteristic function of the wavefront and are suitable for calculation on a high speed computer. Some numerical results for the mean of both processes...

Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: Generalizations of Zernike polynomials

The Zernike aberration theory for constant amplitude circular apertures is extended to annular apertures having a Gaussian-like radial taper. Explicit expressions are obtained for the optimum balanced wave-front aberrations in terms of shifted Jacobi polynomials. Properties of the polynomials (e.g., Rodrigues formula, recurrence relations, derivatives, etc.) are investigated.

Application of the Tichonov regularization algorithm to object restoration

Abstract Object restoration is recognized as an improperly posed problem and an algorithm developed by Tichonov is employed to numerically stabilize the solution. Some numerical results are illustrated.

Statistics of the optical transfer function: Correlated random amplitude and random phase effects

The pupil function of an optical system is taken to have correlated random amplitudes and phases arising from an external cause, such as optical propagation through turbulence. For the general case, where the restriction of isotropy need not apply, the unnormalized random optical transfer function is derived and its first two moments evaluated. Normalization issues are also treated. It is shown that when the cross correlation of random amplitude and random phase is not an even function, a phase shift term is induced. The impact of this shift is discussed in terms of the image of an edge. While isotropy eliminates dependence on the cross correlation of amplitude and phase for the first moment of the transfer function, it does not similarly affect all second-moment behavior.

Time-interval photoelectron counting statistics for multiple spectral lines☆

Abstract Time-interval statistics for photoelectron counting experiments are studied for quasimonochromatic light of arbitrary spectral profile. A numerical scheme is developed for the evaluation of the generating function and its second time derivative. Numerical results for the time-interval statistics are presented for the spectrum which would be obsered in a Brillouin scattering experiment.

Moment estimator approach to the retrieval problem in coherence theory

The purpose of this paper is to suggest a possible approach to the recovery of the spectral density function g(ω) through a knowledge of the first few measured complex zeros of the complex degree of coherence γ(τ). The assumption that γ(τ) is band-limited allows us to express the sums of inverse powers of the complex zeros of γ(τ) in terms of the moments of g(ω). Only the lowest-order moments can be evaluated in this manner with any accuracy for reasons discussed in the text. We use two estimation-type solutions that utilize lower-order moments: beta distribution model and the Shannon maximum entropy model to estimate g(CD). Representative numerical calculations are discussed.

The expected value of the edge spread function in the presence of random wavefronts

Abstract The expected value of the edge spread function imaged by a circular aperture in the presence of a random wavefront is evaluated for both incoherent and coherent illumination.