Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry

Lateral shearing interferometry method based on double-checkerboard grating by suppressing aliasing effect

Ronchi lateral shearing interferometry is a promising wavefront sensing technology with the advantages of simple structure and no reference light, which can realize a high-precision wavefront aberration measurement. To obtain shear information in both directions, the conventional double-Ronchi interferometer sequentially applies two orthogonal one-dimensional Ronchi gratings as the object-plane splitting element of the optics under test. Simultaneously, another Ronchi grating is positioned on the image plane in the same orientation to capture two sets of interferograms, thereby enabling two-dimensional wavefront reconstruction. Mechanical errors will inevitably be introduced during grating conversion, affecting reconstruction accuracy. Based on this, we propose a lateral shearing interferometry applying double-checkerboard grating. Only unidirectional phase shift is needed to obtain shear information in two directions while evading the grating conversion step, aiming to streamline operational processes and mitigate the potential for avoidable errors. We employ scalar diffraction theory to analyze the full optical path propagation process of the double-checkerboard shearing interferometry and introduce a new reconstruction algorithm to effectively extract the two-dimensional shear phase by changing the grating morphology, suppressing the aliasing effect of irrelevant diffraction orders. We reduce the fitting error through iterative optimization to realize high-precision wavefront reconstruction. Compared with conventional Ronchi lateral shearing interferometry, the proposed method exhibits better robustness and stability in noisy environments.

Antiderivative of gradient data by spline model integration

Numerous optical techniques describe the local slope of the functions at their discrete positions but do not report the actual functions. However, many applications require the description of the functions, which must be retrieved from the gradients by an integration process. This study shows a spline model function-based integration technique that can construct original functions from irregularly measured gradient data over general shape domains with high accuracy and speed.

Transform-based phase retrieval techniques from a single off-axis interferogram

Optical phase retrieval (OPR) methods are important because they are used to obtain the transverse phase profile information of a beam. Interference methods are extensively used to convert the phase information into an intensity pattern, which can then be processed further to retrieve the unknown phase. The most widely used interference method involves the interference of the unknown object beam and a known reference beam with an angle between them. There are several algorithms that retrieve the phase information from such a single off-axis interference pattern. For a particular application, the choice of an algorithm for OPR is very important. Therefore, it is necessary to choose between them, depending on the requirements. Three entirely different noniterative, transform-based algorithms, namely the Fourier transform (FT) method, the continuous wavelet transform (CWT) method, and the Hilbert transform (CWT) method, are explained in detail. A quantitative comparison is made using a combination of rms error and standard structural similarity measure. The advantages of using a standard unwrapping algorithm are also validated using the same combination of comparison metrics. We show that the HT method has a better response with object beam with higher spatial frequency content, but with the penalty of affected noise. The FT method and CWT method have better noise immunity, but have the limitation of the spatial frequency range of the object beam. The different constraints, advantages, and some practical limitations of the methods are discussed with the help of a quantitative phase imaging experiment of monodispersed polymethyl methacrylate beads.

Adjustable flexure mount to compensate for deformation of an optic surface

An adjustable mounting structure is proposed to compensate for surface deformation of a mirror caused by the assembly process. The mount adopts a six-point support based on the kinematic mount principle. Three of the support points are adjustable, and they are moved along the axial direction by actuators. Surface deformation is expressed by Zernike coefficients in this paper, and a sensitivity matrix of the surface deformation is established by varying the unit displacement of each adjustment support point and getting the corresponding Zernike coefficient changes. The surface deformation is measured, and the compensation adjustment of each adjustable support point is then obtained by anti-sensitivity calculation. Finally, the feasibility of present support structure design and surface figure compensating method are verified by experiments. The experimental results show that the present structure and method could significantly reduce the surface deformation caused by the assembly process. The surface deformation is 4.6 nm RMS after assembly and it is decreased to 1.3 nm RMS after four iterations of compensation, which is close to the 1.1 nm RMS after optical polishing.

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.

Correction of rotational inaccuracy in lateral shearing interferometry for freeform measurement

A lateral shearing interferometer has an advantage over previous wavefront measuring interferometers since it requires no reference. Therefore the lateral shearing interferometer can be a powerful solution to measure a freeform surface. It, however, has some issues to be resolved before it can be implemented. One of them is the orthogonality problem between two shearing directions in LSI. Previous wavefront reconstruction algorithms assume that the shearing directions are perfectly orthogonal to each other and lateral shear is obtained simultaneously in the sagittal and tangential directions. For practical LSI, however, there is no way to guarantee perfect orthogonality between two shearing directions. Motivated by this, we propose a new algorithm that is able to compensate the rotational inaccuracy. The mathematical model is derived in this paper. Computer simulations and experiments are also displayed to verify our algorithm.

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.

Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms

Four modal methods of reconstructing a wavefront from its difference fronts based on Zernike polynomials in lateral shearing interferometry are currently available, namely the Rimmer-Wyant method, elliptical orthogonal transformation, numerical orthogonal transformation, and difference Zernike polynomial fitting. The present study compared these four methods by theoretical analysis and numerical experiments. The results show that the difference Zernike polynomial fitting method is superior to the three other methods due to its high accuracy, easy implementation, easy extension to any high order, and applicability to the reconstruction of a wavefront on an aperture of arbitrary shape. Thus, this method is recommended for use in lateral shearing interferometry for wavefront reconstruction.

