Method For Phase-shifting Interferometry Research Articles

Dynamic phase measurement based on two-step phase-shifting interferometry with geometric phase grating

Abstract This paper proposes a novel dynamic 2-step phase-shifting interferometry method with a geometric phase grating and direct aberration-free object phase recovery algorithm. By exploring the amplitude-phase modulation property of the geometric phase grating, two-step phase-shifting interferograms with π / 2 phase shift can be obtained in a single shot through two kinds of spatial-multiplexing arrangements. Then the restriction on the background uniformity of the 2-step phase-shifting algorithms and the system aberration can be avoided simultaneously, based on the presented direct and fast object phase demodulation strategy with an object-free state recorded in advance, so an accurate object measurement result can be guaranteed. Accordingly, a detailed analysis of the theoretical model of error sources is conducted by taking the depolarization error of geometric phase grating and phase measurement model error into consideration, with a Least Squares Iteration optimization strategy established to further improve the measurement accuracy. Experiments on various types of phase objects, such as a photo-etched quartz substrate, the spatial light modulator, and a silicon chip, validate the effectiveness and accuracy of our method when compared with the reference phase obtained from 4-step temporal phase-shifting method. The dynamic phase measurement capability is demonstrated by exhibiting the evaporation process of alcohol droplet.

Adjustable accuracy of phase steps in a double-aperture common-path interferometer by rotating and displacing grating at Fourier plane

Detuning errors limit the accuracy of phase extraction in phase-shifting interferometry (PSI). To improve this accuracy, the systems to control the phase shifter dispositive must be highly accurate, involving high technology, exhaustive calibration, high cost, and constant maintenance. This manuscript presents a method for improving the accuracy of the phase step generated at the Fourier plane by rotating and displacing the grating in a double-aperture common-path interferometer. It is demonstrated that both the detuning errors and the phase extraction uncertainty decrease with the grating rotation angle. The experimental implementation is very simple, as it does not require any sophisticated mechanical system to improve the accuracy, no exhaustive calibration, and no additional mechanical and optical dispositive. With the present approach, the PSI method could be achieved very simply and with high accuracy by using a low-accuracy actuator. The theoretical model and experimental results are presented to demonstrate the present proposal.

Robust decorrelation-based iterative algorithm for accurate phase extraction from disturbed fringe patterns.

We propose an accurate and robust phase extraction method for phase-shifting interferometry to reduce the phase ripple error caused by illumination, contrast, phase-shift spatiotemporal variation, and intensity harmonics. In this method, a general physical model of interference fringes is constructed, and the parameters are decoupled using a Taylor expansion linearization approximation. In the iterative process, the estimated illumination and contrast spatial distributions are decorrelated from the phase, thus reducing damage to the algorithm's robustness caused by a large number of linear model approximations. To the best of our knowledge, no method has been able to extract the phase distribution robustly and with high accuracy while considering all of these error sources simultaneously without imposing constraints inconsistent with the practical conditions.

Determination of frequency response of MEMS microphone from sound field measurements using optical phase-shifting interferometry method

Accurate determination of microphone sensitivity is important to build reliable acoustical instruments. The sensitivity is usually determined by calibration. However, because available microphone calibration methods determine the sensitivity from a mathematical model derived from the geometry of a conventional condenser microphone, they cannot be applied to the calibration of microelectromechanical systems (MEMS) microphone straightforwardly. To compromise this geometry difference with the available calibration methods, some authors have proposed the development of adapters that fits the conventional calibration apparatus and modified the calibration procedure. In this paper, we propose a different approach to calibrate the MEMS microphone. The sensitivity is calculated directly from the measurement of the sound field applied to the MEMS microphone and its output voltage. The projection of the sound field is measured by parallel phase-shifting interferometry (PPSI), and sound pressure on the MEMS microphone is obtained by tomographic reconstruction. Experimental calibration of a MEMS microphone was performed and validated using a microphone substitution method to evaluate the discrepancies of the sensitivity result. It is shown that the proposed method can be used to determine the frequency response of the MEMS microphone in the frequency range of 1000 Hz to 12000 Hz.

