Wavefront Reconstruction Algorithm Research Articles

Fast and Highly Accurate Zonal Wavefront Reconstruction from Multi-Directional Slope and Curvature Information Using Subregion Cancelation

The wavefront reconstruction is a crucial step in determining the performance of wavefront detection instruments. The wavefront reconstruction algorithm is primarily evaluated in three dimensions: accuracy, speed, and noise immunity. In this paper, we propose a hybrid zonal reconstruction algorithm that introduces slope and curvature information in the diagonal, anti-diagonal, horizontal, and vertical directions by dividing the neighbor sampling points into subregions in groups of four. By canceling the same parameters in integration equations, an algorithm using multi-directional slope–curvature information is achieved with only two sets of integration equations in each subregion, reducing the processing time. Simulation experiments show that the relative root-mean-square reconstruction error of this algorithm is improved by about 4 orders of magnitude compared with existing algorithms that use multi-directional slope information or slope–curvature information alone. Compared with the hybrid multi-directional slope–curvature algorithm, the proposed algorithm can reduce computation time by about 50% as well as provide better noise immunity and reconstruction accuracy. Finally, the validity of the proposed algorithm is verified by the null test experiment.

Wavefront reconstruction based on ASH-Net with less lenslets SHWFS

Generally, the number of lenslets in Shack–Hartman wavefront sensor (SHWFS) determines its spatial sampling frequency. However, increasing the number of lenslets is limited by the data processing delay time and low sensitivity in the adaptive optics system. In this paper, a wavefront reconstruction method based on ASH-Net for SHWFS is presented to recover the distorted wavefront from the light spots array with high accuracy with a spatial sampling frequency than traditionally required. The results show that the maximum number of Zernike modes recovered by the method is 36, 54, and 70 for 4 × 4, 6 × 6, and 8 × 8 lenslets arrays, respectively. Residual wavefront root mean square values are as low as about 0.02 μm, corresponding to a Strehl Ratio of 0.99. This method breaks the limitation that the number of reconstructed Zernike modes in the traditional wavefront reconstruction algorithm must be less than the number of effective lenslets. Experiments in lab are used to validate the method. Additionally, fewer lenslets is valuable to improve the detection capability of SHWFS for faint targets. This is important for applications of adaptive optics in areas such as astronomy and retinal imaging.

Higher-resolution wavefront sensing based on sub-wavefront information extraction

The limited spatial sampling rates of conventional Shack–Hartmann wavefront sensors (SHWFSs) make them unable to sense higher-order wavefront distortion. In this study, by etching a known phase on each microlens to modulate sub-wavefront, we propose a higher-resolution wavefront reconstruction method that employs a modified modal Zernike wavefront reconstruction algorithm, in which the reconstruction matrix contains quadratic information that is extracted using a neural network. We validate this method through simulations, and the results show that once the network has been trained, for various atmospheric conditions and spatial sampling rates, the proposed method enables fast and accurate high-resolution wavefront reconstruction. Furthermore, it has highly competitive advantages such as fast dataset generation, simple network structure, and short prediction time.

Simulation analysis of iterative wavefront reconstruction algorithm for center-offset radial shearing interference

Simulation analysis of iterative wavefront reconstruction algorithm for center-offset radial shearing interference

Phase retrieval algorithm for wavefront reconstruction using four scattering spots

Phase retrieval algorithm for wavefront reconstruction using four scattering spots

A mathematical framework for nonlinear wavefront reconstruction in adaptive optics systems with Fourier-type wavefront sensing

Advanced adaptive optics (AO) instruments have applications in ophthalmic imaging, free-space optical communications and the future generation of extremely large telescopes. These AO systems are designed to perform real-time corrections of dynamic wavefront aberrations. The corrections can be performed by converting wavefront measurements into deformable mirror (DM) actuator commands. The role of the DM is to mitigate aberrations by restoring a planar wavefront. Optimal DM actuator commands therefore require precise phase measurements across the entire wavefront. Reconstructing a wavefront from wavefront sensor (WFS) data is an inverse problem that depends on the type of WFS implemented. Nonlinear Fourier-type WFSs are included in the design of many current and upcoming AO systems. Conventionally, these sensors perform AO control based on simplifications and linearisations of the underlying models. However, in nonlinear regimes, approximation errors critically degrade image quality. This study looks at overcoming nonlinear wavefront sensing regimes by introducing a nonlinear, iterative algorithm for Fourier-type wavefront reconstruction. The algorithm used is well-known in the field of inverse problems. The underlying mathematical theory for modelling Fourier-type WFSs is provided, along with how these models can be used to perform nonlinear wavefront reconstruction. A significant advantage of the analysis presented is its generalised applicability to any Fourier-type sensor. The only input required is the mathematical expression for the optical element transfer function. The generalised and full mathematical model of Fourier-type WFSs is introduced in a Sobolev space setting. Necessary inputs are derived for the nonlinear iterative algorithms, such as Fréchet derivatives and adjoints. The generalised theory is then expanded to solve the inverse problem of wavefront reconstruction for all Fourier-type WFSs. Moreover, the study concentrates on the pyramid WFS (PWFS)—one of the most well-known Fourier-type WFSs—and shows a Hilbert transform representation of the amplitude of the incoming light on its detector. The developed theory is demonstrated using a simulated PWFS to measure an example wavefront.

