Phase shifts in the Fourier spectra of phase gratings and phase grids: an application for one-shot phase-shifting interferometry

Abstract

Among several techniques, phase shifting interferometry can be implemented with a grating used as a beam divider to attain several interference patterns around each diffraction order. Because each pattern has to show a different phase-shift, a suitable shifting technique must be employed. Phase gratings are attractive to perform the former task due to their higher diffraction efficiencies. But as is very well known, the Fourier coefficients of only-phase gratings are integer order Bessel functions of the first kind. The values of these real-valued functions oscillate around zero, so they can adopt negative values, thereby introducing phase shifts of pi at certain diffraction orders. Because this almost trivial fact seems to have been overlooked in the literature regarding its practical implications, in this communication such phase shifts are stressed in the description of interference patterns obtained with grating interferometers. These patterns are obtained by placing two windows in the object plane of a 4f system with a sinusoidal grating/grid in the Fourier plane. It is shown that the corresponding experimental observations of the fringe modulation, as well as the corresponding phase measurements, are all in agreement with the proposed description. A one-shot phase shifting interferometer is finally proposed taking into account these properties after proper incorporation of modulation of polarization.

Separation of Isochromatics and Isoclinics Phasemaps for the Photoelastic Technique with use Phase Shifting and a Large Number of High Precision Images

“…Sixteen tests are performed, and the correct values of  and  are sought between - and  because the experimental values I j and  j are known [12].…”

“…where M is the number of pixels of the image and  i e and  i e are the exact value calculated by (6)(7)(8)(9)(10)(11)(12) for the disk. The values of  i and  i are calculated by the new equation.…”

Quality Enhancement of a Complex Holographic Display Using a Single Spatial Light Modulator and a Circular Grating

“…Previous reports have proposed several methods that use a spatial light modulator (SLM) to reconstruct a complex hologram. Such complex reconstructed holography with an SLM may use either of two methods: two SLMs [1][2][3], or a single SLM [4][5][6]. The method with two SLMs uses a coupled phase and amplitude for each SLM set up on both sides of a beam splitter, for combination into a complex hologram.…”

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

“…In many other applications of great scientific and technological interest [3][4][5][6][7][8][9][10][11][12] the measurand is linked to the phase sum or the phase difference between the original pattern s(r,) and another one t(r, +), that can be shifted in phase to obtain another series of patterns: (3) where in this case t p (r,, p ) is each pattern of the new series,  p is the phase step between irradiance values and the coefficients b g weight the harmonic contribution. So, 2+ or  can be retrieved with a duplication of the evaluation and unwrapping processes [13] and with a posterior sum or difference of the obtained phases, employing for the phase calculation of the modified pattern a similar expression to (2) (4) being N 2 {t p (r,, p )} and D 2 {t p (r,, p )} combinations of the shifted modified patterns.…”