Spatial-frequency analysis algorithm for in-situ measurement of wavefront

Phase-shifting interferometry with a genetic algorithm is proposed. The correction of unknown phase-shifting error is an important task in general phase-shifting interferometry. Since phase-shifting errors generate twin image noise in a reconstructed image, we can reduce the phase-shifting errors indirectly by trying to eliminate the twin-image noise in the reconstructed image. By Zernike polynomial expansion, the reconstructed image is represented as the evenness and oddness, where the ratio of the evenness and oddness is a measure of the amount of the twin image noise. We employ the genetic algorithm for finding the fittest phase shifts of interferograms by reducing the evenness of the reconstructed image, which leads to reduction of phase-shifting errors. This phase-shifting interferometry with a genetic algorithm is confirmed experimentally.

The phase shift interferometer can be used to detect the wavefront phase of optical system and the piezoelectric transducer is widely used as a phase shifter in phase shift interferometer. However, the displacement of piezoelectric transducer presents hysteresis nonlinearity. Hence, when the equal-step phase shift algorithm (e.g. the Schwider-Hariharan phase-shifting algorithm which obtains five images and sets the phase-shifting between images to 90 degrees) is used in the phase shift interferometer to calculate the wavefront phase, the nonlinearity of piezoelectric transducer will introduce error to the 90 degrees phase-shifting between images. Then the calculated wavefront phase will contain the error. Hence, it is necessary to propose a method to diminish the phase error. The method by choosing an optimized phase-shifting algorithm has been used to mitigate the phase error. However, the method does not directly focus on piezoelectric transducer itself. In the paper, we propose a method by compensating for the hysteresis nonlinearity displacement of piezoelectric transducer to mitigate the phase error. We analyze the relationship between the nonlinearity displacement of piezoelectric transducer and phase shift error and establish the error models to analyze the relationship between the phase shift error and phase error. Furtherly, we propose the Proportion Integration Differentiation (PID) closed-loop feedback control method to compensate for the hysteresis nonlinearity displacement of piezoelectric transducer. The results indicate that, with compensated, the maximum tracking error is reduced from 18.89 % to 5.5 %. And in the ascending stage, the PV error and RMS error with compensated are less 12.35nm and 4.34nm than the PV error and RMS error without compensated. In the descending stage, the PV error and RMS error with compensated are less 19.15nm and 6.76nm than the PV error and RMS error without compensated. Hence, the proposed compensation method can improve the detection accuracy of the phase shift interferometer.

Phase-shifting interferometry has many advantages, and the phase shifting nature of the Liquid Crystal Point Diffraction Interferometer (LCPDI) promises to provide significant improvement over other current OMEGA wavefront sensors. However, while phase-shifting capabilities improve its accuracy as an interferometer, phase-shifting itself introduces errors. Phase-shifting algorithms are designed to eliminate certain types of phase-shift errors, and it is important to chose an algorithm that is best suited for use with the LCPDI. Using polarization microscopy, the authors have observed a correlation between LC alignment around the microsphere and fringe behavior. After designing a procedure to compare phase-shifting algorithms, they were able to predict the accuracy of two particular algorithms through computer modeling of device-specific phase shift-errors.

An algorithm is proposed for extraction of the unknown phase shift steps and correction of the phase shift error in partially coherent illuminated phase-shifting digital holography. The mean contrast of noise (MCN) of the reconstructed amplitude image is developed to evaluate the noise level resulting from phase shift error. Actual phase shift steps are first extracted by minimizing the MCN. Then the extracted actual phase shift steps are used to retrieve the original object wave without phase shift error. The validity of the proposed method is verified by simulations, and then implemented in partially coherent illuminated phase-shifting digital holography where a spatial light modulator with unknown phase modulation characteristic is used as the phase shifter. The results demonstrated that the actual phase shift steps can be extracted and the phase shift error is corrected effectively.

In phase-shifting interferometry experiments, the accuracy of the phase shift is a major issue. Many experimental and data analyses are done to cancel phase-shift errors inherent to the modulation techniques used. We propose to remove most of the phase-shift error by recourse to a frequency-shifting method. This approach can be applied to both holography and interferometry. We validate the idea with a holographic experiment.

