Generalized amplitude-phase-shifting interferometry.

Recently, a novel technique was proposed for recovering the complex amplitude of an optical field from a set of interferograms modulated in both visibility and phase steps for the synchronous case, named amplitude-phase-shifting interferometry (APSI). This requirement leads to errors in the accuracy of phase recovery due to miscalibration or the nonlinear response of the device used to generate the phase steps; in the APSI case, these are polarizers. In this paper, we propose to create a generalized model of the APSI technique where the phase steps can be different from each other. APSI has been successfully implemented in a Mach-Zehnder interferometer, while the present technique has been deployed in a double-aperture common-path interferometer (DACPI). Additionally, it is demonstrated that in a DACPI, the use of experimental elements that may cause aberrations or extra noise in the experimental results is reduced. The aim is to have an even more robust phase and amplitude recovery technique, now named generalized amplitude-phase-shifting interferometry. Experimental results are presented and validated through a numerical noise study.

One-shot carrier fringe polarimeter in a double-aperture common-path interferometer.

This paper presents a novel, simple, robust, and very stable polarimeter which is able to measure in a single-shot the state of polarization (SOP) of an optical field. In this proposal, an optical field with an unknown SOP and an optical field with a reference SOP are superposed by using a double aperture common-path interferometer (DACPI), obtaining two interferograms with adjustable carrier frequency in a single shot. Each interferogram has different visibility and phase-step where the information of the unknown SOP is contained. It is shown that under a typical Fourier analysis any SOP can be successfully measured without the necessity of knowing the carrier frequency. The present proposal is justified mathematically and probed experimentally.

Self-calibrating phase-shifting interferometry of three unequal phase steps by fitting background light to a polynomial of degree K.

This paper presents a novel non-iterative and robust algorithm in phase-shifting interferometry of three unknown and unequal phase steps. First, the modulation light and the object phase are algebraically eliminated; second, the background light is approximated to a 2-D polynomial of degree K and estimated by applying the least squares method; third the phase steps are obtained, and finally, the object phase is computed. This idea is theoretically described and verified numerically and experimentally for several values of K.

Phase retrieval by Euclidean distance in self-calibrating generalized phase-shifting interferometry of three steps.

This paper presents a novel algorithm for phase extraction based on the computation of the Euclidean distance from a point to an ellipse. The idea consists in extracting the intensities from a data row or column in three interferograms to form points of intensity and then fitting them to an ellipse by the method of least squares. The Euclidean distance for each intensity point is computed to find a parametric phase whose value is associated to the object phase. The main advantage of the present method is to avoid the use of tangent function, reducing the error in the desired phase computation.

Phase shifting and phase retrieval with a fully automated laser diode system.

A low-cost and fully automated process for phase-shifting interferometry (PSI) by continuously changing the input voltage of a laser diode (LD) under the scheme of an unbalanced Twyman-Green interferometer (TGI) setup is presented. The input signal of a LD is controlled by a data acquisition (NI-DAQ) device that allows it to change its wavelength according to its tunability features. The automation and data analysis will be done using LabVIEW in combination with MATLAB. The phase map is obtained using the Carré algorithm. Measurements of visibility and phase shift to verify the PSI requirements are shown. It is demonstrated with experimental results and statistical analysis that the phase retrieval can be successfully achieved without calibration and using minimal optical devices.

Fully automated low-cost setup for fringe projection profilometry.

In this paper an alternative low-cost, easy-to-use, and fully automated profilometry setup is proposed. The setup is based on a phase-shifting fringe projection technique with four projected fringe parameters. It uses the well-known triangulation arrangement and low-cost electronic and image acquisition components such as a data acquisition board, a motor controller board, a printer rail, a CMOS webcam, and an LCD projector. The position of the camera, the generation of the fringe pattern, the acquisition of the images, and the calculation of the wrapped and unwrapped phase are all performed in LabVIEW. The setup is portable and can be perfectly adapted to be used in other profilometry techniques such as electronic speckle pattern interferometry and laser scanning profilometry.

Inhomogeneous phase-visibility modulating interferometry by space on-off non-quadrature amplitude modulation.

A new method in interferometry based on on-off non-quadrature amplitude modulation for object phase retrieval is presented. Although the technique introduces inhomogeneous visibility and phase variations in the interferogram, it is shown that the phase retrieval of a given object is still possible. This method is implemented by using three beams and two Mach-Zehnder interferometers in series. One of the arms of the system is used as a probe beam and the other two are used as reference beams, yielding from their sum the conventional reference beam of a two-beam interferometer. We demonstrate that, if there is a phase difference within the range of (0,π) between these two beams, the effect of modulation in both amplitude and phase is generated for the case of on-off non-quadrature amplitude modulation. An analytical discussion is provided to sustain this method. Numerical and experimental results are also shown.

Phase-shifting interferometry based on the lateral displacement of the light source.

