Analysis of lateral shearing interferograms by use of Zernike polynomials

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.

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.

Wavefront restoration from lateral shearing data using spectral interpolation

Although a lateral-shear interferometer is robust against optical component vibrations, its interferogram provides information about differential wavefronts rather than the wavefronts themselves, resulting in the loss of specific frequency components. Previous studies have addressed this limitation by measuring four interferograms with different shear amounts to accurately restore the two-dimensional wavefront. This study proposes a technique that employs spectral interpolation to reduce the number of required interferograms. The proposed approach introduces an origin-shift technique for accurate spectral interpolation, which in turn is implemented by combining two methods: natural extension and least-squares determination of ambiguities in uniform biases. Numerical simulations confirmed that the proposed method accurately restored a two-dimensional wavefront from just two interferograms, thereby indicating its potential to address the limitations of the lateral-shear interferometer.

Wavefront measurements using the Ronchi test for high-NA lithography projection lenses

The Ronchi test is widely used for wavefront measurements in advanced lithography tools, and a physical optics explanation of the Ronchi test based on scalar diffraction theory can be found in numerous publications. However, for high-numerical aperture (high-NA) lithography projection lenses, the vector nature of light should be considered when performing wavefront measurements, especially the effect of polarization aberrations on the wavefront test results. In this paper, a vector model for describing shearing interferometry for high-NA lithography projection lenses is established. In addition to considering the vector nature of light, the vector model also calculates the Ronchigram on the screen of a detector at any distance from a diffraction grating, as opposed to the distance restriction for the Fraunhofer diffraction approximation used by the existing methods. Using the developed mathematical model of the Ronchi test, the Ronchigrams of high-NA lithography projection lenses under non-polarized illumination are simulated, and the effect of the distance between the diffraction grating and the detection screen on the wavefront measurement accuracy are discussed.

Rigorous simulation model of double-Ronchi shearing interferometry on lithographic tools

Double-Ronchi shearing interferometry is widely used in wavefront aberration measurements for advanced lithography projection lens systems. A rigorous simulation model of double-Ronchi shearing interferometry on lithographic tools is the precondition for phase-shifting retrieval algorithm design and error analysis. This paper presents a rigorous simulation model of double-Ronchi shearing interferometry considering the vector nature of light. The model is accurate and can be used in the study of double-Ronchi shearing interferometry.

Interferometric orbital angular momentum mode detection in turbulence with deep learning

Orbital angular momentum (OAM) modes are topical due to their versatility, and they have been used in several applications including free-space optical communication systems. The classification of OAM modes is a common requirement, and there are several methods available for this. One such method makes use of deep learning, specifically convolutional neural networks, which distinguishes between modes using their intensities. However, OAM mode intensities are very similar if they have the same radius or if they have opposite topological charges, and as such, intensity-only approaches cannot be used exclusively for individual modes. Since the phase of each OAM mode is unique, deep learning can be used in conjugation with interferometry to distinguish between different modes. In this paper, we demonstrate a very high classification accuracy of a range of OAM modes in turbulence using a shear interferometer, which crucially removes the requirement of a reference beam. For comparison, we show only marginally higher accuracy with a more conventional Mach-Zehnder interferometer, making the technique a promising candidate towards real-time, low-cost modal decomposition in turbulence.

Fast reconstruction technology of a laser beam spatial transmission characteristic curve

The traditional multiposition method of the M² factor measurement system is a good method, but it is relatively time consuming, so it cannot meet the requirements of the transient test of a Gaussian beam. To solve this problem, a quadriwave lateral shearing interferometer and a wavefront construction method based on a difference Zernike polynomial are analyzed. This interferometer uses a special grating to select four replicas of the wavefront, and the interferogram generated by four replicas includes difference wavefront information. Then the difference Zernike polynomial method is used to analyze quadriwave lateral shearing interferograms. The characteristic parameters are obtained after finding the optimal terms of the Zernike polynomials. As a result, the errors of F-number, beam radius, and radius of curvature are 3.7%, 3.8%, and 0.6%, respectively, which verifies this method to calculate parameters of Gaussian beam. In addition, we also find that the shear amount has influence on the reconstructed wavefront. It is showed that as the amount of shear increases from 20 pixels to 90 pixels, the peak-to-valley (P-V) values and RMS values both gradually decrease with a nonlinear relationship, which could be used to decrease the error of wavefront reconstruction further.

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.

