Use of artificial neural networks for Hartmann-sensor lenslet centroid estimation

For adaptive optical systems to compensate for atmospheric-turbulence effects, the wave-front perturbation must be measured with a wave-front sensor (WFS). A Hartmann WFS typically divides the optical aperture into subapertures and then measures the slope of the wave front within each subaperture. Hartmann WFS slope measurements are based on estimating the location of the centroid of the image that is formed from a guide star within each subaperture. Conventional techniques for centroid estimation involve the use of a linear estimator and conversion tables. Neural networks provide nonlinear solutions to this problem. We address the use of neural networks for estimating the location of the centroid from the subaperture image. We find that neural networks provide more accurate estimates over a larger dynamic range and with less variance than do the conventional linear centroid estimator.     

Laser guide star wavefront sensing for ground-layer adaptive optics on extremely large telescopes

We propose ground-layer adaptive optics (GLAO) to improve the seeing on the 42?m European Extremely Large Telescope. Shack-Hartmann wavefront sensors (WFSs) with laser guide stars (LGSs) will experience significant spot elongation due to off-axis observation. This spot elongation influences the design of the laser launch location, laser power, WFS detector, and centroiding algorithm for LGS GLAO on an extremely large telescope. We show, using end-to-end numerical simulations, that with a noise-weighted matrix-vector-multiply reconstructor, the performance in terms of 50% ensquared energy (EE) of the side and central launch of the lasers is equivalent, the matched filter and weighted center of gravity centroiding algorithms are the most promising, and approximately 10×10 undersampled pixels are optimal. Significant improvement in the 50% EE can be observed with a few tens of photons/subaperture/frame, and no significant gain is seen by adding more than 200 photons/subaperture/frame. The LGS GLAO is not particularly sensitive to the sodium profile present in the mesosphere nor to a short-timescale (less than 100?s) evolution of the sodium profile. The performance of LGS GLAO is, however, sensitive to the atmospheric turbulence profile.     

Shack-Hartmann wavefront sensing with elongated sodium laser beacons: centroiding versus matched filtering

We describe modeling and simulation results for the Thirty Meter Telescope on the degradation of sodium laser guide star Shack-Hartmann wavefront sensor measurement accuracy that will occur due to the spatial structure and temporal variations of the mesospheric sodium layer. By using a contiguous set of lidar measurements of the sodium profile, the performance of a standard centroid and of a more refined noise-optimal matched filter spot position estimation algorithm is analyzed and compared for a nominal mean signal level equal to 1000 photodetected electrons per subaperture per integration time, as a function of subaperture to laser launch telescope distance and CCD pixel readout noise. Both algorithms are compared in terms of their rms spot position estimation error due to noise, their associated wavefront error when implemented on the Thirty Meter Telescope facility adaptive optics system, their linear dynamic range, and their bias when detuned from the current sodium profile.     

Measuring the centroid gain of a Shack-Hartmann quad-cell wavefront sensor by using slope discrepancy

Shack-Hartmann wavefront sensors (SH WFS) are used by many adaptive optics (AO) systems to measure the wavefront. In this WFS, the centroid of the spots is proportional to the wavefront slope. If the detectors consist of 2 x 2 quad cells, as is the case in most astronomical AO systems, then the centroid measurement is proportional to the centroid gain. This quantity varies with the strength of the atmospheric turbulence and the angular extent of the beacon. The benefits of knowing the centroid gain and current techniques to measure it are discussed. A new method is presented, which takes advantage of the fact that, in a SH-WFS-based AO system, there are usually more measurements than actuators. Centroids in the null space of the wavefront reconstructor, called slope discrepancy measurements, contain information about the centroid gain. Tests using the W. M. Keck Observatory AO system demonstrate the accuracy of the algorithm.     

