Phase shifter calibration in phase-shifting interferometry

Virtual temporal phase-shifting phase extraction using generative adversarial networks

In this paper, a virtual temporal phase-shifting method based on generative adversarial networks (GANs) is proposed to realize dynamic measurement, which is referred to as PSNet. The proposed PSNet can produce the virtual phase-shifting fringe patterns from the single-frame fringe pattern, and the standard n-step phase-shifting method is employed to obtain the wrapped phase from the virtual fringe patterns. The wrapped phase can further be unwrapped by the unwrapping algorithm. Simulation analysis shows that the PSNet can produce the virtual phase-shifting fringe patterns without noise, and thus, the denoising process is eliminated. The performance of the method is verified from the captured fringe in the interferometer, which demonstrates the practicability of the proposed method.

Phase-shifting algorithms with known and unknown phase shifts: comparison and hybrid

The phase-shifting interferometry has been intensively studied for more than half a century, and is still actively investigated and improved for more demanding precision measurement requirements. A proper phase-shifting algorithm (PSA) for phase extraction should consider various error sources including (i) the phase-shift errors, (ii) the intensity harmonics, (iii) the non-uniform phase-shift distributions and (iv) the random additive intensity noise. Consequently, a large pool of PSAs has been developed, including those with known phase shifts (abbreviated as kPSA) and those with unknown phase shifts (abbreviated as uPSA). While numerous evaluation works have been done for the kPSAs, there are very few for the uPSAs, making the overall picture of the PSAs unclear. Specifically, there is a lack of (i) fringe pattern parameters' restriction analysis for the uPSAs and (ii) performance comparison within the uPSAs and between the uPSAs and the kPSAs. Thus, for the first time, we comprehensively evaluated the pre-requisites and performance of four representative uPSAs, the advanced iterative algorithm, the general iterative algorithm (GIA), the algorithm based on the principal component analysis and the algorithm based on VU factorization, and then compare the uPSAs with twelve benchmarking kPSAs. From this comparison, the demand for proper selection of a kPSA, and the restriction and attractive performance of the uPSAs are clearly depicted. Due to the outstanding performance of the GIA, a hybrid kPSA-GIA is proposed to boost the performance of a kPSA and relieve the fringe density restriction of the GIA.

Step Surface Profile Measurement Based on Fringe Projection Phase-Shifting Using Selective Sampling

Fringe projection is a non-contact optical method that is widely used in the optical precision measurement of complex stepped surfaces. However, the accuracy of the fringe phase extraction employed has a direct impact on the measurement precision of the surface shape. Where phase-shifting measurement is used, the classical equal step phase extraction algorithm can only be used to measure simple and smooth surfaces, and leads to measurement errors on complex stepped surfaces, which affects the accuracy of the phase extraction. In addition, the iterative process lasts for a long time, resulting in a low efficiency. This paper proposes a step-by-step phase-shifting extraction algorithm based on selective sampling to measure the contour of the stepped surface. Firstly, the fringe pattern is sampled at equal intervals to reduce the iterative calculation time. Finally, the accurate measurement phase is calculated by the alternating iteration method. The phase extraction accuracy and iteration times are compared in experimental measurements between classical iterative algorithms such as four-step phase-shifting algorithms and the variable phase shift phase interpolation algorithm based on selective sampling. It is shown that the variable frequency phase-shifting extraction algorithm based on selective sampling has a shorter operation time, smaller error, and higher accuracy than the traditional iterative algorithm in fringe projection measuring complex stepped surfaces.

Suppressing ripple distortions and spurious pistons in phase-shifting interferometry

In this paper, we present new phase-shifting algorithms (PSAs) that suppress the ripple distortions and spurious pistons in phase-shifting interferometry. These phase errors arise when non-uniform phase-shifting interferograms are processed with PSAs that assume uniform phase shifts. By modeling the non-uniform phase shifts as a polynomial of the unperturbed phase-shift value $\omega_0$ω₀, we show that the conditions for eliminating the ripple distortion and the spurious piston are associated with the $m$mth derivative of the PSA's frequency transfer function (FTF). Thus, we propose an approach to design robust algorithms based on the FTF formalism and we present four ready-to-apply PSAs formulas. Finally, our conclusions are supported by computer simulations.