Average gradient of Zernike polynomials over polygons

Wavefront estimation from slope sensor data is often achieved by fitting measured slopes with Zernike polynomial derivatives averaged over the sampling subapertures. Here we discuss how the calculation of these average derivatives can be reduced to one-dimensional integrals of the Zernike polynomials, rather than their derivatives, along the perimeter of each subaperture. We then use this result to derive closed-form expressions for the average Zernike polynomial derivatives over polygonal areas, only requiring evaluation of polynomials at the polygon vertices. Finally, these expressions are applied to simulated Shack-Hartmann wavefront sensors with 7 and 23 fully illuminated lenslets across a circular pupil, with their accuracy and calculation time compared against commonly used integration methods.

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.

Extended shift-rotation method for absolute interferometric testing of a spherical surface with pixel-level spatial resolution

An improved shift-rotation method for the absolute testing of spherical surfaces is developed to obtain pixel-level spatial resolution and a low noise propagation ratio. The absolute testing process includes multiple rotational tests and two lateral shifting tests with large shifts. A wavefront reconstruction algorithm based on subaperture division and least squares fitting is proposed to reconstruct the surface figure of the test optics. Numerical simulation results show that the method reveals high-frequency figures missed in the traditional Zernike-based shift-rotation method. The algorithm error is lower than 0.4%, and the noise propagation ratio can be reduced by 70% using large shifts. The absolute testing of spherical optics is carried out to verify this method. One spherical surface was tested with the presented absolute testing method and the method using a point diffraction interferometer. The difference of the measurement results based on the two methods showed that the testing uncertainty reached 0.19 nm root mean square (RMS), which indicated that the presented method has potential subnanometer testing uncertainty.

Single-shot 3 × 3 beam grating interferometry for self-imaging free extended range wave front sensing

Crossed grating 3×3 beam lateral shear interferometry for extended range wave front sensing is presented. A Fresnel diffraction pattern of two multiplicatively superimposed linear diffraction gratings each generating three diffraction orders is recorded. A simple solution employs a common crossed binary amplitude Ronchi grating with spatial filtering. Digital processing of a single-shot pattern includes separating multidirectional pairs of orthogonal lateral shear interferograms, retrieving second harmonics of their intensity distribution, and calculating shearing phases. Single-frame automatic fringe pattern processing based on the Hilbert-Huang transform is used for this purpose. Using second harmonics extends the aberration measurement range since they encode self-imaging free two-beam interferograms without contrast modulations. Experimental works corroborate the principle and capabilities of the proposed approach.

Zernike polynomials as a basis for modal fitting in lateral shearing interferometry: a discrete domain matrix transformation method

A Zernike-polynomials-based wavefront reconstruction method for lateral shearing interferometry is proposed. Shear matrices are calculated using matrix transformation instead of mathematical derivation. Simulation results show that the shear matrices calculated using the proposed method are the same as those obtained from mathematical derivation. The advantage of the proposed method is that high order shear matrices can be obtained easily; thus, wavefront reconstruction can be extended to higher order Zernike terms, and reconstruction accuracy can be improved.

Zernike olivary polynomials for applications with olivary pupils

Orthonormal polynomials have been extensively applied in optical image systems. One important optical pupil, which is widely processed in lateral shearing interferometers (LSI) and subaperture stitch tests (SST), is the overlap region of two circular wavefronts that are displaced from each other. We call it an olivary pupil. In this paper, the normalized process of an olivary pupil in a unit circle is first presented. Then, using a nonrecursive matrix method, Zernike olivary polynomials (ZOPs) are obtained. Previously, Zernike elliptical polynomials (ZEPs) have been considered as an approximation over an olivary pupil. We compare ZOPs with their ZEPs counterparts. Results show that they share the same components but are in different proportions. For some low-order aberrations such as defocus, coma, and spherical, the differences are considerable and may lead to deviations. Using a least-squares method to fit coefficient curves, we present a power-series expansion form for the first 15 ZOPs, which can be used conveniently with less than 0.1% error. The applications of ZOP are demonstrated in wavefront decomposition, LSI interferogram reconstruction, and SST overlap domain evaluation.

Diffraction grating three-beam interferometry without self-imaging regime contrast modulations

Achromatic grating shearing interferometry method for wave front sensing is developed. Two Fresnel diffraction patterns formed by grating three lowest diffraction orders are recorded. The beam-splitter grating is displaced laterally by half its period between exposures. Calculating the sum of two patterns results in a two-beam interferogram free of inherent light propagation direction and observation plane contrast modulations imposed by the self-imaging phenomenon (Talbot effect). Single-frame automatic fringe pattern processing provides the interferogram phase distribution. The technique enables continuous shear variations suitable for dynamic range sensing. Experimental works corroborate enhanced capabilities of the proposed approach.

