In optical interferometry are several optimization methods applied to estimate the phase of a closed fringe pattern. There are several optimization techniques which play an important role in finding the phase interferogram parameters. In this work, a Genetic Algorithm in combination with Frequency guided Sequential Demodulation are implemented to estimation the phase of synthetic fringe pattern. The method gives good results in demodulation of closed fringe patterns. Results in processing time are similar to traditional optimization techniques, but the method presents computer simplifications to others algorithms.
A phase demodulation method from a single interferogram with a quadratic phase term is developed. The fringe pattern being analysed may contain circular, elliptic or astigmatic fringes. The Fourier transform of such interferograms is seen to be also a sine or a cosine of a second order polynomial in both the real and imaginary parts. In this work we take a discrete Fourier transform of the fringe patterns and then we take separate inverse discrete transforms of the real and imaginary parts of the frequency spectrum. This results in two new interferograms corresponding to the sine and cosine of the quadratic term of the phase modulated by the sine and cosine of the linear term. The linear term of these interferograms may be recovered with similar procedures of fringe analysis from open fringe interferograms. Once the linear term is retrieved the quadratic phase of the interferogram being analysed can also be calculated. The present approach is also being investigated for interferograms with nearly circularly symmetry given that the phase contains some tilt. The described procedure of Fourier analysis from quadratic phase interferograms of nearly symmetric interferograms could be used instead of complex and time consuming algorithms for phase recovery from fringe patterns with closed fringes. Finally, the method is tested in simulated and real data.
The phase shifting algorithms are a widespread method in the technical and scientific literature for relatively noninvasive measurements of a variety physical variables. PSI (phase-shifting interferometry) techniques consist in acquiring at least three interferograms to obtain the wrapped phase. It is important to emphasize that if these acquired intensity images are inadequate, the measurement would have an error associated. The presence of miscalibration or mechanical vibrations in an interferogram can cause an inaccurate measurement; it is possible to design a phase shifting algorithm with a better performance. The present work proposes a novel methodology in order to determine the existence of a miscalibrated interferogram in a four steps phase shifting algorithm by applying the Radon transform.
A new and simple method for measuring the refractive index of liquid substances is presented. In this method, a laser beam impinges transversely on a glass tube (cylindrical cell) filled with the liquid to be measured. The laser beam incident on the cylindrical cell is deviated when it propagates through the wall of the cell and the liquid contained in it. By measuring the deviation of the principal ray of the laser beam when it emerges from the cylindrical cell, we can determine the refractive index of the liquid. To show the feasibility of the method, we measured the refractive index of pure water with a He-Ne laser.
A phase recovery procedure from several interferograms acquired in highly noisy environments as severe vibrations is
described. This procedure may be implemented when phase shifting techniques may not be applicable due to the high
error in the phase shift due to the vibrations. The phase differences among successive interferograms may contain nonlinear
terms that could lead a sign changes in the supposed constants shift terms among acquired images. This can not be
handled correctly with algorithms that corrects small nonlinearities in the phase shifts due to moderate disturbances
during the phase shifting process. In most interferometric configurations for phase measurements the main effect of
vibrations is to introduce a misalignment in the interferometric setup. Then, the phase differences between each
interferogram may contain piston, tilt, and defocus errors. We observed that the tilt term is often the most dominant of the phase differences terms. In such cases, cosine of the phase differences among interferograms may be recovered. This cosine may be processed with Fourier methods in order to recover the phase differences. Once the phase differences are available the phase encoded in the interferograms may be determined. The proposed algorithm is tested in real interferograms.
This work aims to design a filter to attenuate high- and medium- frequency noise in optical test images without changing
the edges and original characteristics of the test image, generated by traditional filters (spatial or frequential). The noise
produced by the LCD pixels (used as a diffraction grating in the Ronchi test) was analyzed. The diffraction is modulated
by the spherical wavefront of the mirror under test, generating at least two frequency band noise levels. To reduce this
bi-frequential noise, we propose to use an array of filters with the following structure: a low-pass frequential filter LPFF,
a band- pass frequential filter BPFF and a circular mask spatial filter CMSF; thus obtaining the composed filter
CF=LPFF-(BPFF)(CMSF). Various sizes of filters were used to compare their signal-to-noise ratio against simple filters
(low-pass and band-stop).
The aim of this work is to propose the use of printed acetate sheets as quasi-sinusoidal and quasi-triangular diffraction
gratings, as low-cost alternative gratings for application in non-invasive optical tests. Gratings were generated with
Matlab® software and made with various models of laser and inkjet printers. A study of the profile gratings that depend
on the symmetry in the sample was included, gratings were placed in the entrance pupil of a positive lens (illuminated by
a collimated plane wave) to observe their Fourier transforms. It was found that diffraction patterns of various types of
quasi-sinusoidal and quasi-triangular profiles were very close to that of sinusoidal gratings. Gradual change in the size of
printed ink spots was observed in more detail through a magnification of 40x. Additionally, an atomic force microscope
was used to measure the average superficial roughness of the impressions as to observe the behavior of the ink on the
The Ronchi test with a Liquid Crystal Display (LCD) phase grating is used for testing convergent optical systems. The
rulings are computer-generated and displayed on the LCD. We prove that it is possible to make a variable electronically
phase grating by using an LCD. By displaying various phase-shifted rulings and capturing the corresponding
ronchigrams, the phase is obtained with the conventional phase-shifting algorithms. Experimental results are shown.