Sub-nanometer height sensitivity by phase shifting interference microscopy under environmental fluctuations

Phase shifting interferometric (PSI) techniques are among the most sensitive phase measurement methods. Owing to its high sensitivity, any minute phase change caused due to environmental instability results into, inaccurate phase measurement. Consequently, a well calibrated piezo electric transducer (PZT) and highly-stable environment is mandatory for measuring accurate phase map using PSI implementation. Here, we present an inverse approach, which can retrieve phase maps of the samples with negligible errors under environmental fluctuations. The method is implemented by recording a video of continuous temporally phase shifted interferograms and phase shifts were calculated between all the data frames using Fourier transform algorithm with a high accuracy ≤ 5.5 × 10^-4 π rad. To demonstrate the robustness of the proposed method, a manual translation of the stage was employed to introduce continuous temporal phase shift between data frames. The developed algorithm is first verified by performing quantitative phase imaging of optical waveguide and red blood cells using uncalibrated PZT under the influence of vibrations/air turbulence and compared with the well calibrated PZT results. Furthermore, we demonstrated the potential of the proposed approach by acquiring the quantitative phase imaging of an optical waveguide with a rib height of only 2 nm and liver sinusoidal endothelial cells (LSECs). By using 12-bit CMOS camera the height of shallow rib waveguide is measured with a height sensitivity of 4 Å without using PZT and in presence of environmental fluctuations.vn.

Fast and accurate tilt-shift-immune phase-shifting algorithm based on self-adaptive selection of interferogram subblocks and principal component analysis

To eliminate the effect of tilt-shift error on the accuracy of phase-shifting interferometry (PSI), a fast and accurate tilt-shift-immune phase-shifting algorithm based on the self-adaptive selection of interferogram subblocks and principal component analysis (SSPCA) is proposed. First, each interferogram is divided into several subblocks, and principal component analysis and the least-squares method (LSM) are applied to obtain the phase-shift value of each subblock. Next, according to the correlation coefficients between each phase-shift curve, valid and invalid subblocks can be distinguished. Finally, all phase-shift values of the valid subblocks are used to fit the tilt phase-shift plane, and phase results can be obtained using the LSM. Simulations indicate that the accuracy of SSPCA can reach 0.03 rad both for small (1 rad) and large (${2}\pi $2π rad) tilt amplitudes, and it takes only one-tenth or less of the processing time of iterative algorithms. Experiments proved that SSPCA can be applied even without a precision phase shifter and thus provides a low-cost approach for PSI with both high precision and speed.

Phase measurement of nonuniform phase-shifted interferograms using the frequency transfer function

In this paper, we propose a phase measurement method for interferograms with nonuniform phase shifts. First, we measure the phase shifts between consecutive interferograms. Second, we use these values to modify the spectrum of the interferogram data. Then, by analyzing this spectrum, we design a suitable phase-shifting algorithm (PSA) using the frequency transfer function formalism. Finally, we test our PSA with experimental data to estimate the surface of an aluminum thin film. Our result is better than those obtained using the Fourier transform method, the principal component analysis method, and the least-squares PSA.

Phase-stepping algorithms for synchronous demodulation of nonlinear phase-shifted fringes

In optical metrology synchronous phase-stepping algorithms (PSAs) estimate the measured phase of temporal linear-carrier fringes with respect to a linear-reference. Linear-carrier fringes are normally obtained using closed-loop, feedback, optical phase-stepped devices. On the other hand, open-loop phase-stepping devices usually give fringe patterns with nonlinear phase steps. The Fourier spectrum of linear-carrier fringes is composed of Dirac deltas only. In contrast, nonlinear phase-shifted fringes are wideband, spread-spectrum signals. It is well known that using linear-phase reference PSA to demodulate nonlinear phase stepped fringes, one obtains a spurious-piston. The problem with this spurious-piston is that it may wrongly be interpreted as a real thickness in any absolute phase measurement. Here we mathematically find the origin of this spurious-piston and design nonlinear phase-stepping PSAs to cope with nonlinear phase-stepping interferometric fringes. We give a general theory to tailor nonlinear phase-stepping PSAs to synchronously demodulate nonlinear phase-stepped wideband fringes.

