Signal-to-noise ratio limitations for intensity correlation imaging.

Intensity correlation imaging (ICI) is a concept which has been considered for the task of providing images of satellites in geosynchronous orbit using ground-based equipment. This concept is based on the intensity interferometer principle first developed by Hanbury Brown and Twiss. It is the objective of this paper to establish that a sun-lit geosynchronous satellite is too faint a target object to allow intensity interferometry to be used in developing image information about it-at least not in a reasonable time and with a reasonable amount of equipment. An analytic treatment of the basic phenomena is presented. This is an analysis of one aspect of the statistics of the very high frequency random variations of a very narrow portion of the optical spectra of the incoherent (black-body like-actually reflected sunlight) radiation from the satellite, an analysis showing that the covariance of this radiation as measured by a pair of ground-based telescopes is directly proportional to the square of the magnitude of one component of the Fourier transform of the image of the satellite-the component being the one for a spatial frequency whose value is determined by the separation of the two telescopes. This analysis establishes the magnitude of the covariance. A second portion of the analysis considers shot-noise effects. It is shown that even with much less than one photodetection event (pde) per signal integration time an unbiased estimate of the covariance of the optical field's random variations can be developed. Also, a result is developed for the standard deviation to be associated with the estimated value of the covariance. From these results an expression is developed for what may be called the signal-to-noise ratio to be associated with an estimate of the covariance. This signal-to-noise ratio, it turns out, does not depend on the measurement's integration time, Δt (in seconds), or on the optical spectral bandwidth, Δν (in Hertz), utilized-so long as ΔtΔν≫1, which condition it would be hard to violate. It is estimated that for a D=3.16 m diameter satellite, with a pair of D=1.0 m diameter telescopes (which value of D probably represents an upper limit on allowable aperture diameter since the telescope aperture must be much too small to even resolve the size of the satellite) at least N=2.55×10(16) separate pairs of (one integration time, pde count) measurement values must be collected to achieve just a 10 dB signal-to-noise ratio. Working with 10 pairs of telescopes (all with the same separation), and with 10 nearly adjacent and each very narrow spectral bands extracted from the light collected by each of the telescope-so that for each measurement integration time there would be 100 pairs of measurement values available-and with an integration time as short as Δt=1 ns, it would take T=2.55×10(5) s or about 71 h to collect the data for just a single spatial frequency component of the image of the satellite. It is on this basis that it is concluded that the ICI concept does not seem likely to be able to provide a timely responsive capability for the imaging of geosynchronous satellites.

Analytical expressions for the log-amplitude correlation function for spherical wave propagation through anisotropic non-Kolmogorov atmosphere.

An analytical expression for the log-amplitude correlation function based on the Rytov approximation is derived for spherical wave propagation through an anisotropic non-Kolmogorov refractive turbulent atmosphere. The expression reduces correctly to the previously published analytic expressions for the case of spherical wave propagation through isotropic Kolmogorov turbulence. These results agree well with a wave-optics simulation based on the more general Fresnel approximation, as well as with numerical evaluations, for low-to-moderate strengths of turbulence. These results are useful for understanding the potential impact of deviations from the standard isotropic Kolmogorov spectrum.

Analytical expressions for the log-amplitude correlation function for plane wave propagation in anisotropic non-Kolmogorov refractive turbulence.

An analytical expression for the log-amplitude correlation function for plane wave propagation through anisotropic non-Kolmogorov turbulent atmosphere is derived. The closed-form analytic results are based on the Rytov approximation. These results agree well with wave optics simulation based on the more general Fresnel approximation as well as with numerical evaluations, for low-to-moderate strengths of turbulence. The new expression reduces correctly to the previously published analytic expressions for the cases of plane wave propagation through both nonisotropic Kolmogorov turbulence and isotropic non-Kolmogorov turbulence cases. These results are useful for understanding the potential impact of deviations from the standard isotropic Kolmogorov spectrum.

Moment-matching method for extraction of asymmetric beam jitters and bore sight errors in simulations and experiments with actively illuminated satellites of small physical cross section.

Optical returns from remote resident space-based objects such as satellites suffer from pointing and tracking errors. In a previously reported paper [Appl. Opt.46, 5608 (2007)APOPAI0003-693510.1364/AO.46.005608], we developed a moment-matching technique that used the statistics of time series of these optical returns to extract information about bore sight and symmetric beam jitter errors (symmetric here implies that the standard deviations of the jitter measured along two orthogonal axes, perpendicular to the line of sight, are equal). In this paper, we extend that method to cover the case of asymmetric beam jitter and bore sight. The asymmetric beam jitter may be due to the combination of symmetric atmospheric turbulence beam jitter and optical beam train jitter. In addition, if a tracking control system is operating, even the residual atmospheric tracking jitter could be asymmetric because the power spectrum is different for the slewing direction compared to the cross-track direction. Analysis of the problem has produced a set of nonlinear equations that can be reduced to a single but much higher-order nonlinear equation in terms of one of the jitter variances. After solving for that jitter, all the equations can be solved to extract all jitter and bore sight errors. The method has been verified by using simulations and then tested on experimental data. In order to develop this method, we derived analytical expressions for the probability density function and the moments of the received total intensity. The results reported here are valid for satellites of small physical cross section, or else those with retroreflectors that dominate the signal return. The results are, in general, applicable to the theory of noncircular Gaussian speckle with a coherent background.

Deep turbulence effects compensation experiments with a cascaded adaptive optics system using a 3.63 m telescope.

Compensation of extended (deep) turbulence effects is one of the most challenging problems in adaptive optics (AO). In the AO approach described, the deep turbulence wave propagation regime was achieved by imaging stars at low elevation angles when image quality improvement with conventional AO was poor. These experiments were conducted at the U.S. Air Force Maui Optical and Supercomputing Site (AMOS) by using the 3.63 m telescope located on Haleakala, Maui. To enhance compensation performance we used a cascaded AO system composed of a conventional AO system based on a Shack-Hartmann wavefront sensor and a deformable mirror with 941 actuators, and an AO system based on stochastic parallel gradient descent optimization with four deformable mirrors (75 control channels). This first-time field demonstration of a cascaded AO system achieved considerably improved performance of wavefront phase aberration compensation. Image quality was improved in a repeatable way in the presence of stressing atmospheric conditions obtained by using stars at elevation angles as low as 15 degrees.

Moment-matching method for extracting beam jitter and boresight in experiments with satellites of small physical cross section.

The boresight and atmospheric jitter errors in a satellite tracking experiment are currently estimated by matching the probability density function (PDF) of the received signal counts with a set of PDFs of the signal for several combinations of jitter and boresight errors and then the best choice of jitter and boresight error is accepted via the chi-square test. Here a technique that can estimate atmospheric beam jitter and boresight error directly in a satellite active tracking experiment using the moments of the returns off the satellites is proposed. That is, we use the theoretical PDF for the signal return from a small target and compute the corresponding theoretical PDF moments. We can then form a few equations from these moments with only two unknowns, namely, the jitter and boresight. Solving for the unknowns is then unambiguous and very rapid. The method is valid for small physical cross-section targets and has been verified by using simulation and experimental data. Extending the case to asymmetric jitter and asymmetric boresight is possible.