Jose F. Vazquez-Castillo

One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms.

An experimental setup for optical phase extraction from 2-D interferograms using a one-shot phase-shifting technique able to achieve four interferograms with 90 degrees phase shifts in between is presented. The system uses a common-path interferometer consisting of two windows in the input plane and a phase grating in Fourier plane as its pupil. Each window has a birefringent wave plate attached in order to achieve nearly circular polarization of opposite rotations one respect to the other after being illuminated with a 45 degrees linear polarized beam. In the output, interference of the fields associated with replicated windows (diffraction orders) is achieved by a proper choice of the windows spacing with respect to the grating period. The phase shifts to achieve four interferograms simultaneously to perform phase-shifting interferometry can be obtained by placing linear polarizers on each diffraction orders before detection at an appropriate angle. Some experimental results are shown.

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

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.

One-shot phase-shifting interferometry: five, seven, and nine interferograms

To extract phase distributions, which evolve in time using phase-shifting interferometry, the simultaneous capture of several interferograms with a prescribed shift has to be done. Previous interferometric systems aimed to fulfill such a task were reported to get only four interferograms. It is pointed out that more than four suitable interferograms can be obtained with an interferometer that uses two windows in the object plane, a phase grid as a pupil, and modulation of polarization for each diffraction orders in the image plane. Experimental results for five, seven, and nine interferograms are given.

Adjustable lateral-shear single-shot phase-shifting interferometry for moving phase distributions

A grating interferometer to produce n lateral-sheared phase-shifted interferograms simultaneously in collimated light and with adjustable shear is proposed. It consists of a variable lateral-shear interferometer in conjunction with a grating interferometer. This combination enables the use of phase-shifting techniques for measuring moving wavefront slopes. Several shifted interferograms can be achieved with a single shot using two mutually crossed phase gratings of equal frequency, while the required phase shifts can be performed with modulation of polarization. Results of the proposed method for cases of n = 4, 5 and 7 interferograms are reported.

A single-shot phase-shifting radial-shearing interferometer

A grating interferometer to simultaneously produce n radial-sheared phase-shifted interferograms is proposed. It consists basically of an adaptation of a well known radial-shear interferometer in conjunction with a grating interferometer. This combination enables the use of phase-shifting techniques for measuring phase distributions. Several interferograms can be achieved with a double phase grating of a given frequency, and the required phase shift can be performed with modulation of polarization. Experimental results are presented for cases of n = 4, 5, and 7 interferograms. These results are obtained with two configurations of the system, which are discussed.

Optical tomography of phase objects with parallel projection differences and ESPI

An experimental tomographic set-up whose intention is to render phase-distributions slices of transparent objects is presented. It consists of a Mach-Zehnder interferometer and an ESPI system. In one of the interferometer arms, the object under study is placed on a rotatable stage driven by a stepping motor controlled by a PC-computer. Adjacent parallel projections at angles separated by a small amount are used to extract their corresponding phase difference. Well-known reconstruction procedures are used to render an image, which can be related to the angular derivative of the slice. This image enhances the borders of the objects phase distribution. Simulations and preliminary experimental results are shown.

Babinet?s principle for scalar complex objects in the far field

Babinet’s principle is briefly reviewed, especially regarding the zeroth diffraction order of the far field diffraction pattern associated with a given aperture. The pattern is basically described by the squared modulus of the Fourier transform of its amplitude distribution (scalar case). In this paper, complementary objects are defined with respect to complex values and not only with respect to unity in order to include phase objects and phase modulation. It is shown that the difference in complementary patterns can be sometimes a bright spot at the zero order location as is widely known, but also, it can be a gray spot or even a dark one. Conditions of occurrence for each case are given as well as some numerical and experimental examples.

Phase-shifting with translation ruling in common-path interferometer: characterization and experimental results

An experimental setup for phase extraction of two-dimensional phase distributions is presented. The system uses a common-path interferometer consisting of two windows in the input plane and a translating grating as its pupil. In the output, interference of the fields associated with replicated windows is achieved by a proper choice of the windows spacing with respect to the grating period. Because in this type of grating interferometer a grating is placed as a spatial filter, the phase changes which are needed for phase-shifting interferometry can be easily performed with translations of the grating driven by a linear actuator. Some experimental results are shown.

Two references modulation of polarization for phase-shifting holographic interferometry

Phase-shifting interferometry requires of a modulator able to introduce the phase steps needed to produce N+1 interferograms, where N is the number of shifts. The phase mod 2π can be found by solving the (N+1)·(N+1) equation system. Recently, it has been shown that modulation of polarization can be adapted to these techniques with some advantages, as its capability to capture several interferograms simultaneously. In this report, it is shown that this modulation can also be adapted to holographic imagery. To show that, the storage of two slightly different wavefronts is first made with a double exposure heterodyne technique. This means that we use a different reference wave in each recording, thus obtaining a Dndliker-type hologram. This hologram is later reconstructed with both references at the same time, each reference modulated in polarization independently in order to obtain circular polarizations of opposite sign for each reference prior to the respective reconstruction. Observing interference fringes through a linear polarizing, a given shift is determined by the angle ψ of the polarizing filter transmission axis. Capturing the N+1 interferograms generated as described, usual phase-shifting algorithms allow phase extraction. Experimental results for the case N = 3 are shown.

Edge Enhancement of Phase Objects with Hilbert Transforms in Optical Tomography

The Hilbert transform (HT) is implemented in a 4f Fourier‐transform imaging optical system by placing a phase step filter of π radians in the frequency plane.