𝛍m-resolution thickness distribution measurement of transparent glass films by using a multi-wavelength phase-shift extraction method in the large lateral shearing interferometer

In this paper, a precise phase-shift extraction method was introduced for the first time to measure the thickness distribution of transparent glass films. A spatial light modulator modulated the phases of the incident laser in a large lateral shearing interferometer. The phase shifting caused by the thickness of the films can be extracted ranging from 0 to 2π, in a recursive way suitable for real-time implementation. Incident lasers with different wavelengths were utilized to measure the spatial distribution of the thickness of the films, and they can be larger than one wavelength of the incident light. Both artificial and experimental results have verified that the resolution of the thickness measurement reaches up to 1 𝛍m.

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.

Surface shape determination with a stitching Michelson interferometer and accuracy evaluation

Stitching methods are increasingly used for determining the surface shape of large and high precision optical elements used in synchrotron beamlines. They consist in reconstructing the surface topography from multiple measurements on overlapping parts of the optics aperture by various algorithms. This paper is an attempt to investigate how true and accurate such a reconstruction can be. Error sources are identified and evaluated throughout the acquisition and processing steps. The analysis is based on the example SOLEIL Michelson interferometer for nano-topography, a dedicated measurement bench for stitching interferometry. We propose a method for determining the error made on the estimate of the interferometric reference surface from the stitching dataset. This determination is made before and independently of the stitching procedure itself.

Dual-directional shearography based on a modified common-path configuration using spatial phase shift

This paper describes a dual-directional shearography system to address the issue of two-dimensional characterization of the surface strain. A common-path configuration coupled with an additional light path is used to provide the shearing in two directions. One of the three interfering beams is shared by both directional shearograms to improve the light efficiency and enhance the robustness of the system. The two directional shearograms are carried by different spatial carriers to distinguish one from the other. The spatial carrier is introduced by the single-aperture-lens Wollaston prism configuration. Rather than the conventional method in which the aperture is fixed at the front focal point of the imaging lens, a general case is considered by introducing a variable distance between the aperture and the imaging lens. The influence of the aperture-lens distance on the spatial carrier is then analyzed, which enables the separate control of the shearing amount and the spatial carrier. Two types of dual-directional shearography are presented to demonstrate the feasibility and the flexibility of the system. Type I is the simultaneous dual lateral shearography in orthogonal directions, and Type II is the simultaneous lateral and radial shearography. The spatial carrier introduced by the single-aperture-lens Wollaston prism configuration is discussed, and a configuration in which the Wollaston prism and the aperture are located at different sides of the lens is recommended for further shearography applications.

Precise phase demodulation of single carrier-frequency interferogram by pixel-level Lissajous figure and ellipse fitting

Phase demodulation from a single carrier-frequency fringe pattern is becoming increasingly important particularly in areas of optical metrology such as dynamic interferometry, deflectometry and profilometry. The Fourier transform (FT) method and the spatial-carrier phase-shifting technique (SCPS) are two popular and well-established approaches to demodulation. However FT has the drawback of significant edge errors because of the Gibbs effect, whilst detuning errors for the local phase shift occur when SCPS is applied. A novel demodulation method based on pixel-level Lissajous figure and ellipse fitting (PLEF) is presented in this paper. Local demodulation in the spatial domain makes PLEF more flexible than the FT method, without spectral leakage. Based on a more adaptable approach, account is taken of variations in illumination and phase distribution over a few neighboring pixels. The mathematic demodulation model is of interest and has been demonstrated via simulation. Theoretical phase extraction error is as low as 10⁻⁴ rad. Experiments further corroborate the effectiveness of the proposed method. In conclusion, various influencing factors, e.g. variations of background/modulation, phase amplitude, carrier frequency, additive noise that may affect the precision of PLEF are discussed in detail.

