Branch cuts in the phase function.

It is shown that, when the scalar field associated with the propagation of a distorted wave function has nulls in its intensity pattern, the phase function that goes with that scalar field has branch points at the location of these nulls and that there are unavoidable 2pi discontinuities across the associated branch cuts in the phase function. An analytic proof of this supposition is provided. Sample computer-wave optics propagation results are presented that manifest such unavoidable discontinuities. Among other things, the numerical results are organized in a way that demonstrates that for those cases the branch points are unavoidable. It is found in the sample numerical results that the branch cuts can be positioned so that the 2pidiscontinuities are located along lines of minimum intensity. This location tends to minimize the physical significance or importance of the discontinuities, a significant consideration for deformablemirroradaptive optics, for which there is an unavoidable correction error in the vicinity of the branch cut. An algorithm is briefly described that allows the branch cuts to be located automatically and a phase function to be calculated that has discontinuities equal only to 2pidiscontinuities that are located at the branch cuts.

Laser eye safety: the implications of ordinary speckle statistics and of speckled-speckle statistics.

The implications of speckle statistics on laser eye-safety considerations are evaluated. The concept of speckled speckle is introduced, and its statistics are shown to correspond to the K0 function. Speckled speckle is defined in terms of the retinal power density when the eye is viewing an optically rough surface that is illuminated by a laser beam diffused through a ground-glass screen-a situation corresponding to subjective speckle modulated by objective speckle. Extensive numerical results are developed relating the ratio of the average power density on the retina over the eye-damage level to the acceptable probability that speckle statistics will cause the damage level to be exceeded. For ordinary speckle and for speckled speckle, for a probability of 10(-6) (10(-9)) of exceeding the damage level, the average power densities must be 0.072 (0.48) and 0.017 (0.0079) of the damage level, respectively.

Propagation of the mutual coherence function for an infinite plane wave through a turbid medium.

By means of a series of Fourier and inverse Fourier transformations, and by recourse to simple physical arguments concerning statistical stationarity and isotropy and the linearity of the scattering process, we are able to show that the propagation of the mutual coherence function for a plane wave through a turbid medium is governed by a pair of linear, first-order, one-dimensional, simultaneous differential equations. A sample solution to the equation is presented, and it is shown from a limiting form of this solution how the turbidity parameters in the differential equation can be obtained from a single scattering analysis.

Nonlinear resonance scanning.

A new scanning concept is suggested for causing a flat mirror (or telescope) to execute a symmetric sawtooth scan pattern. The concept is based on the use of a pair of control moment gyros whose spin axes are synchronously counterrotated. If these gyros are mounted on the back of the mirror, then through the nonlinear interactions associated with the counterrotation of the spin axes of the pair of gyros, the mirror will be caused to scan back and forth through a symmetric sawtooth pattern without external drive being required. By careful choice of the parameters of the control moment gyros, scan efficiencies of the order of 90% can be achieved. This scanner concept has the virtues not only of high scan efficiency but also of requiring very little power and not reaching back on the scanner mount. It therefore seems particularly well suited for use in space applications. In this paper, the concept is described in detail and an explanation of the principles of operation is presented. An approximate analysis is worked out to provide an insight into the significance of the various design parameters. Finally, an exact analysis is presented and detailed computer calculations of the expected scan performance for a particular case are carried out.

Evaluation of r(o) for Propagation Down Through the Atmosphere.

The concept of the quantity r(o) as a length that measures the effect of atmospheric turbulence on optical systems that are wavefront distortion sensitive is briefly reviewed. It is pointed out that no precise set of data on r(o) for propagation down through the atmosphere has been published. Using astronomical seeing data obtained by Hoag and by Meinel it is shown how statistics for r(o) can be derived, at least for observatory quality sites. We find that the two sets of data are in very good agreement and conclude that r(o) is variable from night to night and distributed according to a log-normal distribution. For 0.55-microm light and zenith propagation, the median value of r(o), is 0.114 m and changes by a factor of 1.36 for occurrences one standard deviation from the median.

Spectral and angular covariance of scintillation for propagation in a randomly inhomogeneous medium.

Based on a multiple transform approach, it is shown that it is possible to compute the statistics of propagation in a randomly inhomogeneous medium without invoking the small scattering angle approximation. This technique also makes it possible to compute the cross-statistics between two waves with different wavelengths or traveling in slightly different directions. The spectral covariance of log-amplitude and the spectral phase, and wave-structure functions are evaluated for horizontal propagation (i.e., for the statistics of turbulence not changing over the path). They are found to vary very slowly with separation of wavenumber. The angular covariance of log-amplitude and the angular-phase and wave-structure functions are formulated, but only the angular variance of log-amplitude is evaluated. It is found that the variance decreases very rapidly. Correlation only extends over an angular range of +/-0.2(4/zk)((1/2)), where z is the path length and k is the wavenumber. Comments concerning the application of these results to several problems are presented. These problems include: (1) the use of spectral diversity techniques in optical communications; (2) the possibility of turbulence garbling optical communications; (3) the cause of chromatic scintillation of stars; and (4) the scintillation and fading of laser illuminated targets, etc.

Bender-bimorph scanner analysis.

The bender-bimorph beam steerer is a piezoelectrically driven optical device capable of pointing a light beam of several centimeters diameter anywhere in a several degree wide field of view with a control bandwidth of several hundred hertz. Its operation depends on the mechanical motion of a piezoelectric bender-bimorph to tilt a mirror, which in turn deflects the light beam. Fundamental relationships governing the operation of such a device constrain the product of mirror diameter, scan amplitude, and control bandwidth so that optimization of a bender-bimorph beam steerer system is a matter of trade-off considerations. A theoretical analysis of bender-bimorph performance is carried out. Expressions are derived for the resonant frequency of a loaded bimorph and its deflection. Graphs of these expressions are presented with several parameters treated as variables. For the numerical calculations, a very light beryllium mirror the same width as the bimorph is assumed. Some experimental data were collected and compared with the predicted performance (i.e., resonant frequency and deflection). The comparison verified the theoretical expressions.

Heterodyne and photon-counting receivers for optical communications.

The relative performance of an optical heterodyne receiver and a photon-counting receiver are compared. These conditions under which the two types of systems perform equally well are defined. Background noise is shown to be an almost negligible consideration. Detector noise is seen to be a much more severe problem. It is concluded that for wavelengths below 1 micro, photon-counting reception is preferred because of the availability of photoemissive detectors. For wavelengths greater than 3 micro detector noise considerations are so severe that it is unlikely that photon counting will be competitive with heterodyne detection. The type of detection preferred in the 1-3 microrange is shown to be a matter of detector development.

The effect of atmospheric scintillation on an optical data channel-laser radar and binary communications.

The effects of scintillation on an optical data channel are analyzed and numerical results presented. Scintillation with a log normal distribution typical of atmospheric optical effects is assumed. The analysis is concerned with the target miss probability of a laser radar and the bit error probability of an optical binary communications link. Results are expressed in terms of a loss factor which is the extra number of dB signal-to-noise ratio necessary to keep the performance in the presence of scintillation up to the level achievable in the absence of scintillation. With even moderately weak scintillation large loss factors are encountered. A brief treatment of the effects of normally distributed scintillation is also presented.