Sky coverage estimates for adaptive optics systems from computations in Zernike space

A sky coverage model for laser guide star adaptive optics systems is proposed. The atmosphere is considered to consist of a finite number of phase screens, which are defined by Zernike basis polynomials, located at different altitudes. These phase screens are transformed to the aperture plane, where they are converted to laser and natural guide star wavefront sensing measurements. These transformations incorporate the cone effect due to guide stars at finite heights, anisoplanatism due to guide stars off axis with respect to the science object, and adaptive optics systems with multiple guide stars. The wavefront error is calculated tomographically with minimum variance estimators derived from the transformation matrices and the known statistical properties of the atmosphere. This sky coverage model provides fast Monte Carlo simulations over random natural guide star configurations, irrespective of telescope diameter. The Monte Carlo simulations outlined show that inclusion of a finite outer scale for the atmosphere significantly reduces the median wavefront error, that increasing the number of laser guide stars in the asterism reduces the median wavefront error, and that a larger natural guide star patrol field provides a smaller median wavefront error when there is a low star density in the field.     

Sky coverage modeling for the whole sky for laser guide star multiconjugate adaptive optics

The scientific productivity of laser guide star adaptive optics systems strongly depends on the sky coverage, which describes the probability of finding natural guide stars for the tip/tilt wavefront sensor(s) to achieve a certain performance. Knowledge of the sky coverage is also important for astronomers planning their observations. In this paper, we present an efficient method to compute the sky coverage for the laser guide star multiconjugate adaptive optics system, the Narrow Field Infrared Adaptive Optics System (NFIRAOS), being designed for the Thirty Meter Telescope project. We show that NFIRAOS can achieve more than 70% sky coverage over most of the accessible sky with the requirement of 191 nm total rms wavefront.     

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.     

Tip-tilt reconstruction with a single dim natural guide star in multiconjugate adaptive optics with laser guide stars

A solution to the problem of detecting the tip-tilt modes in multiconjugate adaptive optics (MCAO) with laser guide stars (LGS) is presented. This solution requires the presence of only a single relatively dim natural guide star (NGS) within the reconstructed field of view (FoV). The dim NGS is used for the reconstruction of the tip-tilt modes on the entire FoV, while the tomographic reconstruction of second-order and higher-order modes is made possible by having an LGS constellation with LGSs at different heights. Due to the relatively low brightness required for the tip-tilt NGS and the large corrected FoV (as compared with the case of conventional adaptive optics) the presented solution provides a means to achieve near-diffraction-limited performance of a 10-m-class telescope in the near infrared over a large portion of the sky. Sky coverage calculations assuming median seeing conditions indicate that this technique could be applied to 75% (95%) of the sky, achieving corrections with an average Strehl ratio approximately 0.42(approximately 0.33) in the 2.2 microm K band across the 1.5' reconstructed FoV.     

Split atmospheric tomography using laser and natural guide stars

Laser guide star (LGS) atmospheric tomography is described in the literature as integrated minimum-variance tomographic wavefront reconstruction from a concatenated wavefront-sensor measurement vector consisting of many high-order, tip/tilt (TT)-removed LGS measurements, supplemented by a few low-order natural guide star (NGS) components essential to estimating the TT and tilt anisoplanatism (TA) modes undetectable by the TT-removed LGS wavefront sensors (WFSs). The practical integration of these NGS WFS measurements into the tomography problem is the main subject of this paper. A split control architecture implementing two separate control loops driven independently by closed-loop LGS and NGS measurements is proposed in this context. Its performance is evaluated in extensive wave optics Monte Carlo simulations for the Thirty Meter Telescope (TMT) LGS multiconjugate adaptive optics (MCAO) system, against the delivered performance of the integrated control architecture. Three iterative algorithms are analyzed for atmospheric tomography in both cases: a previously proposed Fourier domain preconditioned conjugate gradient (FDPCG) algorithm, a simple conjugate gradient (CG) algorithm without preconditioning, and a novel layer-oriented block Gauss-Seidel conjugate gradient algorithm (BGS-CG). Provided that enough iterations are performed, all three algorithms yield essentially identical closed-loop residual RMS wavefront errors for both control architectures, with the caveat that a somewhat smaller number of iterations are required by the CG and BGS-CG algorithms for the split approach. These results demonstrate that the split control approach benefits from (i) a simpler formulation of minimum-variance atmospheric tomography allowing for algorithms with reduced computational complexity and cost (processing requirements), (ii) a simpler, more flexible control of the NGS-controlled modes, and (iii) a reduced coupling between the LGS- and NGS-controlled modes. Computation and memory requirements for all three algorithms are also given for the split control approach for the TMT LGS AO system and appear feasible in relation to the performance specifications of current hardware technology.     

