Modal wavefront estimation from its slopes by numerical orthogonal transformation method over general shaped aperture

Fixed-time observer-based tracking controller for a hysteretic piezoelectric deformable mirror of an adaptive optic system

Piezoelectric deformable mirrors (DM) are benefited from the high accuracy and swift dynamics. The hysteresis phenomenon, which inherently exists in piezoelectric materials, degrades the capability and precision of the adaptive optics (AO) systems. Also, the dynamics of piezoelectric DMs make the controller design more complicated. This research aims to design a fixed-time observer-based tracking controller (FTOTC), which estimates the dynamics, compensates the hysteresis, and ensures tracking to the actuator displacement reference, in the fixed time. Unlike the existing inverse hysteresis operator-based methods, the proposed observer-based controller overcomes the computational burdens and estimates the hysteresis in real-time. The proposed controller tracks the reference displacements, while the tracking error converges in the fixed time. The stability proof is presented by two consecutive theorems. Numerical simulations demonstrate superior tracking and hysteresis compensation by the presented method, from a comparison viewpoint.

Simulation of X-ray Hartmann wavefront sensing with the Synchrotron Radiation Workshop

X-ray wavefront measurement is an important beam diagnostic tool, especially for the diffraction-limited X-ray beam. These in situ diagnostics give a better understanding of beam imperfections, and they enable feedback for possible corrections and/or optical alignment improvements. Hartmann wavefront sensing is one of the promising techniques to perform in situ X-ray wavefront measurements. In this work, a simulation tool of the X-ray Hartmann Wavefront Sensor (HWS) is developed under the Synchrotron Radiation Workshop (SRW) framework. Using this new simulation capability, one can take advantage of the full SRW package to simulate Hartmann wavefront sensing with the beam traveling from the X-ray source to the sample through different X-ray optical components. This SRW HWS simulation tool can help to optimize the wavefront sensor parameters for a specific X-ray energy range. It can also simulate an in situ wavefront measurement experiment with a particular beamline optical layout and predict the expected results of the wavefront measurement under different beamline configurations.

Decoupling control algorithm based on numerical orthogonal polynomials for a woofer-tweeter adaptive optics system

To resolve cooperative control issues of dual wavefront correctors in generalized irregular pupil regions, we propose a decoupling control algorithm based on numerical orthogonal polynomials (NOP). The proposed algorithm firstly deduces NOP from Zernike polynomials in generalized irregular pupil regions. Then, according to wavefront restoration, different spatial frequency aberrations to different wavefront correctors are assigned precisely. Finally, the algorithm calculates and eliminates the cross-coupling between dual wavefront correctors. As observed in numerical simulations and experiments based on a typical woofer-tweeter (W-T) adaptive optics system, NOP decoupling control algorithm restrains the cross-coupling between woofer and tweeter in generalized irregular pupil regions. Moreover, there are obvious advantages over Zernike polynomials decoupling control algorithm in cross-coupling suppression for various scenarios in irregular pupil regions and restoration orders.

Non-propagation fast phase diverse phase retrieval for wavefront measurement with generalized FFT-based basis function

Phase retrieval is an attractive optical testing method with a simple experimental arrangement. The sampling grids wave propagation computation based on the FFT operations is usually involved in each iterative process for the classical phase retrieval model. In this paper, a novel non-propagation optimization phase retrieval technique with the FFT-based basis function is proposed to accelerate wavefront measurement. The sampling grids wave diffraction propagation computation is converted to matrix-vector products that have small dimensions to reduce the computational burden. The diffraction basis function based on generalized numerical orthogonal polynomial and two-step Fresnel propagation is deduced, which is suitable for the generally shaped pupil. This paper provides a universal non-propagation framework to accelerate phase retrieval which is applicable to the arbitrarily shaped wavefront measurement.

