Fractional loop delays in adaptive optics modeling and control.

This paper revisits the problem of optimal (minimum variance) control for adaptive optics (AO) systems when measurement and command applications are asynchronous, resulting in a non-integer servo loop delay. When not properly accounted for, such fractional delays may severely degrade the AO performance, especially in the presence of high-frequency vibrations. We present evidence of this performance degradation thanks to in-lab experimental measurements on the Gran Telescopio Canarias Adaptive Optics (GTCAO) system controlled with standard suboptimal linear quadratic Gaussian (LQG) controllers. A constructive, easy to implement LQG control design is then proposed and validated in a simulation for vibrations affecting the tip-tilt modes. Our methodology is very interesting because it allows a performance assessment for any linear controller in terms of variance, rejection transfer functions, power spectral densities, and stability margins. We also show how the continuous-time disturbance model can be derived from standard discrete-time disturbance data-based modeling.

On the equivalence of binary phase masks optimized for localization or detection in extended depth-of-field localization microscopy.

Binary annular masks have recently been proposed to extend the depth of field (DoF) of single-molecule localization microscopy. A strategy for designing optimal masks has been introduced based on maximizing the emitter localization accuracy, expressed in terms of Fisher information, over a targeted DoF range. However, the complete post-processing pipeline to localize a single emitter consists of two successive steps: detection, where the regions containing emitters are determined, and localization, where the sub-pixel position of each detected emitter is estimated. Phase masks usually optimize only this second step. The presence of a phase mask also affecting detection, the purpose of this paper is to quantify and mitigate this effect. Using a rigorous framework built from a detection-oriented information theoretical criterion (Bhattacharyya distance), we demonstrate that in most cases of practical significance, annular binary phase masks maximizing Fisher information also maximize the detection probability. This result supports the common design practice consisting of optimizing a phase mask by maximizing Fisher information only.

On the validity domain of maximum likelihood estimators for depth-of-field extension in single-molecule localization microscopy.

Localization microscopy approaches with enhanced depth-of-field (EDoF) are commonly optimized using the Cramér-Rao bound (CRB) as a criterion. It is widely believed that the CRB can be attained in practice by using the maximum-likelihood estimator (MLE). This is, however, an approximation, of which we define in this paper the precise domain of validity. Exploring a wide range of settings and noise levels, we show that the MLE is efficient when the signal-to-noise ratio (SNR) is such that the localization standard deviation of a single molecule is less than 20 nm. Thus, our results provide an explicit and quantitative validity boundary for the use of the MLE in EDoF localization microscopy setups optimized with the CRB.

Influence of high numerical aperture on depth-of-field enhancing phase mask optimization in localization microscopy.

The depth-of-field (DoF) of localization microscopes can be extended by placing a phase mask in the aperture stop of the objective. To optimize these masks and characterize their performance, defocus is in general modeled by a simple quadratic pupil phase term. However, this model does not take into account two essential characteristics of localization microscopy setups: an extremely high numerical aperture (NA) and a mismatch between the refractive indices of the immersion liquid and sample. Using the more realistic high NA image formation model of Gibson & Lanni (GL), we show that the DoF extension is simply reduced by a NA-dependent scaling factor. We also show that, provided this scaled DoF extension factor is taken into account, masks optimized with the approximate quadratic model are still nearly optimal in the framework of the GL model. This result is important since it establishes that generic optimized masks can be used in setups with different NA and immersion indices.

Feature issue introduction: applications of adaptive optics.

This feature issue of Optics Express follows the 2020 Imaging and Applied Optics Congress and comprises of articles on the development and use of adaptive optics across the broad range of domains in which the technique has been applied - including atmospheric correction, ophthalmology, vision science, microscopy, optical communications and beam control. This review provides a basic introduction to adaptive optics and a summary of the multidisciplinary articles included in this issue.

Co-designed annular binary phase masks for depth-of-field extension in single-molecule localization microscopy.

Single-molecule localization microscopy has become a prominent approach to study structural and dynamic arrangements of nanometric objects well beyond the diffraction limit. To maximize localization precision, high numerical aperture objectives must be used; however, this inherently strongly limits the depth-of-field (DoF) of the microscope images. In this work, we present a framework inspired by "optical co-design" to optimize and benchmark phase masks, which, when placed in the exit pupil of the microscope objective, can extend the DoF in the realistic context of single fluorescent molecule detection. Using the Cramér-Rao bound (CRB) on localization accuracy as a criterion, we optimize annular binary phase masks for various DoF ranges, compare them to Incoherently Partitioned Pupil masks and show that they significantly extend the DoF of single-molecule localization microscopes. In particular we propose different designs including a simple and easy-to-realize two-ring binary mask to extend the DoF. Moreover, we demonstrate that a simple maximum likelihood-based localization algorithm can reach the localization accuracy predicted by the CRB. The framework developed in this paper is based on an explicit and general information theoretic criterion, and can thus be used as an engineering tool to optimize and compare any type of DoF-enhancing phase mask in high resolution microscopy on a quantitative basis.

