Effect of piezoelectric transducer nonlinearity on phase shift interferometry

Holographic flow scanning cytometry overcomes depth of focus limits and smartly adapts to microfluidic speed

Space-time digital holography (STDH) maps holograms in a hybrid space-time domain to achieve extended field of view, resolution enhanced, quantitative phase-contrast microscopy and velocimetry of flowing objects in a label-free modality. In STDH, area sensors can be replaced by compact and faster linear sensor arrays to augment the imaging throughput and to compress data from a microfluidic video sequence into one single hybrid hologram. However, in order to ensure proper imaging, the velocity of the objects in microfluidic channels has to be well-matched to the acquisition frame rate, which is the major constraint of the method. Also, imaging all the flowing samples in focus at the same time, while avoiding hydrodynamic focusing devices, is a highly desirable goal. Here we demonstrate a novel processing pipeline that addresses non-ideal flow conditions and is capable of returning the correct and extended focus phase contrast mapping of an entire microfluidic experiment in a single image. We apply this novel processing strategy to recover phase imaging of flowing HeLa cells in a lab-on-a-chip platform even when severely undersampled due to too fast flow while ensuring that all cells are in focus.

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.

Correction of PZT scanner errors using a phase compensation method in white-light phase-shifting interferometry

White-light phase-shifting interferometry (WLPSI) is widely used for obtaining surface profiles of inspection targets in liquid-crystal displays (LCD) and semiconductor manufacturing processes. Phase errors caused by irregular and nonlinear scanner movement are inevitable in WLPSI. Moreover, these errors affect the measurement stability. It is difficult to eliminate the mechanical and electrical delay of scanner movement entirely. A novel method, to the best of our knowledge, is proposed that consists of a scanner position tracking system and a phase correction algorithm called compensated peak detection and modified bucket algorithm. The proposed method is experimentally verified and outperforms the conventional phase corrections in both the measurement accuracy and the repeatability.

Linescan Lattice Microscopy: A Technique for the Accurate Measurement and Mapping of Lattice Spacing and Strain with Atomic Force Microscopy

Lateral resolution and accuracy in scanning probe microscopies are limited by the nonideality of piezoelectric scanning elements due to phenomena including nonlinearity, hysteresis, and creep. By taking advantage of the well-established atomic-scale stick-slip phenomenon in contact-mode atomic force microscopy, we have developed a method for simultaneously indexing and measuring the spacing of surface atomic lattices using only Fourier analysis of unidirectional linescan data. The first step of the technique is to calibrate the X-piezo response using the stick-slip behavior itself. This permits lateral calibration to better than 1% error between 2.5 nm and 9 μm, without the use of calibration gratings. Lattice indexing and lattice constant determination are demonstrated in this way on the NaCl(001) crystal surface. After piezo calibration, lattice constant measurement on a natural bulk MoS₂(0001) surface is demonstrated with better than 0.2% error. This is used to measure nonuniform thermal mismatch strain for chemical vapor deposition (CVD)-grown monolayer MoS₂ as small as 0.5%. A spatial mapping technique for the lattice spacing is developed and demonstrated, with absolute accuracy better than 0.2% and relative accuracy better than 0.1%, within a map of 12.5 × 12.5 nm² pixels using bulk highly oriented pyrolytic graphite (HOPG) and MoS₂ as reference materials.

Dynamic displacement measurement beyond half-wavelength in phase-modulated optical interferometer

We demonstrate an interference signal-processing method that extends the measurement range of dynamic displacement beyond half the wavelength of light without deteriorating the measurement accuracy in a phase-modulated optical interferometer. The measurement range extension is realized by the algorithm focusing on peak direction of the interference signal waveform. Deterioration of measurement accuracy is avoided by the data processing that removes noisy data sets. Performance of the proposed system is evaluated by the experiments using a pseudo-vibrator composed of a phase modulator. In the experiments using a lead zirconate titanate (PZT) device as a sample, we successfully measure the dynamic displacement of up to 3127 nm, which is larger than the light source wavelength of 1537 nm. Proof-of-principle simulations, including the evaluation of the measurement error, are also conducted, the results of which show that the measurement error of the proposed method is small in principle.