A new technique for the phase gradient estimation encoded in a single interferogram is proposed. The gradient is
calculated numerically solving a differential equation obtained from the interferogram's derivatives in orthogonal
directions. The phase gradient is assumed to vary almost linearly among adjacent pixels in a small window. A
regularized term is aggregated to the differential equation which enables us to find the solution for the phase derivatives
adjusting a plane in a minimization process. Both phase derivatives terms are obtained simultaneously from a set of
linear equations that results from the minimization process. The algorithm requires a small initial region with the phase,
the phase derivatives, and the sine of the phase already estimated. The calculated values of the sine of the phase from the
initial region are used as a regularized term to solve the differential equation. The phase derivatives solution is then
propagated from the initial region until the whole interferogram field is processed. Each value of the sine of the phase
found is aggregated in the regularized term which makes the solution stable. The initial region may be easily found
applying a band pass filter in the frequency domain as done with the Fourier method. The phase of the interferogram is
calculated with a least square method using the information of the phase derivatives found with the proposed technique.
The feasibility of the described approach for phase gradient reconstruction is tested in simulated and experimental data.
The dynamic angle limited integrated scattering (DALIS) method has been developed to examine optically smooth reflective surfaces with well-defined form. The DALIS system shows advantages over the conventional angle-resolved scattering. We propose a new configuration and results in the DALIS method by using a spherical mirror as a collecting element of the scattered light from the surface of a sample under test. Furthermore, the proposed method improves the detection of the scattered light and is suitable to be applied in workshop inspection during optical polishing processes.
The aim of this work is to propose the use of printed acetate sheets as quasi-sinusoidal diffraction gratings, as low-cost
alternative gratings for application in non-invasive optical tests. Gratings were generated with Matlab® software and
made with various models of laser printers. A study of the discretization effects that depend on the symmetry in the
sample was included, gratings were placed in the entrance pupil of a positive lens (illuminated by a collimated plane
wave) to observe their Fourier transforms. It was found that diffraction patterns of various types of semi-sinusoidal
profiles were very close to that of sinusoidal gratings. Gradual change in the size of printed ink spots was observed in
more detail through a magnification of 40x. Additionally, an atomic force microscope was used to measure the
roughness average of the impressions as to observe the behavior of the ink on the acetate.
An array of light-emitting diodes (LEDs) assembled upon a spherical surface can produce a wider angle distribution of light than a typical array (i.e., an array assembled by mounting LEDs into a flat surface). Arranging each single LED into an optimal placement, the uniformity of the illumination of a target can be improved. We derive approximate formulas and equations for the optimum LED-to-LED angular spacing of several spherical arrangements for uniform far-field irradiance. These design conditions are compact and simple tools that incorporate an explicit dependence on the half-intensity viewing angle (half width half maximum angle) of LEDs.
An algorithm for phase determination from a single interferogram with closed fringes using Fourier transforms is
proposed. Traditionally, the Fourier transform method has been used for interferograms containing only open fringes.
This is done performing a pass-band filtering of one half of the frequency spectrum and taking the arctangent function of
the real and imaginary parts of the filtered spectrum. The proposed method performs multiple pass-band filtering
resulting in several wrapped phases with wrong sign changes corresponding to the orientation of the filters. Among the
different wrapped phases, regions are expected to have the same sign enabling the correct reconstruction of the phase.
The corrections of the sing changes are made by comparing the derivatives of the wrapped phases by means of a
modified process of phase unwrapping taking into account small areas of the interferogram field. This is done under the
assumption of that adjacent areas won't have abrupt changes in their phase derivatives as corresponding to smooth
continuous surfaces. The procedure is robust against noise since low-pass filtering can be easily performed in the
calculation of the wrapped maps and during the modified process of phase unwrapping.
A novel method for phase gradient extraction from fringe patterns is described. We use the sine and cosine of the phase as they are obtained with phase shifting techniques or Fourier methods. The phase gradient is approximated using simultaneously forward and backward phase differences. The resulting set of simultaneous linear equations is solved using iterative procedures. This avoids solving for the phase as a previous step to obtain the searched phase derivatives, decreasing processing time. The method is noise tolerant, incorporating regularization terms to constrain the solution to be smooth. The recovered phase derivatives may be used in surface strain analysis or for the purposes of phase recovery.
We present a novel procedure of phase recovery from undersampled phase-shifted interferograms. First, we use synthetic interferograms to calculate the wrapped phase differences along two orthogonal directions. Second, we unwrap them to recover the true Laplacian of the phase. The final step is to integrate the unwrapped phase differences to give the searched phase. This method may be used with either path independent or path dependent phase unwrapping algorithms. The technique overcomes the sensitivity to noise of previous algorithms when low pass filtering techniques are applied during the calculation of the phase differences and least square methods are employed in the final step of phase integration.
We analyze the detection and interpretation of high fringe-density interferograms with CCD cameras. The high spatial frequencies beyond the Nyquist frequency of the CCD elements are shown in the recorded interferogram as fringes of lesser density and reduced contrast. This is due to the large sampling interval and intensity averaging over the pixel. We develop an analytic expression for the detected modulation intensity. It is seen to contain the phase information in its partial derivatives. We perform a simulation with a set of phase-shifted interferograms detected with finite pixel size to recover the detected modulation intensity. The phase is compared with the analytic expression resulting in the maximum error of less than 0.02 percent demonstrating the feasibility of the proposed model.