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Calibration and Pre-Compensation of Non-Common Path Aberrations for extreme Adaptive Optics

J. -F. Sauvage, T. Fusco, G. Rousset, C. Petit

arXiv:astro-ph/0703077v1astro-ph

TL;DR

NCPA are unseen by the wavefront sensor and limit extreme AO performance. The paper refines phase-diversity measurement and uses iterative pseudo-closed-loop pre-compensation to address model uncertainty. On the ONERA bench, images reached 98.7% Strehl ratio at 633 nm.

  • Problem

    Static NCPA are unseen by the wavefront sensor and therefore limit extreme AO performance.

    Source: §1 · p. 2

  • Method

    The procedure optimizes phase-diversity measurement and uses an iterative pseudo-closed-loop scheme to overcome wavefront-sensor model uncertainty.

    Source: §5 · p. 11, §8 · p. 17

  • Results

    98.7% Strehl ratio at 633 nm was obtained on the ONERA bench, with less than 14 nm residual defects.

    Source: §8 · p. 17

  • Takeaways & Limitations

    The bench residual errors are compatible with the error budget of an extreme AO system like SAXO; using 100 Zernike modes is projected to yield 8 nm residual error and 99.9% Strehl ratio at 1.6 µm.

    Source: §8 · p. 17

    The discussion assumes that the high actuator count removes fitting error, leaving phase-diversity estimation error as the only error source.

    Source: §7 · p. 16

Abstract

from arXiv · show

Non-Common Path Aberrations (NCPA) are one of the main limitations for extreme Adaptive Optics (AO) system. NCPA prevent extreme AO systems to achieve their ultimate performance. These static aberrations are unseen by the wave front sensor and therefore not corrected in closed loop. We present experimental results validating new procedures of measurement and pre-compensation of the NCPA on the AO bench at ONERA. The measurement procedure is based on new refined algorithms of phase diversity. The pre-compensation procedure makes use of a pseudo-closed loop scheme to overcome the AO wavefront sensor model uncertainties. Strehl Ratio obtained in the images reaches 98.7% @ 632.8 nm. This result allows us to be confident of achieving the challenging performance required for extrasolar planet direct observation.

T. Fusco Office National d’tudes et de Recherches Arospatiales

The passage gives a department name and a postal address.

C. Petit Office National d’tudes et de Recherches Arospatiales

The passage gives a department name and a postal address.

1. Introduction

Non-common-path aberrations are unseen by the wavefront sensor and limit extreme adaptive optics performance. The paper proposes an optimized calibration and pre-compensation procedure for high-contrast imaging, targeting residual NCPA wavefront error below 10 nm RMS.

1. Introduction

  • NCPA lie in the wavefront-sensor and imaging paths after beam splitting, so the AO system does not see them in its closed loop.Source: §1 · p. 2
  • A dedicated wavefront-sensing tool and phase-diversity algorithm can measure NCPA for direct pre-compensation in the AO loop.Source: §1 · p. 2
  • The proposed optimized procedure for SPHERE development seeks residual NCPA wavefront error below 10 nm RMS after pre-compensation.Source: §1 · p. 2
  • The paper describes a MAP-based optimized measurement algorithm, a new pre-compensation approach, DM-applied phase diversity, and ONERA bench validation.Source: §1 · p. 2

2. Principle of the NCPA calibration and pre-compensation

The section outlines phase diversity for measuring NCPA and modification of AO-loop references for pre-compensation. It also identifies limitations from diversity implementation, wavefront-sensor model uncertainty, and deformable-mirror fitting.

Phase diversity for NCPA calibration

NCPA pre-compensation in the AO loop

3. Optimization of the NCPA measurement

The phase-diversity algorithm uses a maximum-a-posteriori criterion with non-uniform noise statistics and a phase prior; simulations show gains from both components. The prior-based regularization reduces estimation error, while the non-uniform noise model improves accuracy at high photon flux.

Optimization of the PD algorithm

  • The MAP criterion combines image likelihood, object and phase priors, and non-uniform noise statistics, and is minimized iteratively by conjugate gradient.The phase prior encodes knowledge of NCPA through a covariance matrix whose diagonal variances follow the assumed Zernike-order spectrum.Source: Optimization of the PD algorithm · pp. 6–7

Simulation results

  • Simulations use 128 × 128 images with 45 nm RMS aberrations, photon noise, and uniform 1.6-electron detector noise; the regularized-algorithm test uses a maximum flux of 100 photons.Image SNR is defined as peak focal-plane intensity in photon-electrons divided by detector-noise standard deviation in electrons.Source: Simulation results · p. 7

Gain brought by non-uniform noise model

  • At high photon flux, the non-uniform noise model improves phase-estimation accuracy by 15 to 20% over the uniform model.At low flux, both estimation errors are identical because detector noise limits image SNR.Source: Gain brought by non-uniform noise model · p. 8

