This work presents a very robust and non-iterative algorithm for phase retrieval in phase-shifting interferometry of three unknown and unequal phase-steps under illumination conditions of high spatial non-uniformity. First, the background light is eliminated by subtraction of two interferograms, thereby two secondary patterns are obtained. Second, the object phase is algebraically eliminated from two secondary patterns to obtain only one equation in three unknowns: the modulation light, and the two phase-steps. Third, the square of modulation light is approximated to a polynomial of degree K, and then we demonstrate that it is possible to rewrite the equation in the form of an error function. Forth, the coefficients for the modulation approximation and the phase-steps are computed by applying the least squares method. Some advantages of this approach are its capacity to support high spatial variations in the illumination and the object phase.
The design and implementation of an electronic system to real-time capture and processing speckle interference patterns is presented. Because of the random and instability speckle patterns nature, is very useful a system wich allows obtaining and visualizing interference speckle patterns in the shortest time possible. Proposed system captures the first speckle pattern as steady image while captures subsequent patterns from the same source. Images are electronically transformed separately into value arrays and subtracted to obtain real-time interference speckle patterns, these patterns are automatically archived for later analysis. System consist of a CCD camera, a computer interface that makes capturing, a transparent object and a 4f interferometric system whose source is a laser that passes through diffuser glass in order to obtain speckle effect. Experimental results and analytic explanation is showed bellow.
Numerical results are presented to show the characterization of an electromechanical actuator capable to achieve equally spaced phase shifts and fraction linear wavelength displacements aided by an interface and a computational system. Measurements were performed by extracting the phase with consecutive interference patterns obtained in a Michelson arrangement setup. This paper is based in the use of inexpensive resources on stability adverse conditions to achieve similar results to those obtained with high-grade systems.
An alternative method for phase-shifting interferometry of two steps by using a speckle interferometer is proposed. It is shown that the introduced phase-step could be unknown because the use of an appropriate fringe-pattern processing. The acquired fringe patterns are processed by well established phase-shifting algorithms in order to compare these results with our proposal. Numerical phase difference between two states of phase object is compared with theoretical method using electronic speckle pattern interferometry (ESPI) . Simulated and experimental results are provided.
The fringe-pattern normalization method by parameter estimation is used to relieve the critical filter's requirements in the Fourier transform method for phase demodulation. By the normalization procedure, the zero order spectrum is suppressed allowing a straightforward filtering in the Fourier transform method. Thus, the filtering procedure is carried out by the simple half-plane filter. The benefits of this Fourier normalized-fringe analysis scheme are tested by both computer simulation and optical experiments.
A new method to create elliptically polarized light based on the quadrature amplitude modulation is proposed. This method begins from an unpolarized light and does not use any classical methods, as for instance those using polarizers and retarders or also by refraction, scattering, dichroism or birefringence effects. This method is based on the amplitude modulation of two optical waves in quadrature in order to modulate the field in amplitude and phase. With this idea, we show that any amplitude relation and relative phase difference between components can be created and therefore any polarization state can be carried out. A theoretical model will be shown, and it will be sustained with numerical simulations of several polarization examples
A method to create homogenous polarized light based on non-quadrature amplitude modulation is proposed. This method consists in the addition of two fields out of phase different from mπ and in the variation of their amplitudes only for obtaining a resulting field modulated in both phase and amplitude. This principle is used to modulate the vertical component in both phase and amplitude, while the horizontal component is varied in amplitude only keeping constant its phase, thus any amplitude relation and phase difference between components can be created and therefore any polarization state could be obtained. A theoretical model will be shown, and it will be sustained with numerical simulations of several polarization examples.
