Optimal modal fourier-transform wavefront control

Laboratory demonstration of the prediction of wind-blown turbulence by adaptive optics at 8 kHz with use of LQG control

The low-latency adaptive optical mirror system (LLAMAS) is designed to push the limits on achievable latencies and frame rates. It has 21 subapertures across its pupil. A reformulated version of the linear quadratic Gaussian (LQG) method predictive Fourier control is implemented in LLAMAS; for all modes, it takes just 30 µs to compute. In the testbed, a turbulator mixes hot and ambient air to produce wind-blown turbulence. Wind prediction clearly improves correction when compared to an integral controller. Closed-loop telemetry shows that wind-predictive LQG removes the characteristic "butterfly" and reduces temporal error power by up to a factor of three for mid-spatial frequency modes. Strehl changes seen in focal plane images are consistent with telemetry and the system error budget.

Fast iterative tomographic wavefront estimation with recursive Toeplitz reconstructor structure for large-scale systems

Tomographic wavefront reconstruction is the main computational bottleneck to realize real-time correction for turbulence-induced wavefront aberrations in future laser-assisted tomographic adaptive-optics (AO) systems for ground-based giant segmented mirror telescopes because of its unprecedented number of degrees of freedom, N, i.e., the number of measurements from wavefront sensors. In this paper, we provide an efficient implementation of the minimum-mean-square error (MMSE) tomographic wavefront reconstruction, which is mainly useful for some classes of AO systems not requiring multi-conjugation, such as laser-tomographic AO, multi-object AO, and ground-layer AO systems, but is also applicable to multi-conjugate AO systems. This work expands that by Conan [Proc. SPIE9148, 91480R (2014)PSISDG0277-786X10.1117/12.2054472] to the multi-wavefront tomographic case using natural and laser guide stars. The new implementation exploits the Toeplitz structure of covariance matrices used in an MMSE reconstructor, which leads to an overall O(N log N) real-time complexity compared with O(N²) of the original implementation using straight vector-matrix multiplication. We show that the Toeplitz-based algorithm leads to 60 nm rms wavefront error improvement for the European Extremely Large Telescope laser-tomography AO system over a well-known sparse-based tomographic reconstruction; however, the number of iterations required for suitable performance is still beyond what a real-time system can accommodate to keep up with the time-varying turbulence.

Iterative wave-front reconstruction in the Fourier domain

The use of Fourier methods in wave-front reconstruction can significantly reduce the computation time for large telescopes with a high number of degrees of freedom. However, Fourier algorithms for discrete data require a rectangular data set which conform to specific boundary requirements, whereas wave-front sensor data is typically defined over a circular domain (the telescope pupil). Here we present an iterative Gerchberg routine modified for the purposes of discrete wave-front reconstruction which adapts the measurement data (wave-front sensor slopes) for Fourier analysis, fulfilling the requirements of the fast Fourier transform (FFT) and providing accurate reconstruction. The routine is used in the adaptation step only and can be coupled to any other Wiener-like or least-squares method. We compare simulations using this method with previous Fourier methods and show an increase in performance in terms of Strehl ratio and a reduction in noise propagation for a 40×40 SPHERE-like adaptive optics system. For closed loop operation with minimal iterations the Gerchberg method provides an improvement in Strehl, from 95.4% to 96.9% in K-band. This corresponds to ~ 40 nm improvement in rms, and avoids the high spatial frequency errors present in other methods, providing an increase in contrast towards the edge of the correctable band.

Performance of the Gemini Planet Imager's adaptive optics system

The Gemini Planet Imager's adaptive optics (AO) subsystem was designed specifically to facilitate high-contrast imaging. A definitive description of the system's algorithms and technologies as built is given. 564 AO telemetry measurements from the Gemini Planet Imager Exoplanet Survey campaign are analyzed. The modal gain optimizer tracks changes in atmospheric conditions. Science observations show that image quality can be improved with the use of both the spatially filtered wavefront sensor and linear-quadratic-Gaussian control of vibration. The error budget indicates that for all targets and atmospheric conditions AO bandwidth error is the largest term.

First light of the Gemini Planet imager

The Gemini Planet Imager is a dedicated facility for directly imaging and spectroscopically characterizing extrasolar planets. It combines a very high-order adaptive optics system, a diffraction-suppressing coronagraph, and an integral field spectrograph with low spectral resolution but high spatial resolution. Every aspect of the Gemini Planet Imager has been tuned for maximum sensitivity to faint planets near bright stars. During first-light observations, we achieved an estimated H band Strehl ratio of 0.89 and a 5-σ contrast of 10(6) at 0.75 arcseconds and 10(5) at 0.35 arcseconds. Observations of Beta Pictoris clearly detect the planet, Beta Pictoris b, in a single 60-s exposure with minimal postprocessing. Beta Pictoris b is observed at a separation of 434 ± 6 milliarcseconds (mas) and position angle 211.8 ± 0.5°. Fitting the Keplerian orbit of Beta Pic b using the new position together with previous astrometry gives a factor of 3 improvement in most parameters over previous solutions. The planet orbits at a semimajor axis of [Formula: see text] near the 3:2 resonance with the previously known 6-AU asteroidal belt and is aligned with the inner warped disk. The observations give a 4% probability of a transit of the planet in late 2017.