Zonal-based high-performance control in adaptive optics systems with application to astronomy and satellite tracking.

This paper presents a model-based approach to adaptive optics (AO) control based on a zonal (i.e., pixelized) representation of the incoming atmospheric turbulence. Describing the turbulence on a zonal basis enables the encapsulation of the standard frozen-flow assumption into a control-oriented model. A multilayer zonal model is proposed for single-conjugate AO (SCAO) systems. It includes an edge compensation mechanism involving limited support, which results in a sparser model structure. To further reduce the computational complexity, new resultant zonal models localized in the telescope pupil are proposed, with AR1 or AR2 structures, that match the spatial and temporal cross-correlations of the incoming turbulence. The global performance of the resulting linear quadratic Gaussian (LQG) regulator is evaluated using end-to-end simulations and compared to several existing controllers for two different configurations: a very large telescope SCAO and low earth orbit satellite tracking. The results show the high potential of the new approach and highlight possible trade-offs between the performance and complexity.

Linear controller error budget assessment for classical adaptive optics systems.

Understanding limitations of adaptive optics (AO) systems is crucial when designing new systems. In particular, analyzing the potential of different controllers is of great interest for the upcoming AO systems of the very large telescopes (VLTs) and extremely large telescopes (ELTs). This paper thus details a complete error budget assessment formalism, based on analytic formulas involving the disturbance temporal power spectral density (PSD) and the controller transfer function, and is applicable to any linear controller. This formalism is presented here for the special case of classical AO systems, but can be extended to any closed- or open-loop, single- or multi-conjugated AO configuration. Special attention is paid to the "control-dependent" errors, the importance of which is directly related to the type of control used in the AO system. The proposed method is applied to a NAOS/VLT-type single conjugated AO system, using disturbance PSD derived from a simulated turbulence trajectory or estimated from wavefront sensor measurements, enabling the construction of detailed error budgets for an integrator and different linear quadratic Gaussian controllers. Application to ELT-sized systems is discussed in the conclusion.

Eye-pupil displacement and prediction: effects on residual wavefront in adaptive optics retinal imaging.

This paper studies the effect of pupil displacements on the best achievable performance of retinal imaging adaptive optics (AO) systems, using 52 trajectories of horizontal and vertical displacements sampled at 80 Hz by a pupil tracker (PT) device on 13 different subjects. This effect is quantified in the form of minimal root mean square (rms) of the residual phase affecting image formation, as a function of the delay between PT measurement and wavefront correction. It is shown that simple dynamic models identified from data can be used to predict horizontal and vertical pupil displacements with greater accuracy (in terms of average rms) over short-term time horizons. The potential impact of these improvements on residual wavefront rms is investigated. These results allow to quantify the part of disturbances corrected by retinal imaging systems that are caused by relative displacements of an otherwise fixed or slowy-varying subject-dependent aberration. They also suggest that prediction has a limited impact on wavefront rms and that taking into account PT measurements in real time improves the performance of AO retinal imaging systems.

High-resolution imaging of gunn's dots.

BACKGROUND: In healthy fundi, glistening whitish dots (so-called Gunn's dots) can often be seen, especially in young subjects. They are commonly attributed to the reflectance of Müller cell's footplates. However, despite their potential interest as biomarkers of retinal diseases, Gunn's dots have received little attention in the scientific literature. METHODS: Scanning laser ophthalmoscope reflectance imaging and adaptive optics infrared flood imaging were performed in 18 healthy subjects (age range, 18-58 years) to analyze the localization, density, and shape of Gunn's dots. RESULTS: Gunn's dots were more easily observed in the midperipheral retina along temporal vessels, although in two subjects, they could be detected in the macula. The reflectance of Gunn's dots showed a strong directional variability, which paralleled that of the inner limiting membrane. The mean (±SD) diameter of Gunn's dots was 13.3 µm (±3.5). Their density peaked at ∼120 per square millimeter and decreased with age to become barely detectable after 50 years. CONCLUSION: Gunn's dots are highly anisotropic structures close to the inner limiting membrane. Their density, size, and age-related decline are closer to the characteristics of hyalocytes than those of Müller cells. Further studies are necessary to progress in the determination of their origin and interest as biomarkers of retinal diseases.

Meaning of visualizing retinal cone mosaic on adaptive optics images.