Phase shifting by wavelength modulation in a cube beam splitter interferometer

A low-cost and fully automated process for phase-shifting interferometry by continuously changing the input voltage of a laser diode under the scheme of a cube beam splitter interferometer, performing as a two-window common-path interferometric setup, is presented. The input signal of a laser diode is controlled by a data acquisition device which permits a change in its wavelength according to its tunability features. The automation and data analysis is done using LabVIEW in combination with MATLAB. By using the Carré algorithm, the phase map is obtained. Measurements of visibility and phase shift to verify the phase-shifting interferometry requirements are also shown.

Piezoelectric Actuated Phase Shifter Based on External Laser Interferometer: Design, Control and Experimental Validation

To improve the phase-shifting accuracy, this paper presents a novel integrated framework for design, control and experimental validation of the piezoelectric actuated phase shifter with a trade-off between accuracy and cost. The piezoelectric actuators with built-in sensors are adopted to drive the double parallel four-bar linkage flexure hinge-based mechanisms. Three mechanisms form the tripod structure of the assembled phase shifter. Then, a semi-closed loop controller with inner feedback and outer feedforward loops via the external laser interferometer is developed for accurate positioning of the phase shifter. Finally, experiments related with travel range, step response, linearity and repeatability are carried out. The linearity error is 0.21% and the repeatability error of 10 μm displacement is 3 nm. The results clearly demonstrate the good performance of the developed phase shifter and the feasibility of the proposed integrated framework.

Phase extraction from fringe pattern via light propagation

A phase demodulation method via light propagation is proposed, where one or two fringe patterns are viewed as the superposition of complex amplitudes, and then the phase is reconstructed by separating the light field via light propagation. Simulation and experimental results indicated that the proposed method can extract the phase from a single shot effectively, thereby realizing dynamic phase retrieval. In addition, the accuracy of phase reconstruction can be improved by adding another fringe pattern with an unknown phase shift. The carrier requirement is relatively low, and, thus, the proposed method can be applied to the measurements with an environment disturbance, an inaccurate phase shift, and the requirement of a high speed capture.

Flexible calibration of phase-to-height conversion in fringe projection profilometry

A flexible new technique is presented to calibrate the monocular system of phase-based fringe projection profilometry, which is made of a camera, a projector, and a computer. The proposed algorithm mainly consists of a more flexible phase-to-height conversion model and a minimum norm solution, followed by a nonlinear optimization based on the maximum likelihood criterion. In the whole calibration procedure, this method only requires the camera to capture a few two-dimensional checkerboard target images and several deformed fringe images with at least three different orientations. The proposed technique neither subjects to certain limitations nor measures additional geometry parameters in advance. Also, no highly precise gauge blocks or extra reference phases are involved. In contrast with the existing methods, the proposed technique is easier to use and more flexible. Experiments have been performed to validate the performance of this technique.

Phase extraction from two phase-shifting fringe patterns using spatial-temporal fringes method

Phase extraction from phase-shifting fringe patterns with unknown phase shift values is a valuable but challenging task, especially when there are only two frames of fringes. In this paper, a phase demodulation method based on the spatial-temporal fringes (STF) method is proposed, where two phase shift fringes with linear carrier are fused into one STF image, and then the measured phase can be extracted from its frequency spectrum. The algorithm is deduced by extending the traditional STF theory with at least three frames of fringes to the two frames case. In the simulations, its performance is compared with the classical Fourier Transform method, and the different carrier and phase step conditions are analyzed where the accuracy can be ensured in most cases. The algorithm is also validated by the experiment, where the reliable result can be given even if the phase shift changes within a wide range.

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.

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.

Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology

The accuracy of phase-shifting interferometers (PSI) is crippled by nonlinearity of the phase shifter and instability of the environment such as vibration and air turbulence. A general algorithm, utilizing Lissajous figures and ellipse fitting, of correcting the phase extraction error in the phase shifting interferometry is described in this paper. By plotting N against D, where N and D represent the numerator and denominator terms of the phase extraction function (i.e. an arctangent function) respectively, a Lissajous ellipse is created. Once the parameters of the ellipse are determined by ellipse fitting, one can transform the ellipse to a unit circle (ETC). Through this process the phase extraction error caused by random phase shift errors can be corrected successfully. Proposed method is non-iterated, adapts to all phase shifting algorithms (PSAs), and has high accuracy. Some factors that may affect the performance of proposed method are discussed in numerical simulations. Optical experiments are implemented to validate the effectiveness of proposed algorithm.

Generation of phase-shifting algorithms with N arbitrarily spaced phase-steps

Phase-shifting (PS) is an important technique for phase retrieval in interferometry (and three-dimensional profiling by fringe projection) that requires a series of intensity measurements with known phase-steps. Usual PS algorithms are based on the assumption that the phase-steps are evenly spaced. In practice, however, this assumption is often not satisfied exactly, which leads to errors in the recovered phase. In this work we present a systematic algebraic approach for generating general PS algorithms with N arbitrarily spaced phase-steps, which present advantages (e.g., the PS error can be avoided) over known algorithms that assume equally spaced phase-steps. Simulations are presented.

Dynamic phase imaging of microscopic measurements using parallel interferograms generated from a cyclic shear interferometer

We present a technique which allows us to generate two parallel interferograms with phase shifts of π/2 using a Cyclic Shear Interferometer (CSI) and a polarizing splitter. Because of the use of a CSI, we obtain the derivative phase data map directly, due to its configuration, it is immune to vibrations because the reference wavefront and the object wavefront have a common path; the shearing interferometer is insensitive to temperature and vibration. To obtain the optical phase data map, two interferograms are generated by collocating a polarizing device at the output of the CSI. The optical phase was processed using a Vargas-Quiroga algorithm. Related experimental results obtained for dynamic microscopic transparent samples are presented.

Measuring coalignment of retroreflectors with large lateral incoming-outgoing beam offset

A method based on phase-shifting Fizeau interferometry is presented with which retroreflectors with large incoming-outgoing beam separations can be tested. The method relies on a flat Reference Bar that is used to align two auxiliary mirrors parallel to each other to extend the aperture of the interferometer. The method is applied to measure the beam coalignment of a prototype Triple Mirror Assembly of the GRACE Follow-On Laser Ranging Interferometer, a future satellite-to-satellite tracking device for Earth gravimetry. The Triple Mirror Assembly features a lateral beam offset of incoming and outgoing beam of 600 mm, whereas the acceptance angle for the incoming beam is only about ±2 mrad. With the developed method, the beam coalignment of the prototype Triple Mirror Assembly was measured to be 9 μrad with a repeatability of below 1 μrad.

Generalized phase-shifting algorithm for inhomogeneous phase shift and spatio-temporal fringe visibility variation

A cascade least-squares scheme for wrapped phase extraction using two or more phase-shifted fringe-patterns with unknown and inhomogeneous surface phase shift is proposed. This algorithm is based on the parameter estimation approach to process fringe-patterns where, except for the interest phase distribution that is a function of the space only, all other parameters are functions of both space and time. Computer simulations and experimental results show that phase computing is possible even when an inhomogeneous phase shift is induced by nonlinearity of the piezoelectric materials or miscalibrated phase shifters. The algorithm's features and its operating conditions will been discussed. Due to the useful properties of this algorithm such as the robustness, computational efficiency, and user-free execution, this proposal could be used in automatic applications.

