Method for evaluating lateral shearing interferograms

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.

Self-Calibration of a Large-Scale Variable-Line-Spacing Grating for an Absolute Optical Encoder by Differencing Spatially Shifted Phase Maps from a Fizeau Interferometer

A new method based on the interferometric pseudo-lateral-shearing method is proposed to evaluate the pitch variation of a large-scale planar variable-line-spacing (VLS) grating. In the method, wavefronts of the first-order diffracted beams from a planar VLS grating are measured by a commercial Fizeau form interferometer. By utilizing the differential wavefront of the first-order diffracted beam before and after the small lateral shift of the VLS grating, the pitch variation of the VLS grating can be evaluated. Meanwhile, additional positioning errors of the grating in the lateral shifting process could degrade the measurement accuracy of the pitch variation. To address the issue, the technique referred to as the reference plane technique is also introduced, where the least squares planes in the wavefronts of the first-order diffracted beams are employed to reduce the influences of the additional positioning errors of the VLS grating. The proposed method can also reduce the influence of the out-of-flatness of the reference flat in the Fizeau interferometer by taking the difference between the measured positive and negative diffracted wavefronts; namely, self-calibration can be accomplished. After the theoretical analysis and simulations, experiments are carried out with a large-scale VLS grating to verify the feasibility of the proposed methods. Furthermore, the evaluated VLS parameters are verified by comparing them with the readout signal of an absolute surface encoder employing the evaluated VLS grating as the scale for measurement.

Self-Calibration of a Large-Scale Variable-Line-Spacing Grating for an Absolute Optical Encoder by Differencing Spatially Shifted Phase Maps from a Fizeau Interferometer

A new method based on the interferometric pseudo-lateral-shearing method is proposed to evaluate the pitch variation of a large-scale planar variable-line-spacing (VLS) grating. In the method, wavefronts of the first-order diffracted beams from a planar VLS grating are measured by a commercial Fizeau form interferometer. By utilizing the differential wavefront of the first-order diffracted beam before and after the small lateral shift of the VLS grating, the pitch variation of the VLS grating can be evaluated. Meanwhile, additional positioning errors of the grating in the lateral shifting process could degrade the measurement accuracy of the pitch variation. To address the issue, the technique referred to as the reference plane technique is also introduced, where the least squares planes in the wavefronts of the first-order diffracted beams are employed to reduce the influences of the additional positioning errors of the VLS grating. The proposed method can also reduce the influence of the out-of-flatness of the reference flat in the Fizeau interferometer by taking the difference between the measured positive and negative diffracted wavefronts; namely, self-calibration can be accomplished. After the theoretical analysis and simulations, experiments are carried out with a large-scale VLS grating to verify the feasibility of the proposed methods. Furthermore, the evaluated VLS parameters are verified by comparing them with the readout signal of an absolute surface encoder employing the evaluated VLS grating as the scale for measurement.

Compensated differential Zernike fitting method for wavefront aberration metrology based on grating lateral shearing

Lateral shearing based on the grating is one of the classical configurations when measuring the wavefront aberration of optical systems such as the lithographic projection lens. Because the wavefront under test is spherical, but a detector surface is a plane, the coordinate of the wavefront surface will be distorted on the detector surface. As the numerical aperture (NA) of the optics under test increases, the shear ratios at different positions within the shearing region are significantly different due to the coordinate distortion. Therefore, the reconstructed wavefront from the traditional lateral-shearing reconstruction method designed for a fixed shearing ratio will contain a non-negligible error. In this work, we use the ray-tracing method to calculate the shearing ratio distribution in the shearing region and propose a compensated differential Zernike fitting method to solve the coordinate distortion and shearing ratio variation problem. The relative error of the uncompensated result will increase as the NA increases. This error is around 1% for a 0.1 NA, 10% for a 0.3 NA, and over 100% for an NA above 0.7. Compensation for the shearing ratio variation is necessary when the NA is larger than 0.3. The proposed method has been validated by simulations and experiments.

