How do phase-shifting algorithms work?

Interference is an important phenomenon that plays a key role in understanding the properties of light. However, novice students have difficulties with optical interference experiments because fringe-pattern processing algorithms are not a topic of optics courses. The difficulty increases taking into account that the phase shift is usually induced manually and advanced phase-shifting algorithms must be applied. In this paper, the fundamental principles behind seven well-established phase-shifting algorithms are presented. First, five basic algorithms are analyzed: the three-, four-, and n-steps as well as two least-squares algorithms. Then, the advanced iterative algorithm and the generalized phase-shifting are studied. The operation of the algorithms is illustrated by a typical experiment on the Michelson interferometer. This work could be useful as an introductory lesson to the optics laboratory.

[1]  José A. Ferrari,et al.  Single-element interferometer , 2007 .

[2]  Å. Björck.  Numerical Methods in Matrix Computations , 2014 .

[3]  Rigoberto Juarez-Salazar,et al.  Intensity normalization of additive and multiplicative spatially multiplexed patterns with n encoded phases , 2016 .

[4]  Cruz Meneses-Fabian,et al.  One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms. , 2008, Optics express.

[5]  K. Creath.  V Phase-Measurement Interferometry Techniques , 1988 .

[6]  N. Broderick,et al.  Heterodyne interferometry for the detection of elastic waves: a tutorial and open-hardware project , 2015 .

[7]  Rigoberto Juarez-Salazar,et al.  Phase-unwrapping algorithm by a rounding-least-squares approach , 2014 .

[8]  Pramod Rastogi,et al.  Fringe analysis: Premise and perspectives , 2012 .

[9]  J. Wyant,et al.  Field Guide to Interferometric Optical Testing , 2006 .

[10]  J C Wyant,et al.  Phase shifter calibration in phase-shifting interferometry. , 1985, Applied optics.

[11]  Daniel Malacara,et al.  Interferogram Analysis For Optical Testing, Second Edition , 2005 .

[12]  Michael Frankfurter,et al.  Optical Shop Testing , 2016 .

[13]  Bongtae Han,et al.  Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms. , 2004, Optics letters.

[14]  D. J. Brangaccio,et al.  Digital wavefront measuring interferometer for testing optical surfaces and lenses. , 1974, Applied optics.

[15]  Dong Liu,et al.  Fast and accurate wavefront reconstruction in two-frame phase-shifting interferometry with unknown phase step. , 2018, Optics letters.

[16]  Rigoberto Juarez-Salazar,et al.  Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation. , 2014, Optics express.

[17]  Yasuhiro Oikawa,et al.  Simple, flexible, and accurate phase retrieval method for generalized phase-shifting interferometry. , 2017, Journal of the Optical Society of America. A, Optics, image science, and vision.

[18]  Song Zhang,et al.  Three-dimensional profilometry with nearly focused binary phase-shifting algorithms. , 2011, Optics letters.

[19]  P. Rastogi,et al.  Approaches in generalized phase shifting interferometry , 2005 .

[20]  Simple educational tool for digital speckle shearography , 2012 .

[21]  Song Zhang,et al.  Three-dimensional shape measurement using a structured light system with dual projectors. , 2018, Applied optics.

[22]  T. Eiju,et al.  Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm. , 1987, Applied optics.

[23]  Giuseppe Molesini,et al.  Phase-shift interferometry with a digital photocamera , 2006 .

[24]  D. Malacara.  Optical Shop Testing , 1978 .

[25]  John E. Greivenkamp,et al.  Generalized Data Reduction For Heterodyne Interferometry , 1984 .

[26]  U. Rivera-Ortega,et al.  Computational tool for phase-shift calculation in an interference pattern by fringe displacements based on a skeletonized image , 2015 .

[27]  M. Schmid.  Principles Of Optics Electromagnetic Theory Of Propagation Interference And Diffraction Of Light , 2016 .

[28]  Cruz Meneses-Fabian,et al.  Phase-shifting interferometry based on the lateral displacement of the light source. , 2013, Optics express.

[29]  José A. Gómez-Pedrero,et al.  Algorithm for fringe pattern normalization , 2001 .

[30]  J. Pomarico,et al.  Quantitative phase determination by using a Michelson interferometer , 2007 .

[31]  Sai Siva Gorthi,et al.  Fringe projection techniques: Whither we are? , 2010 .

[32]  J Matías Di Martino,et al.  Generalized phase-shifting algorithms: error analysis and minimization of noise propagation. , 2016, Applied optics.

[33]  Joseph Shamir,et al.  Optical Systems and Processes , 1999 .

[34]  Victor H. Diaz-Ramirez,et al.  Estimation of amplitude and standard deviation of noisy sinusoidal signals , 2017 .

[35]  Pramod Rastogi,et al.  Digital Speckle Pattern Interferometry and Related Techniques , 2000 .

[36]  Max Born,et al.  Principles of optics - electromagnetic theory of propagation, interference and diffraction of light (7. ed.) , 1999 .

[37]  Mario Costantini,et al.  A novel phase unwrapping method based on network programming , 1998, IEEE Trans. Geosci. Remote. Sens..

[38]  C. Cavalcanti,et al.  Software-aided discussion about classical picture of Mach–Zehnder interferometer , 2017 .

[39]  Jason Geng,et al.  Structured-light 3D surface imaging: a tutorial , 2011 .

[40]  Rigoberto Juarez-Salazar,et al.  Theory and algorithms of an efficient fringe analysis technology for automatic measurement applications. , 2015, Applied optics.

[41]  Pramod Rastogi,et al.  Moving ahead with phase , 2007 .