Interferogram Analysis for Optical Testing

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

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.

A low-cost and fully automated process for phase-shifting interferometry by continuously changing the input voltage of a laser diode under the scheme of a cube beam splitter interferometer, performing as a two-window common-path interferometric setup, is presented. The input signal of a laser diode is controlled by a data acquisition device which permits a change in its wavelength according to its tunability features. The automation and data analysis is done using LabVIEW in combination with MATLAB. By using the Carré algorithm, the phase map is obtained. Measurements of visibility and phase shift to verify the phase-shifting interferometry requirements are also shown.

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 theory of phase-detecting algorithms based on Fourier theory was described by Freischlad and Koliopoulos (1990). We present an application of this theory to a discrete sampling. It is shown that Greivenkamp (1984) results for a least-squares fitting of the sampling points can be obtained from this Fourier theory. We also present a new graphical vector description of the conditions required by the sampling weights in synchronous phase detection algorithms. Some applications of this graphical representation are presented.

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.

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.

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

Gauge block parameters are usually measured using Koester interferometer or Twyman interferometer. Conventionally the fraction part of central length of the gauge block within one wavelength may be obtained from static interferogram by fringe analysis. Advanced algorithms should be applied to locate the fringe centers precisely. With phase shifting interferometry (PSI), wavefront reflected from the surfaces of gauge block and its base can be retrieved from several interferograms with a definite phase shift between the adjacent interferograms. Additional hardware, such as PZT phase shifter, should be used to introduce phase shift. In this paper, the gauge block parameters are obtained from the static interferogram with a certain spatial frequency introduced by tilting the reference mirror of the interferometer. The spatial frequency of the actual interferogram can be calculated by FFT algorithm. Then a virtual sinusoidal grating with this spatial frequency is generated in the computer and overlapped on the interferogram to get a moire pattern. By moving the grating 1/4 of the grating period, 4 moire patterns with π/2 phase shift between adjacent patterns are obtained. After removing the high frequency fringes in the above four moire patterns with low pass filtering, four interferograms with accurate π/2 phase shift between the adjacent ones are remained. Then the wavefront is retrieved with 4-bucket algorithm. In this method, the static interferogram is analyzed with PSI algorithm but there is no need to have phase shifter. Besides, the phase shift is introduced mathematically to avoid the associated error. The actual interferograms are processed and results are obtained.

To apply phase-shifting interferometry (PSI) to in-situ measurement, we have proposed an algorithm to detect and suppress phase-shifting error and contrast fluctuation. The phase shift and contrast are analyzed in spatial-frequency domain. The strength of baseband and sideband implies the pattern contrast. The position and phase angle of the sideband indicates the tilt gradients and translational value of phase shift. Thus, the phase shift error and contrast fluctuation could be extracted. A contrast-compensated equation is established to calculate the wavefront phase. The proposed algorithm was applied to the interferograms subjecting to vibration and wavefront phase was calculated. The experimental results show that, under vibration of one micron amplitude and 60Hz frequency, the error of wavefront PV value is less than 0.01wave and the 2σ repeatability is less than 0.01wave. For no hardware is required, the proposed algorithm provides a cost-effective method for wavefront in-situ measurement with PSI.

Phase-shifting interferometry is a highly accurate technique for obtaining phase distribution from the recorded fringe patterns. Generally, phase-shifting interferometry requires recording several fringe patterns with varying phase shifts experimentally and during the acquisition, the object must be stable. Also, the atmospheric turbulence and mechanical conditions should also remain constant during this time. These requirements limit the use of these phase-shifting interferometric techniques in dynamic event studies. In the present work, we introduce Riesz transformed based digital four-step phase-shifting interferometer to obtain phase distribution from a single recorded fringe pattern. All the experimental phase-shifting setups necessary to realize the phase-shifting are removed. The idea is based on the recording of a single fringe pattern, and computes its Riesz transform at first, second and third-orders. The obtained Riesz transform components are combined to generate three π/2 phase-shifted fringe patterns, and then, the phase distribution is obtained from these phase-shifted fringe patterns. The performance of this method is demonstrated first by using numerical simulation and the quantitative appraisal is given by using image quality index. Further, we apply this technique on a real fringe pattern recorded in digital speckle pattern interferometry (DSPI). The obtained results reveal that our method provides a simple and accurate solution for phase evaluation, therefore, makes it suitable for real-time measurements.

For the surface shape measurement of a semiconductor with a highly reflective index, it is im portant to effectively suppress the harmonic signal s from multi ple reflection s. In application, the phase extraction algorithm should have a maximum value when there is no phase -shift miscalibration. In this presentation, a new 4N - 3 phase extraction algorithm , which has the ability to suppress harmonic signal s and exhibits a fringe contrast maximum value when there is no phase -shift error , was derived. This n ew 4N - 3 algorithm consists of a new polynomial window function and a discrete Fourier transform term and has the ability to compensate for 2 nd -order nonlineari ty in the phase shift. The suppression ability of the new polynomial window function is compared with other conventional window functions. The sampling functions of the new 4N - 3 algorithm have much smaller amplitudes in the vicinity of the detection freq uency than does synchronous detection or other phase extraction algorithm s with conventional window functions . Keywords: H ighly reflective index, p hase measurement, nonlinearity, window function

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.

A method for the isolation of two-dimensional (2D) displacement components by using one phase map obtained by phase-shifting electronic speckle pattern interferometry (ESPI) is presented. When the typical ESPI is used for displacement measurement, a mixed phase distribution of deformation is measured. If the deformation of the object is symmetrical, two components of deformation can be separated from each other by using the mixed phase distribution. We turn over the mixed phase map first to obtain the second phase map, and then overlap them. Two displacement components can be separated from each other by boundary alignment and algebraic calculation between the two phase maps. This method has been proved feasible by a typical three-point bending experiment. Some experimental results are offered and compared with the results obtained by a dual-beam symmetrical illuminations experiment. This technique presented provides an alternative approach to 2D deformation measurement.

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.