Robust adaptive phase-shifting demodulation for testing moving wavefronts

Optical interferometers are very sensitive when environment perturbations affect its optical path. The wavefront under test is not static at all. In this paper, it is proposed a novel and robust phase-shifting demodulation method. This method estimates the interferogram's phase-shifting locally, reducing detuning errors due to environment perturbations like vibrations and/or miscalibrations of the Phase-Shifting Interferometry setup. As we know, phase-shifting demodulation methods assume that the wavefront under test is static and there is a global phase-shifting for all pixels. The demodulation method presented here is based on local weighted least-squares, letting each pixel have its own phase-shifting. This is a different and better approach, considering that all previous works assume a global phase-shifting for all pixels of interferograms. Seeing this method like a black box, it receives an interferogram sequence of at least 3 interferograms and returns the modulating phase or wavefront under test. Here it is not necessary to know the phase shifts between the interferograms. It does not assume a global phase-shifting for the interferograms, is robust to the movements of the wavefront under test and tolerates miscalibrations of the optical setup.

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.

3D shape reconstruction of large specular surface

A novel three-dimensional (3D) reconstruction method based on fringe reflection technique for shape measurement of large specular surfaces is presented in this paper, which effectively integrates path integration technique with zonal wavefront reconstruction algorithm. The height information of specular surface obtained from cross-path integration can then be used as the initial value in a zonal wavefront reconstruction algorithm. This method not only has the advantages of global integration, but also enables user-friendly, high-speed operation. A specific iterative algorithm is adopted to improve the antinoise capability of the measuring system, which accelerates the rate of convergence significantly and even improves the accuracy of the reconstructed 3D surface. Moreover, the proper use of boundary contour extraction of the acquired images reduces the computational load of 3D reconstruction dramatically and hence achieves high reconstruction accuracy and enhances the surface integrity at the boundary. An ultraprecision, diamond-turned planar mirror with diameter of 150 mm has been employed to implement the system calibration. The reconstruction results of simulated and actual hyperbolic surfaces and the gauge blocks identify the validity of this new method. It is demonstrated that the measurement error is about 50 μm with reconstruction points of 150×560 pixels of gauge blocks.

Characterization of the phase modulation property of a free-space electro-optic modulator by interframe intensity correlation matrix

Characterization of a phase modulator or phase shifter has always been an integral part of phase-modulating or phase-adjusting applications. We propose a simplified approach to characterize a phase modulator by investigating the performance of phase shifts from grabbed interferograms using the phase extraction method. After reviewing some phase analysis techniques, the interframe intensity correlation (IIC) matrix method is introduced to the investigation. The proposed strategy is illustrated by the measurement of a free-space electro-optic modulator (EOM). Placing the modulator in one arm of a Michelson interferometer, the global phase shifts are estimated by the IIC method from the phase-stepped interferograms. Experimental results demonstrate the tested EOM has a phase modulation response of at least 2π  rad with a π/20  rad modulation precision for λ=1064  nm. In addition, our method is applicable to various types of phase modulator or phase shifter calibration, e.g., electro-optic phase modulator, spatial light modulator, or piezoelectric transducer (PZT).

Diffractive generalized phase contrast for adaptive phase imaging and optical security

We analyze the properties of Generalized Phase Contrast (GPC) when the input phase modulation is implemented using diffractive gratings. In GPC applications for patterned illumination, the use of a dynamic diffractive optical element for encoding the GPC input phase allows for on-the-fly optimization of the input aperture parameters according to desired output characteristics. For wavefront sensing, the achieved aperture control opens a new degree of freedom for improving the accuracy of quantitative phase imaging. Diffractive GPC input modulation also fits well with grating-based optical security applications and can be used to create phase-based information channels for enhanced information security.

Two-wavelength in-line phase-shifting interferometry based on polarizing separation for accurate surface profiling

We describe a configuration that can be used for two-wavelength phase-shifting in-line interferometry based on polarizing separation. The experiment is conducted on a sample with a step height of 1.34 μm nominally. In this paper, five- and seven-phase step algorithms have been compared for their effectiveness in reducing the noise in the phase maps. The noise is further reduced by the application of the flat fielding method. The recorded interferograms are processed using seven-phase step algorithm to obtain the phase map for each wavelength separately. The independent phase maps are subtracted and a phase map for the beat-wavelength is obtained and converted to height map. The results extracted from the seven-phase step algorithm have been compared with the results extracted from the single shot off-axis geometry and the results are in agreement.

Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method

A method combining the principal component analysis (PCA) and the least squares method (LSM) is proposed to extract the phase from interferograms with random phase shifts. The method estimates the initial phase by PCA, and then determines the correct global phase sign and reduces the residual phase error by LSM. Some factors that may influence the performance of the proposed method are analyzed and discussed, such as the number of frames used, the number of fringes in interferogram and the amplitude of random phase shifts. Numerical simulations and optical experiments are implemented to verify the effectiveness of this method. The proposed method is suitable for randomly phase-shifted interferograms.

Integrating fault tolerance algorithm and circularly polarized ellipsometer for point-of-care applications

A circularly polarized ellipsometer was developed to enable real-time measurements of the optical properties of materials. Using a four photo-detector quadrature configuration, a phase modulated ellipsometer was substantially miniaturized which has the ability to achieve a high precision detection limit. With a proven angular resolution of 0.0001 deg achieved by controlling the relative positions of a triangular prism, a paraboloidal and a spherical mirror pair, this new ellipsometer possesses a higher resolution than traditional complex mechanically controlled configurations. Moreover, the addition of an algorithm, FTA (fault tolerance algorithm) was adopted to compensate for the imperfections of the opto-mechanical system which can decrease system measurement reliability. This newly developed system requires only one millisecond or less to complete the measurement task without having to adopt any other modulation approach. The resolution achieved can be as high as 4x10(-7) RIU (refractive index unit) which is highly competitive when compared with other commercially available instruments. Our experimental results agreed well with the simulation data which confirms that our quadrature-based circularly polarized ellipsometer with FTA is an effective tool for precise detection of the optical properties of thin films. It also has the potential to be used to monitor the refractive index change of molecules in liquids.

Statistical phase-shifting step estimation algorithm based on the continuous wavelet transform for high-resolution interferometry metrology

We propose a statistical phase-shifting estimation algorithm for temporal phase-shifting interferometry (PSI) based on the continuous wavelet transform (CWT). The proposed algorithm explores spatial information redundancy in the intraframe interferogram dataset using the phase recovery property on the power ridge of the CWT. Despite the errors introduced by the noise of the interferogram, the statistical part of the algorithm is utilized to give a sound estimation of the phase-shifting step. It also introduces the usage of directional statistics as the statistical model, which was validated, so as to offer a better estimation compared with other statistical models. The algorithm is implemented in computer codes, and the validations of the algorithm were performed on numerical simulated signals and actual phase-shifted moiré interferograms. The major advantage of the proposed algorithm is that it imposes weaker conditions on the presumptions in the temporal PSI, which, under most circumstances, requires uniform and precalibrated phase-shifting steps. Compared with other existing deterministic estimation algorithms, the proposed algorithm estimates the phase-shifting step statistically. The proposed algorithm allows the temporal PSI to operate under dynamic loading conditions and arbitrary phase steps and also without precalibration of the phase shifter. The proposed method can serve as a benchmark method for comparing the accuracy of the different phase-step estimation methods.

Phase-shift extraction for phase-shifting interferometry by histogram of phase difference

We propose a non-iterative approach to extract the unknown phase shift in phase shifting interferometry without the assumption of equal distribution of measured phase in [0,2π]. According to the histogram of the phase difference between two adjacent frames, the phase shift can be accurately extracted by finding the bin of histogram with the highest frequency. The main factors that influence the accuracy of the proposed method are analyzed and discussed, such as the random noise, the quantization bit of CCD, the number of fringe patterns used and the bin width of histogram. Numerical simulations and optical experiments are also implemented to verify the effectiveness of this method.

