Establishing a precise connection between imaged intensity and the incident wavefront is essential for emerging applications in adaptive optics, holography, computational microscopy, and non-line-of-sight imaging. While prior work has shown that breaking symmetries in pupil design enables wavefront recovery from a single intensity measurement, there is little guidance on how to design a pupil that improves wavefront estimation. In this work we introduce a quantitative asymmetry metric to bridge this gap and, through an extensive empirical study and supporting analysis, demonstrate that increasing asymmetry enhances wavefront recoverability. We analyze the trade-offs in pupil design, and the impact on light throughput along with performance in noise. Both large-scale simulations and optical bench experiments are carried out to support our findings.
Figures & tables
Figure 1: Our work investigates the role of pupil design in wavefront estimation, proposes an asymmetry metric to gauge pupil performance, and provides thorough empirical evidence to support our findings. It directly impacts fields like adaptive optics and microscopy, where accurate wavefront estimation is crucial.
Figure 2: Visualization of trivial ambiguities. Different phase aberrations (or any combinations of them) yield the same point spread function (PSF), resulting in a many-to-one relationship for the wavefront estimation problem.
Figure 3: The asymmetry metric α is defined as the maximum non-overlapping area between the pupil and its flip about its center. A pupil can therefore be decomposed into two parts: a symmetric part that is invariant over flipping, and an asymmetric part. The symmetric part of the pupil contributes to the ambiguities in the intensity measurement, while the asymmetric parts encode distinguishable intensities.
Figure 4: Overview of the pipeline of this paper. A dataset of pupils is generated to uniformly cover a range of asymmetry values. A large dataset of phase-pupil-PSF triplets is used to train a network in a supervised fashion.
Figure 5: Qualitative light throughput and noise results. With our simulation, more asymmetric pupils yield lower PSNR even when the noise variance ( σ2 ) is fixed, reflecting a physics-grounded light throughput simulation. The PSNR of each PSF is noted, with the average PSNR for that specific pupil and noise sigma in parentheses.
Figure 6: (a) Wavefront estimation MSE loss as a function of pupil asymmetry (lower is better), and (b) Corrected wavefront Strehl ratio as a function of pupil asymmetry (higher is better). Results reported in this figure use testing data.
Figure 7: The normalized difference between the PSF and the PSF of the conjugate flip. More asymmetric pupils encode higher PSF difference, and that correlates with more distinguishability, and therefore lower wavefront estimation error, and higher Strehl ratios.
Figure 8: Mean Strehl ratio and MSE for noiseless training and testing sets. Each point represents a pupil performance averaged over 5000 (train) and 1500 (test) phase aberrations. A total of 30 million datapoints (6000 pupils × 5000 phases), and 1.5 million (1000 pupils × 1500 phases) were calculated for training and testing, respectively. Consistent training results across asymmetry levels confirm that the network is unbiased, while testing performance highlights the superior recoverability and generalizability of asymmetrical pupils.
Figure 9: Visualization of pupil asymmetry effect on wavefront estimation loss and the Strehl ratio of the corrected wavefront. More asymmetry yields lower error and a higher Strehl ratio, implying minimizing the MSE loss is a meaningful objective function to maximize Strehl ratio. The loss increases and Strehl decreases with more noise, but the asymmetry trend is consistent. All plots are on the same scale for easier interpretation. The colors represent increasing asymmetry.
Figure 10: Comparison of wavefront estimation performance in different aberration scenarios (0.25 to 2.0) from the noise-free case. As the aberration scale increases, symmetric pupils exhibit higher error and degradation in the Strehl ratio. The MSE is shown in log scale for visibility between different aberration strengths.
Figure 11: The Strehl ratio against the reference circular pupil diffraction limit in the presence of different measurement noise.
Figure 12: Wavefront correction simulated results with low and high noise. The target is blurred with the aberration PSF produced by the phase in the first row. The corrected image is shown for different levels of noise, with the aberrated and corrected images in the left and right insets, respectively. The image in the background is the corrected version.
Figure 13: The collimated light source illuminates the SLM. The reflected light passes through the focusing lens. The PSF is then captured by a camera. L1 is the collimating lens. L2 is the focusing lens. BS is a 50:50 beamsplitter. Examples from the real data acquisition (c) with phase pattern (second row) and the captured PSFs (third row).
Pupil
Asymmetry ( α )
RMSE (rad)
MSE (rad 2 )
Pupil 1
0.0654
0.379285
0.143857
Pupil 2
0.1893
0.353183
0.124739
Pupil 3
0.2320
0.323678
0.104768
Pupil 4
0.2524
0.315149
0.099319
Pupil 5
0.2827
0.305383
0.093259
Table 1: Real data average wavefront estimation error on a testing subset. The more asymmetric the pupil, the lower the error. Highest asymmetry and lowest error are highlighted in boldface. Corresponding pupils are shown in Figure 13 .