Methods for correcting tilt anisoplanatism in laser-guide-star-based multiconjugate adaptive optics

Multiconjugate adaptive optics (MCAO) is a technique for correcting turbulence-induced phase distortions in three dimensions instead of two, thereby greatly expanding the corrected field of view of an adaptive optics system. This is accomplished with use of multiple deformable mirrors conjugate to distinct ranges in the atmosphere, with actuator commands computed from wave-front sensor (WFS) measurements from multiple guide stars. Laser guide stars (LGSs) must be used (at least for the forseeable future) to achieve a useful degree of sky coverage in an astronomical MCAO system. Much as a single LGS cannot be used to measure overall wave-front tilt, a constellation of multiple LGSs at a common range cannot detect tilt anisoplanatism. This error alone will significantly degrade the performance of a MCAO system based on a single tilt-only natural guide star (NGS) and multiple tilt-removed LGSs at a common altitude. We present a heuristic, low-order model for the principal source of tilt anisoplanatism that suggests four possible approaches to eliminating this defect in LGS MCAO: (i) tip/tilt measurements from multiple NGS, (ii) a solution to the LGS tilt uncertainty problem, (iii) additional higher-order WFS measurements from a single NGS, or (iv) higher-order WFS measurements from both sodium and Rayleigh LGSs at different ranges. Sample numerical results for one particular MCAO system configuration indicate that approach (ii), if feasible, would provide the highest degree of tilt anisoplanatism compensation. Approaches (i) and (iv) also provide very useful levels of performance and do not require unrealistically low levels of WFS measurement noise. For a representative set of parameters for an 8-m telescope, the additional laser power required for approach (iv) is on the order of 2 W per Rayleigh LGS.     

Monostatic and bistatic schemes and an optimal algorithm for tilt correction in ground-based adaptive telescopes

It has been pointed out that the two terms effective scattering volume and laser guide star are scientific synonyms. The first term was introduced by the end of 1970 by specialists in atmospheric optics and laser sounding. Information has also been given concerning fluctuations of waves reflected from an object and waves passed twice through atmospheric inhomogeneities. A corresponding mathematical apparatus allows one to estimate correlation characteristics of a laser guide star. The mutual correlation function of random angular displacements of a plane-wave image and the centroid displacements of a focused Gaussian laser beam are considered. The algorithm of optimal correction is introduced in which a priori information is used. Based on this algorithm, the variance of residual jitter of a star image obtained with the use of a laser guide star is evaluated.     

High-resolution lidar observations of mesospheric sodium and implications for adaptive optics

Observations of sodium density variability in the upper mesosphere/lower thermosphere, obtained using a high-resolution lidar system, show rapid fluctuations in the sodium centroid altitude. The temporal power spectrum extends above 1 Hz and is well-fit by a power law having a slope that is -1.95±0.12. These fluctuations produce focus errors in adaptive optics systems employing continuous-wave sodium laser guide stars, which can be significant for large-aperture telescopes. For a 30 m aperture diameter, the associated rms wavefront error is approximately 4 nm per meter of altitude change and increases as the square of the aperture diameter. The vertical velocity of the sodium centroid altitude is found to be ~23 ms(-1) on a 1 s time scale. If these high-frequency fluctuations arise primarily from advection of horizontal structure by the mesospheric wind, our data imply that variations in the sodium centroid altitude on the order of tens of meters occur over the horizontal scales spanned by proposed laser guide star asterisms. This leads to substantial differential focus errors (~107 nm over a 1 arc min separation with a 30 m aperture diameter) that may impact the performance of wide-field adaptive optics systems. Short-lasting and narrow sodium density enhancements, more than 1 order of magnitude above the local sodium density, occur due to advection of meteor trails. These have the ability to change the sodium centroid altitude by as much as 1 km in less than 1 s, which could result in temporary disruption of adaptive optics systems.     

Laboratory demonstration of accurate and efficient nanometer-level wavefront control for extreme adaptive optics

A 32 x 32 microelectricalmechanical systems mirror is controlled in a closed-loop adaptive optics test bed with a spatially filtered wavefront sensor (WFS), Fourier transform wavefront reconstruction, and calibration of references with a high-precision interferometer. When correcting the inherent aberration of the mirror, 0.7 nm rms phase error in the controllable band is achieved. When correcting an etched phase plate with atmospheric statistics, a dark hole 10(3) deeper than the uncontrollable phase is produced in the phase power spectral density. Compensation of the mirror's influence function is done with a Fourier filter, which results in improved loop convergence. Use of the spatial filter is shown to reduce the gain variability of the WFS in a quadcell configuration.     

Tomographic wavefront error using multi-LGS constellation sensed with Shack-Hartmann wavefront sensors

Noise effects induced by laser guide star (LGS) elongation have to be considered globally in a multi-LGS tomographic reconstruction analysis. This allows a fine estimation of performance and the comparison of different launching options. We present a modal analysis of the wavefront error with Shack-Hartmann wavefront sensors based on quasi-analytical matrix formalism. Including spot elongation and the Rayleigh fratricide effect, edge launching produces similar performance to central launching and avoids the risk of possible underestimation of fratricide scatter. Performance improves slightly with an optimized centroid estimator and is not affected by a slight field-of-view truncation of the subapertures. Finally we discuss detector characteristics for a LGS Shack-Hartmann wavefront sensor.     

