Santasri R. Bose-Pillai

Simulation of Anisoplanatic Imaging through Optical Turbulence Using Numerical Wave Propagation with New Validation Analysis

Abstract. We present a numerical wave propagation method for simulating imaging of an extended scene under anisoplanatic conditions. While isoplanatic simulation is relatively common, few tools are specifically designed for simulating the imaging of extended scenes under anisoplanatic conditions. We provide a complete description of the proposed simulation tool, including the wave propagation method used. Our approach computes an array of point spread functions (PSFs) for a two-dimensional grid on the object plane. The PSFs are then used in a spatially varying weighted sum operation, with an ideal image, to produce a simulated image with realistic optical turbulence degradation. The degradation includes spatially varying warping and blurring. To produce the PSF array, we generate a series of extended phase screens. Simulated point sources are numerically propagated from an array of positions on the object plane, through the phase screens, and ultimately to the focal plane of the simulated camera. Note that the optical path for each PSF will be different, and thus, pass through a different portion of the extended phase screens. These different paths give rise to a spatially varying PSF to produce anisoplanatic effects. We use a method for defining the individual phase screen statistics that we have not seen used in previous anisoplanatic simulations. We also present a validation analysis. In particular, we compare simulated outputs with the theoretical anisoplanatic tilt correlation and a derived differential tilt variance statistic. This is in addition to comparing the long- and short-exposure PSFs and isoplanatic angle. We believe this analysis represents the most thorough validation of an anisoplanatic simulation to date. The current work is also unique that we simulate and validate both constant and varying Cn2(z) profiles. Furthermore, we simulate sequences with both temporally independent and temporally correlated turbulence effects. Temporal correlation is introduced by generating even larger extended phase screens and translating this block of screens in front of the propagation area. Our validation analysis shows an excellent match between the simulation statistics and the theoretical predictions. Thus, we think this tool can be used effectively to study optical anisoplanatic turbulence and to aid in the development of image restoration methods.

A fast and efficient method for producing partially coherent sources

A fast, flexible and efficient method for generating partially coherent sources is presented. It is shown that the Schell-model (uniformly correlated) and non-uniformly correlated sources can be produced quickly using a fast steering mirror and low-actuator-count deformable mirror, respectively. The statistical optics theory underpinning the proposed technique is presented and discussed. Simulation results of two Schell-models and one non-uniformly correlated source are presented and compared to the theory to test the new approach.

Estimation of turbulence from time-lapse imagery

Abstract. Atmospheric turbulence parameters are estimated for an imaging path based on time-lapse imaging results. Atmospheric turbulence causes frame-to-frame shifts of the entire image as well as parts of the image. The statistics of these shifts encode information about the turbulence strength (as characterized by Cn2, the refractive index structure function constant) along the optical path. The shift variance observed is simply proportional to the variance of the tilt of the optical field averaged over the area being tracked and averaged over the camera aperture. By presuming this turbulence follows the Kolmogorov spectrum, weighting functions, which relate the turbulence strength along the path to the shifts measured, are derived. These weighting functions peak at the camera and fall to zero at the object. The larger the area observed, the more quickly the weighting function decays. One parameter we would like to estimate is r0 (the Fried parameter or atmospheric coherence diameter.) The weighting functions derived for pixel sized or larger parts of the image all fall faster than the weighting function appropriate for estimating the spherical wave r0. If we were to presume that Cn2 is constant along the path, then an estimate for r0 could be obtained for each area tracked, but since the weighting function for r0 differs substantially from that for every realizable tracked area, it can be expected that this approach would yield a poor estimate. Instead, the weighting functions for a number of different patch sizes can be combined through the Moore–Penrose pseudoinverse to create a weighting function that yields the least-squares optimal linear combination of measurements for the estimation of r0. This approach is carried out for one example and is shown to give noisy results. A modified version of this approach that creates larger patches by averaging several smaller patches together solves this noise issue. This approach can also work to estimate other atmospheric parameters.

Partially coherent sources with circular coherence: comment

In [Opt. Lett.42, 1512 (2017)OPLEDP0146-959210.1364/OL.42.001512], the authors present a new class of non-uniformly correlated sources with circular coherence. They also describe a basic experimental setup for synthesizing this class of sources, which uses the Van Cittert-Zernike theorem. Here, we present an alternative way to analyze these sources and a different way to generate them.

Modeling random screens for predefined electromagnetic Gaussian–Schell model sources

In a previous paper [Opt. Express22, 31691 (2014)] two different wave optics methodologies (phase screen and complex screen) were introduced to generate electromagnetic Gaussian Schell-model sources. A numerical optimization approach based on theoretical realizability conditions was used to determine the screen parameters. In this work we describe a practical modeling approach for the two methodologies that employs a common numerical recipe for generating correlated Gaussian random sequences and establish exact relationships between the screen simulation parameters and the source parameters. Both methodologies are demonstrated in a wave-optics simulation framework for an example source. The two methodologies are found to have some differing features, for example, the phase screen method is more flexible than the complex screen in terms of the range of combinations of beam parameter values that can be modeled. This work supports numerical wave optics simulations or laboratory experiments involving electromagnetic Gaussian Schell-model sources.