Tukey's robust M-estimator for phase demodulation of interferograms with nonuniform shifts

In this paper, we introduce an iterative scheme for phase demodulation of interferograms with nonuniformly spaced phase shifts. Our proposal consists of two stages: first, the phase map is obtained through a least squares fitting; second, the phase steps are retrieved using a statistical robust estimator. In particular, we use Tukey's biweighted M-estimator because it can cope with both noisy data and outliers in comparison with the ordinary least squares estimator. Furthermore, we provide the frequency description of the algorithm and the phase demodulation allowing us to analyze the procedure and estimation according to the frequency transfer function (FTF) formalism for phase-shifting algorithms. Results show that our method can accurately retrieve the phase map and phase shifts, and it converges by the 10th iteration.

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.

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.

Spectral analysis for the generalized least squares phase-shifting algorithms with harmonic robustness

We introduce the frequency transfer function (FTF) formalism for generalized least squares phase-shifting algorithms (GLS-PSAs), whose phase shifts are nonuniformly spaced. The GLS-PSA's impulsive response is found by computing the Moore-Penrose pseudoinverse. FTF theory allows analyzing these GLS-PSAs spectrally, as well as easily finding figures of merit such as signal-to-noise ratio (SNR) and harmonic rejection capabilities. We show simulations depicting that the SNR slightly decreases as the harmonic rejection robustness improves.

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.

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.

An Optical Crack Growth Sensor Using the Digital Sampling Moiré Method

High-accuracy crack growth measurement is crucial for the health assessment of concrete structures. In this work, an optical crack growth sensor using the digital sampling moiré (DSM) method is developed for two-dimensional (2D) crack growth monitoring. The DSM method generates moiré fringes from a single image through digital image processing, and it measures 2D displacements using the phase difference of moiré fringes between motion. Compared with the previous sensors using traditional photogrammetric algorithms such as the normalized cross-correlation (NCC) method, this new DSM-based sensor has several advantages: First, it is of a higher sensitivity and lower computational cost; second, it requires no prior calibration to get accurate 2D displacements which can greatly simplify the practical application for multiple crack monitoring. In addition, it is more robust to the change of imaging distance, which is determined by the height difference between two sides of a concrete crack. These advantages break the limitation of the NCC method and broaden the applicability of the crack growth sensor. These advantages have been verified with one numerical simulation and two laboratory tests.

Complete characterization of ultrashort optical pulses with a phase-shifting wedged reversal shearing interferometer

The ability of an interferometer to characterize the spatial information of a light beam is often limited by the temporal profile of the beam, with femtosecond pulse characterization being particularly challenging. In this study, we developed a simple, stable, controllable shearing and vectorial phase-shifting wedged reversal shearing interferometer that is able to characterize all types of coherent and partially coherent light beams. The proposed interferometer consists of only a single beam splitter cube with one wedged entrance face and is insensitive to environmental vibration due to its common path configuration. A near zero-path length difference of the proposed interferometer ensures its operation for ultrashort pulses, providing, for the first time, a simple and stable interferometric tool to fully characterize sub-100  fs laser pulses. All common beam characterization can be carried out with the interferometer, such as the amplitude, phase, polarization, wavelength, and pulse duration. Furthermore, this technique is sensitive to the wavefront tilt and can be used for precise beam alignment. Therefore, this interferometer can be an essential tool for beam characterization, optical imaging, and the testing required for a wide range of applications, including astronomy, biomedicine, ophthalmology, optical testing and imaging systems, and adaptive optics.

