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

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.

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.

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.

Noise-immune phase unwrapping by use of calculated wrap regions

We implement a proposed method for noise-immune phase unwrapping through the calculation of what may be called wrap regions. The theory of this method is reformulated and discussed with regard to random wrapping, a phenomenon that occurs with diffusely reflecting objects if the optical phase change caused by a deformation is calculated by subtraction of the phases calculated before and after the deformation. Two methods of eliminating random wrapping are presented. The algorithm for phase unwrapping is described, and experimental results are presented.

Noise-immune method for locating wrap regions in phase-step interferometry

A method is presented that can be applied to phase-step interferometry to locate the regions within which integral multiples of pi should be added to the wrapped phase that results from arctangent calculations. Unlike previous methods, this one is immune to the noise-related errors that confuse simple phase-unwrap schemes. It makes use of the fact that the noise-related errors occur at fixed locations in the phase map, whereas the location of the wrap regions depends on the way in which the arctangent is calculated.