Improved difference model applied in the Fourier-transform-based integration method based on Taylor theory

The two-dimensional Fourier-transform-based integration algorithm is widely used in shape or wavefront reconstruction from gradients. However, its reconstruction accuracy is limited by the truncation error of the difference model. The truncation error is affected by the distribution of the sampling points. It increases when the sampling points are unevenly distributed and arranged irregularly. For improving, a novel way to calculate the difference is proposed based on Taylor expansion theory of binary functions. The first-order partial derivative terms are used to estimate the second- and third-order partial derivative terms for reducing the truncation error. The proposed difference model is applied to Fourier-transform-based integration. The reconstruction results show that it can get better results when the sampling points are irregularly distributed.

Generalized shift-rotation absolute measurement method for optical surface shapes with polygonal apertures based on migration recognition by Radon transform

A generalized shift-rotation absolute measurement method for optical surface shapes with polygonal apertures based on migration recognition by Radon transform is proposed. The rotation angles and translation distances of the test surface, measured three times, are calculated through migration recognition. The absolute shape of the test surface with the polygonal aperture is fitted by orthogonal Zernike polynomials. Compared to the existing absolute measurement method for polygonal apertures, our method ensures test surface measurement accuracy without high-precision attitude control and repeated adjustments. The measurement is simple and coherent, which reduces the measurement time and improves the efficiency.

Fast and non-iterative zonal estimation for the non-rectangular data in the transparent surface reconstruction from polarization analysis

In the method of surface reconstruction from polarization, the reconstructed area is generally non-rectangular and contains a large number of sampling points. There is a difficulty that the coefficient matrix in front of the height vector changes with the shape of the measured data when using the zonal estimation. The traditional iterative approaches consume more time for the reconstruction of this type of data. This paper presents a non-iterative zonal estimation to reduce the computing time and to accurately reconstruct the surface. The index vector is created according to the positions of both the valid and invalid elements in the difference and gradient matrices. It is used to obtain the coefficient matrix corresponding to the general data. The heights in the non-rectangular area are calculated non-iteratively by the least squares method. At the same time, the sparse matrix is applied for handling the large-scale data quickly. The simulation and the experiment are designed to verify the feasibility of the proposed method. The results show that the proposed method is highly efficient and accurate in the reconstruction of the non-rectangular data.

Wavefront propagation based on the ray transfer matrix and numerical orthogonal Zernike gradient polynomials

The aberrated wavefront propagates along its normal. Both the magnitude and boundary change after the propagation. Wavefronts characterized by Zernike coefficients and a normalized pupil radius can also be represented by a bundle of feature rays normal to the local surface. A ray transfer matrix parameterized by the pupil radius and propagation distance is proposed to transfer these feature rays to obtain the slope and position data of the propagated feature rays. Numerical orthogonal Zernike gradient polynomials are derived to reconstruct the wavefront from the discrete data by using a numerical method. Two aberrated wavefronts are performed as examples to validate the accuracy and flexibility of the proposed numerical method.

In situ noncontact measurement system and two-step compensation strategy for ultra-precision diamond machining

Ultra-precision diamond machining is a promising technique for non-rotationally symmetrical surfaces with sub-micrometer form accuracy. The measurement and compensation processes in the fabrication process must be conducted carefully to achieve high form accuracy. However, significant challenges remain to improve the measurement accuracy and machining efficiency. Because of the remounting process, the off-machine measurements would reduce the efficiency. On the other hand, contact-type measurements can cause physical damage to some soft materials. To overcome these problems, a noncontact on-machine measurement (OMM) system is developed using two optical probes, and a two-step compensation strategy is proposed to generate a modified tool path. To verify the accuracy of the proposed measurement system, OMMs were performed on a spherical mirror using this system and were later compared with off-machine measurements. To evaluate the compensation strategy, an off-axis paraboloid mirror was diamond-machined and compensated using the proposed method. The results show that the OMM system and compensation strategy are effective for improving the form accuracy while simultaneously enhancing the machining efficiency.