A Method Used to Improve the Dynamic Range of Shack-Hartmann Wavefront Sensor in Presence of Large Aberration.

With the successful application of the Shack–Hartmann wavefront sensor in measuring aberrations of the human eye, researchers found that, when the aberration is large, the local wavefront distortion is large, and it causes the spot corresponding to the sub-aperture of the microlens to shift out of the corresponding range of the sub-aperture. However, the traditional wavefront reconstruction algorithm searches for the spot within the corresponding range of the sub-aperture of the microlens and reconstructs the wavefront according to the calculated centroid, which leads to wavefront reconstruction errors. To solve the problem of the small dynamic range of the Shack–Hartmann wavefront sensor, this paper proposes a wavefront reconstruction algorithm based on the autocorrelation method and a neural network. The autocorrelation centroid extraction method was used to calculate the centroid in the entire spot map in order to obtain a centroid map and to reconstruct the wavefront by matching the centroid with the microlens array through the neural network. This method breaks the limitation of the sub-aperture of the microlens. The experimental results show that the algorithm improves the dynamic range of the first 15 terms of the Zernike aberration reconstruction to varying degrees, ranging from 62.86% to 183.87%.

파면 분석을 위한 편광 격자 기반 층밀림 간섭계

In this investigation, we propose a simple and effective lateral shearing interferometer based on a polarization grating. In the lateral shearing device, an incident beam is split into two beams by a polarization grating, and the returning beams can be laterally shifted after reflecting off a flat mirror and passing through the polarization grating again. These two beams are not only laterally shifted, but also their polarization states are orthogonal to each other as circular polarizations. With a single image obtained by a pixelated polarization CMOS camera, the proposed LSI can obtain the phase map corresponding to the x-sheared interferogram, and the other phase map can be calculated from another single image obtained by 90° rotation of the shearing device. Then, the original wavefront corresponding to the surface figure of the specimen can be reconstructed by wavefront reconstruction algorithms. In the experiments, various wavefronts generated by concave mirrors and a deformable mirror were measured and compared with those of a commercial Shack-Hartmann sensor.

Hybrid wavefront reconstruction from multi-directional slope and full curvature measurements using integral equations with higher-order truncation errors for wavefront sensors

Wavefront reconstruction is one of the most important steps in zonal wavefront sensors and has great influence on measurement accuracy. An improved hybrid wavefront reconstruction algorithm that adds two diagonal slope and curvature values at each grid point is proposed to improve reconstruction accuracy and noise immunity. Hybrid reconstruction equations with smaller truncation errors in diagonal and anti-diagonal directions are deduced. Furthermore, multi-directional hybrid reconstruction equations incorporating vertical, horizontal, diagonal and anti-diagonal directions are also formulated. Numerical experiments show that the relative reconstruction errors of the proposed algorithms are reduced by about four orders of magnitude compared with the existing hybrid reconstruction algorithm with noise ignored. Considering the measurement noise, the proposed algorithms have better noise immunity and reconstruction accuracy, especially for the reconstruction of high-order aberrations.