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.     

Analysis of the improvement in sky coverage for multiconjugate adaptive optics systems obtained using minimum variance split tomography

The scientific utility of laser-guide-star-based multiconjugate adaptive optics systems depends upon high sky coverage. Previously we reported a high-fidelity sky coverage analysis of an ad hoc split tomography control algorithm and a postprocessing simulation technique. In this paper, we present the performance of a newer minimum variance split tomography algorithm, and we show that it brings a median improvement at zenith of 21 nm rms optical path difference error over the ad hoc split tomography control algorithm for our system, the Narrow Field Infrared Adaptive Optics System for the Thirty Meter Telescope. In order to make the comparison, we also validated our previously developed sky coverage postprocessing software using an integrated simulation of both high- (laser guide star) and low-order (natural guide star) loops. A new term in the noise model is also identified that improves the performance of both algorithms by more properly regularizing the reconstructor.     

Adaptive optics with four laser guide stars: correction of the cone effect in large telescopes

We study the performance of an adaptive optics (AO) system with four laser guide stars (LGSs) and a natural guide star (NGS). The residual cone effect with four LGSs is obtained by a numerical simulation. This method allows the adaptive optics system to be extended toward the visible part of the spectrum without tomographic reconstruction of three-dimensional atmospheric perturbations, resolving the cone effect in the visible. Diffraction-limited images are obtained with 17-arc ms precision in median atmospheric conditions at wavelengths longer than 600 nm. The gain achievable with such a system operated on an existing AO system is studied. For comparison, performance in terms of achievable Strehl ratio is also computed for a reasonable system composed of a 40 x 40 Shack-Hartmann wave-front sensor optimized for the I band. Typical errors of a NGS wave front are computed by use of analytical formulas. With the NGS errors and the cone effect, the Strehl ratio can reach 0.45 at 1.25 microm under good-seeing conditions with the Nasmyth Adaptive Optics System (NAOS; a 14 x 14 subpupil wave-front sensor) at the Very Large Telescope and 0.8 with a 40 x 40 Shack-Hartmann wave-front sensor.     

Including outer scale effects in zonal adaptive optics calculations

Mellin transform techniques are applied to evaluate the covariance of the integrated turbulence-induced phase distortions along a pair of ray paths through the atmosphere from two points in a telescope aperture to a pair of sources at finite or infinite range. The derivation is for the case of a finite outer scale and a von Karman turbulence spectrum. The Taylor hypothesis is assumed if the two phase distortions are evaluated at two different times and amplitude scintillation effects are neglected. The resulting formula for the covariance is a power series in one variable for the case of a fixed atmospheric wind velocity profile and a power series in two variables for a fixed wind-speed profile with a random and uniformly distributed wind direction. These formulas are computationally efficient and can be easily integrated into computer codes for the numerical evaluation of adaptive optics system performance. Sample numerical results are presented to illustrate the effect of a finite outer scale on the performance of natural and laser guide star adaptive optics systems for an 8-m astronomical telescope. A hypothetical outer scale of 10 m significantly reduces the magnitude of tilt anisoplanatism, thereby improving the performance of a laser guide star adaptive optics system if the auxiliary natural star used for full-aperture tip/tilt sensing is offset from the science field. The reduction in higher-order anisoplanatism that is due to a 10-m outer scale is smaller, and the off-axis performance of a natural guide star adaptive optics system is not significantly improved.     