Random two-frame phase-shifting interferometry via minimization of coefficient of variation

Random two-frame phase-shifting interferometry (PSI) is an advanced technique to retrieve the phase information from as few as two interferograms with unknown phase steps. Because of the advantages of no requirement for accurate phase shifters and much less time for data acquisition and processing, random two-frame PSI is attracting more and more interest in fast and high-precision optical metrology. However, reconstructing the phase from only two interferograms is challenging because it is an ill-posed problem in essence, especially when the phase step is unknown. Although some solutions have been proposed for this problem to date, most of them require complicated preprocessing or special usage preconditions for interferograms to be demodulated. In this letter, we developed an elegant phase reconstruction method for random two-frame PSI, which is much different from frameworks of existing methods. In the proposed approach, the phase of random two-frame PSI can be accurately reconstructed using the phase step value which minimizes the coefficient of variation (CV) of the modulation term of interferograms. Sufficient numerical simulations and experimental data demonstrate the high accuracy and high efficiency of this CV minimization (CVM) method. Moreover, its performance is not limited by the number of fringes in interferograms, in contrast to existing state-of-the-art approaches. We anticipate extensive applications of the CVM method in random two-frame PSI in the future.

Universal phase reconstruction approach of self-calibrating phase-shifting interferometry.

Self-calibrating phase-shifting interferometry (PSI) reconstructs the phase map from three-frame or more phase-shifting interferograms without the need for knowing accurate phase steps. The existing phase reconstruction methods for self-calibrating PSI still have many constraints in terms of the required number of interferograms, special usage preconditions, etc. In this Letter, a universal, accurate, and efficient phase reconstruction method for self-calibrating PSI is proposed. In this approach, we search the solution space of phase shifts to obtain the modulation amplitude of interferograms with the minimum coefficient of variation (CV). Then the phase is reconstructed through the searched phase shifts. Numerical and experimental studies demonstrate that this method can realize highly accurate phase reconstruction consistently, compared to the currently popular methods in their own usable ranges. We anticipate that this Letter may provide a universal and powerful solution for the phase reconstruction of self-calibrating PSI.

Simple phase shifting by polarizer rotations in a cube beam-splitter interferometer.

A simple and low-cost method for phase-shifting interferometry by the rotation of a polarizer is presented. This proposal takes advantage of the polarization aberration in a cube beam splitter due to its geometry, to the angular dependence with the coating, and to the polarization angle of the input beam. The interferometric setup performs as a two-window common-path interferometer in which the added phase shifting is achieved by simply rotating a polarizer at the interferometer output. The viability of the proposal is sustained with experimental results in which the phase-shift value and the resulting wavefront are calculated with Farrel's technique and the three-step PSI algorithm, respectively.

Photothermoplastic recording media and its application in the holographic method of determination of the refractive index of liquid objects.

The photothermoplastic medium based on the films of photosensitive polymeric composites with semiconductor properties is developed for application in optical information recording and storage, in holographic interferometry, as well as for medical purposes. This medium was used in the modified holographic device for determination of changes of the refractive index of homogeneous and inhomogeneous liquid objects. The technique and holographic equipment were modified by employing the specially developed and produced transparent cuvette of special shape and the phase shifting interferometry method. Experimentally demonstrated precision of the measurements is not less than 10-5.

Development of phase-shifting device for implementation of three-step interferometric method with arbitrary phase shifts of reference wave