Phase sensitivity evaluation and its application to phase shifting interferometry

In quantitative phase imaging, sensitivity is a key measure of system reproducibility. Despite continuous experimental breakthroughs in achieving highly sensitive detection, in-depth studies of theoretical constraints on sensitivity are inadequate and comparisons between different techniques are difficult. In this paper, we introduce the method to evaluate the sensitivity of phase shifting interferometry which is a major category of quantitative phase imaging techniques. The method discusses in detail several key concepts of sensitivity evaluation, including a general three-level evaluation framework, a complete interference signal model, Cramér-Rao bound and algorithm sensitivity, algorithm and system efficiencies, as well as energy efficiency of an algorithm. In discussions of specific phase shifting algorithms, we focus on the shot noise-limited model. This simplified model not only reflects the rapid developments in modern detectors that are often dominated by shot noise, but also permits the calculation of theoretical sensitivities based on measured data, which is important in evaluating experimental performance. As examples, we study several common phase shifting interferometric techniques. The results of different techniques are compared to provide insights into algorithm optimization and energy efficiency of sensitivity. A normalized algorithm sensitivity table is also provided for readers to conveniently estimate their system's algorithm sensitivity and compare against experiments.

Multi-region phase calibration of liquid crystal SLM for holographic display

The liquid crystal spatial light modulator (SLM) is able to provide flexible wavefront control, whereas the initial phase and its response distortions will heavily influence the modulation accuracy. The currently existing calibration methods are tedious and time consuming. A novel multi-region calibration method for minimizing those distortions is proposed. The entire panel is divided into several local regions based on the similarity of phase response characteristic. The nonlinear phase response and static phase distortion of each local region are calibrated in the iterative division procedure. The calibration method is theoretically analyzed and experimentally verified. For the Jasper 4 K SLM panel, when five local regions are built, the root mean error of linear phase shifts is reduced to 0.1 rad and the compensation accuracy of the static phase distortion reaches 0.24 wavelength. The calibrated SLM is applied for the color holographic display and the results show that the reconstructed image quality is improved significantly. The proposed method is simpler and faster because of the reasonable regional division and lower calibration complexity. It could be used for the calibration of various phase only or complex modulators with high space bandwidth product in the future.

Non-contact single shot elastography using line field low coherence holography

Optical elastic wave imaging is a powerful technique that can quantify local biomechanical properties of tissues. However, typically long acquisition times make this technique unfeasible for clinical use. Here, we demonstrate non-contact single shot elastographic holography using a line-field interferometer integrated with an air-pulse delivery system. The propagation of the air-pulse induced elastic wave was imaged in real time, and required a single excitation for a line-scan measurement. Results on tissue-mimicking phantoms and chicken breast muscle demonstrated the feasibility of this technique for accurate assessment of tissue biomechanical properties with an acquisition time of a few milliseconds using parallel acquisition.

Measurement of the surface form error of large aperture plane optical surfaces with a polarization phase-shifting liquid reference reflection Fizeau interferometer

A polarization phase-shifting liquid reference reflection Fizeau interferometer has been proposed. A polarization cyclic path optical configuration along with a concave telescope mirror is used to produce a pair of expanded, collimated p and s polarized beams with a small angular separation between them. The collimated beams are deflected along a vertical direction toward a Fizeau interferometer cavity formed between a liquid surface that acts as a reference surface and a plane test surface. Either the p or s polarized beam is allowed to strike the liquid surface normally and the orientation of the test surface is adjusted to reflect the other beam, having orthogonal linear polarization, in the direction of the normally reflected reference beam from the liquid surface. A combination of a quarter-wave plate and linear polarizer is used to apply polarization phase shift between the test and reference beams, and quantitative surface form error is measured by applying phase-shifting interferometry. A method for elimination of the residual system aberration is discussed. Results obtained for an optically polished BK-7 disk of clear aperture diameter ≈160  mm are presented.

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.

Determination of the surface form error of a spherical mirror with phase shifting Sagnac interferometer

A polarization Sagnac interferometer (SI) is used to produce two laterally separated, identical, convergent emergent beams with linear orthogonal polarizations. The emergent p-polarized and s-polarized beams converge toward their respective focal points. The test and reference spherical mirrors are placed at confocal positions with respect to the s and p focal points so as to normally reflect back the test and reference beams through the SI that recombines the test and reference waves. Polarization phase shifting interferometry is applied to obtain the surface form error of the test surface with respect to the reference surface. A two-step measurement procedure eliminates the system aberrations. Results obtained for a concave spherical test surface with respect to a convex spherical reference surface are presented. The optical configuration is relatively less susceptible to external mechanical vibration.