Gain brought by phase regularization

  • The least-squares estimate has 46 nm total error for 133 coefficients, and coefficients above a36 have signal-to-noise ratios below one.For those high-order coefficients, estimation error exceeds the signal to be estimated.Source: Gain brought by phase regularization · p. 8
  • 22 nm total error with prior-based regularization, versus 29 nm when estimating only the first 33 coefficients; low-order error was 12 nm versus 16 nm.The regularized estimate covers the first 133 coefficients using an (n + 1)^−2 spectrum prior.Source: Gain brought by phase regularization · p. 9
  • The (n + 1)^−2 prior lowers total error across 133 coefficients to 22 nm, with high-order errors tending toward the phase itself rather than noise amplification.The unregularized 133-coefficient estimate has 46 nm total error, similar to the introduced wavefront error.Source: Gain brought by phase regularization · pp. 8–9

4. Optimization of the NCPA pre-compensation

The paper proposes a pseudo-closed loop that measures residual NCPA with phase diversity and iteratively updates AO wavefront-sensor references to address model uncertainties. Compensation is limited by deformable-mirror fitting, so experiments use the first 25 Zernike polynomials.

Pseudo-closed loop process

  • The pseudo-closed loop remeasures residual NCPA with phase diversity after updating AO wavefront-sensor references, repeating until convergence.Errors in the AO wavefront-sensor model slow convergence rather than preventing it; after initial pre-compensation, the images may have higher SNR for coefficient estimation.Source: Pseudo-closed loop process · pp. 9–10

Number of compensated modes

  • Considering Zernike polynomials of radial degree larger than 6 leads to significant fitting errors, larger than 45% of input standard deviation; most Section 6 experiments used the first 25 polynomials, from defocus Z4 up to Z28, for NCPA compensation.Source: Number of compensated modes · p. 11
  • The first 25 Zernike polynomials reduce NCPA amplitudes while minimizing coupling between compensated modes caused by the deformable mirror’s limited actuator count.The authors state that this choice is not an important limitation in final performance because NCPA amplitude is expected to decrease with Zernike order.Source: Number of compensated modes · p. 11

5. DM application of a phase diversity

The deformable mirror can apply phase diversity by modifying the AO wavefront-sensor references, keeping the imposed aberration stable in closed loop. Uncertainty in the wavefront-sensor model makes the known diversity imperfect and biases estimation of the matching aberration coefficient.

5. DM application of a phase diversity

  • Applying phase diversity through modified AO wavefront-sensor references is a software procedure that keeps the imposed defocus stable in closed loop.Applying it directly through deformable-mirror voltages without closing the loop is susceptible to DM creeping; other low-order aberrations can also be used.Source: §5 · p. 11
  • Uncertainty in the AO wavefront-sensor model makes the introduced phase diversity imperfect, with uncertainty in its amplitude as the main effect.Source: §5 · p. 11
  • The defocus-estimation error varies linearly with the applied known-defocus distance, and only a4 is biased when defocus is the diversity.Source: §5 · p. 11
  • A 10 nm known-defocus error produces an error close to 10 nm in measured a4, while errors in the other modes total less than 1 nm.With a 10 nm known-astigmatism error, the corresponding astigmatism error is 10 nm and the other polynomials total 1 nm.Source: §5 · pp. 11–12
  • The pseudo-closed loop cannot remove this systematic phase-diversity bias, so defocus requires image-quality optimization or measurement with another diversity mode.Source: §5 · p. 12

6. Laboratory results

ONERA bench experiments validate pseudo-closed-loop pre-compensation and phase-diversity refinements, with performance depending on the measured and corrected modes and the estimation conditions. The 42-mode test reached a Strehl Ratio of 98.7% at 633 nm, using a Kalman-filter AO loop.

6. Laboratory results

  • The ONERA bench used a 633 nm fibered laser source, a 9-by-9 deformable mirror with 69 valid actuators, and a 270 Hz Shack–Hartmann sensor.The imaging camera had 4 e-/pixel/frame read-out noise; image-quality comparisons relied on Strehl Ratio estimates whose error bars required special treatment.Source: §6 · p. 12

Test of the “pseudo-closed loop” process

  • After four pseudo-closed-loop iterations, residual phase on corrected modes was below 1 nm RMS, while the 50 uncorrected higher-order modes remained nearly unchanged.Total residual error across the 78 measured modes was 24 nm.Source: Test of the “pseudo-closed loop” process · p. 13

Defocus determination

  • A defocus estimate from phase diversity can be biased by uncertainty in the supposedly known aberration, so the authors proposed two post-convergence ways to determine it.These were Strehl optimization and a one-shot measurement using astigmatism phase diversity.Source: Defocus determination · p. 14

SR optimization

  • The Strehl Ratio maximum occurred at a4 = 34 nm, rather than the phase-diversity estimate of 43 nm, yielding about 1% gain.Source: SR optimization · p. 14