Cylinder gauge measurement using a position sensitive detector

A position sensitive detector (PSD) has been used to determine the diameter of cylindrical pins based on the shift in a laser beam's centroid. The centroid of the light beam is defined here as the weighted average of position by the local intensity. A shift can be observed in the centroid of an otherwise axially symmetric light beam, which is partially obstructed. Additionally, the maximum shift in the centroid is a unique function of the obstructing cylinder diameter. Thus to determine the cylinder diameter, one only needs to detect this maximum shift as the cylinder is swept across the beam.     

Widening the effective field of view of adaptive-optics telescopes by deconvolution from wave-front sensing: average and signal-to-noise ratio performance

A fundamental problem of adaptive-optics systems is the very narrow corrected field of view that can be obtained because turbulence is extended in altitude throughout the atmosphere. The correctable field of view is of the order of 5-10 μrad at visible wavelengths and increases as the wavelength increases. Previous concepts to broaden the corrected field of view have been hardware oriented, requiring multiple wave-front sensor (WFS) measurements to control multiple deformable mirrors. We analyze the average and the signal-to-noise-ratio performance of an image measurement and postprocessing technique that uses simultaneous measurements of a short-exposure compensated image measured in an off-axis direction; an additional WFS measurement is taken in the off-axis direction. Results are presented for infinite-altitude WFS beacons driving both the WFS for the adaptive optics and the WFS looking in the off-axis direction, a variety of seeing and WFS light-level conditions, and off-axis angles from two to six times the isoplanatic angle. This technique improves the average effective transfer function out to a field angle of at least six times the isoplanatic angle while providing a higher signal-to-noise ratio in the spatial frequency domain.     

Correlation processing for correction of phase distortions in subaperture imaging

Ultrasonic subaperture imaging combines synthetic aperture and phased array approaches and permits low-cost systems with improved image quality. In subaperture processing, a large array is synthesized using echo signals collected from a number of receive subapertures by multiple firings of a phased transmit subaperture. Tissue inhomogeneities and displacements in subaperture imaging may cause significant phase distortions on received echo signals. Correlation processing on reference echo signals can be used for correction of the phase distortions, for which the accuracy and robustness are critically limited by the signal correlation. In this study, we explore correlation processing techniques for adaptive subaperture imaging with phase correction for motion and tissue inhomogeneities. The proposed techniques use new subaperture data acquisition schemes to produce reference signal sets with improved signal correlation. The experimental test results were obtained using raw radio frequency (RF) data acquired from two different phantoms with 3.5 MHz, 128-element transducer array. The results show that phase distortions can effectively be compensated by the proposed techniques in real-time adaptive subaperture imaging.     

Laser beam projection with adaptive array of fiber collimators. I. Basic considerations for analysis

We present a mathematical model and provide an analysis of optical beam director systems composed of adaptive arrays of fiber collimators (subapertures), referred to here as conformal optical systems. Performances of the following two system architectures are compared: A conformal-beam director with mutually incoherent output laser beams transmitted through fiber collimators (beamlets), and a corresponding coherent system whose beamlets can be coherently combined (phase locked) at a remote target plane. The effect of the major characteristics of the conformal systems on the efficiency of laser beam projection is evaluated both analytically and through numerical simulations. The characteristics considered here are the number of fiber collimators and the subaperture and conformal aperture fill factors, as well as the accuracy of beamlet pointing.     

Subaperture processing for ultrasonic imaging

Ultrasonic subaperture processing using aperture synthesis and beam space interpolation is presented. The number of beam lines scanning the image plane for a given transmit-receive subaperture combination is chosen according to the spatial sampling criteria for that combination. On each beam line, echo signals over the entire array are collected through electronic multiplexing of array channels, where the transmit subaperture at the transducer center is fired K successive times, with K equal to the number of nonoverlapping receive subapertures. For every receive subaperture, the number of beam lines is increased through digital interpolation using a linear filter with spatial frequency band associated with the subaperture. Interpolated beam lines from all receive subapertures are then added to obtain a high resolution sector image. The efficiency of subaperture processing for different system configurations is tested on experimental rf data acquired from two different phantoms using a 3.5 MHz, 128-element transducer array. The proposed subaperture processing reduces the number of firings for data acquisition, and thus allows real-time imaging with very low susceptibility to motion artifacts.     

