Efficient reconstruction of spatially limited phase distributions from their sheared representation

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.

Optimization of wavefront reconstruction accuracy for conjugate shift differential absolute testing

The conjugate shift differential method, based on Fourier transforms, is critical for surface error testing of high-precision optical elements. However, this common approach is also prone to periodic spectrum loss. As such, this paper proposes conjugate double shift differential (CDSD) absolute testing, which can effectively compensate for spectrum loss and achieve accurate wavefront reconstructions. Spectrum loss in the single shift differential method is analyzed through a study of the Fourier reconstruction process. A calculation model for the proposed CDSD method is then established and constraint conditions for shift quantities are provided by analyzing double shear effects observed in transverse shear interference. Finally, the reconstruction accuracies of various spectrum compensation methods are compared. Results showed that spectrum loss became more evident with increasing shift amounts. However, the CDSD method produced the smallest measurement error compared with conventional direct zero filling and adjacent point averaging, suggesting our approach could effectively improve absolute shape measurement accuracy for planar optical elements.

Double-frequency grating shearing interferometer with built-in phase-shifting function for robust multi-level phase retrieval

In this paper, we propose and demonstrate a novel interferometer and signal process to retrieve two-dimensional signals with multilevel phases. The interferometer is based on a shearing interferometry with double-frequency grating, and phase-shifting interferometry is derived as a built-in function of the lateral displacement of the grating. The interferometer not only retrieves the multilevel phase signals but also eliminates slow-varying phase errors wherever they occur. Owing to the common path algorithm, the new interferometer is more robust in dynamic circumstances for optical testing and data processing. We propose a pre-integral signal process for two-dimensional (2D) data processing to prevent post-phase-integral due to shearing interferometry. The simulation and experiment showed that the proposed interferometer with a pre-integral process has various advantages in signal processing for multilevel phase retrieval, and will be useful for higher data rates in optical data storage and free-space communication.

Iterative deformation calibration of a transmission flat via the ring-point support on a 300-mm-aperture vertical Fizeau interferometer

In the development of a high-precision vertical Fizeau interferometer with a 300-mm aperture, the deformation of the transmission flat because of clamping and gravity must be considered. In this paper, we proposed a ring-point support scheme for the deformation calibration of a large-diameter transmission flat. The calibration theory of the ring-point support system with elastic deformation was derived. The changes in the surface and stress field of the transmission flat were analyzed quantitatively by finite element method modeling, leading to the optimization of the support structure. To validate the proposed calibration approach, we performed an absolute test of the transmission flat using a liquid reference. The test result was compared to a measurement of the Zygo interferometer demonstrating the effectiveness of the proposed ring-point support design. Finally, with the iterative deformation calibration, the accuracy of the transmission flat reached λ/25 (Peak Valley, PV) for a 300-mm aperture.

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.

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.

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

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