Iterative zonal wave-front estimation algorithm for optical testing with general-shaped pupils

Effective selection of shears in variable lateral shearing holography

The efficiency of reconstruction of complex wavefields in digital holography through shear interferometry has a direct correlation with the shears selected for image acquisition. Although studies to investigate the effect of shears have shown correlations between the selected shear set and the spatial and frequency contents of the reconstructed complex wavefield, to our best knowledge, not much information is available to provide a guide on how to select these shears optimally and what factors to be considered during this selection procedure. In this paper, we study the effect of shear parameters on the phase error through a series of simulations using a synthetic object wavefield and provide a range of shear parameters for optimal reconstruction. Further, we correlated the data by comparing the results with corresponding frequency information density maps.

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 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.

Three-Dimensional Shape Measurements of Specular Objects Using Phase-Measuring Deflectometry

The fast development in the fields of integrated circuits, photovoltaics, the automobile industry, advanced manufacturing, and astronomy have led to the importance and necessity of quickly and accurately obtaining three-dimensional (3D) shape data of specular surfaces for quality control and function evaluation. Owing to the advantages of a large dynamic range, non-contact operation, full-field and fast acquisition, high accuracy, and automatic data processing, phase-measuring deflectometry (PMD, also called fringe reflection profilometry) has been widely studied and applied in many fields. Phase information coded in the reflected fringe patterns relates to the local slope and height of the measured specular objects. The 3D shape is obtained by integrating the local gradient data or directly calculating the depth data from the phase information. We present a review of the relevant techniques regarding classical PMD. The improved PMD technique is then used to measure specular objects having discontinuous and/or isolated surfaces. Some influential factors on the measured results are presented. The challenges and future research directions are discussed to further advance PMD techniques. Finally, the application fields of PMD are briefly introduced.

High spatial resolution zonal reconstruction with modified multishear method in frequency domain

An exact multishear zonal algorithm is proposed to reconstruct two-dimensional wavefronts in frequency domain. The algorithm maintains the advantage of fast Fourier transform and loosens the "natural extension" requirement that the shear amounts must be divisors of sampling points N; therefore, it can be rapidly executed for large data arrays. The effect of tilt errors in multishear interferometry is analyzed and compensated in our method. The presented algorithm is applicable for a general aperture shape by using an iterative method. Application of large shears is allowed, and high resolution of the reconstructed wavefront can be achieved. Results of numerical simulations demonstrate the capability of our method.

Exact recovery of wavefront from multishearing interferograms in spatial domain

An exact algorithm based on the multishearing interferograms has been proposed to reconstruct a two-dimensional wavefront. It allows large shears and high resolution of the reconstructed wavefront to be achieved. In this paper, we use simultaneous linear equations to express the relationship between difference wavefronts and the unknown original wavefront, and then the least-squares method is applied to reconstruct the wavefront. To solve the memory problem, an improved wavefront reconstruction algorithm based on virtual subaperture stitching was proposed to improve the calculation efficiency. Lastly, numerical simulations are implemented and the proposed algorithm is compared with another modal and zonal method. The results indicate that the proposed algorithm is capable of reconstructing continuous or discontinuous wavefronts exactly with a large grid. Numerical simulation also shows high accuracy recovery capability of the proposed method in the existence of mixed noise.

Improved method for rapid shape recovery of large specular surfaces based on phase measuring deflectometry