Phase-shifts n pi/2 calibration method for phase-stepping interferometry

A novel phase-shifts n pi/2 calibration method for phase-stepping interferometry, in which the sum intensities and the squared sum of a series of phase-shifted interferograms are firstly calculated, and then by the minimization of the variances of these sums perform the n pi/2 phase-shifts calibration, is proposed in this paper. The proposed method can overcome effectively the effect of the variations of background and modulation intensities in interferograms with any phase structure, and it is also insensitive to the nonlinearity of phase shifter. Numerical simulation and experiments are implemented to verify the effectiveness of this method.

Easy and straightforward construction of wideband phase-shifting algorithms for interferometry

We show a practical way for building wideband phase-shifting algorithms for interferometry. The idea presented combines first- and second-order quadrature filters to obtain wideband phase-shifting algorithms. These first- and second-order quadrature filters are analogous to the first- and second-order filters commonly used in communications engineering, named building blocks. We present a systematic way to develop phase-shifting algorithms with large detuning robustness or large bandwidth. In general, the approach presented here gives a powerful frequency analysis and design tool for phase-shifting algorithms robust to detuning for interferometry.

High speed interferometric ellipsometer

A novel high speed interferometric ellipsometer (HSIE) is proposed and demonstrated. It is based on a novel differential-phase decoder which is able to convert the phase modulation into amplitude modulation in a polarized heterodyne interferometer. Not only high detection sensitivity but also fast response ability on ellipsometric parameters (EP) measurements based on amplitude-sensitive method is constructed whereas different amplitudes with respect to P and S polarized heterodyne signals in this phase to amplitude modulation conversion is discussed. The ability of HSIE was verified by testing a quarter wave plate while a real time differential-phase detection of a liquid crystal device versus applied voltage by using HSIE was demonstrated too. These results confirm that HSIE is able to characterize the optical property of specimen in terms of EP at high speed and high detection sensitivity experimentally.

Interference fringe analysis based on centroid detection

Phase measurement of interference fringes is an integral part of several fields in optics. Using simple straight sinusoidal fringe patterns, we describe the relationship between fringe position or phase to the centroid position when these fringes are incident on a position sensitive detector. With detailed descriptions and some experimental results, we show that a phenomenal sensitivity is possible in principle with what we believe is a new approach of phase measurement, and excellent sensitivity is readily achieved.

Phase-step calibration technique based on a two-run-times-two-frame phase-shift method

A novel phase-step calibration technique is presented on the basis of a two-run-times-two-frame phase-shift method. First the symmetry factor M is defined to describe the distribution property of the distorted phase due to phase-shifter miscalibration; then the phase-step calibration technique, in which two sets of two interferograms with a straight fringe pattern are recorded and the phase step is obtained by calculating M of the wrapped phase map, is developed. With this technique, a good mirror is required, but no uniform illumination is needed and no complex mathematical operation is involved. This technique can be carried out in situ and is applicable to any phase shifter, whether linear or nonlinear.

High-precision methods and devices for in situ measurements of thermally induced aberrations in optical elements

An optical system that comprises two devices for remote measurements, a broadband optical interferometer and a scanning Hartmann sensor, is described. The results of simultaneous measurements with both devices and the results of numerical modeling of sample surface heating are presented.

Adaptive-optics system with liquid-crystal phase-shift interferometer

We develop an adaptive-optics system based on a Mach-Zehnder radial shearing interferometer with liquid-crystal-device (LCD) phase-shift interferometry (PSI). Using accurate phase calibration and transient nematic driving of the LCD, the developed three-step PSI procedure can be achieved in a time of 5 ms. The proposed Mach-Zehnder radial shearing PSI method reconstructs the phase information using a digital signal processor (DSP). The DSP then computes appropriate control signals to drive a deformable mirror in such a way as to eliminate the wavefront distortion. The current adaptive-optics system is capable of suppressing low-frequency thermal disturbances with a signal-to-noise ratio improvement of more than 20 dB and a steady-state phase error of less than 0.02pi root mean square when the control loop is operated at a frequency of 30 Hz.