In this day of digitalization, you can work within the technology of optics without having to fully understand the science behind it. However, for those who wish to master the science, rather than merely be its servant, it's essential to learn the nuances, such as those involved with studying fringe patterns produced by optical testing interferometers.When Interferogram Analysis for Optical Testing originally came to print, it filled the need for an authoritative reference on this aspect of fringe analysis. That it was also exceptionally current and highly accessible made its arrival even more relevant. Of course, any book on something as cutting edge as interferogram analysis, no matter how insightful, isn't going to stay relevant forever. The second edition of Interferogram Analysis for Optical Testing is designed to meet the needs of all those involved or wanting to become involved in this area of advanced optical engineering. For those new to the science, it provides the necessary fundamentals, including basic computational methods for studying fringe patterns. For those with deeper experience, it fills in the gaps and adds the information necessary to complete and update one's education. Written by the most experienced researchers in optical testing, this text discusses classical and innovative fringe analysis, principles of Fourier theory, digital image filtering, phase detection algorithms, and aspheric wavelength testing. It also explains how to assess wavefront deformation by calculating slope and local average curvature.

To mitigate the influence of phase-shift errors in wavelength-tuning phase-shifting interferometry, it is essential to enhance the precision of the phase-shifting steps and develop algorithms that are less sensitive to such errors. Leveraging the robust recognition capabilities of neural networks, we propose an Artificial Neural Network Phase-shifting Algorithm (ANNPA) for phase-shifting interferometry, detailing the network's design and training methods while optimizing calculation steps. By specially designing the training dataset, we fundamentally suppress high-order phase-shift errors. Simulations validate the algorithm's excellent insensitivity to both linear phase-shift errors and random noise in phase-shifting. Finally, we conducted a phase-shifting interferometry experiment using a Fizeau interferometer and employed ANNPA for calculations, with results clearly demonstrating its ability to perform multi-surface phase-shifting interferometry under non-ideal phase-shifting conditions.

Phase based imaging and sensing have been for decades effective optical methodologies for high resolution surface profiling. Several techniques for acquiring the phase information encoding the surface topography have been developed; the most prominent is phase shift interferometry (PSI) in which several phase shifted interference images are usually acquired in a sequence and are algebraically combined to extract the phase information. However, phase imaging is limited both by the 2π phase modulo limiting the ability to map structures with heights only up to half the source's wavelength i.e. several hundreds of nm, and also by error induced by the movements of the sample between the acquisitions of phase shifted interference images. Several approached for dealing with these limitations have been developed that provided only a limited solution, e.g. using a beat wavelength interferogram by a two wavelength illumination but that is more sensitive to phase noise and thus less accurate and parallel PSI in which all phase shifted images are acquired simultaneously but that does not resolve the height limitation. We have developed a combined and improved technique for parallel PSI and three wavelength illumination enabling overcoming both limitations without elevating phase noise sensitivity and have set up two prototypes: the first allowing video rate 3D imaging of moving samples such as biological live samples or high throughput scanning of metrology samples with nm scale resolution, and the second allowing single point very high speed axial motion tracking and vibrometry with sub-nm scale resolution and max step height of 30µm.

A phase shift selection method is proposed to design algorithms immune against phase shift errors in two-step generalized phase-shifting interferometry. A general formula for wavefront reconstruction error is derived, and its specific expressions for two common errors are also given. Calculation results suggest that the proper range of phase shift for general application is about from 0.5 to 2.0 rad for both the fixed and linear phase shift errors. Computer simulations have demonstrated the effectiveness of this phase shift selection method by decreasing the wave reconstruction errors to one-fifth.

A new method for derivation of phase-shifting algorithms is presented. The cost function is defined and it is minimized using descending gradient method. The family of algorithms is derived as the result. The properties of the algorithms do not change within the family. Examples of algorithms for constant phase-step as well as for randomly disturbed phase- steps are given and analyzed.

A new method of determining arbitrary phase shifts in Phase Shifting Interferometry (PSI) is presented. The method uses a unique property of Localized Cross Spectral Analysis (LCSA) that constitutes a part of the Stransform to estimate the unknown parameter. The computed phase shifts are then used as an input to the generalized PSI algorithm to obtain accurate phase distribution information. The effectiveness of this method is then verified by means of computer simulations and experimentally acquired data.