Impact of CMOS Pixel and Electronic Circuitry in the Performance of a Hartmann-Shack Wavefront Sensor

This work presents a numerical simulation of a Hartmann-Shack wavefront sensor (WFS) that assesses the impact of integrated electronic circuitry on the sensor performance, by evaluating a full detection chain encompassing wavefront sampling, photodetection, electronic circuitry and wavefront reconstruction. This platform links dedicated C algorithms for WFS to a SPICE circuit simulator for integrated electronics. The complete codes can be easily replaced in order to represent different detection or reconstruction methods, while the circuit simulator employs reliable models of either off-the-shelf circuit components or custom integrated circuit modules. The most relevant role of this platform is to enable the evaluation of the applicability and constraints of the focal plane of a given wavefront sensor prior to the actual fabrication of the detector chip. In this paper, we will present the simulation results for a Hartmann-Shack wavefront sensor with an orthogonal array of quad-cells (QC) integrated along with active-pixel (active-pixel sensor (APS)) circuitry and analog-to-digital converters (ADC) on a “complementary metal oxide semiconductor” (CMOS) process and deploying a modal wavefront reconstructor. This extended simulation capability for wavefront sensors enables the test and verification of different photosensitive and circuitry topologies for position-sensitive detectors combined with the simulation of sampling microlenses and reconstruction algorithms, with the goal of enhancing the accuracy in the prediction of the wavefront-sensor performance before a detector CMOS chip is actually fabricated.

Orthonormal vector general polynomials derived from the Cartesian gradient of the orthonormal Zernike-based polynomials

The concept of orthonormal vector circle polynomials is revisited by deriving a set from the Cartesian gradient of Zernike polynomials in a unit circle using a matrix-based approach. The heart of this model is a closed-form matrix equation of the gradient of Zernike circle polynomials expressed as a linear combination of lower-order Zernike circle polynomials related through a gradient matrix. This is a sparse matrix whose elements are two-dimensional standard basis transverse Euclidean vectors. Using the outer product form of the Cholesky decomposition, the gradient matrix is used to calculate a new matrix, which we used to express the Cartesian gradient of the Zernike circle polynomials as a linear combination of orthonormal vector circle polynomials. Since this new matrix is singular, the orthonormal vector polynomials are recovered by reducing the matrix to its row echelon form using the Gauss-Jordan elimination method. We extend the model to derive orthonormal vector general polynomials, which are orthonormal in a general pupil by performing a similarity transformation on the gradient matrix to give its equivalent in the general pupil. The outer form of the Gram-Schmidt procedure and the Gauss-Jordan elimination method are then applied to the general pupil to generate the orthonormal vector general polynomials from the gradient of the orthonormal Zernike-based polynomials. The performance of the model is demonstrated with a simulated wavefront in a square pupil inscribed in a unit circle.

Shape reconstruction based on zero-curl gradient field estimation in a fringe reflection technique

A novel shape reconstruction method based on zero-curl gradient field estimation is presented in this paper. Zero-curl field estimation makes the most of curl information to obtain the ideal gradient data, and achieves the reconstruction with the quality map path integration method. In the estimation process, an algebraic approach is adopted to enforce integrability, which maintains the local information well. Moreover, we use the residual gradients of surface obtained from the Southwell zonal reconstruction algorithm as the raw gradient data in zero-curl field estimation, which has a stable tradeoff between smoothness and local shape confinement. The performance of the proposed method over antinoise capability is discussed and demonstrated by the simulations. The measurement experiment of an ultraprecision sphere mirror identifies the validity over general shapes, and the reconstruction results of hyperbolic surface with a local shape map demonstrate the better performance on local details retention. Therefore, this method performs well in handling complex objects with local mutation regions and high accuracy requirement of local information in practical measurement.

Wavefront control in adaptive microscopy using Shack-Hartmann sensors with arbitrarily shaped pupils

In adaptive optical microscopy of thick biological tissue, strong scattering and aberrations can change the effective pupil shape by rendering some Shack-Hartmann spots unusable. The change of pupil shape leads to a change of wavefront reconstruction or control matrix that should be updated accordingly. Modified slope and modal wavefront control methods based on measurements of a Shack-Hartmann wavefront sensor are proposed to accommodate an arbitrarily shaped pupil. Furthermore, we present partial wavefront control methods that remove specific aberration modes like tip, tilt and defocus from the control loop. The proposed control methods were investigated and compared by simulation using experimentally obtained aberration data. The performance was then tested experimentally through closed-loop aberration corrections using an obscured pupil.