Richard B. Holmes

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.

Investigation of the Cauchy–Riemann equations for one-dimensional image recovery in intensity interferometry

A method of image recovery using noniterative phase retrieval is proposed and investigated by simulation. This method adapts the Cauchy-Riemann equations to evaluate derivatives of phase based on derivatives of magnitude. The noise sensitivity of the approach is reduced by employing a least-mean-squares fit. This method uses the analytic properties of the Fourier transform of an object, the magnitude of which is measured with an intensity interferometer. The solution exhibits the degree of nonuniqueness expected from root-flipping arguments for the one-dimensional case, but a simple assumption that restricts translational ambiguity also restricts the space of solutions and permits essentially perfect reconstructions for a number of non-symmetric one-dimensional objects of interest. Very good reconstructions are obtained for a large fraction of random objects, within an overall image flip, which may be acceptable in many applications. Results for the retrieved phase and recovered images are presented for some one-dimensional objects and for different noise levels. Extensions to objects of two dimensions are discussed. Requirements for signal-to-noise ratio are derived for intensity interferometry with use of the proposed processing.

High angular resolution imaging with stellar intensity interferometry using air Cherenkov telescope arrays

Optical stellar intensity interferometry with air Cherenkov telescope arrays, composed of nearly 100 telescopes, will provide means to measure fundamental stellar parameters and also open the possibility of model-independent imaging. In addition to sensitivity issues, a main limitation of image recovery in intensity interferometry is the loss of phase of the complex degree of coherence during the measurement process. Nevertheless, several model-independent phase reconstruction techniques have been developed. Here we implement a Cauchy-Riemann based algorithm to recover images from simulated data. For bright stars (mv � 6) and exposure times of a few hours, we find that scale features such as diameters, oblateness and overall shapes are reconstructed with uncertainties of a few percent. More complex images are also well reconstructed with high degrees of correlation with the pristine image. Results are further improved by using a forward algorithm.

Analytical expressions for the log-amplitude correlation function of a plane wave through anisotropic atmospheric refractive turbulence

The effect of anisotropic Kolmogorov turbulence on the log-amplitude correlation function for plane-wave fields is investigated using analysis, numerical integration, and simulation. A new analytical expression for the log-amplitude correlation function is derived for anisotropic Kolmogorov turbulence. The analytic results, based on the Rytov approximation, agree well with a more general wave-optics simulation based on the Fresnel approximation as well as with numerical evaluations, for low and moderate strengths of turbulence. The new expression reduces correctly to previously published analytic expressions for isotropic turbulence. The final results indicate that, as asymmetry becomes greater, the Rytov variance deviates from that given by the standard formula. This deviation becomes greater with stronger turbulence, up to moderate turbulence strengths. The anisotropic effects on the log-amplitude correlation function are dominant when the separation of the points is within the Fresnel length. In the direction of stronger turbulence, there is an enhanced dip in the correlation function at a separation close to the Fresnel length. The dip is diminished in the weak-turbulence axis, suggesting that energy redistribution via focusing and defocusing is dominated by the strong-turbulence axis. The new analytical expression is useful when anisotropy is observed in relevant experiments.

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.

Two-dimensional image recovery in intensity interferometry using the Cauchy-Riemann relations

Intensity interferometry utilizes measurements of the squared-magnitude of the Fourier transform of an object using relatively simple, phase-insensitive hardware, and therefore holds the promise of extremely high spatial resolution in astronomy and various branches of physics. However, this promise has not been realized due to signal-noise-ratio (SNR) issues and due to the maturity of image recovery algorithms. To recover an image, the phase of the Fourier transform must be determined in addition to its magnitude. In a recent paper, relatively good one-dimensional (1-D) image recoveries were obtained with a fast non-iterative algorithm utilizing the Cauchy-Riemann relations and a mild constraint on the symmetry of the object. In this paper, the approach is extended to two spatial dimensions by combining multiple 1-D reconstructions, and ensuring mutual consistency between 1-D slices. Mutual consistency is enforced using several different approaches, including phase retrieval. This use of the Cauchy-Riemann approach combined with imposition of mutual consistency is found to reduce noise sensitivity significantly. Three approaches are evaluated for image quality for different objects using sparse Fourier-plane sampling, showing good reconstruction of images at SNRs as low as 7 at the origin in the Fourier plane (and thus even lower SNRs at higher angular frequencies).

Toward a revival of stellar intensity interferometry

Building on technological developments over the last 35 years, intensity interferometry now appears a feasible option by which to achieve diffraction-limited imaging over a square-kilometer synthetic aperture. Upcoming Atmospheric Cherenkov Telescope projects will consist of up to 100 telescopes, each with ~100m2 of light gathering area, and distributed over ~1km2. These large facilities will offer thousands of baselines from 50m to more than 1km and an unprecedented (u,v) plane coverage. The revival of interest in Intensity Interferometry has recently led to the formation of a IAU working group. Here we report on various ongoing efforts towards implementing modern Stellar Intensity Interferometry.

Phase screen simulations of laser propagation through non-Kolmogorov atmospheric turbulence

Phase Screen simulations for laser propagation through non-Kolmogorov turbulence are presented and the results for scintillation index and correlation functions for the intensity are compared with the theory at low turbulence levels at selected non-Kolmogorov exponents. Additional simulation results are presented the strong turbulence region. In particular, effects of transitioning from Kolmogorov to non-Kolmogorov turbulence using their spectral equivalence at the Fresnel scale (as suggested in the literature) on the scintillation index and correlation functions at the receiver are examined for two example paths.

Intensity interferometry experiments and simulations

Intensity Interferometry is a form of imaging developed in the 1950’s by Hanbury Brown and Twiss, which gave very early results for estimates of the diameters of stellar discs. It relies on the statistical properties of light to form an image by correlating the electronic signals measured independently and simultaneously at two or more separate collection telescopes. Its benefits are that it can provide very high resolution, can be very low in cost, does not require precision path matching, and is insensitive to atmospheric effects. Its disadvantages are that it has relatively poor SNR properties for larger telescope separations. An experiment is performed with three telescopes in Kihei, HI to investigate the potential for large-separation, high-resolution, multi-telescope operation. Simulations were performed to address key issues related to the experiment. Correlations were measured during lab checkouts, and also for early field testing. A compression scheme was developed to archive the raw data. The compression process had the added advantage of eliminating spurious electronic interference signals.

Stellar intensity interferometry: Imaging capabilities of air Cherenkov telescope arrays

Sub milli-arcsecond imaging in the visible band will provide a new perspective in stellar astrophysics. Even though stellar intensity interferometry was abandoned more than 40 years ago, it is capable of imaging and thus accomplishing more than the measurement of stellar diameters as was previously thought. Various phase retrieval techniques can be used to reconstruct actual images provided a sufficient coverage of the interferometric plane is available. Planned large arrays of Air Cherenkov telescopes will provide thousands of simultaneously available baselines ranging from a few tens of meters to over a kilometer, thus making imaging possible with unprecedented angular resolution. Here we investigate the imaging capabilities of arrays such as CTA or AGIS used as Stellar Intensity Interferometry receivers. The study makes use of simulated data as could realistically be obtained from these arrays. A Cauchy-Riemann based phase recovery allows the reconstruction of images which can be compared to the pristine image for which the data were simulated. This is first done for uniform disk stars with different radii and corresponding to various exposure times, and we find that the uncertainty in reconstructing radii is a few percent after a few hours of exposure time. Finally, more complex images are considered, showing that imaging at the sub-milli-arc-second scale is possible.