Adaptive optics with advanced phase-contrast techniques. I. High-resolution wave-front sensing

High-resolution phase-contrast wave-front sensors based on phase spatial light modulators and micromirror/ liquid-crystal arrays are introduced. Wave-front sensor performance is analyzed for atmospheric-turbulence-induced phase distortions described by the Kolmogorov and the Andrews models. A high-resolution phase-contrast wave-front sensor (nonlinear Zernike filter) based on an optically controlled liquid-crystal phase spatial light modulator is experimentally demonstrated. The results demonstrate high-resolution visualization of dynamically changing phase distortions within the sensor time response of approximately 10 ms.     

Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a shack-hartmann wave-front sensor

We examine wave-front distortion caused by high-power lasers on transmissive optics using a Shack-Hartmann wave-front sensor. The coupling coefficient for a thermally aberrated Gaussian beam to the TEM(00) mode of a cavity was determined as a function of magnitude of the thermally induced aberration. One wave of thermally induced phase aberration between the Gaussian intensity peak and the 1/e(2) radius of the intensity profile reduces the power-coupling coefficient to the TEM(00) mode of the cavity to 4.5% with no compensation. With optimal focus compensation the power coupling is increased to 79%. The theoretical shape of the thermally induced optical phase aberration is compared with measurements made in a neutral-density filter glass, Faraday glass, and lithium niobate. The agreement between the theoretical and the measured thermal aberration profiles is within the rms wave-front measurement sensitivity of the Shack-Hartmann wave-front sensor, which is a few nanometers.     

Spatial spectrum analysis of wave-front correction with a segmented mirror

An expression is derived for the spatial power spectrum of wave-front errors after correction with a segmented mirror. This includes estimates of the spectral contributions of segment piston and tilt corrections and spatial aliasing by a regular array of segments. The approach allows rapid computation of wave-front error spectra in systems with highly segmented mirrors.     

Geometric view of adaptive optics control

The objective of an astronomical adaptive optics control system is to minimize the residual wave-front error remaining on the science-object wave fronts after being compensated for atmospheric turbulence and telescope aberrations. Minimizing the mean square wave-front residual maximizes the Strehl ratio and the encircled energy in pointlike images and maximizes the contrast and resolution of extended images. We prove the separation principle of optimal control for application to adaptive optics so as to minimize the mean square wave-front residual. This shows that the residual wave-front error attributable to the control system can be decomposed into three independent terms that can be treated separately in design. The first term depends on the geometry of the wave-front sensor(s), the second term depends on the geometry of the deformable mirror(s), and the third term is a stochastic term that depends on the signal-to-noise ratio. The geometric view comes from understanding that the underlying quantity of interest, the wave-front phase surface, is really an infinite-dimensional vector within a Hilbert space and that this vector space is projected into subspaces we can control and measure by the deformable mirrors and wave-front sensors, respectively. When the control and estimation algorithms are optimal, the residual wave front is in a subspace that is the union of subspaces orthogonal to both of these projections. The method is general in that it applies both to conventional (on-axis, ground-layer conjugate) adaptive optics architectures and to more complicated multi-guide-star- and multiconjugate-layer architectures envisaged for future giant telescopes. We illustrate the approach by using a simple example that has been worked out previously [J. Opt. Soc. Am. A 73, 1171 (1983)] for a single-conjugate, static atmosphere case and follow up with a discussion of how it is extendable to general adaptive optics architectures.     

Kalman filtering to suppress spurious signals in adaptive optics control

In many scenarios, an adaptive optics (AO) control system operates in the presence of temporally non-white noise. We use a Kalman filter with a state space formulation that allows suppression of this colored noise, hence improving residual error over the case where the noise is assumed to be white. We demonstrate the effectiveness of this new filter in the case of the estimated Gemini Planet Imager tip-tilt environment, where there are both common-path and non-common-path vibrations. We discuss how this same framework can also be used to suppress spatial aliasing during predictive wavefront control assuming frozen flow in a low-order AO system without a spatially filtered wavefront sensor, and present experimental measurements from Altair that clearly reveal these aliased components.     