Orthonormal polynomials for elliptical wavefronts with an arbitrary orientation

We generalize the analytical form of the orthonormal elliptical polynomials for any arbitrary aspect ratio to arbitrary orientation and give expression for them up to the 4th order. The utility of the polynomials is demonstrated by obtaining the expansion up to the 8th order in two examples of an off-axis wavefront exiting from an optical system with a vignetted pupil.

Orthonormal curvature polynomials over a unit circle: basis set derived from curvatures of Zernike polynomials

Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.

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.

Adaptive optics technology for high-resolution retinal imaging

Adaptive optics (AO) is a technology used to improve the performance of optical systems by reducing the effects of optical aberrations. The direct visualization of the photoreceptor cells, capillaries and nerve fiber bundles represents the major benefit of adding AO to retinal imaging. Adaptive optics is opening a new frontier for clinical research in ophthalmology, providing new information on the early pathological changes of the retinal microstructures in various retinal diseases. We have reviewed AO technology for retinal imaging, providing information on the core components of an AO retinal camera. The most commonly used wavefront sensing and correcting elements are discussed. Furthermore, we discuss current applications of AO imaging to a population of healthy adults and to the most frequent causes of blindness, including diabetic retinopathy, age-related macular degeneration and glaucoma. We conclude our work with a discussion on future clinical prospects for AO retinal imaging.

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.

Two-dimensional wavefront reconstruction from lateral multi-shear interferograms

We propose and demonstrate multiple shearing interferometry for measuring two-dimensional phase object. Multi-shear interference can effectively eliminate the problem of spectral leakage that results from the single-shear interference. The Fourier coefficients of a two-dimensional wavefront are computed from phase differences obtained from multiple shearing interferograms, which are acquired by a shearing interferometer, and the desired phase is then reconstructed. Numerical and optical tests have confirmed that the multiple shearing interferometry has a higher recovery accuracy than single-shear interferometry and the reconstruction precision increases as the number of shear steps increases.

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.

Method for designing phase-calculation algorithms for two-dimensional grating phase-shifting interferometry

We propose a design method of phase-analysis algorithms based on two-dimensional grating phase shifting for Talbot interferometry, Talbot-Lau imaging, or the Ronchi test. These algorithms are designed to separate the two orthogonal shearing wavefronts and eliminate error effects of unwanted diffraction orders, simultaneously. Taking the effect of multidiffraction into account, moving the two-dimensional grating along a certain pass leads to a series of phase-shifted interfrograms, from which two orthogonal shearing wavefronts are derived, for the tested wavefront to be retrieved. The designing process is demonstrated, and the residual errors are analyzed via simulation works and experimental comparison.

Exact wavefront recovery with tilt from lateral shear interferograms

A novel method is presented for one-dimensional wavefront recovery on the basis of difference measurements from two shearing interferograms with varying tilt. The method uses large shears and obtains high lateral resolution. Furthermore, the wavefront under test can be recovered exactly up to an arbitrary constant and straight line at all evaluation points with suitably chosen shears of two shearing interferograms.

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.

Spatially resolved wavefront aberrations of ophthalmic progressive-power lenses in normal viewing conditions

PURPOSE

To measure the wavefront aberration at different locations in progressive-power lenses (PPL's) isolated and in situ (PPL's plus eye).

METHODS

A Hartmann-Shack wavefront sensor was used to measure progressive-power lenses and human eyes either independently or in combination. In each selected zone, the lens was placed and tilted accordingly to simulate natural viewing conditions. We measured 21 relevant locations across an isolated PPL (plano lens of power addition of 2 D). In six of the locations, the wavefront aberration of the eye plus PPL were obtained in two ways: (1) by direct measurement of the system and (2) by adding the individual wavefront aberrations of the eye and the lens for each appropriate zone. In every case, we obtained the wavefront aberration as Zernike polynomials expansions, the root mean square error, the point-spread function, and the Strehl ratio.

RESULTS

Along the corridor of the PPL, third-order coma and trefoil, and astigmatism were the dominant aberrations. In areas of the PPL outside the corridor, astigmatism increased, whereas other aberrations remained similar to the lens center. Small differences were found between the direct and calculated methods used to obtain the wavefront aberration of the eye with the lens, and the possible sources of errors were discussed. In some lenses zones, the aberrations of the lens may be compensated by the particular aberrations of the eye, yielding improved optical performance over that present in the lens alone.

