The main purpose of this paper is to present a method to design tunable quadrature filters in phase shifting interferometry. The algorithm is obtained from a generalized Fourier transform of a symmetrical quadrature filter. This formalism allows us to represent the detuning phase shift error and bias modulation as geometrical conditions. Therefore, the design of the filter becomes a set of solvable linear equations. Hence, to prove our method, several general tunable filters, like three and four frame algorithms, are obtained. Finally, from our results we reproduce particular symmetrical four frame algorithms reported in literature.
The detuning phase shift error is a common systematic error observed in temporal phase shifting (TPS) algorithms. This error, generally due to miscalibration of the phase shifter, is solved by using a quadrature filter insensitive to this detuning error. To compare algorithms, this error is frequently analyzed numerically. However, in this work we present an exact and analytical expression to calculate such error which is applicable to any kind of filters with real or complex frequency response. Finally, a table with the detuning error for several algorithms is reported.
The following explicit model, valid for high aperture refraction with homogenous and isotropic materials, encompasses all explicit solutions of the first-order nonlinear differential equation representing the perfect image-forming process of any axial object point into its axial image point. Solutions include well-known cases, such as flats, spheres, prolate ellipsoids, prolate hyperboloids, and other sections of nondegenerate Cartesian ovals of revolution, now classified according to the recurrent explicit solution introduced herein. We also present some series expansions, given in cylindrical coordinates z(r), for more efficient computation. Explicit solutions allow accurate and expedite thickness calculation as compared to the regular series, parametric, or implicit solutions commonly used. The results of this study are useful in the design of centered optical systems that are perfectly aligned.
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