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

Self-calibration strategies for reducing systematic slope measurement errors of autocollimators in deflectometric profilometry

Deflectometric profilometers are used to precisely measure the form of beam shaping optics of synchrotrons and X-ray free-electron lasers. They often utilize autocollimators which measure slope by evaluating the displacement of a reticle image on a detector. Based on our privileged access to the raw image data of an autocollimator, novel strategies to reduce the systematic measurement errors by using a set of overlapping images of the reticle obtained at different positions on the detector are discussed. It is demonstrated that imaging properties such as, for example, geometrical distortions and vignetting, can be extracted from this redundant set of images without recourse to external calibration facilities. This approach is based on the fact that the properties of the reticle itself do not change - all changes in the reticle image are due to the imaging process. Firstly, by combining interpolation and correlation, it is possible to determine the shift of a reticle image relative to a reference image with minimal error propagation. Secondly, the intensity of the reticle image is analysed as a function of its position on the CCD and a vignetting correction is calculated. Thirdly, the size of the reticle image is analysed as a function of its position and an imaging distortion correction is derived. It is demonstrated that, for different measurement ranges and aperture diameters of the autocollimator, reductions in the systematic errors of up to a factor of four to five can be achieved without recourse to external measurements.

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.

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.

Effects and elimination of image grating defocusing on a double-Ronchi shearing interference field

Double-Ronchi shearing interferometry is a promising wavefront aberration measurement system for advanced lithography projection lens systems. The image grating defocusing is a key systematic error of the interferometer. However, the effects and elimination of this error have not been systematically researched. In this work, the interference field effects caused by the image grating defocusing are analyzed based on the theories of scalar diffraction, and a method to eliminate the effects is proposed. The theoretical analysis has been verified by a simulation and experiments. The results show that the error of image grating defocusing is mainly expressed as the Zernike Z4 term and Z9 term in the reconstructed wavefront, and the coefficients of Z4, and Z9, respectively, are related to NA2, NA4, and the defocus distance z. When the numerical aperture (NA) of the under-test projection lens is 0.6, 99.8384% of the errors caused by the image grating defocusing can be removed. When the NA is reduced to 0.3, 99.9854% of the errors can be removed. Additionally, when the NA is less than 0.1, almost all the errors can be eliminated.

Exact formula to relate optical path differences and transversal aberration components

An exact equation to relate the optical path differences (OPD) with its transversal aberration components (TAC) is determined. The OPD-TAC equation reproduces the Rayces formula and introduces the coefficient for the longitudinal aberration. The defocus orthonormal Zernike polynomial (Z _DF) is not a solution for the OPD-TAC equation since the obtained longitudinal defocus depends on the ray height on the exit pupil, meaning that it cannot be interpreted as a defocus. In order to find an exact expression for OPD defocus, first, a general relationship between the wavefront shape and its OPD is established. Second, an exact formula for the defocus OPD is established. Finally, it is proved that only the exact defocus OPD is an exact solution of the exact OPD-TAC equation.

Effects of illumination non-uniformity on the double-Ronchi lateral shearing interference field

Double-Ronchi shearing interferometry is a promising technique for insitu wavefront aberration measurement of the projection lens in photolithography systems. In practice, the non-uniformity of illumination is an important issue affecting the interference field, which has not been systematically researched. In this work, the interference field errors caused by non-uniform illumination distributions are analyzed utilizing the theories of scalar diffraction. The theoretical analysis has been verified by simulation and fundamental experiments. Results show that the uniformity requirements for the abrupt annular, Gaussian, and uniform random illumination distribution (RD) are 0.9434, 0.8439, and 0.2751, respectively, with a shear ratio of 5% and a relative wavefront reconstruction error of 1%. The uniformity of the three distributions is reduced to 0.6513, 0.5864, and 0.1234, respectively, with the shear ratio shrunk to 3%. When the shear ratio is less than 1%, there is no specific requirement for illumination uniformity.

Self-calibration interferometric stitching test method for cylindrical surfaces

The surface figure accuracy requirement of cylindrical surfaces widely used in rotors of gyroscope, spindles of ultra-precision machine tools and high-energy laser systems is nearly 0.1 µm. Cylindricity measuring instrument that obtains 1-D profile result cannot be utilized for deterministic figuring methods. Interferometric stitching test for cylindrical surfaces utilizes a CGH of which the system error will accumulated to unacceptable extent for large aperture/angular aperture that require many subapertures. To this end, a self-calibration interferometric stitching method for cylindrical surfaces is proposed. The mathematical model of cylindrical surface figure and the completeness condition of self-calibration stitching test of cylindrical surfaces were analyzed theoretically. The effects of shear/stitching motion error and the subapertures lattice on the self-calibration test results were analyzed. Further, a self-calibration interferometric stitching algorithm that can theoretically recover all the necessary components of the system error for testing cylindrical surfaces was proposed. Simulations and experiments on a shaft were conducted to validate the feasibility.