Two basic approaches to the implementation of procedures of a wavefront stepwise and smooth phase shifts in a two-beam optical interferometer are considered. Advantages and lacks of both procedures are analyzed. It is noted that the smooth phase shift procedure removes the oscillations of a phase shifting element (PSE) that occur during the stepwise procedure. A phase-shifting device that is used to realize the three-step phase shifting interferometry (PSI) method with two arbitrary phase shifts of the reference wave within the angular interval (0, π) is developed. This device does not need the calibration procedure, because the three step PSI method allows defining any phase shift angle by calculating the correlation coefficient between two recorded interferograms. The correlation coefficient can be considered as a normalized scalar product of two interferograms that can be represented by centered multidimensional vectors or as a cosine of a phase shift angle between these vectors. Therefore, the developed device is much simpler and cheaper than its calibrated prototypes. It contains a PSE consisting of a piezoelectric transducer (PZT) and a mirror attached to the PZT, and an electronic unit for smooth linear motion of the mirror in the reference beam of the interferometer. Basic characteristics of the developed electronic unit are considered. The operating principle of the phase-shifting device electronic unit is to provide the smooth rising voltage supply initiated a smooth change of the PZT size. Therefore, the mirror attached to the PZT also moves smoothly. The PSE is located in the reference beam of the two-beam interferometer. If the reference beam in the interferometer enters the mirror in the direction perpendicular to the plane of the mirror, the beam wavefront moves to a distance twice that of the mirror. To verify the reliability of the developed phase-shifting device, we have elaborated the experimental setup dedicated for formation, registration and processing of interferograms of test surfaces and based on the Twyman-Green interferometer architecture. In this setup, the three-step PSI method with arbitrary phase shifts of a reference wave is fulfilled by recording of three interferograms during smooth phase shift of the mirror attached to the PZT. Comparative analysis of stepwise and smooth phase shift procedures for interferograms recording in the experimental setup has shown that their modulation transfer functions (MTFs) are similar for small exposures of interferograms. In particular, it is shown that for interferogram’s small exposure times, whose ratio to the voltage increase time in the angular interval (0, π) does not exceed 0.06, the MTF of the interferogram recorded with the smooth phase shift of the reference beam does not differ from the interferogram MTF obtained by stepwise phase shift. Key words: phase shifting interferometry, phase shift, phase shifting element, interferogram, modulation transfer function

Hyper ellipse fitting in subspace method for phase-shifting interferometry: practical implementation with automatic pixel selection

This paper presents a method of significantly improving the previously proposed simple, flexible, and accurate phase retrieval algorithm for the random phase-shifting interferometry named HEFS [ K. Yatabe , J. Opt. Soc. Am. A34, 87 (2017)]. Although its effectiveness and performance were confirmed by numerical experiments in the original paper, it is found that the algorithm may not work properly if observed fringe images contains untrusted (extremely noisy) pixels. In this paper, a method of avoiding such untrusted pixels within the estimation processes of HEFS is proposed for the practical use of the algorithm. In addition to the proposal, an experiment of measuring a sound field in air was conducted to show the performance for real data, where the proposed improvement is critical for that situation. MATLAB codes (which can be downloaded from http://goo.gl/upcsFe) are provided within the paper to aid understanding the main concept of the proposed methods.

Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information.

Phase extraction methods based on the principal component analysis (PCA) can extract objective phase from phase-shifted fringes without any prior knowledge about their shift steps. Although it is fast and easy to implement, many fringe images are needed for extracting the phase accurately from noisy fringes. In this paper, a simple extension of the PCA method for reducing extraction error is proposed. It can effectively reduce influence from random noise, while most of the advantages of the PCA method is inherited because it only modifies the construction process of the data matrix from fringes. Although it takes more time because size of the data matrix to be decomposed is larger, computational time of the proposed method is shown to be reasonably fast by using the iterative singular value decomposition algorithm. Numerical experiments confirmed that the proposed method can reduce extraction error even when the number of interferograms is small.

Phase shifting interferometry using a spatial light modulator to measure optical thin films.

This work describes a process for measuring thin film steps, using phase shifting interferometry (PSI). The phase shifts are applied only in the region where the thin film steps are located. The phase shift is achieved by displaying different gray levels on a spatial light modulator (SLM Holoeye LC2012) placed in one arm of a Twyman-Green (T-G) interferometer. Before measuring the thin film steps, it was necessary to quantify the phase shifts achieved with this SLM by measuring the fringe shifts in experimental interferograms. The phase shifts observed in the interference patterns were produced by displaying the different gray levels on the SLM one by one, from 0 to 255. The experimental interferograms and the thicknesses of the thin film steps were successfully quantified, proving that this method can be used to measure thin films by applying the PSI method only on the region occupied by them.