Signal-to-noise comparison of deconvolution from wave-front sensing with traditional linear and speckle image reconstruction

It is well known that atmospheric turbulence severely degrades the performance of ground-based imaging systems. Techniques to overcome the effects of the atmosphere have been developing at a rapid pace over the past 10 years. These techniques can be grouped into two broad categories: predetection and postdetection techniques. A recent newcomer to the postdetection scene is deconvolution from wave-front sensing (DWFS). DWFS is a postdetection image-reconstruction technique that makes use of one feature of predetection techniques. A wave-front sensor (WFS) is used to record the wave-front phase distortion in the pupil of the telescope for each short-exposure image. The additional information provided by the WFS is used to estimate the system's point-spread function (PSF). The PSF is then used in conjunction with the ensemble of short-exposure images to obtain an estimate of the object intensity distribution through deconvolution. With the addition of DWFS to the suite of possible postdetection image-reconstruction techniques, it is natural to ask "How does DWFS compare with both traditional linear and speckle image-reconstruction techniques?" In the results we make a direct comparison based on a frequency-domain signal-to-noise-ratio performance metric. This metric is applied to each technique's image-reconstruction estimator. We find that DWFS nearly always results in improved performance over the estimators of traditional linear image reconstruction such as Wiener filtering. On the other hand, DWFS does not always outperform speckle-imaging techniques, and in cases that it does the improvement is small.     

大气相位屏的协方差冲激函数产生法

提出正交基为冲激函数的协方差法,在Matlab中仿真实现符合大气湍流统计分布(vonKarman谱和Kolmogorov谱)的大气相位屏。协方差冲激函数产生法对连续波前相位进行冲激函数正交分解,根据正交分解系数的协方差模拟平行光通过大气后的瞬时波前相位。计算多帧相位屏的波前结构函数,与理论值比对,验证产生大气相位屏的正确性。根据比对结果得,该方法产生的相位屏精度高,对于Kolmogorov谱,该方法产生的大气相位屏的波前结构函数误差约为理论波前结构函数均值的1%。     

Modeling the combined effect of static and varying phase distortions on the performance of adaptive optical systems

The end-to-end performance achieved by an adaptive optical (AO) imaging system is determined by a combination of the residual time-varying phase distortions associated with atmospheric turbulence and the quasi-static unsensed and uncorrectable aberrations in the optical system itself. Although the effects of these two errors on the time-averaged Strehl ratio and the time-averaged optical transfer function (OTF) of the AO system are not formally separable, such an approximation is found to be accurate to within a few percent for a range of representative residual wave-front errors. In these calculations, we combined static optical system aberrations and time-varying residual phase distortion characteristics of a deformable mirror fitting error, wave-front sensor noise, and anisoplanatism. The static aberrations consist of focus errors of varying magnitudes as well as a combination of unsensed and uncorrectable mirror figure errors derived from modeling by the Gemini 8-Meter Telescopes Project. The overall Strehl ratios and OTF's that are due to the combined effect of these error sources are well approximated as products of separate factors for the static and time-varying aberrations, as long as the overall Strehl ratio that is due to both errors is greater than approximately 0.1. For lower Strehl ratios, the products provide lower bounds on the actual values of the Strehl ratio and the OTF. The speckle transfer function is also well approximated by a product of two functions, but only where AO compensation is sufficiently good that speckle imaging techniques are usually not required.     

Rician distribution to describe speckle statistics in adaptive optics

Adaptive optics systems allow us to retrieve high-spatial-frequency information that is preserved in the wave fronts distorted by the atmosphere. Although wave-front correction should be as complete as possible, only partial compensation is attainable in the visible. We provide a procedure that uses the Rician distribution to predict the intensity statistics of the light at the image center as a function of the number of corrected Zernike polynomials.     

Tip/tilt estimation from defocused images

In astronomical imaging, the errors in the wave-front slope are a significant cause of aberrations in the detected image. We investigate how the slope can be estimated optimally using an intensity measurement of the propagated wave front. We show that the optimal location for detection of wave-front tilt is the focal plane, and we quantify the error in using defocused images, such as would be obtained from a curvature sensor, for estimating the wave-front tilt. The effect of using broadband light is also quantified.     