Open-loop phase shifting for fast acquisition of interferograms in low light levels

Phase shifting interferometry relies on sets of interferograms taken at multiple known phase offsets to deduce the instantaneous phase of a quasi-static fringe pattern. The traditional method for introducing these phase shifts has been either to step a mirror, and measure the fringe pattern at each step, or to scan a mirror, integrating the fringe pattern for discrete time intervals while the fringes "move" on the detector. A stepping mirror eliminates this fringe smear but has typically required a closed-loop controller to ensure that the optical path introduced is accurately known. Furthermore, implementing rapid stepping of a moderately sized optic can prove difficult if the fringe phase needs to be measured on a short time scale. We report results demonstrating very fast (>100 Hz) and precise phase shifting using a piezomodulated mirror operated in open-loop without any position feedback. Our method exploits the use of a synthetic driving waveform that is optimized to match the complex frequency response of the modulator and its supported optic. For phase measurements in the near-infrared at 2.15 μm, and with a time between steps as small as 0.2 ms, we report errors below λ/100 in the desired position of our optic, i.e., an effective optical path difference error of ~λ/55. For applications in near-infrared stellar interferometry, this implies an enhancement in the fringe-tracking sensitivity of roughly 20% (in the photon-limited regime) over that which is conventionally realized using a swept mirror.

Three-dimensional phase step profilometry with a multicore optical fiber

This paper demonstrates the feasibility of using phase stepping and a multicore optical fiber to calculate an object's depth profile. An interference pattern is projected by an optical fiber onto the object. The distorted interference pattern containing the object information is captured by a CCD camera and processed using a phase step interferometry method. The phase step method is less computationally intensive compared to two-dimensional Fourier transform profilometry and provides more accuracy when measuring objects of high frequency spatial variations.

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.

Invited review article: measurement uncertainty of linear phase-stepping algorithms

Phase retrieval techniques are widely used in optics, imaging and electronics. Originating in signal theory, they were introduced to interferometry around 1970. Over the years, many robust phase-stepping techniques have been developed that minimize specific experimental influence quantities such as phase step errors or higher harmonic components of the signal. However, optimizing a technique for a specific influence quantity can compromise its performance with regard to others. We present a consistent quantitative analysis of phase measurement uncertainty for the generalized linear phase stepping algorithm with nominally equal phase stepping angles thereby reviewing and generalizing several results that have been reported in literature. All influence quantities are treated on equal footing, and correlations between them are described in a consistent way. For the special case of classical N-bucket algorithms, we present analytical formulae that describe the combined variance as a function of the phase angle values. For the general Arctan algorithms, we derive expressions for the measurement uncertainty averaged over the full 2π-range of phase angles. We also give an upper bound for the measurement uncertainty which can be expressed as being proportional to an algorithm specific factor. Tabular compilations help the reader to quickly assess the uncertainties that are involved with his or her technique.

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.

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.

Systematic errors of phase-shifting speckle interferometry

A theoretical and numerical investigation of the systematic phase errors in phase-shifting speckle interferometry is presented. The theoretical investigation analyzes the behavior of some systematic error induced by intensity variations in two cases of data-computing techniques. The first case deals with the technique in which the phase change is computed, unwrapped, and then linearly filtered; the second case deals with the technique in which the data are linearly filtered before the arctangent calculation and then unwrapped. With the first filtering technique it is shown that it is preferable when the phase change is of relatively low spatial frequency, leading to a particularly accurate method. With the second case it is demonstrated that an important parameter of speckle interferometry is the modulation factor of the interference frame that induces phase errors when the data are filtered before the arctangent calculation. We show that this technique is better than the first when the phase change is composed of high-spatial-frequency variations. The theoretical investigation of the two techniques is compared with numerical simulations, considering the frequency characteristics of the phase change, and this shows a good match between theory and simulations.

Multiphase homodyne interferometry: analysis of some error sources

Some sources of error with multiphase homodyne interferometry are reviewed. A major advantage over the classic two-channel approach is that the inaccuracies that originate from laser-power fluctuations and drifts are shown to be automatically compensated for by proper adjustment of the light beams.

Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting

We present a new computational algorithm of phase-shifting interferometry that can effectively eliminate the uncertainty errors of reference phases encountered when we obtain multiple interferograms. The algorithm treats the reference phases as additional unknowns and we determine their exact values by analyzing interferograms using the numerical least-squares technique. A series of simulations prove that this algorithm can improve measuring accuracy because it is unaffected by the nonlinear and random errors of phase shifters.

Phase-measuring interferometry: new methods and error analysis

New methods that can be used to determine phase in phase-stepping interferometry are presented. It is shown that a combination of some of these methods can be used to reduce the error introduced by phase-stepper miscalibration and nonlinearity. Moreover these new algorithms can also be used to detect the presence of miscalibration or phase-shifter nonlinearity. A simplified approach to understanding the error introduced by miscalibration and nonlinearity of the phase stepper and its reduction in phase-shifting interferometry is also presented.

Precise method of determining systematic errors in phase-shifting interferometry on Fizeau interferences

Fizeau phase-stepping interferometry permits multiple-beam interferences to be evaluated exactly with a resolution in the subnanometer range when a special four-bucket algorithm is applied. The step width, which is set at 2π/4 for the four-bucket algorithm, has to be maintained precisely to avoid erroneousphase determination. A method is described by which deviations from the exact value of the step width can be quantitatively determined. In numerical calculations the phase dependence of the step-width error is determined and confirmed by accompanying experiments. Step-width errors recognized by the phase dependence of a difference measurement can be corrected by means of a general four-bucket algorithm. The topography of Fabry-Perot plates can therefore be measured with a high degree of accuracy, of λ/200.

Linear approximation for measurement errors in phase shifting interferometry

This paper shows how measurement errors in phase shifting interferometry (PSI) can be described to a high degree of accuracy in a linear approximation. System error sources considered here are light source instability, imperfect reference phase shifting, mechanical vibrations, nonlinearity of the detector, and quantization of the detector signal. The measurement inaccuracies resulting from these errors are calculated in linear approximation for several formulas commonly used for PSI. The results are presented in tables for easy calculation of the measurement error magnitudes for known system errors. In addition, this paper discusses the measurement error reduction which can be achieved by choosing an appropriate phase calculation formula.

Fiber optic interferometer for testing conic section surfaces

A null test method for testing aspheric optical surfaces based on geometric considerations of conic sections is presented. The basic principles have been investigated experimentally by designing and building a modified Mach-Zehnder interferometer. By using optical fibers the inconvenience of alignment has partly been overcome. Phase shifting has been implemented and the measurements are performed with the help of a microcomputer. The performance of the interferometer has been investigated by testing an elliptical toroid mirror. Further improvement of the test procedure by Zernike polynomial decomposition is discussed and performed. Finally, an even more compact conic section interferometer is proposed.

Fiber optic phase stepping system for interferometry

A closed loop phase control system using an all-fiber optical configuration has been developed for use in phase stepping interferometry. This system drives the relative phase of two interfering beams through a sequence of pi/2 rad increments so that the initial relative phase of these beams can be determined. This phase stepping system uses optical fibers to provide spatially uniform phase steps from a flexible, easily aligned optical configuration. In addition, this system uses phase feed back to eliminate phase modulator errors and to compensate for phase drifts caused by environmental disturbances.

Interferometric measurement of fiber strain hysteresis and nonlinearity for sensitivity optimization

A simple and reliable fringe-counting method is used to determine the fiber static elastic response in fiber-optic sensors and modulators. Large strain hysteresis and nonlinearity have been observed in the commonly employed configuration of bonding a bare fiber onto a strain element. Jacketed fibers show reduced hysteresis and nonlinearity in their elastic response. Reductions in hysteresis from 23% to 1% of the maximum strain and in the nonlinearity by a factor of 3 have been measured.

Effect of spurious reflection on phase shift interferometry

The phase errors caused by spurious reflection in Twyman-Green and Fizeau interferometers are studied. A practical algorithm effectively eliminating the error is presented. Two other algorithms are reviewed, and the results obtained using the three algorithms are compared.