Comparison of processing speed of typical wavefront reconstruction methods for lateral shearing interferometry

Lateral shearing interferometry is widely applied in wavefront sensing, optical components testing, and defect inspection. The procedure of reconstructing the wavefront is the most specific difference between lateral shearing interferometry and other classical methods such as the Fizeau and Twyman interferometers. The speed and accuracy are two main features to evaluate the performance of one wavefront reconstruction method. In this work, optimized procedures for three typical wavefront reconstruction methods-the iterative FFT wavefront reconstruction method (FFT method), the partial differential least-squares method (LSQ method), and the difference Zernike polynomial fitting method (DZF method)-are designed. The calculation speeds of the three wavefront reconstruction methods are evaluated with different GPUs and CPUs. According to the test results, the DZF method is the fastest method both in the GPUs and CPUs. The shortest processing times of the DZF, FFT, and LSQ methods are 100, 449, and 494 ms, respectively, with the wavefront size of 1024×1024pixels. The calculation speeds of the FFT method and the LSQ method are similar in the CPUs, and the FFT method is faster in the GPUs. The relationship between the consumed time and the wavefront size is an exponential function in the CPUs and a power function in the GPUs. Generally speaking, GPUs' processing speeds are faster than CPUs'. But CPUs can be faster than GPUs when the test wavefront sizes are smaller than 64×64pixels. Besides, the differences between the consumed times of different CPUs are relatively smaller than those of the GPUs.

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.

Method for designing phase-retrieval algorithms for Ronchi phase-shifting lateral-shearing interferometry

We propose a general method of designing phase-analysis algorithms for Ronchi phase-shifting lateral-shearing interferometry. Based on the expression and method, three new phase-shifting algorithms are designed to eliminate negative error effects of interference from unwanted diffraction orders, which limits the accuracy of wavefront aberration measurement. The proposed eight-, 10-, and 13-frame algorithms can eliminate the effects of the first ±5, ±9, and ±15 multi-diffraction orders, respectively. Simulation works and experimental results verify the general expression and corresponding designed method.

Improvement in error propagation in the Shack-Hartmann-type zonal wavefront sensors

Estimation of the wavefront from measured slope values is an essential step in a Shack-Hartmann-type wavefront sensor. Using an appropriate estimation algorithm, these measured slopes are converted into wavefront phase values. Hence, accuracy in wavefront estimation lies in proper interpretation of these measured slope values using the chosen estimation algorithm. There are two important sources of errors associated with the wavefront estimation process, namely, the slope measurement error and the algorithm discretization error. The former type is due to the noise in the slope measurements or to the detector centroiding error, and the latter is a consequence of solving equations of a basic estimation algorithm adopted onto a discrete geometry. These errors deserve particular attention, because they decide the preference of a specific estimation algorithm for wavefront estimation. In this paper, we investigate these two important sources of errors associated with the wavefront estimation algorithms of Shack-Hartmann-type wavefront sensors. We consider the widely used Southwell algorithm and the recently proposed Pathak-Boruah algorithm [J. Opt.16, 055403 (2014)JOOPDB0150-536X10.1088/2040-8978/16/5/055403] and perform a comparative study between the two. We find that the latter algorithm is inherently superior to the Southwell algorithm in terms of the error propagation performance. We also conduct experiments that further establish the correctness of the comparative study between the said two estimation algorithms.

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.

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.

Quantitative phase imaging in flows with high resolution holographic diffraction grating

This paper proposes quantitative phase imaging by using a high resolution holographic grating for generating a four-wave shearing interferogram. The high-resolution holographic grating is designed in a "kite" configuration so as to avoid parasitic mixing of diffraction orders. The selection of six diffraction orders in the Fourier spectrum of the interferogram allows reconstructing phase gradients along specific directions. The spectral analysis yields the useful parameters of the reconstruction process. The derivative axes are exactly determined whatever the experimental configurations of the holographic grating. The integration of the derivative yields the phase and the optical thickness. Demonstration of the proposed approach is carried out for the case of the analysis of the supersonic flow of a small vertical jet, 5.56mm in diameter. The experimental results compared with those obtained with digital holography exhibit a very good agreement.