Incorporating the modal and zonal estimation approaches into a unifying scheme, we introduce an improved three-dimensional shape reconstruction version of specular surfaces based on phase measuring deflectometry in this paper. The modal estimation is first implemented to derive the coarse height information of the measured surface as initial iteration values. Then the real shape can be recovered utilizing a modified zonal wavefront reconstruction algorithm to simultaneously achieve consistently high accuracy and dramatically rapid convergence. Moreover, the iterative process based on an advanced successive over-relaxation technique shows a consistent rejection of measurement errors, guaranteeing the stability and robustness in practical applications. The reconstruction results of numerical examples of the sphere, hyperbolic, and arbitrary surfaces, as well as an experimentally measured sphere mirror demonstrate the validity and efficiency of the proposed improved method. In the simulations, the proposed method increases the rate of convergence by fourfold compared with the existing zonal approach and realizes three orders of magnitude improvement in reconstruction accuracy compared with the modal technique when handling the sample points of 401×401  pixels of a sphere surface. Furthermore, the computation time decreases approximately 74.92% in contrast to the zonal estimation, and the surface error is about 6.68 μm with reconstruction points of 391×529  pixels of an experimentally measured sphere mirror. In general, this new method can be conducted with fast convergence speed and high accuracy, providing an efficient, stable, and real-time approach for shape reconstruction in practical situations.

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

Wavefront estimation from the slope-based sensing metrologies zis important in modern optical testing. A numerical orthogonal transformation method is proposed for deriving the numerical orthogonal gradient polynomials as numerical orthogonal basis functions for directly fitting the measured slope data and then converting to the wavefront in a straightforward way in the modal approach. The presented method can be employed in the wavefront estimation from its slopes over the general shaped aperture. Moreover, the numerical orthogonal transformation method could be applied to the wavefront estimation from its slope measurements over the dynamic varying aperture. The performance of the numerical orthogonal transformation method is discussed, demonstrated and verified by the examples. They indicate that the presented method is valid, accurate and easily implemented for wavefront estimation from its slopes.

High spatial resolution zonal wavefront reconstruction with improved initial value determination scheme for lateral shearing interferometry

In a recent paper [J. Opt. Soc. Am. A 29, 2038 (2012)], we proposed a generalized high spatial resolution zonal wavefront reconstruction method for lateral shearing interferometry. The test wavefront can be reconstructed with high spatial resolution by using linear interpolation on a subgrid for initial values estimation. In the current paper, we utilize the difference between the Zernike polynomial fitting method and linear interpolation in determining the subgrid initial values. The validity of the proposed method is investigated through comparison with the previous high spatial resolution zonal method. Simulation results show that the proposed method is more accurate and more stable to shear ratios compared with the previous method. A comprehensive comparison of the properties of the proposed method, the previous high spatial resolution zonal method, and the modal method is performed.

Three-dimensional shape measurement of a highly reflected, specular surface with structured light method

This paper proposes a mathematical measurement model of a highly reflected, specular surface with structured light method. In the measurement, an auxiliary fringe pattern named amplitude perturbation is adopted to be projected onto the measured surface. The amplitude perturbation can ease the procedure of searching the corresponding points between the phase map of the measured surface and that of the reference plane by locking up the most reliable point as the starting unwrapping point whose true phase can be calculated accurately. The proposed method is also suitable for measuring the step surfaces such as gauge blocks with different heights. Furthermore, the image segmentation technology is introduced in the phase unwrapping procedure to increase the speed. Based on the unwrapped phase map, zonal wave-front reconstruction algorithm is implemented to realize three-dimensional, highly reflected, specular surface reconstruction. Experimental studies show that the developed methodology displays accuracy and high stability for highly reflected, specular surface measurement.

Improving wavefront boundary condition for in vivo high resolution adaptive optics ophthalmic imaging

An ophthalmic adaptive optics (AO) imaging system is especially affected by pupil edge effects due to the higher noise and aberration level at the edge of the human pupil as well as the impact of head and eye motions on the pupil. In this paper, a two-step approach was proposed and implemented for reducing the edge effects and improving wavefront slope boundary condition. First, given an imaging pupil, a smaller size of sampling aperture can be adopted to avoid the noisy boundary slope data. To do this, we calibrated a set of influence matrices for different aperture sizes to accommodate pupil variations within the population. In step two, the slope data was extrapolated from the less noisy slope data inside the pupil towards the outside such that we had reasonable slope data over a larger aperture to stabilize the impact of eye pupil dynamics. This technique is applicable to any Neumann boundary-based active /adaptive modality but it is especially useful in the eye for improving AO retinal image quality where the boundary positions fluctuate.