Use of numerical orthogonal transformation for the Zernike analysis of lateral shearing interferograms

A numerical orthogonal transformation method for reconstructing a wavefront by use of Zernike polynomials in lateral shearing interferometry is proposed. The difference fronts data in two perpendicular directions are fitted to numerical orthonormal polynomials instead of Zernike polynomials, and then the orthonormal coefficients are used to evaluate the Zernike coefficients of the original wavefront by use of a numerical shear matrix. Due to the fact that the dimensions of the shear matrix are finite, the high-order terms of the original wavefront above a certain order have to be neglected. One of advantages of the proposed method is that the impact of the neglected high-order terms on the outcomes of the lower-order terms can be decreased, which leads to a more accurate reconstruction result. Another advantage is that the proposed method can be applied to reconstruct a wavefront on an aperture of arbitrary shape from its difference fronts. Theoretical analysis and numerical simulations shows that the proposed method is correct and its reconstruction error is obviously smaller than that of Rimmer-Wyant method.

Measuring the complex amplitude of wave fields by means of shear interferometry

This paper presents a treatise on the determination of the complex amplitude of a monochromatic wave field from measurements obtained by a lateral shear interferometer. Both amplitude and phase distributions are recovered from the same set of measurements. Special consideration is given to the case of measurements with large shear. Here, the state of the art in the reconstruction of discontinuous wavefronts is extended by introducing a two-step process. In the first step, the phasors of the underlying wavefront are reconstructed across specific subsets of the measurement grid. In the second step, the individual reconstructions are combined by a novel (to the best of the author's knowledge) convolution approach in the Fourier domain, called residual phasor separation.

Direct and inverse discrete Zernike transform

An invertible discrete Zernike transform, DZT is proposed and implemented. Three types of non-redundant samplings, random, hybrid (perturbed deterministic) and deterministic (spiral) are shown to provide completeness of the resulting sampled Zernike polynomial expansion. When completeness is guaranteed, then we can obtain an orthonormal basis, and hence the inversion only requires transposition of the matrix formed by the basis vectors (modes). The discrete Zernike modes are given for different sampling patterns and number of samples. The DZT has been implemented showing better performance, numerical stability and robustness than the standard Zernike expansion in numerical simulations. Non-redundant (critical) sampling along with an invertible transformation can be useful in a wide variety of applications.

Orthonormal polynomials in wavefront analysis: error analysis

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.

Efficient reconstruction of spatially limited phase distributions from their sheared representation

We present a method that allows the reconstruction of smooth phase distributions from their laterally sheared representation. The proposed approach is efficient in the sense that only one sheared distribution is needed to completely restore the signal. A mandatory requirement is that the phase distribution is spatially limited. The method is exemplified by means of a synthetic signal, and in addition a practical algorithm is given. Finally, experimental results are presented. The deformation of a metallic surface is investigated by both speckle shearography and electronic speckle pattern interferometry (ESPI) respectively. To give proof of the proposed technique, the phase distribution reconstructed from the shearographic measurement is shown to match the results obtained by the ESPI.

Exact two-dimensional wave-front reconstruction from lateral shearing interferograms with large shears

A method is proposed for exact discrete reconstruction of a two-dimensional wave front from four suitably designed lateral shearing experiments. The method reconstructs any wave front at evaluation points of a circular aperture exactly up to an arbitrary constant for noiseless data, and it shows excellent stability properties in the case of noisy data. Application of large shears is allowed, and high resolution of the reconstructed wave front can be achieved. Results of numerical experiments are presented that demonstrate the capability of the method.

Solution to the shearing problem

Lateral shearing interferometry is a promising reference-free measurement technique for optical wave-front reconstruction. The wave front under study is coherently superposed by a laterally sheared copy of itself, and from the interferogram difference measurements of the wave front are obtained. From these difference measurements the wave front is then reconstructed. Recently, several new and efficient algorithms for evaluating lateral shearing interferograms have been suggested. So far, however, all evaluation methods are somewhat restricted, e.g., assume a priori knowledge of the wave front under study, or assume small shears, and so on. Here a new, to our knowledge, approach for the evaluation of lateral shearing interferograms is presented, which is based on an extension of the difference measurements. This so-called natural extension allows for reconstruction of that part of the underlying wave front whose information is contained in the given difference measurements. The method is not restricted to small shears and allows for high lateral resolution to be achieved. Since the method uses discrete Fourier analysis, the reconstructions can be efficiently calculated. Furthermore, it is shown that, by application of the method to the analysis of two shearing interferograms with suitably chosen shears, exact reconstruction of the underlying wave front at all evaluation points is obtained up to an arbitrary constant. The influence of noise on the results obtained by this reconstruction procedure is investigated in detail, and its stability is shown. Finally, applications to simulated measurements are presented. The results demonstrate high-quality reconstructions for single shearing interferograms and exact reconstructions for two shearing interferograms.

Phase-step calibration for phase-stepped interferometry

A novel method to set the proper phase steps, as used in phase-stepped interferometry, is presented. It is indicated how and when this method can be used. With only two images one can deduce the relative phase step between them by calculating the correlation between the two images. The error of the proposed method is shown to be smaller than 0.1%.