3D surface mapping of freeform optics using wavelength scanning lateral shearing interferometry

Freeform optics have emerged as promising components in diverse applications due to the potential for superior optical performance. There are many research fields in the area ranging from fabrication to measurement, with metrology being one of the most challenging tasks. In this paper, we describe a new variant of lateral shearing interferometer with a tunable laser source that enables 3D surface profile measurements of freeform optics with high speed, high vertical resolution, large departure, and large field-of-view. We have verified the proposed technique by comparing our measurement result with that of an existing technique and measuring a representative freeform optic.

Mutual interferometric characterization of a pair of independent electric fields

We demonstrate a novel interferometric characterization scheme that allows the complete reconstruction of two interfering electric fields. The phase profiles of both beams, and their relative phase, can be retrieved simultaneously as a function of any degree of freedom in which it is possible to shear one of the beams. The method has applications in wavefront sensing or ultrashort-pulse measurement, especially also in the domain of extreme light sources where it is difficult to generate a reference field or to replicate the beam in order to perform a self-referencing measurement. We demonstrate the technique experimentally by measuring simultaneously two ultrashort pulses in a single laser shot.

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.

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.

Quantitative phase imaging with a scanning transmission x-ray microscope

We obtain quantitative phase reconstructions from differential phase contrast images obtained with a scanning transmission x-ray microscope and 2.5 keV x rays. The theoretical basis of the technique is presented along with measurements and their interpretation.

In situ optical testing of infrared lenses for high-performance cameras

We propose to evaluate infrared lenses with a dedicated analyzer having the same mechanical interface as the usual cameras. The proposed analysis is based on a wavefront measurement and allows a diagnostic of possible internal defects of the analyzed lens. The infrared lens analyzer described is constituted with a quadriwave lateral shearing interferometer and works with a blackbody light. We describe the response of this interferometer and an innovative method to obtain the wavefront under test. We finally present the experimental analysis of long-wavelength infrared lenses and the particular case of a modified lens that generates a large spherical aberration.

Multigrid algorithm for least-squares wavefront reconstruction

Multigrid (MG) methods are presented for fast, efficient, flexible, and robust least-squares wavefront reconstruction in extremely high-resolution conventional adaptive optics, or ExAO. We demonstrate that MG can robustly handle a variety of sensor-actuator geometries, and it can accommodate deformable mirror influence function models that are more realistic than the common piecewise bilinear model. With MG one can also easily incorporate additional penalty, or regularization, terms to damp out the waffle mode in Fried geometry and to damp out instabilities due to actuators near the pupil boundary with poorly sensed influence. We present closed-loop simulation results that suggest that only one or two MG iterations per time step are needed to control an ExAO system.

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

An iterative zonal wave-front estimation algorithm for slope or gradient-type data in optical testing acquired with regular or irregular pupil shapes is presented. In the mathematical model proposed, the optical surface, or wave-front shape estimation, which may have any pupil shape or size, shares a predefined wave-front estimation matrix that we establish. Owing to the finite pupil of the instrument, the challenge of wave front shape estimation in optical testing lies in large part in how to properly handle boundary conditions. The solution we propose is an efficient iterative process based on Gerchberg-type iterations. The proposed method is validated with data collected from a 15 x 15-grid Shack-Hartmann sensor built at the Nanjing Astronomical Instruments Research Center in China. Results show that the rms deviation error of the estimated wave front from the original wave front is less than lambda/130-lambda/150 after approximately 12 iterations and less than lambda/100 (both for lambda = 632.8 nm) after as few as four iterations. Also, a theoretical analysis of algorithm complexity and error propagation is presented.