Effect of a spoke surface error on a phase mask in a computational imaging system

The effect of a spoke surface error on a phase mask in a computational imaging system was analyzed by combining the similarity of the point spread function (PSF) and the peak signal-to-noise ratio (PSNR) of de-convoluted images. The spoke surface error was applied on a phase mask with different peak-to-valley (P-V) values with various numbers of spoke rings in simulation. The minimum requirement of PSF similarity will be determined by a given PSNR threshold, which relates the defocus aberration. Finally, it can be concluded that a low-spatial frequency surface error is critical for a cubic phase mask in a computational imaging system with lower P-V error.     

Decoupled stochastic parallel gradient descent optimization for adaptive optics: integrated approach for wave-front sensor information fusion

A new adaptive wave-front control technique and system architectures that offer fast adaptation convergence even for high-resolution adaptive optics is described. This technique is referred to as decoupled stochastic parallel gradient descent (D-SPGD). D-SPGD is based on stochastic parallel gradient descent optimization of performance metrics that depend on wave-front sensor data. The fast convergence rate is achieved through partial decoupling of the adaptive system's control channels by incorporating spatially distributed information from a wave-front sensor into the model-free optimization technique. D-SPGD wave-front phase control can be applied to a general class of adaptive optical systems. The efficiency of this approach is analyzed numerically by considering compensation of atmospheric-turbulence-induced phase distortions with use of both low-resolution (127 control channels) and high-resolution (256 x 256 control channels) adaptive systems. Results demonstrate that phase distortion compensation can be achieved during only 10-20 iterations. The efficiency of adaptive wave-front correction with D-SPGD is practically independent of system resolution.     

Shack Hartmann wave-front measurement with a large F-number plastic microlens array

A new plastic microlens array, consisting of 900 lenslets, has been developed for the Shack Hartmann wave-front sensor.The individual lens is 300 μm × 300μm and has a focal length of 10 mm, which provides the same focal size, 60 μm in diameter, with a constant peak intensity. One can improve thewave-front measurement accuracy by reducing the spot centroiding error by averaging a few frame memories of an image processor. A deformable mirror for testing the wave-front sensor gives anappropriate defocus and astigmatism, and the laser wave front is measured with a Shack Hartmann wave-front sensor. The measurement accuracy and reproducibility of our wave-front sensor are better than λ/20 and λ/50 (λ = 632.8 nm),respectively, in rms.     

Noise propagation in wave-front sensing with phase diversity

The phase diversity technique is studied as a wave-front sensor to be implemented with widely extended sources. The wave-front phase expanded on the Zernike polynomials is estimated from a pair of images (in focus and out of focus) by use of a maximum-likelihood approach. The propagation of the photon noise in the images on the estimated phase is derived from a theoretical analysis. The covariance matrix of the phase estimator is calculated, and the optimal distance between the observation planes that minimizes the noise propagation is determined. The phase error is inversely proportional to the number of photons in the images. The noise variance on the Zernike polynomials increases with the order of the polynomial. These results are confirmed with both numerical and experimental validations. The influence of the spectral bandwidth on the phase estimator is also studied with simulations.     

Atmospheric-turbulence-induced power-fade statistics for a multiaperture optical receiver

To estimate the probability distributions of power fades, we consider two basic types of disturbance in electromagnetic wave propagation through atmospheric turbulence: wave-front intensity fluctuations and wave-front distortion. We assess the reduction in the cumulative probability of losses caused by these two effects through spatial diversity by using a multiaperture receiver configuration. Degradations in receiver performance are determined with fractal techniques used to simulate the turbulence-induced wave-front phase distortion, and a log normal model is assumed for the collected power fluctuations.     