Figure S1 : Results from experiments with MLP architecture: (a) Wavefront estimation MSE loss as a function of pupil asymmetry (lower is better), and (b) Corrected wavefront Strehl ratio as a function of pupil asymmetry (higher is better). Results reported in this figure use testing data.
Figure S2 : Results under different noise conditions with the main U-Net architecture: (a) Wavefront estimation MSE loss as a function of pupil asymmetry (lower is better), and (b) Corrected wavefront Strehl ratio as a function of pupil asymmetry (higher is better). Results reported in this figure use testing data.
Figure S3 : Visualization of how progressively more asymmetric pupils are located on the MSE loss plot.
Figure S4 : The architecture of the U-Net model.
Figure S5 : The architecture of the MLP model.
Figure S6 : Comparison of wavefront estimation performance in different aberration scenarios (0.25 to 2.0) under different noise conditions. As the aberration scale increases, symmetric pupils exhibit higher error and degradation in the Strehl ratio. The MSE is shown in log scale for visibility between different aberration strengths.
Figure S7 : Wavefront estimation results from PSFs captured by our optical setup.
Figure S8 : Visualization of PSFs constructed from the estimated wavefront.
Computational imaging combines optical and sensor design with algorithms to recover or infer scene information from indirect, compressed, or noisy measurements.
Extreme adaptive optics (AO) is necessary for high contrast astronomy at scales of the habitable zone of nearby systems. We seek to evaluate wavefront sensors that approach fundamental limits of wavefront sensing, enabling adaptive optics systems to run faster or on fainter targets. We present the phase-induced amplitude apodisation Zernike wavefront sensor (PIAA-ZWFS): an adaptation of the conventional Zernike wavefront sensor (ZWFS) that leverages lossless apodisation of the pupil to concentrate the starlight in the focal plane. We optimise and evaluate the sensor with a differentiable modelling framework, drawing on concepts from Bayesian experimental design to minimise the variance of a maximum likelihood estimator that uses the system in the high Strehl regime. Our architecture shows state-of-the-art performance in simulation for different apertures, bandwidths, photon fluxes and source sizes, closing the gap to the fundamental limit by a factor 10 (2.5) compared to the conventional ZWFS (optimised ZWFS) in a typical photon-limited case. For extended sources, we show that even an ideal point source sensor rapidly becomes sub-optimal, and our system outperforms it for stellar diameters larger than 0.8λ/D. We verify that these gains do not come at the cost of dynamic range with either linear or non-linear reconstructors. Finally, we present a proof that there must be a trade-off between the information gained about amplitude and phase errors for any wavefront sensor. The PIAA-ZWFS is a viable wavefront sensor operating near the fundamental sensitivity limits.
A. K. Taras, S. Y. Haffert, L. Desdoigts
Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands
The efficiency of high-power laser systems is limited by wavefront distortions in the beam, particularly non-common path aberrations, which reduce the peak intensity at the focal plane. Compensating for these aberrations requires the calibration of the adaptive optics system. Conventional calibration methods rely on a time-consuming iterative optimization that is highly sensitive to initial conditions. While deep learning-based models offer high speed, they often demonstrate insufficient accuracy. In this work, we present a hybrid wavefront reconstruction method that combines a convolutional neural network to generate an initial estimate of the wavefront distortions, with the L-BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno) algorithm for its subsequent refinement. In numerical simulations, the method achieved an efficiency of ∼0.99 in 80% of the cases for a root-mean-square (RMS) of wavefront distortions ranging from 0 to 1.3λ. In a physical experiment, for initial wavefront distortions with RMS values from 0.15 to 0.6λ, the method achieved an efficiency of ∼0.75. As a result, focusing with a Strehl ratio of 0.96±0.02 was attained within 2 to 4 iterations of the algorithm, confirming the applicability of the method for the fast and accurate calibration of adaptive optics systems under real experimental conditions.
Y. Rodimkov, A. Kotov, K. Burdonov +4
Lobachevskii State University of Nizhni Novgorod, pr. Gagarina 23, Nizhni Novgorod, 603022 Russia · Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod 603951, Russia
This paper describes a linear solution method for near-light photometric stereo by exploiting symmetric light source arrangements. Unlike conventional non-convex optimization approaches, by arranging multiple sets of symmetric nearby light source pairs, our method derives a closed-form solution for surface normal and depth without requiring initialization. In addition, our method works as long as the light sources are symmetrically distributed about an arbitrary point even when the entire spatial offset is uncalibrated. Experiments showcase the accuracy of shape recovery accuracy of our method, achieving comparable results to the state-of-the-art calibrated near-light photometric stereo method while significantly reducing requirements of careful depth initialization and light calibration.
Lilika Makabe, Heng Guo, Hiroaki Santo +2
Graduate School of Information Science and Technology, Osaka University, Japan.