Wave-front reconstruction from multidirectional phase derivatives generated by multilateral shearing interferometers

To increase the accuracy of wave-front evaluation, we propose to exploit the natural capability of multiple lateral shearing interferometers to measure simultaneously more than two orthogonal phase derivatives. We also describe a method, based on Fourier-transform analysis, that uses this multiple information to reconstruct the wave-front under study.

Stroke amplifier for deformable mirrors

We demonstrate a simple optical configuration that amplifies the usable stroke of a deformable mirror. By arranging for the wavefront to traverse the deformable mirror more than once, we correct it more than once. The experimental implementation of the idea demonstrates a doubling of 2.0 and 2.04 by two different means.

Phase-shifting shearing interferometer

A single-element phase-shifting interferometer has been developed based on the lateral shearing interferometer. This new interferometer requires no precise alignment, and the phase is continuously varied by changes in the voltage across a commercially available liquid-crystal phase retarder.

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.

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.

Modified Fourier transform method for interferogram fringe pattern analysis

A modified Fourier transform method for interferogram fringe pattern analysis is proposed. While it retains most of the advantages of the Fourier transform method, the new method overcomes some drawbacks of the previous method. It eliminates the assumptions of slowly varying phase variation in the test section and the constant spatial carrier frequency. It also extends the frequency bandwidth and avoids phase distortion caused by discreteness of the sampling frequency. Both numerical simulation and experimental examination are performed to evaluate the performance of the method.

Analysis of lateral shearing interferograms by use of Zernike polynomials

A modal phase-reconstruction method for wave-front analysis in lateral shearing interferometry is presented. Pseudo-Zernike polynomial functions describe the differential wave fronts and are related to a Zernike polynomial description of the original wave front. We show that this reconstruction is robust for shear ratios in the range 0.15-0.50. The error propagation properties of this differential Zernike polynomial matrix-inversion method are discussed on the basis of both analysis and simulation. It is concluded that the method allows wave-front analysis with an absolute inaccuracy of 2 mλ rms for diffraction-limited wave fronts and with 1% relative inaccuracy for more strongly aberrated wave fronts.

Simple algorithm for large-grid phase reconstruction of lateral-shearing interferometry

A simple zonal approach is proposed for estimating phase distribution on large grids. The estimation is based on phase differences that are precisely measured in two orthogonal directions by a lateral-shearing interferometer. It requires only O(N(2)) operations for reconstructing a phase distribution on an N × N grid. Computer simulation and experimental results are demonstrated to show the effectiveness of the new algorithm.

Polynomial fitting of interferograms with Gaussian errors on fringe coordinates. 1: Computer simulations

A series of simulations were made for an ideal Twyman-Green interferogram of equally spaced straight fringes having tilt only about x. It was found that fitting polynomials to the interferometric data resulted in biased estimates of some of the fitting coefficients to the optical path difference. The acceptance of the Seidel aberrations grows with the noise level and diminishes when the number of fringes is increased.

Scanning differential interferometer to measure index heterogeneity

The index heterogeneity of rectangular glass samples is measured to a repeatability of 2 x 10(-8) by a scanning differential interferometer. The noise-limited instrument resolution is 2 nm of optical path length. The surface figure is decoupled from bulk inhomogeneity by a thin film of index-matching liquid, which is located by surface tension between the interferometer cavity and the test sample. An algorithm based on Poisson's equation reconstructs the integrated optical path length profile from data in differential form with a minimal integration of noise.

Analytical representation of the phase and its mode components reconstructed according to the wave-front slopes

A precise analytical solution of the problem of phase reconstruction and its mode components is obtained based on the measurements of wave-front slopes in shear interferometers and Hartmann sensors. The algorithm has been developed, and a numerical experiment on reconstruction of the phase and its expansion coefficients as a sum of the Zernike polynomials has been carried out.