THE ACCURACY OF ZERNIKE RECONSTRUCTION OF OCULAR ABERRATIONS WITH MEASUREMENT NOISE

The ocular aberrations (wavefront) can be used to describe the vision defects, and wavefront sensing devices (aberrometers) are used to measure the ocular aberrations (OAs). The Zernike polynomials (ZPs) have been successfully used in wavefront reconstruction algorithms of aberrometers for years. As the aberrometer measurements consist with noise, the measurements observed using aberrometers are not always accurate. It is therefore necessary to take multiple measurements from the patient in each sitting. However, taking multiple measurements also causes to induce the variations on measurements, and this variability of measurements is significant, and leads errors in clinical practice. Therefore, the variability should be taken into account during wavefront reconstruction. Consequently, unlike prior work, the accuracy of Zernike representation for the ocular aberrations with noise (due to the system noise and the variability of measurements) is studied. In the study, the noisy data is fitted using ZPs. Data sets are created using extracted data from arbitrary OAs, that is, synthetic data sets are used. Normally distributed very small random numbers are added into the data sets in order to create measurement noise. The magnitude of added noisy has been changed to range the signal-to-noise ratio (SNR) from 55 dB to 20 dB. The corresponding Zernike coefficients for each noised wavefront are computed, and visual acuity is used to examine the deviation of the reconstructed wavefront from the data set. The study concludes that the Zernike polynomials are not good enough to represent the wavefront with noisy level less than certain value of SNR which is greater than the reference SNR values of aberrometers, and note that the aberrometer SNR is always between 20-30 dB.

Spherical wavefront measurement on modified cyclic radial shearing interferometry.

We propose a radial shearing interferometric approach to measure spherical wavefronts as both of the reflective and transmissive optical configurations. The modified cyclic radial shearing interferometer uses a single lens in the optical layout, which can conveniently adjust the radial shearing ratio between two shearing spherical wavefronts, and the use of a polarization camera enables to reconstruct the wavefront by a single image. The wavefront mapped onto the camera plane can be identified and quantified throughout an optimized wavefront reconstruction algorithm. In the experiments, plano-convex lenses and concave mirrors were used to generate spherical wavefronts, and the proposed system was able to reconstruct the surface figures after system characterization and calibration. Further investigations were performed to evaluate the system measurement accuracy by the radius of curvature comparison with design value and a commercial Shack-Hartmann wavefront sensor.

Fast, precise, and shape-flexible registration of wavefronts

A fast and precise algorithm for wavefront reconstruction by the registration of wavefront segments is presented. If the wavefront exceeds the sensor aperture or the dynamic range of the sensor, a Shack-Hartmann sensor can measure only segments of an optical wavefront. The algorithm registers the wavefront segments in parallel, where they are simultaneously transformed to minimize their overlap mismatch for precise reconstruction of the entire wavefront. The original nonlinear optimization problem is approximated by a convex optimization problem that can be solved more efficiently. A simulation-based analysis of the algorithm and a comparison to a previously proposed parallel registration (PR) algorithm as well as to the iterative closest point (ICP) algorithm are presented. It is shown that despite measurement noise, the algorithm can precisely register plane as well as divergent wavefronts with root mean square registration errors smaller than 10 nm. Particularly for the divergent wavefront, this enables a reduction of the registration error by a factor of up to 750 as compared to the established algorithms. Analysis and comparison to the ICP and PR algorithm also show that the computation time of the proposed algorithm can be from one to three orders of magnitude smaller.

A B-spline based fast wavefront reconstruction algorithm

Traditional schemes for Shack-Hartmann wavefront reconstruction can be classified into zonal and modal methods. The zonal methods are good at reconstructing the local details of the wavefront, but are sensitive to the noise in the slope data. The modal methods are much more robust to the noise, but they have limited capability of recovering the local details of the wavefront. In this paper, a B-spline based fast wavefront reconstruction algorithm in which the wavefront is expanded to the linear combination of bi-variable B-spline curved surfaces is proposed. Then, a method based on successive over relaxation (SOR) algorithm is proposed to fast reconstruct the wavefront. Experimental results show that the proposed algorithm can recover the local details of the wavefront as good as the zonal methods, while is much more robust to the slope noise.

Adaptive beam clean-up of high power slab lasers using least-squares wavefront reconstruction algorithm with performance-based filtering

The edge effects of high-power slab lasers cause severe wavefront distortions with large amplitude and gradients, which is a great challenge for the adaptive optics systems for improving the beam qualities. The standard least-squares wavefront reconstruction method would inherently assign higher priorities to correcting the edge aberrations which contribute little to the laser power in the near diffraction-limit areas in the far-field, while the other aberrations are not well corrected. To solve the problems, we propose a least-squares reconstruction algorithm with performance-based filtering, in which the correction capability of the deformable mirror is analyzed in the spatial frequency domain, and then the input phase distortions are filtered accordingly. The uncorrectable components of input distortions are ignored so that the residuals of the correctable components could be further minimized. Experiments with a 20 kW Nd:YAG slab laser indicate that the proposed method is effective to improve the beam quality of high-power slab laser with edge effect.