Profiling of atmospheric turbulence along a path using two beacons and a Hartmann turbulence sensor

The Hartmann Turbulence Sensor (HTS) is an optical system capable of estimating several atmospheric turbulence parameters, such as Greenwood frequency, Fried’s coherence diameter and inner scale of turbulence. It primarily comprises of a 40 cm Meade telescope, a 32 x 32 Shack- Hartmann lenslet array, and a high-speed camera. The HTS estimates the turbulence parameters by measuring the local tilts of the aberrated wavefront coming from a laser source and incident at the pupil plane of the telescope. At the Air Force Institute of Technology (AFIT), a technique has been developed to measure the distribution of turbulence along an experimental path using the HTS and two laser sources of the same wavelength. By measuring the variances of the difference in wavefront tilts due to the two sources sensed by a pair of Hartmann subapertures with varying separations, turbulence information along the path can be extracted. The method relies on deriving a set of weighting functions, each weighting function dipping to zero at a range where the two sensing paths from the beacons to the subapertures intersect, thus canceling out the effect of turbulence at this location on the differential tilt signal. The analytical expression for the path weighting functions has been derived here. The technique has been applied to experimental data collected over a 500 m grassy path and the profiling results have been compared to a co-located scintillometer. This work will eventually aid in obtaining a better understanding of turbulence in the lower atmosphere and how it varies with height.

Characterizing atmospheric turbulence over long paths using time-lapse imagery

In recent times, there has been a growing interest in measuring atmospheric turbulence over long paths. Irradiance based techniques such as scintillometry, suffer from saturation and hence commercial scintillometers have limited operational ranges. In the present work, a method to estimate path weighted Cn2 from turbulence induced random, differential motion of extended features in the time-lapse imagery of a distant target is presented. Since the method is phase based, it can be applied to longer paths. The method has an added advantage of remotely sensing turbulence without the need for deployment of sensors at the target location. The imaging approach uses a derived set of path weighting functions that drop to zero at both ends of the imaging path, the peak location depending on the size of the imaging aperture and the relative sizes and separations of the features whose motions are being tracked. For sub-aperture sized features and separations, the peaks of the weighting functions are closer to the target end of the path. For bigger features and separations, the peaks are closer to the camera end. Using different sized features separated by different amounts, a rich set of weighting functions can be obtained. These weighting functions can be linearly combined to produce a desired weighting function such as that of a scintillometer or that of r0. The time-lapse measurements can thus mimic the measurements of a scintillometer or any other instrument. The method is applied to both simulated and experimentally obtained imagery and some validation results with a scintillometer is shown as well.

Global tilt removal on a Hartmann turbulence sensor

A Hartmann Turbulence Sensor (HTS) was used to quantify the atmospheric turbulence along a 1 km near ground-level path. This study examines the effect of removing the average tilt over all subapertures from each subaperture in the data analysis. The HTS captures a laser beam projected along a path of interest with a telescope; a lenslet array in the detector system breaks the beam up into 700 subapertures spread across the telescope pupil, and then forms images of the laser source from each of these subapertures onto a fast camera. Turbulence along the path induces tilts in the laser wavefront which are captured as centroid motion of the many laser spots in the camera image. This motion is used to quantify the turbulence. The raw spot positions contain undesired image motion due to telescope motion and vibration. This motion can be removed from the data by subtracting the average centroid motion of all subapertures from each subaperture. This subtraction changes the data in other ways, and this detail must be included in the analysis. The result can be exactly represented as weighted sum over the differential tilt variances between subapertures pairs. The tilt-removed variance averaged over all the subapertures is shown to be one-half of the average variance over all subaperture pairs. This work also resolves some discrepancies in previous results involving the expected variances of these differential tilts.

Simulating time-evolving non-cross-spectrally pure schell-model sources

Investigating the time-evolution of partially-coherent sources is necessary for certain optical coherence effects. Several simulation approaches have been developed, many of which can only treat cross-spectrally pure sources. However, some significant source types are not cross-spectrally pure. This paper reviews two methods for the synthesis of time-evolving sources which need not be pure. Both involve filtering matrices of uncorrelated Gaussian random numbers. One method requires control of both amplitude and phase, while the other only requires phase control. The utility of the methods for non-cross-spectrally pure sources is demonstrated for the first time. The source is generated by passing coherent light through two different diffusers which move at the same speed but in opposite directions. Simulation results for the time-evolving field are shown. Further, the coherence functions of the synthesized field are compared to theory for validation.

Fresnel spatial filtering of quasihomogeneous sources for wave optics simulations

High-spatial-frequency optical fields or sources are often encountered when simulating directed energy, active imaging, or remote sensing systems and scenarios. These spatially broadband fields are a challenge in wave optics simulations because the sampling required to represent and then propagate these fields without aliasing is often impractical. To address this, two spatial filtering techniques are presented. The first, called Fresnel spatial filtering, finds a spatially band-limited source that, after propagation, produces the exact observation plane field as the broadband source over a user-specified region of interest. The second, called statistical or quasihomogeneous spatial filtering, finds a spatially band-limited source that, after propagation and over a specified region of interest, yields an observation plane field that is statistically representative of that produced by the original broadband source. The pros and cons of both approaches are discussed in detail. A wave optics simulation of light transiting a ground glass diffuser and then propagating to an observation plane in the near-zone is performed to validate the two filtering approaches.