Paul Diament

Photodetection of Low-Level Radiation through the Turbulent Atmosphere*

Photoelectron-counting distributions are obtained for stochastic light that is caused to scintillate by passage through a random medium. The result is applied specifically to transmission of amplitude-stabilized radiation, with and without independent additive background, and of chaotic radiation, through the turbulent atmosphere. The counting distribution is found to broaden markedly and its peak occurs at decreasing count numbers for increasing turbulence. The cases studied here are of particular interest for low-level direct optical communications and radar using single-mode lasers, multimode lasers, thermal sources, and scattering targets. The results are obtained by extending the usual formulation of photoelectron counting; a product of two random variables, the source intensity and the effect of the turbulent medium, rather than the usual single stochastic irradiance, is considered. Plots of the counting distributions for various degrees of turbulence and for several signal-to-noise ratios are presented. A possible explanation is given for the observed decrease of the log-amplitude variance to values below the saturation value in long path length and high turbulence experiments.

Multiply stochastic representations for K distributions and their Poisson transforms

The K distribution is used in a number of areas of scientific endeavor. In optics, it provides a useful statistical description for fluctuations of the irradiance (and the electric field) of light that has been scattered or transmitted through random media (e.g., the turbulent atmosphere). The Poisson transform of the K distribution describes the photon-counting statistics of light whose irradiance is K distributed. The K-distribution family can be represented in a multiply stochastic (compound) form whereby the mean of a gamma distribution is itself stochastic and is described by a member of the gamma family of distributions. Similarly, the family of Poisson transforms of the K distributions can be represented as a family of negative-binomial transforms of the gamma distributions or as Whittaker distributions. The K distributions have heretofore had their origins in random-walk models; the multiply stochastic representations provide an alternative interpretation of the genesis of these distributions and their Poisson transforms. By multiple compounding, we have developed a new transform pair as a possibly useful addition to the K-distribution family. All these distributions decay slowly and are difficult to calculate accurately by conventional formulas. A recursion relation, together with a generalized method of steepest descent, has been developed to evaluate numerically the photon-counting distributions and their factorial moments with excellent accuracy.

Two‐Photon Counting Statistics for Laser and Chaotic Radiation

The relation between the two‐quantum photocounting distribution p(n, T) and the intensity fluctuations of the radiation incident on the detector is obtained and compared with the one‐quantum results. The actual statistics are evaluated for several cases of interest, including a chaotic source, an amplitude‐stabilized wave, and the output of a single‐mode laser (Van der Pol oscillator). The two‐quantum count rate 〈n〉/T is found to depend on the mean‐square intensity of the radiation, in contrast to the one‐quantum count rate which is proportional to the mean intensity. The effects of photon correlations in the radiation beam become apparent since the two‐quantum distributions manifest a distinctly more positive second derivative than the corresponding one‐quantum distributions. Both the low‐and the high‐count probabilities are therefore increased at the expense of counts near the mean. The quantum‐theoretical treatment is found to be equivalent to the semiclassical treatment for density operators possessin...

Evolution of the statistical properties of photons passed through a traveling-wave laser amplifier

The authors determine the evolution of the photon statistics of a light beam as it passes through a traveling-wave laser amplifier, modeled as a birth-death immigration (BDI) medium. The relationship between the input and output probability distributions and probability generating functions with given (but possibly varying) birth, death, and immigration rates for arbitrary input statistics is obtained. The case of constant birth, death, and immigration rates is considered in particular detail. The photon statistics at the output of a general BDI traveling-wave amplifier are always broader than those at the input, and they can take many forms. The most general solution can be applied when the input distribution to the amplifier takes the form of a negative-binomial transform. >

Photoelectron-Counting Distributions for Irradiance-Modulated Radiation*

Photoelectron-counting distributions are obtained for modulated radiation sources, with arbitrary modulation waveforms and depths. Cases treated in detail include chaotic, amplitude-stabilized, and Risken sources, with square-wave, triangular, and sinusoidal modulation. Modulation is shown to broaden the counting distributions, an effect interpretable as accentuated photon bunching. The broadening is not marked for the chaotic source, but the shape of the distribution changes drastically for the amplitude-stabilized source. Varying the radiation statistics and the modulation waveforms can produce a variety of counting distributions, from double peaked to extremely flat. Modulation may serve to accentuate the distinctions between counting distributions associated with sources of different radiation statistics.

SNR in photocounting images of rough objects in partially coherent light

Image irradiance distributions from objects illuminated with partially coherent, quasi-monochromatic light, viewed against a spatially uniform background and received with a photosensitive detector are analyzed. A general expression for the SNR at the detector output is obtained as a function of the coherence of the illuminating light, the object surface roughness, the width of the telescope point spread function, and the aperture and integration time of the detector. The expression is evaluated for several limiting cases of coherence of illumination and of object surface roughness.

Wave Propagation along Warm Plasma Columns

Dispersion relations for waves propagating along homogeneous, isotropic, warm plasma columns surrounded by a dielectric medium are obtained. A steady‐state linearized model with temperature introduced through a scalar pressure is analyzed both quasistatically and more rigorously from Maxwells equations. The results show effects of temperature on the cold plasma surface wave and predict a new series of propagating modes above the plasma frequency related to the Tonks‐Dattner scattering resonances near cutoff and to the Bohm‐Gross longitudinal plasma waves for large wave numbers. Warm and cold plasmas, large and small columns, and quasistatic and rigorous predictions are contrasted. Regions of the Brillouin diagram corresponding to different bound and radiating wave types are delineated and related to their physical properties.

Relative refractoriness in visual information processing

Pulse-interval distributions are obtained for a counting system in which there is a gradual, rather than abrupt, increase in excitability following the registration of a pulse (relative refractoriness). The results are applicable to systems in which Poisson counting would be observed in the absence of such effects, and in which the memory reaches back at most one pulse. Choosing a particular functional form for the recovery function, the theory fits the experimentally measured distribution for the maintained discharge in the cats retinal ganglion cell. It is also consistent with the notion that Webers Law emerges from refractoriness in the visual system, as first proposed by van der Velden.

Flat counting distribution for triangularly-modulated poisson process

Abstract Photocounting distributions for an amplitude-stabilized radiation source with triangularly-modulated intensity should exhibit the unusual property of extreme flatness over a tunable region of counts. The result applies to any Poisson process whose mean can be linearly swept.

Maximum angular accuracy of pulsed laser radar in photocounting limit

To estimate the angular position of targets with pulsed laser radars, their images may be sensed with a fourquadrant noncoherent detector and the image photocounting distribution processed to obtain the angular estimates. The limits imposed on the accuracy of angular estimation by signal and background radiation shot noise, dark current noise, and target cross-section fluctuations are calculated. Maximum likelihood estimates of angular positions are derived for optically rough and specular targets and their performances compared with theoretical lower bounds.