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.     

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.     

Tilt angular anisoplanatism and a full-aperture tilt-measurement technique with a laser guide star

A method is presented for sensing atmospheric wave-front tilt from a laser guide star (LGS) by observing a laser beacon with auxiliary telescopes. The analysis is performed with a LGS scatter model and Zernike polynomial expansion of wave-front distortions. It is shown that integration of the LGS image over its angular extent and the position of the auxiliary telescope in an array reduce the tilt sensing error associated with the contribution from the downward path. This allows us to single out only the wave-front tilt of the transmitted beam on the uplink path that corresponds to the tilt for the scientific object. The tilt angular correlation is analyzed in the atmosphere with a finite turbulence outer scale. The tilt correlation angle depends on the angular size of the telescope and the outer scale of turbulence. The tilt sensing error increases with the auxiliary telescope diameter, suggesting that an auxiliary telescope must be small. The Strehl ratio associated with the contribution from the downward path is in the range from 0.1 to 0.9 when the relative telescope diameter D/r(0) varies from 4 to 93 and the turbulence outer scale is in the 10-150-m range. Tilt correction increases the Strehl ratio compared with the uncorrected image for all the system parameters and seeing conditions considered. The method discussed gives a higher performance than the conventional technique, which uses an off-axis natural guide star. A scheme for measuring tilt with a beam projected from a small aperture is described. This scheme allows us to avoid phosphorescence of the main optical train for a sodium LGS.     

Fundamental limits on isoplanatic correction with multiconjugate adaptive optics

We investigate the performance of a general multiconjugate adaptive optics (MCAO) system in which signals from multiple reference beacons are used to drive several deformable mirrors in the optical beam train. Taking an analytic approach that yields a detailed view of the effects of low-order aberration modes defined over the metapupil, we show that in the geometrical optics approximation, N deformable mirrors conjugated to different ranges can be driven to correct these modes through order N with unlimited isoplanatic angle, regardless of the distribution of turbulence along the line of sight. We find, however, that the optimal deformable mirror shapes are functions of target range, so the best compensation for starlight is in general not the correction that minimizes the wave-front aberration in a laser guide beacon. This introduces focal anisoplanatism in the wave-front measurements that can be overcome only through the use of beacons at several ranges. We derive expressions for the number of beacons required to sense the aberration to arbitrary order and establish necessary and sufficient conditions on their geometry for both natural and laser guide stars. Finally, we derive an expression for the residual uncompensated error by mode as a function of field angle, target range, and MCAO system geometry.     

Adaptive-optics performance of Antarctic telescopes

The performance of natural guide star adaptive-optics systems for telescopes located on the Antarctic plateau is evaluated and compared with adaptive-optics systems operated with the characteristic mid-latitude atmosphere found at Mauna Kea. A 2-m telescope with tip-tilt correction and an 8-m telescope equipped with a high-order adaptive-optics system are considered. Because of the large isoplanatic angle of the South Pole atmosphere, the anisoplanatic error associated with an adaptive-optics correction is negligible, and the achievable resolution is determined only by the fitting error associated with the number of corrected wave-front modes, which depends on the number of actuators on the deformable mirror. The usable field of view of an adaptive-optics equipped Antarctic telescope is thus orders of magnitude larger than for a similar telescope located at a mid-latitude site; this large field of view obviates the necessity for multiconjugate adaptive-optics systems that use multiple laser guide stars. These results, combined with the low infrared sky backgrounds, indicate that the Antarctic plateau is the best site on Earth at which to perform high-resolution imaging with large telescopes, either over large fields of view or with appreciable sky coverage. Preliminary site-testing results obtained recently from the Dome Concordia station indicate that this site is far superior to even the South Pole.     

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.     

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.     

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.     

Optical performance of a diffraction-limited molded-glass biaspheric lens

Optical performance results are reported for a molded-glass biaspheric lens. The 6-mm optical diam lens is intended for use in a laser-based optical disk application. The design with fabrication tolerances has an expected on-axis transmitted wave-front performance of 0.06-wave rms optical path difference (OPD) when tested at 0.6328 ,4m and a numerical aperture of 0.45. Transmitted wave-front aberrations were measured on actual molded lenses using a heterodyne interferometer. Typical performance was 0.05-0.08-wave rms- OPD. Experimental results involving mold rotation indicate that lens performance is primarily limited by a surface figure accuracy of one of the molds.     