A simple and inexpensive optical setup to phase-shifting interferometry is proposed. This optical setup is based on the Twyman-Green Interferometer where the phase shift is induced by the lateral displacement of the point laser source. A theoretical explanation of the induced phase by this alternative method is given. The experimental results are consistent with the theoretical expectations. Both, the phase shift and the wrapped phase are recovered by a generalized phase-shifting algorithm from two or more interferograms with arbitrary and unknown phase shift. The experimental and theoretical results show the feasibility of this unused phase-shifting technique.

Double aperture common-path phase-shifting interferometry by translating a ruling at the input plane.

A method for introducing phase steps in an interferogram based on translating a ruling at the input plane of a double aperture common-path interferometer is presented. The setup is built on a 4f optical system consisting of two apertures at the input plane and a Ronchi ruling in the Fourier plane, where at each aperture a Ronchi ruling is also placed. By filtering at the Fourier plane a single diffraction order of the spectrum from the rulings in the object plane, we demonstrate that a phase step is generated when one of the rulings in the input plane is translated. The principal advantage of this proposal lies in improving the resolution in the phase step. We develop a theoretical model and show experimental results.

Directional edge enhancement in optical tomography of thin phase objects.

In this paper, we make a proposal to obtain the Hilbert-transform for each entry of the projection data leaving the slice of a thin phase object. These modified projections are stacked in such a way that they form a modified sinogram called Hilbert-sinogram. We prove that the inverse Radon-transform of this sinogram is the directional Hilbert-transform of the slice function, and the reconstructed image is the directional edge enhancement of the distribution function on the slice. The Hilbert-transform is implemented by a 4f optical Fourier-transform correlator and a spatial filter consisting of a phase step of π radians. One important feature of this proposal is to perform a turn of 180° in the spatial filter at a certain value of the projection angle within the range [0°, 360°]. The desired direction of enhancement can be chosen by the proper selection of such turning angle. We present both the mathematical modeling and numerical results.

Isotropic edge-enhancement by the Hilbert-transform in optical tomography of phase objects.

In optical tomography, isotropic edge-enhancement of phase-object slices under the refractionless limit approximation can be reconstructed using spatial filtering techniques. The optical Hilbert-transform of the transmittance function leaving the object at projection angles ϕ∈(0°,360°), is one of these techniques with some advantages. The corresponding irradiance of the so modified transmittance is considered as projection data, and is proved that they share two properties with the Radon transform: its symmetry property and its zeroth-moment conservation. Accordingly, a modified sinogram able to reconstruct edge-enhanced phase slices is obtained. In this paper, the theoretical model is amply discussed and illustrated both with numerical and experimental results.

Phase-shifting interferometry by wave amplitude modulation.

A new method for phase-shifting interferometry based on wave amplitude modulation is proposed and discussed. This proposal is based on the interference of three waves, where two waves attend as two reference waves and the other wave attends as a probe wave. Thereby, three interference terms are obtained, but because a phase difference of π/2 between the two references is kept constant, one of the three terms will be dropped, while the two remaining will be put in quadrature. Under these conditions, the resulting pattern is mathematically modeled by an interferogram of two waves, where an additional phase is given by the amplitude variations of the reference waves. In this Letter, both a theoretical model and some numerical simulations are presented.

Carrier fringes in the two-aperture common-path interferometer.

We present a method to introduce a linear phase into an interference pattern. This phase is introduced when a grating is placed outside of the Fourier plane of a two-aperture common-path interferometer, which is built using a 4f optical imaging system. Based on an analysis of near-field diffraction, the introduction of the carrier fringes is mathematically justified. It is important to note that no tilt between the two beams is requested to produce this effect, and it turns out to be simpler, easier, and more versatile than other existing methods. The main attributes and advantages of the setup will be discussed and illustrated in detail with experimental fringe patterns.

Experimental generating the partially coherent and partially polarized electromagnetic source.

The technique for generating the partially coherent and partially polarized source starting from the completely coherent and completely polarized laser source is proposed and analyzed. This technique differs from the known ones by the simplicity of its physical realization. The efficiency of the proposed technique is illustrated with the results of physical experiment in which an original technique for characterizing the coherence and polarization properties of the generated source is employed.

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 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.

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.

Optical tomography of transparent objects with phase-shifting interferometry and stepwise-shifted Ronchi ruling.

An experimental setup for tomographic inspection of phase objects 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. With a rotating object in one window and a plane wave in the second one, the phase distribution of each projection is encoded as a corresponding digital image row, which, in turn, constructs a composite interferogram over the plane of a traditional sinogram. Phase stepping of composite interferograms can be achieved by a proper translation of the grating in order to obtain the unwrapped phase distribution as the corresponding sinogram. This sinogram allows tomographic reconstruction of phase slices by standard procedures. Composite interferograms and reconstructions for some transparent samples are shown.