The Monte Carlo method as a tool for statistical characterisation of differential and additive phase shifting algorithms

Several metrological applications base their measurement principle in the phase sum or difference between two patterns, one original s(r,) and another modified t(r,+Δ). Additive or differential phase shifting algorithms directly recover the sum 2+Δ or the difference Δ of phases without requiring prior calculation of the individual phases. These algorithms can be constructed, for example, from a suitable combination of known phase shifting algorithms. Little has been written on the design, analysis and error compensation of these new two-stage algorithms. Previously we have used computer simulation to study, in a linear approach or with a filter process in reciprocal space, the response of several families of them to the main error sources. In this work we present an error analysis that uses Monte Carlo simulation to achieve results in good agreement with those obtained with spatial and temporal methods.

[1]  Benito V. Dorrío,et al.  Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration , 2009 .

[2]  Higinio González-Jorge,et al.  Error propagation in differential phase evaluation. , 2010, Optics express.

[3]  B. V. Dorrio,et al.  Error‐phase compensation properties of differential phase‐shifting algorithms for Fizeau fringe patterns , 2008 .

[4]  Masato Shibuya,et al.  Simultaneous formation of four fringes by using a polarization quadrature phase-shifting interferometer with wave plates and a diffraction grating. , 2008, Applied optics.

[5]  J. Schwider,et al.  Digital wave-front measuring interferometry: some systematic error sources. , 1983, Applied optics.

[6]  Ignacio Lira,et al.  Whole-field analysis of uniaxial tensile tests by Moiré interferometry , 2005 .

[7]  Guillermo H. Kaufmann,et al.  A portable digital speckle pattern interferometry device to measure residual stresses using the hole drilling technique , 2006 .

[8]  K A Stetson,et al.  Fourier-transform evaluation of phase data in spatially phase-biased TV holograms. , 1996, Applied optics.

[9]  Anand Asundi,et al.  Algorithm for directly retrieving the phase difference: a generalization , 2003 .

[10]  B V Dorrío,et al.  REVIEW ARTICLE: Phase-evaluation methods in whole-field optical measurement techniques , 1999 .

[11]  Jérôme Molimard,et al.  Uncertainty analysis of temporal phase-stepping algorithms for interferometry , 2007 .

[12]  D. Malacara,et al.  Interferogram Analysis for Optical Testing , 2018 .

[13]  Mahendra P. Kothiyal,et al.  A comparative study of phase-shifting algorithms in digital speckle pattern interferometry , 2008 .

[14]  A. Gustavo González,et al.  Estimation of the uncertainty of indirect measurements from the propagation of distributions by using the Monte-Carlo method: An overview , 2005 .

[15]  Marta Miranda,et al.  Fourier analysis of two-stage phase-shifting algorithms. , 2010, Journal of the Optical Society of America. A, Optics, image science, and vision.

[16]  Maurice G. Cox,et al.  The use of a Monte Carlo method for evaluating uncertainty and expanded uncertainty , 2006 .

[17]  Cruz Meneses-Fabian,et al.  Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for one-shot phase-shifting interferometry. , 2008, Optics express.

[18]  Peter M. Harris,et al.  An Outline of Supplement 1 to the Guide to the Expression of Uncertainty in Measurement on Numerical Methods for the Propagation of Distributions , 2005 .

[19]  S. Standard.  GUIDE TO THE EXPRESSION OF UNCERTAINTY IN MEASUREMENT , 2006 .

[20]  C. S. Vikram,et al.  Algorithm for phase-difference measurement in phase-shifting interferometry. , 1993, Applied optics.

[21]  Matt Novak,et al.  Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer. , 2005, Applied optics.

[22]  W. R. Brohinsky,et al.  Electrooptic holography and its application to hologram interferometry. , 1985, Applied optics.

[23]  Benito V. Dorrío,et al.  Design and assessment of Differential Phase-Shifting Algorithms by means of their Fourier representation , 2009 .

[24]  Salah Karout,et al.  Two-dimensional phase unwrapping , 2007 .

[25]  M. Miranda,et al.  Error behaviour in differential phase-shifting algorithms , 2008, Optical Systems Design.

[26]  M. Owner-Petersen.  Digital speckle pattern shearing interferometry: limitations and prospects. , 1991, Applied optics.

[27]  J Burke,et al.  Complex division as a common basis for calculating phase differences in electronic speckle pattern interferometry in one step. , 1998, Applied optics.

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