A new method to generate inhomogeneous spatial phase variation and phase retrieval is proposed. This method is based under the scheme of three beams forming two Mach-Zehnder interferometers in series, where one arm is considered as the probe beam, while the other two will compose the reference beam where a phase difference within the range of (0,π ) is kept and this remains spatially variable in opposition to the classical PSI method. The generation of phase-shifts is done by on-off nonquadrature amplitude modulation. Although the phase difference between the reference beams are spatially variable having as a consequence that the visibility and phase variations are inhomogeneous, it will be demonstrated that it is still possible to retrieve the phase of the object and also to measure the phase variation. An analytic discussion and experimental results will be shown to sustain the proposed method.
A faster and robust generalized phase-shifting interferometry suitable to automatic real-time applications is presented. The proposal is based on the parameter estimation by the least squares method. The algorithm can retrieve the wrapped phase from two or more phase shifted interferograms with unknown phase steps between 0 and π rad. Moreover, by the multiple or single parameter estimation approach of this algorithm, interferograms with variable visibility both spatial and temporally can be processed, overcoming the restrictions and drawbacks from usual variable spatial visibility approaches. By both computer simulation and a optical experiment, the algorithm's feasibility is illustrated.
The simple filtering procedure, high spatial resolution, and low computation time benefits of Fourier normalized-fringe analysis are verified. For this, both the fringe-pattern normalization method by parameter estimation using the least squares method and the standard Fourier transform method are implemented. This proposal, or any Fourier normalized-fringe analysis scheme, has the advantage that the filter’s properties are not very critical because the zero-order spectrum is suppressed by the normalization stage. Then, the simple half-plane filter is applied in the filtering procedure which, in addition, increases the spatial resolution. Both a computer simulation and the experimental results show the functionality and feasibility of the suggested scheme.
A new method for phase-shifting interferometry based on the 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. Whereby, three interference terms are obtained, but because of 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 paper, both a theoretical model and some numerical simulations are presented
In this work we propose the simulation of a fringe
pattern in analogy with a oscilating pendulum with adaptive
parameters. This technique will be used in future to recovery
the phase of interferograms by mean a dynamical model.
We reproduce a reference fringe pattern solving the dynamical
model of a pendulum with adaptive length parameter, by
computer simulation. We obtained good preliminary results
with this dynamical system. The differential equation (DE) was
simulated with Simulink Toolbox of MATLAB. We show our
results.
In optical tomography of parallel projections, the light rays that cross the slice of the
object are experimentally approached to suffer minimal refraction, i.e. take refractional limits.
Generally, a media is used for immersion whose refractive index rate tied the environment to
study, but the geometry of the containment vessels also affects refraction and may be the case
that the approach is not subject performed. In this work we make a numerical study of the
refraction of a ray of light that enters a typical experimental system for studying the
thermodynamic behaviour of a paraffinic wax around their cloud point. Since it has special
properties in the heat capacity and refractive index near the phase transition, these results will
be used to characterize the transition and is intended to give tomographic information to the
study of thermal properties obtained using the T-History calorimetric technique. In this study,
we simulate the behaviour of the refraction of parallel rays crossing the T-History test system
to find the optimal values of the dimensions of the containment vessels and the index of
refraction of the medium for immersion, considering that the optical properties of the sample
under study vary with temperature. Thus, we obtain the optimum conditions of minimum
refraction technique for which reconstruction of a tomographic slice parallel projection can be
applied. The distribution of the linear attenuation coefficient on the slice of the object,
typically, is obtained by applying the filtered backprojection algorithm to the set of projections
(sinogram) obtained experimentally, which constitutes a way to detect mobile interfacial
boundaries in real time. The projections are sequentially measuring the intensity of the wave
emerging from the slice of the object at different angles.
A novel phase shifting interferometry method based on the variation of the electric field under the scheme of a
three beams interferometer is proposed. One beam contains the object under study, that makes this beam the
probe beam; the other two will be consider as the reference beams with a phase difference of π 2 . Due to
this, one of the three resulting interference terms will be cancelled and the two remaining will be in
quadrature. Applying some trigonometric identities, we show that the resulting interference pattern could
become modeled by the interfering of two beams with an additional phase term; we obtain that the tangent
function of the additional phase depends on the division of the amplitude of the third field divided by the
amplitude of the first, and it is possible to group the sum of the squares of these fields in a square amplitude.