Astigmatism phase-diversity

  • Astigmatism phase diversity estimated a4 = 35 nm, compatible with the Strehl-optimization estimate, while transferring the diversity-related bias to a5.Source: Astigmatism phase-diversity · p. 14

Test of optimized algorithms

  • At low SNR (10^2), phase regularization raised Strehl Ratio to 91.9%, compared with about 72.1% without regularization.The reported 10% gain from the non-uniform noise model at low SNR was a one-shot result, not an average over many trials; at high SNR its gain was slight.Source: Test of optimized algorithms · p. 15

Number of modes

  • With the PCL, 42 of 75 measured Zernike modes were compensated; the resulting SR was 98.7%, with 13 nm total residual error across 75 coefficients, versus 24.5 nm in Figure 7.These performance were obtained using a Kalman filter in the AO loop. The same 98.7% SR was not attainable with an integrator under the same conditions.Source: Number of modes · pp. 15–16
  • Correcting 42 rather than 25 modes reduced residual error to 13 nm across 75 coefficients, but DM fitting and aliasing slightly reduced pre-compensation efficiency on lower modes.The partial correction of higher-order modes produced a net gain despite that aliasing loss.Source: Number of modes · pp. 15–16

7. Discussion

The SAXO error budget calls for correction of at least 100 Zernike modes with 8 nm WFE, and the discussion argues that the proposed PCL and optimized algorithms can meet this target.

7. Discussion

  • SAXO specifies compensation of at least 100 Zernike modes, with a goal of 200, and allocates 8nm WFE to those modes.The system has a 41 by 41 actuator deformable mirror and an 80nm WFE budget after extreme AO correction.Source: §7 · p. 16
  • The discussion assumes that deformable-mirror fitting is not a limiting error source, leaving phase-diversity estimation error as the assumed source.Source: §7 · p. 16
  • A SNR of 104 is sufficient for 0.4nm2 precision on each of 100 measured Zernike modes, yielding 6nm residual WFE.The passage states that this SNR can be achieved by averaging infrared-camera images.Source: §7 · p. 17

8. Conclusion

The work validates optimized measurement and pre-compensation procedures for NCPA on the ONERA AO bench. It reports high Strehl ratio and residual errors, and finds the results compatible with extreme-AO requirements.

8. Conclusion

  • The approach combines optimized phase-diversity algorithms, using an accurate noise model and phase regularization, with iterative pseudo-closed-loop pre-compensation to overcome model uncertainties.Source: §8 · p. 17
  • 98.7% SR at 633 nm was obtained on the ONERA AO bench, with less than 14nm residual defects, less than 4.5nm RMS on corrected modes, and less than 12.5nm RMS on uncorrected aberrations.Source: §8 · p. 17
  • Using 100 Zernike modes, the authors project 8nm residual error and 99.9% SR at 1.6µm, compatible with an extreme-AO error budget such as SAXO's.The experiments are presented as support for confidence in achieving performance required for direct extrasolar-planet detection.Source: §8 · p. 17

A. Strehl ratio estimation

The appendix describes Fourier-domain Strehl-ratio estimation and assesses how instrumental uncertainties affect its accuracy. It also compares image-based SR with an estimate derived from measured Zernike coefficients.

Strehl Ratio estimation in focal plan images

  • The procedure estimates SR in the Fourier domain from aberrated and aberration-free optical transfer functions, accounting for background, pixel scale, CCD transfer, and a partially resolved calibration source.Instrumental limitations can affect accuracy even with an optimized algorithm, motivating error estimates.Source: Strehl Ratio estimation in focal plan… · pp. 17–18

Influence of residual background

  • Residual background equal to 1% of image flux modifies SR by 1%; the measured residual background is ±0.1% of total flux, giving ±0.1% SR-estimation accuracy.Source: Influence of residual background · p. 19

Influence of an uncertainty on the pixel scale

  • Pixel-scale uncertainty is a stable bias: it matters for absolute SR, but its influence is dramatically reduced when comparing relative SR evolution.The pixel scale also plays an important role in NCPA estimation through phase diversity.Source: Influence of an uncertainty on the… · p. 19
  • A 0.4% random relative error on pixel scale for 128 frequency pixels yields a 0.8% relative SR-estimation error, a bias that is 0.8% in absolute SR when SR is approximately 100%.The same measured pixel scale, e = λ/4.1D, was used to process all experiments.Source: Influence of an uncertainty on the… · p. 19

SR accuracy in experimental data

SR estimated using the measured Zernike coefficients

  • The Zernike-based SR estimate is a lower bound at relatively low SR and a good approximation at high SR, where it slightly overestimates SR because it includes only the first Nmax Zernike modes.It is computed from residual phase variance using phase-diversity-estimated Zernike coefficients.Source: SR estimated using the measured… · p. 20
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