Design and evaluation of laser sources with high-quality wave fronts

The design and evaluation of laser sources that have high-quality wave fronts with minimal residual aberrations and irregularities are described. Two such sources have been developed-a collimated reference source and a point reference source. The collimated reference source provides a collimated wave front with a wave-front irregularity of <0.01 wave rms over a 6-mm clear aperture. Collimation is better than 0.067 wave peak to valley, with an angular ray departure of <1.75 x 10(-5) rad. The point reference source provides a diverging wave front with a wave-front irregularity of <0.01 wave rms over a 0.60 numerical aperture and 0.005 wave rms over a 0.35 numerical aperture. Source wavelengths are 670, 780, and 820 nm. These laser sources may be characterized in terms of their fourth-order wave-front polynomial coefficients by using the testing methods developed in this paper. Once characterized, the sources are used as a known input for testing the optical response of unknown systems. This enables complete characterization of an unknown system in terms of aberration polynomials or other criteria of interest when used in conjunction with a wave-front sensing interferometer. These sources are used to evaluate an equal-path, phase-modulated Mach-Zehnder interferometer. Applications include general-purpose laboratory sources as well as tools for evaluating the accuracy of optical systems.

Phase measuring Ronchi test

Use of synchronous phase detection in the conventional Ronchi test is discussed to measure a large amount of wavefront aberration or an aspheric wavefront. This method provides a simple but powerful tool for aspheric surface evaluation. By moving the Ronchi grating sideways, a periodic phase shift in the Ronchigram is introduced for synchronous phase detection. A theoretical interpretation of the Ronchigram and the procedure for an analysis are presented, and experimental results are shown.

Heterodyne Fizeau interferometer for testing flat surfaces

A heterodyne Fizeau interferometer, which uses a rotating radial grating to achieve the required optical frequency shift, is described. An analysis of the effects of grating ruling errors shows that they may be nearly eliminated by averaging the interferometer phase readings over integral numbers of grating revolutions. Experimental tests indicate that the interferometer is capable of measuring with a reproducibility of lambda/200 (lambda = 632.8 nm), limited by temperature effects.

Laser beam divergence utilizing a lateral shearing interferometer

An analysis of the lateral shearing interferometer is given for laser beams. The results enable a simple determination of the local radius of curvature of the wavefront. In addition, the presence of phase distortion in the beam may be ascertained.

Use of an ac heterodyne lateral shear interferometer with real-time wavefront correction systems

An analysis is performed to determine the accuracy with which an ac heterodyne lateral shear interferometer can measure wavefront aberrations if a white light extended source is used with the interferometer, and shot noise is the predominate noise source. The analysis shows that for uniform circular or square sources larger than a derived minimum size, the wavefront measurement accuracy depends only upon the radiance of the source and not upon the angular subtense of the source. For a 1-msec integration time, a 25-cm(2) collecting area, and a source radiance of 10 W/m(2)-sr the rms wavefront error is approximately 1/30 wave, assuming the signal is shot noise limited. It is shown that for both uniform circular and square sources an optimum shear distance is approximately (1/2) the aperture diameter required to resolve the light source. Comments are made on the optimum shear for nonuniform radiance distributions.

Evaluation of large aberrations using a lateral-shear interferometer having variable shear

A variable shear lateral shearing interferometer consisting of two holographically produced crossed diffraction gratings is used to test nonrotationally symmetric wavefronts having aberrations greater than 100 wavelengths and slope variations of more than 400 wavelengths/diameter. Comparisons are made with results of Twyman-Green interferometric tests for wavefront aberrations of up to thirty wavelengths. The results indicate that small wavefront aberrations can be measured as accurately with the lateral-shear interferometer as with the Twyman-Green interferometer and that aberrations that cannot be measured at all with a Twyman-Green interferometer can be measured to about 1% accuracy or better.