Fast computation of an optimal controller for large-scale adaptive optics

The linear quadratic Gaussian regulator provides the minimum-variance control solution for a linear time-invariant system. For adaptive optics (AO) applications, under the hypothesis of a deformable mirror with instantaneous response, such a controller boils down to a minimum-variance phase estimator (a Kalman filter) and a projection onto the mirror space. The Kalman filter gain can be computed by solving an algebraic Riccati matrix equation, whose computational complexity grows very quickly with the size of the telescope aperture. This "curse of dimensionality" makes the standard solvers for Riccati equations very slow in the case of extremely large telescopes. In this article, we propose a way of computing the Kalman gain for AO systems by means of an approximation that considers the turbulence phase screen as the cropped version of an infinite-size screen. We demonstrate the advantages of the methods for both off- and on-line computational time, and we evaluate its performance for classical AO as well as for wide-field tomographic AO with multiple natural guide stars. Simulation results are reported.

Cumulative Reconstructor: fast wavefront reconstruction algorithm for Extremely Large Telescopes

The Cumulative Reconstructor (CuRe) is a new direct reconstructor for an optical wavefront from Shack-Hartmann wavefront sensor measurements. In this paper, the algorithm is adapted to realistic telescope geometries and the transition from modified Hudgin to Fried geometry is discussed. After a discussion of the noise propagation, we analyze the complexity of the algorithm. Our numerical tests confirm that the algorithm is very fast and accurate and can therefore be used for adaptive optics systems of Extremely Large Telescopes.

Closed-loop control of a woofer-tweeter adaptive optics system using wavelet-based phase reconstruction

A novel closed-loop control technique for adaptive optics (AO) systems based on a wavelet-based phase reconstruction technique and a woofer-tweeter controller is presented. The wavelet-based reconstruction technique is based on obtaining a Haar decomposition of the phase screen directly from gradient measurements and has been extended here with the use of a Poisson solver to improve performance. This method is O(N) (i.e., a linear computation cost as number of actuators increases) and is the fastest of the known O(N) reconstruction techniques. The controller configuration is based on the woofer-tweeter controller to control low- and high-spatial-frequency aberrations, respectively. The separation of the woofer and tweeter signals is done using a computationally efficient method that is based on the availability of a low-spatial-resolution reconstruction during the wavelet synthesis process. The performance of the proposed technique is evaluated using a simulated AO system and phase screens generated to reflect atmospheric turbulence with various dynamic characteristics. Results indicate that the combination of the wavelet-based phase reconstruction and woofer-tweeter controller leads to very good results with respect to speed and accuracy.

Performance comparison of wavefront reconstruction and control algorithms for Extremely Large Telescopes

Extremely Large Telescopes (ELTs) are very challenging with respect to their adaptive optics (AO) requirements. Their diameters and the specifications required by the astronomical science for which they are being designed imply a huge increment in the number of degrees of freedom in the deformable mirrors. Faster algorithms are needed to implement the real-time reconstruction and control in AO at the required speed. We present the results of a study of the AO correction performance of three different algorithms applied to the case of a 42-m ELT: one considered as a reference, the matrix-vector multiply (MVM) algorithm; and two considered fast, the fractal iterative method (FrIM) and the Fourier transform reconstructor (FTR). The MVM and the FrIM both provide a maximum a posteriori estimation, while the FTR provides a least-squares one. The algorithms are tested on the European Southern Observatory (ESO) end-to-end simulator, OCTOPUS. The performance is compared using a natural guide star single-conjugate adaptive optics configuration. The results demonstrate that the methods have similar performance in a large variety of simulated conditions. However, with respect to system misregistrations, the fast algorithms demonstrate an interesting robustness.