Experimental study on measurement of free-form surface with wavefront reconstruction algorithm

In this paper, a triangulation interpolation wavefront reconstruction algorithm is proposed to measure the full aperture map of the free-form surface. Based on the method we proposed, the free-form surface can be reconstructed with the scanning white light interferometry (SWLI). Applying the triangulation interpolation wavefront reconstruction algorithm, we provide an experimental demonstration by testing a 318 mm * 288 mm free-form surface in a rectangular shape. By comparing reconstruction results with computer-generated hologram (CGH) compensation testing result, it proves that the surface map can be got with high precision by using the wavefront reconstruction algorithm. In addition, we also test the free-form surface with a probe in contact measurement mode. It shows that with the white light scanning interferometry method, the surface map can be reconstructed with greater efficiency.

Robust wavefront segment registration based on a parallel approach.

This paper presents a robust registration algorithm for wavefront reconstruction from multiple partial measurements. Wavefronts exceeding the dynamic range or size of the Shack-Hartmann sensor can be measured as a set of segments. The wavefront is reconstructed by parallel registration of these wavefront segments, enabling compensation for sensor misalignment as well as for phase differences. For registration, a global mismatch metric is minimized by rigid body transformations and propagation of the wavefront segments. Apart from the description of the algorithm, a simulation-based evaluation and comparison to the iterative closest point (ICP) algorithm is performed. It is shown that in the case of a noisy data set, the parallel approach enables reconstruction errors that are a factor of 10 smaller than the result obtained with the ICP algorithm.

Wavefront Parameters Recovering by Using Point Spread Function

At various stages of the life cycle of optical systems, one of the most important tasks is quality of optical system elements assembly and alignment control. The different wavefront reconstruction algorithms have already proven themselves to be excellent assistants in this. Every year increasing technical capacities opens access to the new algorithms and the possibilities of their application. The paper considers an iterative algorithm for recovering the wavefront parameters. The parameters of the wavefront are the Zernike polynomials coefficients. The method involves using a previously known point spread function to recover Zernike polynomials coefficients. This work is devoted to the research of the defocusing influence on the convergence of the algorithm. The method is designed to control the manufacturing quality of optical systems by point image. A substantial part of the optical systems can use this method without additional equipment. It can help automate the controlled optical system adjustment process.

Compressed off-axis digital holography

The Fourier transform method (FTM), which can reconstruct a wavefront from a single hologram, is widely used in digital holography (DH) as a reconstruction technique. However, the space-bandwidth product (SBP) is limited because of the frequency filtering required. We propose a wavefront reconstruction algorithm based on compressed sensing that can obtain a high SBP for use instead of the FTM. The proposed method can reconstruct a high-SBP wavefront by solving the hologram recording model with total variation regularization. We demonstrate the proposed method’s effectiveness via simulations and experiments. The method can be applied to conventional off-axis DH without alteration and thus can be applied in various fields.

Single-frame far-field wavefront retrieval method based on Walsh function modulation

The reconstruction of wavefront from single far-field image data has unique advantages in simplicity of structure. However, the traditional wavefront reconstruction algorithm has multiple solutions based on single far-field image, its iterative process easily falls into stagnation. In this paper, based on the analysis of the multi-solution problem of single-frame phase retrieval method, a wavefront reconstruction method based on Walsh function two-dimensional discrete phase modulation is proposed. This method can effectively break the symmetry of near-field wavefront and overcome problem of multiple solutions. The simulation results show that the method can accurately reconstruct wavefront aberration with only one far-field image.

Wavefront reconstruction of a Shack-Hartmann sensor with insufficient lenslets based on an extreme learning machine.

In a standard Shack-Hartmann wavefront sensor, the number of effective lenslets is the vital parameter that limits the wavefront restoration accuracy. This paper proposes a wavefront reconstruction algorithm for a Shack-Hartmann wavefront sensor with an insufficient microlens based on an extreme learning machine. The neural network model is used to fit the nonlinear corresponding relationship between the centroid displacement and the Zernike model coefficients under a sparse microlens. Experiments with a 6×6 lenslet array show that the root mean square (RMS) relative error of the proposed method is only 4.36% of the initial value, which is 80.72% lower than the standard modal algorithm.