Generalization of the Matsumoto–Tonomura approximation for the phase shift within an open aperture

Transmission optical coherence tomography (tOCT) is gaining interest for imaging biological samples; however, so far, quantitative phase information has not been directly obtained. This study introduces SOPHAST-OCT, an approach for integrated 2D phase recovery in Off-Axis Full-Field Swept-Source tOCT. It uses phase-shifting to leverage OCT’s inherent phase changes between recorded interferograms. SOPHAST-OCT demonstrates promising phase agreement with the simulated ground truth phase. This approach represents a first step toward quantitative imaging in tOCT. Full Text: PDF References Y. Park, C. Depeursinge, G. Popescu, "Quantitative phase imaging in biomedicine", Nature Photon 12, 578 (2018). CrossRef K. Alm, H. Cirenajwis, L. Gisselsson, A.G. Wingren, B. Janicke, A. Mölder, et al., in Holography, Research and Technologies (IntechOpen, 2011). CrossRef W. Drexler, J. Fujimoto, editors , Optical Coherence Tomography: Technology and Applications, 2nd ed. (Springer Cham, 2015). CrossRef W. Krauze, P. Ossowski, M. Nowakowski, M. Szkulmowski, M. Kujawińska, "Enhanced QPI functionality by combining OCT and ODT methods", Quantitative Phase Imaging VII, 11653, 19 (2021). CrossRef B.C. Wilson, S.L. Jacques, "Optical reflectance and transmittance of tissues: principles and applications", IEEE J. Quantum Electron. 26, 2186 (1990). CrossRef J. van der Horst, A.K. Trull, J. Kalkman, "Deep-tissue label-free quantitative optical tomography", Optica 7, 1682 (2020). CrossRef P. Hosseini, Y. Sung, Y. Choi, N. Lue, Z. Yaqoob, P. So, "Scanning color optical tomography (SCOT)", Opt. Expr. 23, 19752 (2015). CrossRef P. de Groot, "Phase Shifting Interferometry" in Optical Measurement of Surface Topography, edited by R. Leach (Springer, Berlin, Heidelberg, 2011). CrossRef M. Wojtkowski, A. Kowalczyk, R. Leitgeb, A.F. Fercher, "Full range complex spectral optical coherence tomography technique in eye imaging", Opt. Lett. 27, 1415 (2002). CrossRef Q. Zhang, S. Zhong, J. Lin, W. Chen, M. Luo, J. Zhong, Y. Yu, Z. Peng, S. Cheng, "Full-range Fourier-domain optical coherence tomography based on Mach–Zehnder interferometer", Opt. Lasers Engin. 124, 105794 (2020). CrossRef R. Juarez-Salazar, C. Robledo-Sanchez, F. Guerrero-Sanchez, A. Rangel-Huerta, "Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation", Opt. Express 22, 4738 (2014). CrossRef R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. Arévalo Aguilar, "Generalized phase-shifting interferometry by parameter estimation with the least squares method", Opt. Lasers Engin. 51, 626 (2013). CrossRef M.A. Herráez, D.R. Burton, M.J. Lalor, M.A. Gdeisat, "Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path", Appl. Opt. 41, 7437 (2002). CrossRef Z. Wang, A.C. Bovik, "A universal image quality index", IEEE Signal Proc. Lett. 9, 81 (2002). CrossRef

In this work, we deduce explicit conditions to design phase shifting algorithms (PSAs) in order to suppress errors in phase measurement, the double-frequency ripple distortion and the spurious piston. These errors arise when non-uniform phase-shifting interferograms are processed with conventional PSAs which assume uniform phase shifts. By modeling the non-uniform phase shifts as a polynomial of the unperturbed phase shift value !0, we show that the conditions for eliminating those errors are associated to the m-th derivative of the PSA's frequency transfer function (FTF). Thus, we propose an approach to design robust algorithms based on the FTF formalism, and we present a ready-to-apply eight-frame PSA. Finally, our conclusions are supported by computer simulations.

Accurate phase retrieval algorithm based on linear correlation in self-calibration phase-shifting interferometry with blind phase shifts