Multiconjugate adaptive optics for large telescopes: analytical control of the mirror shapes

We present an analytical algorithm for deriving the shapes of the deformable mirrors to be used for multiconjugate adaptive correction on a large telescope. The algorithm is optimal in the limit where the overlap of the wave-front contributions from relevant atmospheric layers probed by the guide stars is close to the size of the pupil. The fundamental principle for correction is based on a minimization of the sum of the residual power spectra of the phase fluctuations seen by the guide stars after correction. On the basis of the expressions for the mirror shapes, so-called layer transfer functions describing the distribution of the correction of a single atmospheric layer among the deformable mirrors and the resulting correction of that layer have been derived. It is shown that for five guide stars distributed in a regular cross, two- and three-mirror correction will be possible only up to a maximum frequency defined by the largest separation of the conjugate altitudes of the mirrors and by the angular separation of the guide stars. The performance of the algorithm is investigated in the K band by using a standard seven-layer atmosphere. We present results obtained for two guide-star configurations: a continuous distribution within a given angular radius and a five-star cross pattern with a given angular arm length. The wave-front fluctuations are subjected to correction using one, two, and three deformable mirrors. The needed mirror dynamic range is derived as required root-mean-square stroke and actuator pitch. Finally the performance is estimated in terms of the Strehl ratio obtained by the correction as a function of field angle. No noise has been included in the present analysis, and the guide stars are assumed to be at infinity.     

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.     

Evaluation of least-squares phase-diversity technique for space telescope wave-front sensing

Because of mechanical aspects of fabrication, launch, and operational environment, space telescope optics can suffer from unforeseen aberrations, detracting from their intended diffraction-limited performance goals. We give the results of simulation studies designed to explore how wave-front aberration information for such near-diffraction-limited telescopes can be estimated through a regularized, low-pass filtered version of the Gonsalves (least-squares) phase-diversity technique. We numerically simulate models of both monolithic and segmented space telescope mirrors; the segmented case is a simplified model of the proposed next generation space telescope. The simulation results quantify the accuracy of phase diversity as a wave-front sensing (WFS) technique in estimating the pupil phase map. The pupil phase is estimated from pairs of conventional and out-of-focus photon-limited point-source images. Image photon statistics are simulated for three different average light levels. Simulation results give an indication of the minimum light level required for reliable estimation of a large number of aberration parameters under the least-squares paradigm. For weak aberrations that average a 0.10lambda pupil rms, the average WFS estimation errors obtained here range from a worst case of 0.057lambda pupil rms to a best case of only 0.002lambda pupil rms, depending on the light level as well as on the types and degrees of freedom of the aberrations present.     

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.     

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.     

Modulation effect of the atmosphere in a pyramid wave-front sensor

The pyramid wave-front sensor in its original form works with a mechanical modulation that adapts the linear range of the sensor to seeing and sensing conditions. For adaptive optics systems working in an astronomical context, the way in which the aberrations produced by the atmospheric turbulence, which are not seen by the sensor owing to its limited temporal bandwidth, act as modulators is shown. These aberrations have the same effect of increasing the linear range and localizing the measurement as does mechanical modulation. The effect of residual wave-front aberrations is estimated for some example conditions of telescope diameter, system bandwidth, wind velocity, and Fried parameter.     

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.     

Wave-front tilt sensor with two quadrant detectors and its application to a laser beam pointing system

A method that uses two quadrant detectors (QDs) for sensing wave-front tilt is described. The detectors are set at the front and back of the focal plane of the focusing optics. When the intensity distribution at the aperture of a telescope is not uniform because of atmospheric turbulence, the wave-front tilt can be measured more accurately with our two-QD method than with a conventional single-focal-plane QD. We proved this method to be effective by using it for fixed-star observation. Application of the method to a ground-to-satellite laser beam pointing system showed that it increases the intensity of the laser beam that is transmitted to a satellite.     

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.     

Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography

By 'atmospheric tomography' we mean the estimation of a layered atmospheric turbulence profile from measurements of the pupil-plane phase (or phase gradients) corresponding to several different guide star directions. We introduce what we believe to be a new Fourier domain preconditioned conjugate gradient (FD-PCG) algorithm for atmospheric tomography, and we compare its performance against an existing multigrid preconditioned conjugate gradient (MG-PCG) approach. Numerical results indicate that on conventional serial computers, FD-PCG is as accurate and robust as MG-PCG, but it is from one to two orders of magnitude faster for atmospheric tomography on 30 m class telescopes. Simulations are carried out for both natural guide stars and for a combination of finite-altitude laser guide stars and natural guide stars to resolve tip-tilt uncertainty.