Brent L. Ellerbroek, "Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques," J. Opt. Soc. Am. A 19, 1803-1816 (2002)

The complexity of computing conventional matrix multiply wave-front reconstructors scales as $O({n}^{3})$ for most adaptive optical (AO) systems, where n is the number of deformable mirror (DM) actuators. This is impractical for proposed systems with extremely large n. It is known that sparse matrix methods improve this scaling for least-squares reconstructors, but sparse techniques are not immediately applicable to the minimum-variance reconstructors now favored for multiconjugate adaptive optical (MCAO) systems with multiple wave-front sensors (WFSs) and DMs. Complications arise from the nonsparse statistics of atmospheric turbulence, and the global tip/tilt WFS measurement errors associated with laser guide star (LGS) position uncertainty. A description is given of how sparse matrix methods can still be applied by use of a sparse approximation for turbulence statistics and by recognizing that the nonsparse matrix terms arising from LGS position uncertainty are low-rank adjustments that can be evaluated by using the matrix inversion lemma. Sample numerical results for AO and MCAO systems illustrate that the approximation made to turbulence statistics has negligible effect on estimation accuracy, the time to compute the sparse minimum-variance reconstructor for a conventional natural guide star AO system scales as $O({n}^{3/2})$ and is only a few seconds for $n=3500,$ and sparse techniques reduce the reconstructor computations by a factor of 8 for sample MCAO systems with 2417 DM actuators and 4280 WFS subapertures. With extrapolation to 9700 actuators and 17,120 subapertures, a reduction by a factor of approximately 30 or 40 to 1 is predicted.

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Reconstructor Performance versus WFS Measurement Noise for an Order $8\times 8$
NGS AO Systema

WFS Noise(arc sec)

CMMR ${\sigma}^{2}$
(μm^{2})

SRA ${\sigma}^{2}$
(μm^{2})

0.02

0.01304

0.01306

0.04

0.01734

0.01737

0.08

0.03218

0.03229

0.16

0.07834

0.07941

This table compares the residual mean square wave-front error ${\sigma}^{2}$
due to the combined effects of fitting error and WFS measurement noise for the conventional matrix multiply reconstruction (CMMR) implementation of the minimum-variance estimator and the sparse reconstruction algorithm (SRA) described in this paper. The approximations made in modeling atmospheric turbulence statistics for the latter algorithm have only a very modest effect on the mean square wave-front estimation error ${\sigma}^{2}.$

Table 4

SRA Wave-Front Fitting Error versus Order of Correction for a Conventional NGS AO Systema

This table summarizes the performance of the SRA for a conventional NGS AO scenario where the only significant source of wave-front error is the finite spatial resolution of the DM actuators and the WFS subapertures. The NGS is coincident with the evaluation direction, and the noise equivalent angle for the WFS is an almost negligible 0.02 arc sec. The DM actuator spacing is held constant, so the AO order of correction is proportional to the telescope aperture diameter. The SRA estimation error is virtually identical with the minimum-variance CMMR for the case of an order $16\times 16$
AO system and increases fairly gradually with increasing telescope aperture diameter. The computation times and the memory requirements for the SRA grow much less rapidly than the $O({n}^{3})$
and $O({n}^{2})$
scaling laws that apply for the case of the CMMR.

Table 5

Results and Scaling Law Predictions for CMMR and SRA Performance for MCAO Systemsa

The simulated MCAO parameters are summarized in Table 2. The parenthesized values are extrapolations based upon the standard $O({n}^{2})$
and $O({n}^{3})$
power laws for the CMMR and two-point power-law curve fits for the SRA.

Tables (5)

Table 1

Atmospheric Turbulence Profile Used for Simulationsa

Reconstructor Performance versus WFS Measurement Noise for an Order $8\times 8$
NGS AO Systema

WFS Noise(arc sec)

CMMR ${\sigma}^{2}$
(μm^{2})

SRA ${\sigma}^{2}$
(μm^{2})

0.02

0.01304

0.01306

0.04

0.01734

0.01737

0.08

0.03218

0.03229

0.16

0.07834

0.07941

This table compares the residual mean square wave-front error ${\sigma}^{2}$
due to the combined effects of fitting error and WFS measurement noise for the conventional matrix multiply reconstruction (CMMR) implementation of the minimum-variance estimator and the sparse reconstruction algorithm (SRA) described in this paper. The approximations made in modeling atmospheric turbulence statistics for the latter algorithm have only a very modest effect on the mean square wave-front estimation error ${\sigma}^{2}.$

Table 4

SRA Wave-Front Fitting Error versus Order of Correction for a Conventional NGS AO Systema

This table summarizes the performance of the SRA for a conventional NGS AO scenario where the only significant source of wave-front error is the finite spatial resolution of the DM actuators and the WFS subapertures. The NGS is coincident with the evaluation direction, and the noise equivalent angle for the WFS is an almost negligible 0.02 arc sec. The DM actuator spacing is held constant, so the AO order of correction is proportional to the telescope aperture diameter. The SRA estimation error is virtually identical with the minimum-variance CMMR for the case of an order $16\times 16$
AO system and increases fairly gradually with increasing telescope aperture diameter. The computation times and the memory requirements for the SRA grow much less rapidly than the $O({n}^{3})$
and $O({n}^{2})$
scaling laws that apply for the case of the CMMR.

Table 5

Results and Scaling Law Predictions for CMMR and SRA Performance for MCAO Systemsa

The simulated MCAO parameters are summarized in Table 2. The parenthesized values are extrapolations based upon the standard $O({n}^{2})$
and $O({n}^{3})$
power laws for the CMMR and two-point power-law curve fits for the SRA.