PURPOSE: To explore the anatomic correlation of the retinal cone mosaic on adaptive optics images. DESIGN: Retrospective nonconsecutive observational case series. METHODS: A retrospective review of the multimodal imaging charts of 6 patients with focal alteration of the cone mosaic on adaptive optics was performed. Retinal diseases included acute posterior multifocal placoid pigment epitheliopathy (n = 1), hydroxychloroquine retinopathy (n = 1), and macular telangiectasia type 2 (n = 4). High-resolution retinal images were obtained using a flood-illumination adaptive optics camera. Images were recorded using standard imaging modalities: color and red-free fundus camera photography; infrared reflectance scanning laser ophthalmoscopy, fluorescein angiography, indocyanine green angiography, and spectral-domain optical coherence tomography (OCT) images. RESULTS: On OCT, in the marginal zone of the lesions, a disappearance of the interdigitation zone was observed, while the ellipsoid zone was preserved. Image recording demonstrated that such attenuation of the interdigitation zone co-localized with the disappearance of the cone mosaic on adaptive optics images. In 1 case, the restoration of the interdigitation zone paralleled that of the cone mosaic after a 2-month follow-up. CONCLUSION: Our results suggest that the interdigitation zone could contribute substantially to the reflectance of the cone photoreceptor mosaic. The absence of cones on adaptive optics images does not necessarily mean photoreceptor cell death.

First on-sky SCAO validation of full LQG control with vibration mitigation on the CANARY pathfinder.

Adaptive optics provides real time correction of wavefront disturbances on ground based telescopes. Optimizing control and performance is a key issue for ever more demanding instruments on ever larger telescopes affected not only by atmospheric turbulence, but also by vibrations, windshake and tracking errors. Linear Quadratic Gaussian control achieves optimal correction when provided with a temporal model of the disturbance. We present in this paper the first on-sky results of a Kalman filter based LQG control with vibration mitigation on the CANARY instrument at the Nasmyth platform of the 4.2-m William Herschel Telescope. The results demonstrate a clear improvement of performance for full LQG compared with standard integrator control, and assess the additional improvement brought by vibration filtering with a tip-tilt model identified from on-sky data, thus validating the strategy retained on the instrument SPHERE at the VLT.

Distributed Kalman filtering compared to Fourier domain preconditioned conjugate gradient for laser guide star tomography on extremely large telescopes.

This paper discusses the performance and cost of two computationally efficient Fourier-based tomographic wavefront reconstruction algorithms for wide-field laser guide star (LGS) adaptive optics (AO). The first algorithm is the iterative Fourier domain preconditioned conjugate gradient (FDPCG) algorithm developed by Yang et al. [Appl. Opt.45, 5281 (2006)], combined with pseudo-open-loop control (POLC). FDPCG's computational cost is proportional to N log(N), where N denotes the dimensionality of the tomography problem. The second algorithm is the distributed Kalman filter (DKF) developed by Massioni et al. [J. Opt. Soc. Am. A28, 2298 (2011)], which is a noniterative spatially invariant controller. When implemented in the Fourier domain, DKF's cost is also proportional to N log(N). Both algorithms are capable of estimating spatial frequency components of the residual phase beyond the wavefront sensor (WFS) cutoff frequency thanks to regularization, thereby reducing WFS spatial aliasing at the expense of more computations. We present performance and cost analyses for the LGS multiconjugate AO system under design for the Thirty Meter Telescope, as well as DKF's sensitivity to uncertainties in wind profile prior information. We found that, provided the wind profile is known to better than 10% wind speed accuracy and 20 deg wind direction accuracy, DKF, despite its spatial invariance assumptions, delivers a significantly reduced wavefront error compared to the static FDPCG minimum variance estimator combined with POLC. Due to its nonsequential nature and high degree of parallelism, DKF is particularly well suited for real-time implementation on inexpensive off-the-shelf graphics processing units.

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.

Tip-tilt disturbance model identification for Kalman-based control scheme: application to XAO and ELT systems.

Adaptive optics (AO) systems have to correct tip-tilt (TT) disturbances down to a fraction of the diffraction-limited spot. This becomes a key issue for very or extremely large telescopes affected by mechanical vibration peaks or wind shake effects. Linear quadratic Gaussian (LQG) control achieves optimal TT correction when provided with the temporal model of the disturbance. We propose a nonsupervised identification procedure that does not require any auxiliary system or loop opening and validate it on synthetic profile as well as on experimental data.

Minimum-variance control for woofer-tweeter systems in adaptive optics.

