Marta Miranda

Fourier analysis of two-stage phase-shifting algorithms

Differential phase-shifting algorithms (DPSAs) and sum phase-shifting algorithms (SPSAs) recover directly the phase difference and the phase sum, respectively, encoded in two patterns. These algorithms can be obtained, for instance, by an appropriate combination of phase-shifting algorithms (PSAs), which makes unnecessary the previous calculation and subtraction or addition of each individual optical phase by means of conventional PSAs. A filtering process in the frequency domain is presented that allows us to obtain in a simple and elegant manner a qualitative characterization with a Fourier description of the two-stage phase-shifting evaluation that reveals possible phase shifter miscalibration errors and unexpected harmonics in the signal.

Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration

This paper describes the uncertainty contribution of nine different phase-shifting algorithms (PSAs) to a gauge block calibration evaluated using a Monte Carlo method. The phase map and its standard deviation, codified in the output distribution obtained with each PSA, are used as input parameters in the exact fractions method for the final calculation of the gauge block length. Results obtained show the behaviour of each PSA versus different types of error sources. Uncertainty evaluation fits well to a Gaussian distribution for all algorithms tested with more than 104 trials and the central limit theorem is satisfied. As expected, the Symmetric 5 + 1 PSA shows the best behaviour with the lower uncertainty contribution and appears among PSAs used as the best algorithm for this technical application. As the selected PSAs are representative for the main PSA families, the protocol employed can be used for any other specifically tailored algorithm.

Error propagation in differential phase evaluation.

In many metrological applications the data being measured is associated to the phase difference codified in two fringe patterns. This phase difference can be recovered directly with what are called Differential Phase Shifting Algorithms (DPSAs) by using a combination of irradiance values from both patterns in the arctangent argument. Use of such algorithms requires characterisation mechanisms to inform of their sensitivity to the various random and systematic error sources, which is the same as for well-studied Phase Shifting Algorithms (PSAs). Thus, we present a new analysis of error propagation for DPSAs taking into account the frequency shifting property of the employed arctangent function. The general analysis is verified for significant specific cases associated to large errors that appear during phase difference evaluation using the Monte Carlo method, which provides a characterisation of a DPSAs sensitivity; this is an alternative to spatial and temporal techniques but has wholly coinciding results. Monte Carlo simulation opens up the possibilities for the analysis of other error types for any DPSA.

Error‐phase compensation properties of differential phase‐shifting algorithms for Fizeau fringe patterns

Many researches have been reported on error‐compensating Phase‐Shifting Algorithms (PSAs) that eliminate or minimize the main error sources during the corresponding phase‐evaluation process. However this kind of analyses have not been carried out in a similar way with respect to the Differential Phase‐Shifting Algorithms (DPSAs) that provide directly the phase‐difference between two fringe patterns, being unnecessary the previous calculation of each individual optical phase or even the phase‐unwrapping process, if the phase‐difference is small. In previous works we have studied, by numerical simulation and in a linear approximation, the quantitative response of several families of DPSAs to detuning errors, nonlinearities of detector and harmonics of the signal, achieving satisfactory results that improve those obtained subtracting the two phases calculated separately with PSAs and that provide valuable information related with the sensibilities of the analysed DPSAs. In this work the combined effect of th...

Design of an interferometric system for gauge block calibration

Abstract. We have developed an interferometer for gauge block calibration based on phase shifting algorithms. The measurement process can provide flatness, parallelism, and length. Wavelength values need to be corrected according to the refractive index of air. This correction is obtained indirectly using Edlén’s equation. High-resolution sensors provide the temperature, pressure, and relative humidity readings. To preserve stability, the interferometer is encapsulated in a chamber with active temperature control. Its design, measurement principle, calibration, stability, and reproducibility are analyzed. Since one goal is to employ robust and cheap diode lasers as light sources, we describe the system developed to stabilize a red laser diode using a mode locking technique with a reference gas cell. The instruments and assembly are used to avoid the Doppler effect in the gas cell, which would limit wavelength resolution. Several experiments are carried out to restrict the influence of environmental changes, which affect laser diode frequency.

Measurement of beating effects in narrowband multimode Lamb wave displacement fields in aluminum plates by pulsed TV Holography

Narrowband ultrasonic surface acoustic waves are of the greatest current interest for the nondestructive testing of thin-walled members and shell structures like plates, pipes, bridge girders, cans and many others. The measurement and characterization of ultrasonic displacement fields of Lamb waves by pulsed TV holography (TVH) is presented. Narrowband ultrasound is generated in a few millimeters thick aluminum plate by the prismatic coupling block method using a tone-burst excitation signal in the range of 1MHz. At this frequency, the plate supports only a few Lamb wave modes, mainly the A0 and S0 ones. The simultaneous presence of these modes produces a beating clearly detectable as a spatial amplitude modulation. Our self-developed TVH system performs the optical phase evaluation by the Spatial Fourier Transform Method and renders the instantaneous out-of-plane mechanical displacement field along the whole inspected area. From this field, the wavenumber of each Lamb mode can be obtained and, by combining them with the value of the ultrasound frequency and with the Rayleigh-Lamb theoretical frequency spectrum, information about the elastic constants of the specimen material is obtained.

Design of new phase shifting algorithms according to the interferometer sensitivities

Correct metrological calibration is the key to avoiding the main systematic error sources in an interferometer configuration in order to obtain high resolutions and repeatability. The calibration procedure can consist of the acquisition of two fringe patterns with a known phase shift between them, which means the correct adjustment of this phase difference fits the system. Differential phase shifting algorithms, built by combining two known phase shifting algorithms in a non-linear way, obtain this phase difference directly. In this sense, the two-dimensional characteristic polynomial is an innovative tool for qualitatively characterizing the properties of the differential phase shifting algorithms and, particularly, for determining the accuracy of the system. In previous works, it was demonstrated that sensitivities are inherited from precursor phase shifting algorithms. This mechanism of error propagation allows us to design new differential phase shifting algorithms according to standard requirements in each system in order to cancel or at least minimize this known source of errors.

Laval nozzle flow characterization by Fourier-transform Mach-Zehnder interferometry

The role of the assist gas blown by a nozzle during the laser cutting process of ceramics is very important as a complex flow field is created by the interaction with the material. Flow visualization provides valuable information related with the properties and characteristics of the process in order to clear up misconceptions and as a first step towards the nozzle optimization. Optical methods as Schlieren or interferometry are suitable techniques for this task as well-known non-intrusive full-field methods utilized for analysing fast transient flow phenomena. In this work a Mach-Zehnder interferometer is employed in order to characterize a Laval nozzle by studying the shock wave patterns in a free gas jet as a previous step in order to study the complex interaction of the gas flow against a transparent model of the processed material. The optical phase is extracted from the obtained high-frequency fringe pattern by the Fourier Transform Phase-Difference Method. Disturbing effects are cancelled by proper combination of high-frequency fringe patterns obtained with and without gas flow. The shock wave pattern is analyzed for different geometrical configurations and operating pressures.