Comparison of curvature-based and Shack-Hartmann-based adaptive optics for the Gemini telescope

We present the results of independent numerical simulations of adaptive optics systems for 8-m astronomical telescopes that use both Shack-Hartmann and wave-front curvature sensors. Four differents codes provided consistency checks and redundancy. All four simulate a complete system and model noise and servo-lag effects. A common atmospheric turbulence generator was used for consistency. We present the main characteristics of the codes, and we report the system performance in term of Strehl ratio and full width at half-maximum versus the magnitude of the (on-axis) guide star. We show that a Shack-Hartmann plus stacked actuator mirror system with 10 x 10 subapertures or a curvature plus bimorph mirror system with 56 subapertures yields a 50% Strehl ratio at 1.6 mum for a m(R) = 14.7 magnitude star, with almost equivalent performance at both brighter and dimmer light levels.     

Closed-loop stability and performance analysis of least-squares and minimum-variance control algorithms for multiconjugate adaptive optics

Recent progress has been made to compute efficiently the open-loop minimum-variance reconstructor (MVR) for multiconjugate adaptive optics systems by a combination of sparse matrix and iterative techniques. Using spectral analysis, I show that a closed-loop laser guide star multiconjugate adaptive optics control algorithm consisting of MVR cascaded with an integrator control law is unstable. Tosolve this problem, a computationally efficient pseudo-open-loop control (POLC) method was recently proposed. I give a theoretical proof of the stability of this method and demonstrate its superior performance and robustness against misregistration errors compared with conventional least-squares control. This can be accounted for by the fact that POLC incorporates turbulence statistics through its regularization term that can be interpreted as spatial filtering, yielding increased robustness to misregistration. For the Gemini-South 8-m telescope multiconjugate system and for median Cerro Pachon seeing, the performance of POLC in terms of rms wave-front error averaged over a 1-arc min field of view is approximately three times superior to that of a least-squares reconstructor. Performance degradation due to 30% translational misregistration on all three mirrors is approximately a 30% increased rms wave-front error, whereas a least-squares reconstructor is unstable at such a misregistration level.     

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.     

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.     

Impact of sodium laser guide star fratricide on multi-conjugate adaptive optics systems

Laser beams projected from the ground to form sodium layer laser guide stars (LGSs) for adaptive optics (AO) systems experience scattering and absorption that reduce their intensity as they propagate upward through the atmosphere. Some fraction of the scattered light will be collected by the other wavefront sensors and causes additional background in parts of the pupil. This cross-talk between different LGS wavefront sensors is referred to as the fratricide effect. In this paper we quantify the magnitude of four different sources of scattering/absorption and backscattering, and we evaluate their impact on performance with various zenith angles and turbulence profiles for one particular AO system. The resulting wavefront error for the Thirty Meter Telescope (TMT) multi-conjugate AO (MCAO) system, NFIRAOS, is on the order of 5 to 20 nm RMS, provided that the mean background from the fratricide effect can be calibrated and subtracted with an accuracy of 80%. We also present the impact on system performance of momentary variations in LGS signal levels due to variations in cirrus absorption or laser power, and we show that this affects the performance more than does an equal variation in the level of the fratricide.     

Centroid gain compensation in Shack-Hartmann adaptive optics systems with natural or laser guide star

In an adaptive optics system with an undersampled Shack-Hartmann wave-front sensor (WFS), variations in seeing, laser guide star quality, and sodium layer thickness and range distance all combine to vary WFS centroid gain across the pupil during an exposure. While using the minimum of 4 pixels per WFS subaperture improves frame rate and read noise, the WFS centroid gain uncertainty may introduce static aberrations and degrade servo loop phase margin. We present a novel method to estimate and compensate WFS gains of each subaperture individually in real time for both natural and laser guide stars.     

Fast calibration of high-order adaptive optics systems

We present a new method of calibrating adaptive optics systems that greatly reduces the required calibration time or, equivalently, improves the signal-to-noise ratio. The method uses an optimized actuation scheme with Hadamard patterns and does not scale with the number of actuators for a given noise level in the wavefront sensor channels. It is therefore highly desirable for high-order systems and/or adaptive secondary systems on a telescope without a Gregorian focal plane. In the latter case, the measurement noise is increased by the effects of the turbulent atmosphere when one is calibrating on a natural guide star.     