Applicability of the Van Cittert-Zernike theorem in a Ronchi shearing interferometer

Ronchi shearing interferometry is a promising technique for in situ wavefront aberration measurement of the projection lens in advanced photolithography systems. The Van Cittert-Zernike theorem has been used to analyze the interference signal of a Ronchi shearing interferometer in the effective interference area (overlapping area of the ±1st diffraction orders produced by the image grating). However, the applicability of this theorem has not been systematically studied. In this work, the analytical expression of the Ronchigram in this area is derived based on the theory of scalar diffraction and incoherent imaging. The results show that only the object and image grating with infinite diffraction orders can fully satisfy the Van Cittert-Zernike theorem. In the finite diffraction orders case, the theorem can be considered approximately applicable in the overlapping area of the ±3rd diffraction orders produced by the image grating. The applicable area extends to the overlapping area of the ±2nd diffraction orders under a shear ratio of less than 1%, which accounts for 97% of the effective interference area. The theoretical analysis has been verified by simulation and fundamental experiments.

Spherical wavefront measurement on modified cyclic radial shearing interferometry

We propose a radial shearing interferometric approach to measure spherical wavefronts as both of the reflective and transmissive optical configurations. The modified cyclic radial shearing interferometer uses a single lens in the optical layout, which can conveniently adjust the radial shearing ratio between two shearing spherical wavefronts, and the use of a polarization camera enables to reconstruct the wavefront by a single image. The wavefront mapped onto the camera plane can be identified and quantified throughout an optimized wavefront reconstruction algorithm. In the experiments, plano-convex lenses and concave mirrors were used to generate spherical wavefronts, and the proposed system was able to reconstruct the surface figures after system characterization and calibration. Further investigations were performed to evaluate the system measurement accuracy by the radius of curvature comparison with design value and a commercial Shack-Hartmann wavefront sensor.

Phase retrieval using bidirectional interference

In this paper we describe a phase retrieval algorithm using constraints given by diffraction patterns and phase difference obtained from bidirectional interference. Wave propagation and linear phase ramps are used to connect the recordings. At least three patterns are recorded and processed (two diffraction patterns and one interference pattern). The quality of the results can be improved when recording and processing more patterns. The method works well with non-sparse samples and short (few millimeter) recording distances. Simulations, comparisons with other methods, and experimental validations are presented.

Focal Spot and Wavefront Sensing of an X-Ray Free Electron laser using Ronchi shearing interferometry

The Linac Coherent Light Source (LCLS) is an X-ray source of unmatched brilliance, that is advancing many scientific fields at a rapid pace. The highest peak intensities that are routinely produced at LCLS take place at the Coherent X-ray Imaging (CXI) instrument, which can produce spotsize at the order of 100 nm, and such spotsizes and intensities are crucial for experiments ranging from coherent diffractive imaging, non-linear x-ray optics and high field physics, and single molecule imaging. Nevertheless, a full characterisation of this beam has up to now not been performed. In this paper we for the first time characterise this nanofocused beam in both phase and intensity using a Ronchi Shearing Interferometric technique. The method is fast, in-situ, uses a straightforward optimization algoritm, and is insensitive to spatial jitter.

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.

Quadriwave lateral shearing interferometric microscopy with wideband sensitivity enhancement for quantitative phase imaging in real time

Real-time quantitative phase imaging has tremendous potential in investigating live biological specimens in vitro. Here we report on a wideband sensitivity-enhanced interferometric microscopy for quantitative phase imaging in real time by employing two quadriwave lateral shearing interferometers based on randomly encoded hybrid gratings with different lateral shears. Theoretical framework to analyze the measurement sensitivity is firstly proposed, from which the optimal lateral shear pair for sensitivity enhancement is also derived. To accelerate the phase retrieval algorithm for real-time visualization, we develop a fully vectorized path-independent differential leveling phase unwrapping algorithm ready for parallel computing, and the framerate for retrieving the phase from each pair of two 4 mega pixel interferograms is able to reach 47.85 frames per second. Experiment results demonstrate that the wideband sensitivity-enhanced interferometric microscopy is capable of eliminating all the periodical error caused by spectral leaking problem and reducing the temporal standard deviation to the half level compared with phase directly retrieved by the interferogram. Due to its high adaptability, the wideband sensitivity-enhanced interferometric microscopy is promising in retrofitting existing microscopes to quantitative phase microscopes with high measurement precision and real-time visualization.

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.