Error budget analysis for an adaptive optics optical coherence tomography system

The combination of adaptive optics (AO) technology with optical coherence tomography (OCT) instrumentation for imaging the retina has proven to be a valuable tool for clinicians and researchers in understanding the healthy and diseased eye. The micrometer-isotropic resolution achieved by such a system allows imaging of the retina at a cellular level, however imaging of some cell types remains elusive. Improvement in contrast rather than resolution is needed and can be achieved through better AO correction of wavefront aberration. A common tool for assessing and ultimately improving AO system performance is the development of an error budget. Specifically, this is a list of the magnitude of the constituent residual errors of an optical system so that resources can be directed towards efficient performance improvement. Here we present an error budget developed for the UC Davis AO-OCT instrument indicating that bandwidth and controller errors are the limiting errors of our AO system, which should be corrected first to improve performance. We also discuss the scaling of error sources for different subjects and the need to improve the robustness of the system by addressing subject variability.

Wavefront reconstruction over a circular aperture using gradient data extrapolated via the mirror equations

Methods for extrapolating gradient data outside a circular aperture from measurements obtained within a circular aperture are presented. The proposed methods are required to be computationally efficient and to avoid the excitation of additional waffle modes in Fried alignment. It is shown that, using an octagon as an intermediate step from the circle to the square in the extrapolation process, the computations or residual reconstruction error can be reduced. The resulting computational cost is as low as O(N(1/2)), where N is the number of measurement points. The performances of the extrapolation methods are studied in connection with a recently developed O(N) wavefront reconstruction algorithm based on wavelet filter banks [IEEE J. Sel. Top. Signal Process. 2, 781 (2008)] Experiments indicate that, as expected, there is a significant reconstruction error if no extrapolation is used. Further, the proposed extrapolation techniques lead to a reconstruction with data that are marginally different from a pupil masked reconstruction using data from a square aperture.

Experimental verification of the frozen flow atmospheric turbulence assumption with use of astronomical adaptive optics telemetry

We use closed-loop deformable mirror telemetry from Altair and Keck adaptive optics (AO) to determine whether atmospheric turbulence follows the frozen flow hypothesis. Using telemetry from AO systems, our algorithms (based on the predictive Fourier control framework) detect frozen flow >94% of the time. Usually one to three layers are detected. Between 20% and 40% of the total controllable phase power is due to frozen flow. Velocity vector RMS variability is less than 0.5 m/s (per axis) on 10-s intervals, indicating that the atmosphere is stable enough for predictive control to measure and adapt to prevailing atmospheric conditions before they change.

Linear-quadratic-Gaussian control for adaptive optics systems using a hybrid model

This paper presents a linear-quadratic-Gaussian (LQG) design based on the equivalent discrete-time model of an adaptive optics (AO) system. The design model incorporates deformable mirror dynamics, an asynchronous wavefront sensor and zero-order hold operation, and a continuous-time model of the incident wavefront. Using the structure of the discrete-time model, the dimensions of the Riccati equations to be solved are reduced. The LQG controller is shown to improve AO system performance under several conditions.

Woofer-tweeter control in an adaptive optics system using a Fourier reconstructor

Future adaptive optics systems seeking a higher degree of correction may need to split the command between two deformable mirrors (DMs), the woofer and the tweeter. We first present a method to specify the order of correction of the woofer DM for a given tweeter stroke requirement using an analytical and Monte Carlo approach. We then develop a computationally efficient algorithm to split the command between the two DMs in the case where a Fourier reconstructor is used. We find that the computational efficiency can be dramatically improved by splitting the command in Fourier space and by sending to the woofer only the modes it can reproduce accurately.

Predictive wavefront control for adaptive optics with arbitrary control loop delays

We present a modification of the closed-loop state space model for adaptive optics control that allows delays that are a noninteger multiple of the system frame rate. We derive the new forms of the predictive Fourier control Kalman filters for arbitrary delays and show that they are linear combinations of the whole-frame delay terms. This structure of the controller is independent of the delay. System stability margins and residual error variance both transition gracefully between integer-frame delays.

Optimal Fourier control performance and speckle behavior in high-contrast imaging with adaptive optics

High-contrast imaging with adaptive optics (AO) for planet detection requires a sophisticated AO control system to provide the best possible performance. We evaluate the performance improvements in terms of residual error and point-spread function intensity provided by optimal Fourier control using detailed end-to-end simulation. Intensity, however, is not the final measure of system performance. We explore image contrast through analysis and simulation results, showing that speckles caused by atmospheric errors behave very differently in a temporal fashion from speckles caused by wavefront sensor noise errors.

Wavefront expansion basis functions and their relationships

Wavefront expansion basis functions are important in representing ocular aberrations and phase perturbations due to atmospheric turbulence. A general discussion is presented for the conversions of the coefficients between any two sets of basis functions. Several popular sets of basis functions, namely, Zernike polynomials, Fourier series, and Taylor monomials, are discussed and the conversion matrices between any two of these basis functions are derived. Some analytical and numerical examples are given to demonstrate conversion of coefficients of different basis function sets.