Optimal reconstruction for closed-loop ground-layer adaptive optics with elongated spots

The design of the laser-guide-star-based adaptive optics (AO) systems for the Extremely Large Telescopes requires careful study of the issue of elongated spots produced on Shack-Hartmann wavefront sensors. The importance of a correct modeling of the nonuniformity and correlations of the noise induced by this elongation has already been demonstrated for wavefront reconstruction. We report here on the first (to our knowledge) end-to-end simulations of closed-loop ground-layer AO with laser guide stars with such an improved noise model. The results are compared with the level of performance predicted by a classical noise model for the reconstruction. The performance is studied in terms of ensquared energy and confirms that, thanks to the improved noise model, central or side launching of the lasers does not affect the performance with respect to the laser guide stars' flux. These two launching schemes also perform similarly whatever the atmospheric turbulence strength.     

Signal-to-noise ratio of phase sensing telescope interferometers

The paper described is the third part of a trilogy dealing with the principles, performance, and limitations of what the author named "telescope-interferometers" (TIs). The basic idea consists in transforming one telescope into a wavefront error (WFE) sensing device. This can be achieved in two different ways, namely, off-axis and phase-shifting TIs. In both cases the point-spread function measured in the focal plane of the telescope carries information about the transmitted WFE, which is retrieved by fast and simple algorithms suitable to an adaptive optics (AO) regime. The uncertainties of both types of TIs are evaluated in terms of noise and systematic errors. Numerical models are developed to establish the dependence of driving parameters such as useful spectral range, angular size of the observed star, or detector noise on the total WFE measurement error. The latter is found particularly sensitive to photon noise, which rapidly governs the achieved accuracy for telescope diameters higher than 10 m. A few practical examples are studied, showing that the TI method is applicable to AO systems for telescope diameters ranging from 10 to 50 m, depending on seeing conditions and magnitude of the observed stars. Also discussed is the case of a space-borne coronagraph, where the TI technique provides high sampling of the input WFE map.     

Numerical simulations of multiconjugate adaptive optics wave-front reconstruction on giant telescopes

We present sample Monte Carlo simulation results to illustrate the trends in multiconjugate adaptive optics (MCAO) performance as the telescope aperture diameter increases from 8 to 32 m with all other first-order system parameters held constant. The MCAO system considered includes three deformable mirrors, a 1-arc min square field of view, and five wave-front-sensing references consisting of either natural guide stars or laser guide stars at a range of either 30 or 90 km. The rms residual wave-front error decreases slowly with increasing aperture diameter with natural guide stars, whereas performance degrades significantly with increasing aperture diameter for laser guide stars at 30 km if the number of guide stars is held fixed. Performance with laser guide stars at 90 km is a weak function of telescope aperture diameter in the range from 8 to 32 m, with rms wave-front errors no more than 20% greater than the corresponding natural guide-star case for the same level of wave-front sensor's measurement noise.     

Incorporating higher-order modal measurements in tilt estimation: natural and laser guide star applications

Tilt compensation performance is generally suboptimal when phase measurements from natural or laser guide stars are used as the conjugate phase in an adaptive optics system. Optimal compensation is obtained when the conjugate-phase coefficients are estimated from beacon measurements, given knowledge of the correlation between the on-axis object phase and the beacon measurements. We apply optimal compensation theory to tilt correction for the case of an off-axis beacon. Because off-axis higher-order modes are correlated with the on-axis tilt components, a performance gain can be realized when the tilt estimator includes higher-order modal measurements. For natural guide star compensation, it is shown that equivalent tilt compensation can be achieved at beacon offsets that are three times larger when higher-order modes through Zernike 15 are used in the tilt estimator. For a laser guide star, although tilt information cannot be measured directly because of beam reciprocity, off-axis higher-order modal measurements can be used to estimate tilt components, leading to a maximum Strehl ratio of approximately 0.3 for the relative aperture diameter D/r(0) = 4 and the relative turbulence outer scale L(0)/D = 10.