Application of advanced shearing techniques to the calibration of autocollimators with small angle generators and investigation of error sources

The application of advanced error-separating shearing techniques to the precise calibration of autocollimators with Small Angle Generators (SAGs) was carried out for the first time. The experimental realization was achieved using the High Precision Small Angle Generator (HPSAG) of TUBITAK UME under classical dimensional metrology laboratory environmental conditions. The standard uncertainty value of 5 mas (24.2 nrad) reached by classical calibration method was improved to the level of 1.38 mas (6.7 nrad). Shearing techniques, which offer a unique opportunity to separate the errors of devices without recourse to any external standard, were first adapted by Physikalisch-Technische Bundesanstalt (PTB) to the calibration of autocollimators with angle encoders. It has been demonstrated experimentally in a clean room environment using the primary angle standard of PTB (WMT 220). The application of the technique to a different type of angle measurement system extends the range of the shearing technique further and reveals other advantages. For example, the angular scales of the SAGs are based on linear measurement systems (e.g., capacitive nanosensors for the HPSAG). Therefore, SAGs show different systematic errors when compared to angle encoders. In addition to the error-separation of HPSAG and the autocollimator, detailed investigations on error sources were carried out. Apart from determination of the systematic errors of the capacitive sensor used in the HPSAG, it was also demonstrated that the shearing method enables the unique opportunity to characterize other error sources such as errors due to temperature drift in long term measurements. This proves that the shearing technique is a very powerful method for investigating angle measuring systems, for their improvement, and for specifying precautions to be taken during the measurements.

Objective shearing digital holography for removing aberration from optical system

We propose a new digital holography based on the lateral shearing interference concept to remove the total aberrations from the reference wave, illumination wave, and the optical elements. It uses three mutually shifted image holograms of the object that are divided from each other to obtain phase differences. The phase aberration can be removed and the original sample phase can be reconstructed by the phase differences. Then, the influence of the stage moving imprecision on the reconstruction quality is analyzed. Optical experiments verified that the proposed method can totally remove the phase aberrations. As a result, the proposed method could be used for ultra-precise optical measurement through eliminating optical phase aberration to increase the measurement accuracy.

Recovery of wavefront from multi-shear interferograms with different tilts

An improved multi-shear algorithm is proposed to reconstruct a two-dimensional wavefront from multiple phase differences measured by lateral shearing interferograms with different tilts. The effects of the tilt errors in the wavefront are analyzed and a compensation method is developed. Unbiased estimators are added to Fourier coefficients of the phase differences to eliminate the tilt errors adaptively. The algorithm is immune to the tilt errors and the wavefront under test can be recovered exactly. Computer simulation and optical test demonstrated that the proposed algorithm has higher recovery accuracy than the existing multi-shear algorithms.

Wave field sensing by means of computational shear interferometry

In this paper, we present a method to recover the complex amplitude of speckle fields from measurements performed by a shear interferometer. It is based on the optimization of an objective function using the steepest descent gradient technique in combination with a heuristic initial guess. In contrast to already existing methods, the algorithm finds a local minimum least-squares solution even in the presence of Poissonian and Gaussian noise.

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.

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.

Three-wave shearing interferometer based on spatial light modulator

The use of a SLM for the three-wave lateral shearing interference is proposed, and an eight-step phase-shifting scheme is developed for extracting phase information from three-wave interferograms. The two-dimensional phase of object is reconstructed from two phase differences which are calculated from two orthogonal sheared interferograms. The flexibility of SLM can be fully utilized in the sense of dynamical controlling of the direction and amount of shear, as well as phase shift. The numerical simulation and optical experiment are carried out to demonstrate the feasibility and reliability of the proposed scheme.

Wavefront reconstruction from its gradients

Wavefronts reconstructed from measured gradients are composed of a straightforward integration of the measured data, plus a correction term that disappears when there are no measurement errors. For regions of any shape, this term is a solution of Poisson's equation with Dirichlet conditions (V = 0 on the boundaries). We show that for rectangular regions, the correct solution is not a periodic one, but one expressed with Fourier cosine series. The correct solution has a lower variance than the periodic Fourier transform solution. Similar formulas exist for a circular region with obscuration. We present a near-optimal solution that is much faster than fast-Fourier-transform methods. By use of diagonal multigrid methods, a single iteration brings the correction term to within a standard deviation of 0.08, two iterations, to within 0.0064, etc.

Comment on "Redundancy of phase-space distribution functions in complex field recovery problems"

In a recent publication [Appl. Opt. 42, 1932 (2003)], redundancies of phase-space representations were studied. In particular, signal recovery from a single section of the ambiguity function was explored. It is shown that this signal-recovery method can be associated with interferometry.

Wave-front reconstruction from shear phase maps by use of the discrete Fourier transform

Fast wave-front reconstruction methods are becoming increasingly important, for example, in large astronomical adaptive optics systems and high spatial resolution shear interferometry, where pseudoinverse matrix methods scale poorly with problem size. Wave-front reconstruction from difference measurements can be achieved by use of fast implementations of the discrete Fourier transform (DFT), obtaining performance comparable with that of the pseudoinverse in terms of the noise propagation coefficient. Existing methods that are based on the use of the DFT give exact results (in the absence of noise) only for the particular case in which the shear is a divisor of the number of samples to be reconstructed. We present two alternate solutions for the more general case when the shear is any integer. In the first solution the dimensions of the problem are enlarged, and in the second the problem is subdivided into a set of smaller problems with shear amplitude equal to one. We also show that the retrieved solutions have minimum norm and calculate the noise propagation coefficient for both methods. The proposed algorithms are implemented and timed against pseudoinverse multiplication. The results show a speed increase by a factor of 50 over the pseudoinverse multiplication for a grid with N = 3 x 10(3) samples.