To recover the phase by using the phase shifting interferometry techniques it is necessary to keep constant
the visibility of the interference pattern, at first sight we can think that this is not possible because the
variations of the field amplitude affect the visibility of the patterns. However this problem is solved if the values
of the amplitude corresponding to the fields one and three are seen as an ordered pair contained over an arc
segment at the first quadrant. We justify the viability of this method by a theoretical analysis and a numerical
simulation of the interference of three beams under the conditions mentioned above.
Using a 4f optical system, the Hilbert transform [1] of the projections of the object slice is implemented to reconstruct its
edge enhancement using the parallel projection tomography. The projection going out of the object is considered as the
entrance to the 4f optical system, the filter employed in the Fourier plane consisting of a phase step of π radians, in the
image plane, as it is well known, the output function is given by the convolution of the impulse response and the input
function, then in the image plane the Hilbert transform of the projection is obtained. We show that the reconstruction in
this case is the Hilbert transform in vertical direction of the slice function, which is a special case. Also study in detail
the synthesis of the filter in order to obtain the reconstruction the edge enhanced in an arbitrary direction, which is
general case.
A novel method of phase differences extraction is applied in order to study vibrations on a homogenous cantilever. It is
achieved considering fringe patterns with carrier frequency and slow varying respect to the time. The technique is based
on the Fourier-transform method and by the quotient of the two adjacent frames of the patterns. Fringe pattern is
obtained by fringe projection on a fixed end homogenous cantilever, undergoing harmonic vibration. With this
technique, we can obtain profiles of different vibrating modes of the cantilever for frequencies less than that of the
camera frame rate (33.3 ms). In this communication theoretical analysis and experimental profile modes are shown.
An experimental setup for phase extraction of 2-D 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 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 as well as applications on Optical Tomography are shown.
We design a tomographic system to the slices study of transparent objects with the using parallel projection. The tomographic system is based in a common-path interferometer and this is equipped with a phase-shifting system and a rotate system. Rotate system, sample position at particular angle of projection. Through a visual interface, the projections arriving of the slice object are captured and storage. In this paper, we shown a novel technique of projections capture, which we get a both advantage, it reduces time of capture and diminishes the memory space of computer. Therefore, notably diminishes the processing time. Finally, the optical tomograph construction is discussed and some interfero-sinograms of transparent samples are shown.
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.
Optical tomography of phase objects could be carried out with optical derivatives at the phase detecting stage. A particularly convenient related technique is the use of a fractional derivative of fractional order 1/2, which consists of a spatial filter with square-root amplitude transmittance. An advantage of this type of filter is that it can be implemented as a real filter when dealing with phase objects. Irradiance is proportional to the phase
derivative of order one, and it can be obtained. If a tomographic detection system is equiped with such a filter, phase gradient of projections can be determined, and directional derivatives of phase slices can be displayed. Slice images resulting from an experimental tomographic system for parallel projections includes a spatial
filter. with square-root amplitude transmittance made from a liquid crystal display operating in amplitude modulation are shown.
An experimental setup for tomographic inspection of phase objects is presented. The system is based on an interferometer which uses two windows in the input plane and a translating grating as its pupil. In the output, interference of the fields associated with replicated windows can be achieved by a proper choice of the spacing between windows with respect to the grating period. Placing a rotating object in one window and with a reference crossing through the second one, the phase of projections results encoded in a composite interference pattern over the plane of the traditional sinogram. Phase stepping of such composite interferograms can be achieved by lateral translation of the grating to obtain the unwrapped phase distribution as the corresponding sinogram of the object slide under inspection. The sinogram allows tomographic reconstruction of slices by standard procedures. Composite interferograms and preliminary reconstructions for some transparent samples are shown.
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 object's phase distribution. Simulations and preliminary experimental results are shown.
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