Noel-Ivan Toto-Arellano

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.

4D profile of phase objects through the use of a simultaneous phase shifting quasi-common path interferometer

Modulation of polarization is commonly employed in optical interferometry through the use of polarizers and quarter-wave retarders. Phase shifts between interfering beams can be easily controlled with such techniques. This communication describes some details of modulation of polarization which are useful in phase shifting interferometry applied to the study of phase objects. As an application, the case of a two-beam phase grating interferometer is discussed on the grounds of polarization analysis as an example. The configuration presented does not require micro-polarizer arrays or additional software to eliminate noise caused by vibration. This system does not use a double window, and generates two beams, the separation of which can be varied according to the characteristics of the grid used. Experimental results are also given.

Parallel two-step phase shifting interferometry using a double cyclic shear interferometer.

A parallel two-step polarizing phase shifting interferometer based on a Double Cyclic Shear Interferometer (DCSI) is proposed in this paper for quantitative phase imaging. The system has the advantage of retrieving the derivative phase data map directly. Due to its configuration, it presents better stability against external configurations than other types of interferometers. The DCSI generates two π-shifted interferograms, which are recorded by the CCD camera in a single-shot. The separation between parallel interferograms can be varied in the two axes for convenience. To obtain the optical phase data map, a parallel phase shift between interferograms is obtained by rotating a half wave plate retarder. We analyzed the cases of four patterns with shifts of π/2 captured in two shots; the optical phase was processed by a four-step algorithm. Related experimental results obtained for microscopic transparent samples are presented.

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.

Dynamic temperature field measurements using a polarization phase-shifting technique

Abstract. In this study, an optical system capable of simultaneously grabbing three phase-shifted interferometric images was developed for dynamic temperature field measurements of a thin flame. The polarization phase-shifting technique and a Michelson interferometer that is coupled to a 4-f system with a Ronchi grating placed at the frequency plane are used. This configuration permits the phase-shifted interferograms to be grabbed simultaneously by one CCD. The temperature field measurement is based on measuring the refraction index difference by solving the inverse Abel transform, which requires information obtained by the fringe order localization. The phase map is retrieved by a three-step algorithm. Experimental results of a dynamic thin flame are presented.

Adjustable-window grating interferometer based on a Mach-Zehnder configuration for phase profile measurements of transparent samples

This communication describes some details of polarization modulation that are useful in phase-shifting interferometry when applied to phase profile measurements of phase objects. Since non-destructive optical techniques allow surface measurement with high accuracy, a Mach-Zehnder configuration coupled to a 4-f arrangement using phase gratings placed on the Fourier plane was implemented to analyze phase objects. Each beam of the interferometer goes through a birefringent wave plate in order to achieve nearly circular polarization of opposite rotations, with respect to each other. The interference of the fields associated with replicated beams, centered on each diffraction order, is achieved varying the spacing of windows with respect to the grating period. Experimental results are presented for cases of four and nine simultaneously captured interferograms.

Dynamic phase imaging of microscopic measurements using parallel interferograms generated from a cyclic shear interferometer

We present a technique which allows us to generate two parallel interferograms with phase shifts of π/2 using a Cyclic Shear Interferometer (CSI) and a polarizing splitter. Because of the use of a CSI, we obtain the derivative phase data map directly, due to its configuration, it is immune to vibrations because the reference wavefront and the object wavefront have a common path; the shearing interferometer is insensitive to temperature and vibration. To obtain the optical phase data map, two interferograms are generated by collocating a polarizing device at the output of the CSI. The optical phase was processed using a Vargas-Quiroga algorithm. Related experimental results obtained for dynamic microscopic transparent samples are presented.