Full-field two-dimensional least-squares method for phase-shifting interferometry

The most known and used phase shifting interferometry (PSI) demodulation methods are one-dimensional temporal linear systems. These methods use the information of the interferogram sequence at a single pixel to recover the modulating phase. Accordingly, scanning all pixels, we obtain the two-dimensional (2-D) modulated phase sought. As PSI demodulation methods do not take into account spatial information, these methods cannot remove unwanted harmonics or noise from the interferogram image space (spatial domain). To remove these unwanted artifacts from the image space, spatial information must be included in the demodulation model. We are going to show that the well-known least-squares system for PSI can be used as a full-field 2-D linear system that uses the temporal and spatial information in conjunction in order to recover the modulating phase while removing noise, unwanted harmonics, and interpolating small empty sections of the image space all in the same process with a low computational time.

An effective method in calibrating nonlinear errors for phase-shifting interferometry

In phase measurement or digital holography for phase-shifting interferometry, the key role is the variation of reference light wave and recover algorithm based on interferograms and reference phase, so the calculation result is directly affected by phase-shift accuracy. However, because of the errors of nonlinear and other random factors, it is difficult to control the actual phase-shifting amount accurately. In this paper, we aim to propose an efficient method for phase-shifting interferometry which does not require accurate initial estimation of phase-shift amounts, only a few pixels with several randomly shifted interferograms are sufficient for accurate extraction of phase information. This method has reduced the dependence of reference phase, and can obtain phase-shifting amount directly without using complex revised algorithm for correcting phase-shifting nonlinear errors.

Opto-Mechanical Properties of PTFE Fibres Using Phase Shifting Interferometry

Polytetrafluoroethylene (PTFE) is one of a class of polymer that has many unique properties, which make it valuable in scores of applications. Here, the molecular alignment effect in fibre drawing and its relation with nonlinear optical properties of the PTFE fibres were studied by means of interferometry. For this propose, the phase shifting Mach-Zehnder interferometer was utilized. The fibre drawing device was attached to the interferometer for measuring the opto-mechanical properties of PTFE fibres. From the recorded interferomgrams the wrapped phase map and its contour line were determined using the phase shifting interferometry (PSI) method. From the contour lines the refractive index, refractive index profile, birefringence, polaryzability and orientation function of the investigated sample were determined at different draw ratios. The PSI method showed good results for enhancing the noisy fringe pattern and for obtaining contour line microinterferograms. This study offers the correlation between the mechanical properties and the structure properties of fibres. Thus, a better structure-property correlation can improve the technical and industry performance of the fibres.

Frequency domain phase extraction algorithm for phase-shifting interferometry with random phase-shifting amount and low-sampling rate

To extract the exact phase of the measured object in the environmental conditions with random noise, a novel kind of frequency domain phase extraction method for phase-shifting interferometry with random phase-shifting amount and low-sampling rate is proposed, in which the time-varied intensity signal of an arbitrary pixel of phase-shifting interferograms is cut into a lot of sub-sections intensity signals with the same length, and then the frequency domain weighted stack operation is performed for the corresponding Fourier spectrum of these sub-section intensity signals. Importantly, the phase obtained from the proposed method coincides well with the conventional four-step phase-shifting algorithm, but the requirement for the environmental conditions in the former are greatly lowered than that in the latter, moreover, while the same number of phase-shifting interferograms is employed to perform the phase extraction, the phase measuring accuracy in the proposed method is greatly higher than that in the time domain Fourier transform (TFT) method.

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.

Comparison of phase retrieval methods for optical diffraction tomography

We compare three different phase retrieval methods for applications in optical diffraction tomography (ODT). The first method is based on Gabor holography and is limited by the 'twin-image' problem. The second is an iterative approach for which the uniqueness of the retrieved phase is questionable. In the third method the phase is measured by phase shifting interferometry (PSI), but this technique is more demanding experimentally. Using realistic computer simulations, which include noise and lens aberrations, we show that the PSI method determines the phase accurately. The iterative method gives equally accurate results in most cases. But in some cases this method produces a false phase. This problem is demonstrated using both simulated and experimental intensity data. The images reconstructed from the false phases are found to be severely distorted. We also show that the uniqueness problem can be avoided by increasing the distance between the scattering object and the measurement line.