An absolute test for axicon surfaces

We present a method for absolute testing of axicon surfaces in a null test setup. The absolute test exploits the symmetry of axicons, which allows us to introduce a shift of the surface under test in both the axial and rotational directions while still maintaining the null test condition. With two shifts of the surface under test and four measurements, the interferometer and null optics error can be removed. The absolute surface local deviation can be obtained by wavefront reconstruction with a double-side spiral-path direct integration method. A simulation of the method, including typical systematic as well as statistical errors as input, is presented to estimate the error propagation. Experimental absolute test results of a 90° axicon surface are given.

Amplitude filter and Zernike polynomial expansion method for quality control of microlens arrays

This paper deals with a computer simulation and an experimental realization of an optical setup for automatic quality control of microlens arrays. The method is based on a 4f coherent light correlator setup with an amplitude filter placed in the Fourier plane. The output intensity signal is proportional to the first derivative of the distortion of the input wavefront. An analysis can be carried out with the use of the Zernike polynomial expansion method. It must be carried out separately for each lens, but it allows for a more precise, quantitative assessment of their quality. What is important is that the analysis is computer-based and performed on the basis of the initial single optical measurement.

Modeling the eye's optical system by ocular wavefront tomography

PURPOSE

Ocular wavefront tomography (OWT) is the process of using wavefront aberration maps obtained along multiple lines-of-sight (LoS) to determine the shape and position of the major refracting elements of an eye. One goal of OWT is to create a customized schematic model eye that is anatomically similar and functionally equivalent to the individual eye over a large field of view.

METHODS

Wavefront aberration maps along multiple LoS were used as design goals for configuring a generic, multi-surface model eye with aberrations that match the measurements. The model was constrained by gross anatomical dimensions and optimized to mimic the measured eye. The method was evaluated with two test cases: (1) a physical model eye with a doublet lens measured with a clinical wavefront aberrometer along six LoS between -31 deg and +29 deg eccentricities, and (2) a mathematical model of the myopic eye for which wavefront aberrations were computed by ray tracing.

RESULTS

In case 1, the OWT algorithm successfully predicted the structure of the doublet model eye from the experimental on- and off-axis aberration measurements. In case 2, the algorithm started with a symmetric five surface model eye and optimized it to generate the on- and off-axis aberrations of a GRIN myopia model eye. The adjusted model closely mimicked the physical parameters and optical behavior of the expected myopia model eye over a large field of view. The maximum discrepancy between aberrations of the OWT optimized model and measurements was 0.05 microns RMS for test case 1 and 0.2 microns RMS for test case 2.

CONCLUSION

Our implementation of OWT is a valid, feasible, and robust method for constructing an optical model that is anatomically and functionally similar to the eye over a wide field of view.

Quantifications of error propagation in slope-based wavefront estimations

We discuss error propagation in the slope-based and the difference-based wavefront estimations. The error propagation coefficient can be expressed as a function of the eigenvalues of the wavefront-estimation-related matrices, and we establish such functions for each of the basic geometries with the serial numbering scheme with which a square sampling grid array is sequentially indexed row by row. We first show that for the wavefront estimation with the wavefront piston value determined, the odd-number grid sizes yield better error propagators than the even-number grid sizes for all geometries. We further show that for both slope-based and difference-based wavefront estimations, the Southwell geometry offers the best error propagators with the minimum-norm least-squares solutions. Noll's theoretical result, which was extensively used as a reference in the previous literature for error propagation estimates, corresponds to the Southwell geometry with an odd-number grid size. Typically the Fried geometry is not preferred in slope-based optical testing because it either allows subsize wavefront estimations within the testing domain or yields a two-rank deficient estimations matrix, which usually suffers from high error propagation and the waffle mode problem. The Southwell geometry, with an odd-number grid size if a zero point is assigned for the wavefront, is usually recommended in optical testing because it provides the lowest-error propagation for both slope-based and difference-based wavefront estimations.