Sampling error analysis of Shack–Hartmann wavefront sensor with variable subaperture pixels

Abstract

The sampling error of a Shack–Hartmann wavefront sensor with variable subaperture pixels is analysed under the consideration of various threshold values and detecting dynamic ranges. A generalized expression, which is used for fitting the sampling error of a Shack-Hartmann wavefront sensor with variable subaperture pixels, is presented. The computational results of the sampling error of a Shack–Hartmann wavefront sensor with different pixel numbers per subaperture, different detecting dynamic ranges, different atmospheric coherence length, different extended degree of the object and the different threshold values are also given. The results indicate that the sampling error of the Shack–Hartmann wavefront sensor is sensitive to the dynamic range of the subaperture, the pixel numbers per subaperture, the extended degree of the object and the coherent length of atmosphere, but not sensitive to the threshold value.     

Iterative algorithm for subaperture stitching test with spherical interferometers

Recently we have proposed an iterative algorithm for subaperture stitching interferometry. It was referred to as the subaperture stitching and localization (SASL) algorithm. The limitation of the algorithm is that three-dimensional Cartesian coordinates are required, whereas the standard spherical interferometer can read out only the phase differences on the pixels. On the basis of the SASL algorithm, we propose an iterative algorithm for a spherical subaperture stitching test. It deals with data directly from the spherical interferometer. Unknown radii of best-fit spheres for a null test of subapertures are included in the optimization variables. The developed algorithm inherits the advantages of the SASL algorithm.     

Experimental demonstration of an optical phased array antenna for laser space communications

The feasibility of an optical phased array antenna applicable for spaceborne laser communications was experimentally demonstrated. Heterodyne optical phase-locked loops provide for a defined phase relationship between the collimated output beams of three single-mode fibers. In the far field the beams interfere with a measured efficiency of 99%. The main lobe of the interference pattern can be moved by phase shifting the subaperture output beams. The setup permitted agile beam steering within an angular range of 1 mr and a response time of 0.7 ms. We propose an operational optical phased array antenna fed by seven lasers, featuring high transmit power and redundance.     

Self-calibrated subaperture stitching test of hyper-hemispheres using latitude and longitude coordinates

Limited by the f-number of the transmission sphere, it is impossible to test the whole surface of a hyper-hemisphere using a standard interferometer directly. This paper presents an extension of the subaperture stitching test method to hyper hemispheres. The stitching algorithm is based on the coordinate mapping from local measurement frame to a global frame, and overlapping correspondence is calculated by virtue of coordinates of latitude and longitude. The reference surface error is represented by Zernike polynomials and self-calibrated during the stitching to achieve higher accuracy. Then the stitched surface error distribution is presented by map projection. To realize accessibility to the whole surface of a hyper-hemisphere, we also propose a design for the subaperture test platform, according to the subaperture lattice design. Finally, a hemisphere and a full sphere are tested and figured, respectively, to validate the method and the experimental setup.     

Optical sparse aperture imaging

The resolution of a conventional diffraction-limited imaging system is proportional to its pupil diameter. A primary goal of sparse aperture imaging is to enhance resolution while minimizing the total light collection area; the latter being desirable, in part, because of the cost of large, monolithic apertures. Performance metrics are defined and used to evaluate several sparse aperture arrays constructed from multiple, identical, circular subapertures. Subaperture piston and/or tilt effects on image quality are also considered. We selected arrays with compact nonredundant autocorrelations first described by Golay. We vary both the number of subapertures and their relative spacings to arrive at an optimized array. We report the results of an experiment in which we synthesized an image from multiple subaperture pupil fields by masking a large lens with a Golay array. For this experiment we imaged a slant edge feature of an ISO12233 resolution target in order to measure the modulation transfer function. We note the contrast reduction inherent in images formed through sparse aperture arrays and demonstrate the use of a Wiener-Helstrom filter to restore contrast in our experimental images. Finally, we describe a method to synthesize images from multiple subaperture focal plane intensity images using a phase retrieval algorithm to obtain estimates of subaperture pupil fields. Experimental results from synthesizing an image of a point object from multiple subaperture images are presented, and weaknesses of the phase retrieval method for this application are discussed.