CONCLUSIONS

We designed and built a wavefront sensor to perform spatially resolved aberration measurements in ophthalmic lenses, in particular in PPL's, either isolated or in combination with the eye. The aberrations appearing in the PPL were compared with those in normal aged eyes.

Improved Saunders method for the analysis of lateral shearing interferograms

An interferogram obtained by use of ordinary interferometers, such as Fizeau and Twyman-Green interferometers, will show a contour map of the wave front under test. A lateral-shearing interferogram, however, will show a contour map of the difference between the wave front under test and a sheared wave front, that is, a contour map of the derivative of the wave front under test. Therefore one can reconstruct the shape of the wave front under test by analyzing that difference. Many methods for reconstructing a wave front have been proposed. The Saunders method reconstructs a wave front; rapidly however the wave-front data are reconstructed only at intervals of the amount of shear along the direction of the shear. Therefore the method has low spatial resolution. A method for reconstructing a wave front that is based on the Saunders method and has high spatial resolution is proposed. The method analyzes the differences that are produced by shearing of the wave front under test in many directions. This method requires a large number of interferograms for reconstructing the wave front. Here the method is described, and its validity is confirmed by simulation.

Experimental study of a shearing interferometer concept for at-wavelength characterization of extreme-ultraviolet optics

We describe the experimental evaluation of a shearing interferometer concept for at-wavelength testing of extreme-ultraviolet optics. The concept is based on the Ronchi test, which has been modified by a new design for entrance and exit gratings to suppress disturbing higher-order interference patterns. The interferometer concept has been tested on an experimental setup, of which all relevant parameters have been scaled from extreme-ultraviolet to visible-light wavelengths. A Twyman-Green interferometer has been integrated into the setup for comparison with the improved Ronchi test. A systematic difference of 7-12 mlambda rms has been found between wave fronts measured with the improved Ronchi test and with the Twyman-Green interferometer. Possible error sources have been analyzed. The accuracy of the interferometer is estimated to be 10 mlambda rms.

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.

High-precision analysis of a lateral shearing interferogram by use of the integration method and polynomials

Interferograms obtained with ordinary interferometers, such as the Fizeau interferometer or the Twyman-Green interferometer, show the contour maps of a wave front under test. On the other hand, lateral shearing interferograms show the difference between a wave front under test and a sheared wave front, that is, the inclination of the wave front. Therefore the shape of the wave front under test is reconstructed by means of analyzing the difference. To reconstruct the wave front, many methods have been proposed. An integration method is usually used to reconstruct the wave front under test rapidly. However, this method has two disadvantages: The analysis accuracy of the method is low, and part of the wave front cannot be measured. To overcome these two problems, a new, to our knowledge, integration method, improved by use of polynomials, is proposed. The validity of the proposed method is evaluated by computer simulations. In the simulations the analysis accuracy achieved by the proposed method is compared with the analysis accuracy of the ordinary integration method and that of the method proposed by Rimmer and Wyant. The results of the simulations show that the analysis accuracy of the newly proposed method is better than that of the integration method and that of the Rimmer-Wyant method.

Position and displacement sensing with shack-hartmann wave-front sensors

The use of a Shack-Hartmann wave-front sensor as a position-sensing device is proposed and demonstrated. The coordinates of a pointlike object are determined from the modal Zernike coefficients of the wave fronts emitted by the object and detected by the sensor. The position of the luminous centroid of a moderately extended incoherent flat object can also be measured with this device. Experimental results with off-the-shelf CCD cameras and conventional relay optics as well as inexpensive diffractive microlens arrays show that axial positioning accuracies of 74 microm rms at 300 mm and angular accuracies of 4.3 microrad rms can easily be achieved.

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.

Ronchi test with a square grid

We use a square grid in the Ronchi test. This grid allows processing of both the X and the Y directions when calculating optical path difference. We use trapezoidal integration to analyze the new patterns, since it does not have the smoothing drawback at the edges of the wave front.

Wave-front analysis with high accuracy by use of a double-grating lateral shearing interferometer

A phase-stepped double-grating lateral shearing interferometer to be used for wave-front analysis is presented. The resulting interference patterns are analyzed with a differential Zernike polynomial matrix-inversion method. Possible error sources are analyzed in the design stage, and it is shown that the inaccuracy can be kept within 2-5 mλ rms. The apparatus was tested and evaluated in practice. Comparison with a phase-stepped Twyman-Green interferometer demonstrates that the accuracy of the two methods is comparable. Lateral shearing interferometry scores better on reproducibility, owing to the stability and robustness of the method.