Wind-induced deformations in a segmented mirror

A Zernike expansion of wind-induced deformations in a segmented mirror is described. The wind model is a frozen turbulent field with a Kolmogorov spectrum for scales smaller than the outer scale and a flat spectrum for scales larger than the outer scale. The approach allows a mode-by-mode comparison of the wave-front error contributions from atmospheric phase distortions, wind-induced deformations, and the mirror control system noise. This is used to design a controller that minimizes the mirror surface errors by application of corrections based on edge sensor measurements and wave-front measurements on a guide star.     

Theory of singular-phase reconstruction for an optical speckle field in the turbulent atmosphere

Analytical expressions are derived and computational algorithms are constructed for retrieving optical-field phase distribution under strong scintillation. The input data for the phase reconstruction are the wave-front slopes registered by a Hartmann sensor or shearing interferometer. The theory is based on representing the slope-vector field as the sum of its potential and solenoid components; it introduces the concept of phase-source and phase-vortex density and uses strict integral expressions relating these quantities to the wave-front slopes. To overcome the difficulties arising from the singular character of phase distribution, use is made of regularization of the wave-front slopes. The slopes can be measured with an ideal point wave-front sensor. It is shown that the slopes measured at the output of a nonideal sensor can be treated as regularized values of these slopes. Numerical simulation of phase unwrapping from the reference values of the wave-front slopes has shown that the algorithm designed for visualization of local phase singularities and those for phase reconstruction are very helpful in eliminating the measurement noise.     

Binary adaptive optics: atmospheric wave-front correction with a half-wave phase shifter

We describe a binary approach to adaptive wave-front correction, especially suitable for narrow band applications, which would be simpler than conventional adaptive technology. Appropriate parts of the aberrant wave front are phase retarded by half a wavelength to ensure that none of the image-forming rays add together destructively. Simulations for monochromatic light show that the residual wave-front errors, in the absence of other errors, would result in Strehl ratios of ~40% with diffraction-limited widths at visible wavelengths. We simulate the imaging performance of such a system and describe a possible implementation that uses a ferroelectric liquid-crystal spatial light modulator.     

Monochromatic aberrations provide an odd-error cue to focus direction

Monochromatic aberrations that exist in the human eye will cause differences in the appearance of the point-spread function (PSF) depending on whether there is positive or negative defocus. We establish whether it is possible to use these differences in the PSF to distinguish the direction of defocus. The monochromatic aberrations of eight subjects were measured with a Hartmann-Shack wave-front sensor. Subjects also performed a forced-choice psychophysical task in which they decided whether a blurred target was defocused in front of or behind the retina. The optical system for the psychophysical task was designed to isolate the blur due to monochromatic aberrations as the only odd-error cue to the direction of defocus. Shack-Hartmann measurements showed that monochromatic aberrations increase as the pupil size increases. On average, the correct/incorrect responses for discriminating differences in the PSF for different directions of defocus were 54/46 for a 1-mm pupil and 83/17 for a 5-mm pupil, representing more than an eight-fold increase in discriminability. This discriminability extended for large amounts of defocus and also for more complex targets, such as letters. Sensitivity to the differences in the PSF for different directions of defocus increased as monochromatic aberrations increased, particularly for the even-order aberrations, which give rise to an odd-error focus cue. It was found that the ability to discriminate PSFs for different directions of defocus varied among individuals but, in general, depended on the magnitude of monochromatic aberrations.     

Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor

The performance of ground-based optical imaging systems is severely degraded from the diffraction limit by the random effects of the atmosphere. Adaptive-optics techniques have been used to compensate for atmospheric-turbulence effects. A critical component in the adaptive-optics system is the wave-front sensor. At present, two types of sensors are common: the Hartmann-Shack wave-front sensor and the shearing interferometer wave-front sensor. In this paper we make a direct performance comparison of these two sensors. The performance calculations are restricted to common configurations of these two sensors and the fundamental limits imposed by shot noise and atmospheric effects. These two effects encompass the effects of extended reference beacons and sensor subaperture spacings larger than the Fried parameter r(0). Our results indicate comparable performance for good seeing conditions and small beacons. However, for poor seeing conditions and extended beacons, the Hartmann sensor has lower error levels than the shearing interferometer.