The woofer-tweeter concept in adaptive optics consists in correcting for the turbulent wavefront disturbance with a combination of two deformable mirrors (DMs). The woofer corrects for temporally slow-evolving, spatially low-frequency, large-amplitude disturbances, whereas the tweeter is generally its complement, i.e., corrects for faster higher-order modes with lower amplitude. A special feature is that in general both are able to engender a common correction space. In this contribution a minimum-variance solution for the double stage woofer-tweeter concept in adaptive optics systems is addressed using a linear-quadratic-Gaussian approach. An analytical model is built upon previous developments on a single DM with temporal dynamics that accommodates a double-stage woofer-tweeter DM. Monte Carlo simulations are run for a system featuring an 8×8 actuator DM (considered infinitely fast), mounted on a steering tip/tilt platform (considered slow). Results show that it is essential to take into account temporal dynamics on the estimation step. Besides, unlike the other control strategies considered, the optimal solution is always stable.

Wide field adaptive optics laboratory demonstration with closed-loop tomographic control.

HOMER, the new bench developed at ONERA devoted to wide field adaptive optics (WFAO) laboratory research, has allowed the first experimental validations of multi-conjugate adaptive optics (MCAO) and laser tomography adaptive optics (LTAO) concepts with a linear quadratic Gaussian (LQG) control approach. Results obtained in LTAO in closed loop show the significant gain in performance brought by LQG control, which allows tomographic reconstruction. We present a calibration and model identification strategy. Experimental results are shown to be consistent with end-to-end simulations. These results are very encouraging and demonstrate robustness of performance with respect to inevitable experimental uncertainties. They represent a first step for the study of very large telescope (VLT) and extremely large telescopes (ELT) instruments.

On the optimal reconstruction and control of adaptive optical systems with mirror dynamics.

In adaptive optics (AO) the deformable mirror (DM) dynamics are usually neglected because, in general, the DM can be considered infinitely fast. Such assumption may no longer apply for the upcoming Extremely Large Telescopes (ELTs) with DM that are several meters in diameter with slow and/or resonant responses. For such systems an important challenge is to design an optimal regulator minimizing the variance of the residual phase. In this contribution, the general optimal minimum-variance (MV) solution to the full dynamical reconstruction and control problem of AO systems (AOSs) is established. It can be looked upon as the parent solution from which simpler (used hitherto) suboptimal solutions can be derived as special cases. These include either partial DM-dynamics-free solutions or solutions derived from the static minimum-variance reconstruction (where both atmospheric disturbance and DM dynamics are neglected altogether). Based on a continuous stochastic model of the disturbance, a state-space approach is developed that yields a fully optimal MV solution in the form of a discrete-time linear-quadratic-Gaussian (LQG) regulator design. From this LQG standpoint, the control-oriented state-space model allows one to (1) derive the optimal state-feedback linear regulator and (2) evaluate the performance of both the optimal and the sub-optimal solutions. Performance results are given for weakly damped second-order oscillatory DMs with large-amplitude resonant responses, in conditions representative of an ELT AO system. The highly energetic optical disturbance caused on the tip/tilt (TT) modes by the wind buffeting is considered. Results show that resonant responses are correctly handled with the MV regulator developed here. The use of sub-optimal regulators results in prohibitive performance losses in terms of residual variance; in addition, the closed-loop system may become unstable for resonant frequencies in the range of interest.

Statistical performance modeling for superresolution: a discrete data-continuous reconstruction framework.

We address performance modeling of superresolution (SR) techniques. Superresolution consists in combining several images of the same scene to produce an image with better resolution and contrast. We propose a discrete data-continuous reconstruction framework to conduct SR performance analysis and derive a theoretical expression of the reconstruction mean squared error (MSE) as a compact, computationally tractable function of signal-to-noise ratio (SNR), scene model, sensor transfer function, number of frames, interframe translation motion, and SR reconstruction filter. A formal expression for the MSE is obtained that allows a qualitative study of SR behavior. In particular we provide an original outlook on the balance between noise and aliasing reduction in linear SR. Explicit account for the SR reconstruction filter is an original feature of our model. It allows for the first time to study not only optimal filters but also suboptimal ones, which are often used in practice.

Linear quadratic Gaussian control for adaptive optics and multiconjugate adaptive optics: experimental and numerical analysis.

We present a comprehensive analysis of the linear quadratic Gaussian control approach applied to adaptive optics (AO) and multiconjugated AO (MCAO) based on numerical and experimental validations. The structure of the control law is presented and its main properties discussed. We then propose an extended experimental validation of this control law in AO and a simplified MCAO configuration. Performance is compared with end-to-end numerical simulations. Sensitivity of the performance regarding tuning parameters is tested. Finally, extension to full MCAO and laser tomographic AO (LTAO) through numerical simulation is presented and analyzed.