Grating-based real-time polarization phase-shifting interferometry: error analysis

A phase Ronchi grating-based real-time polarization phase-shifting method can be efficiently used for dynamic phase measurement in optical interferometry. A thorough error analysis is required for exhibiting how error sources influence phase-measurement results. We analyze the phase-measurement errors that are induced by the retardation error and azimuth angle error of the quarter-wave plate, the azimuth angle error of polarizers, the phase and intensity aberrations of diffractive wave fronts, and pixel mismatch of the interferometric patterns. The results will also be useful for evaluating the phase-measurement accuracy of other similar systems.     

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.     

All-fiber interferometry: design and analysis

This paper presents a design procedure for a fiber interferometer, the optical system, and its associated electronic control. Analog and digital circuits were optimized to achieve an inexpensive and compact system. The lock-in amplifier required for phase control was designed using a field programmable gate array that was also configured to carry out the required phase stepping. The interferometer was built into two stages. The first stage used only one wavelength to measure samples with step heights in the hundreds of nanometers, with improvements in accuracy through averaging of data. The next stage included two measurement wavelengths to extend the measurement range to a few hundreds of micrometers. The paper conducts a systematic study of sources of errors. A least squares method that minimizes measurement time but retains accuracy has been used to measure height.     

Local error analysis in 3-D panel methods

In this paper a local error analysis is given of the approximation of the integrals used in boundary integral equation methods for solving Laplace's equation. Using Taylor series expansions appriximate expressions are derived for the velocity and velocity-potential integrals. The asymptotic behaviour of the local truncation error is given in terms of the panel grid size.     

Mueller matrix error correction for a fringe-free interferometry system

We present an automated surface profiling system based on a shearing interferometer, in which precise measurement of the polarization states eliminates fringe ambiguity. A full error correction based on Mueller matrices allows comparatively inaccurate but rapidly switchable liquid-crystal wave plates to be used, enabling unambiguous profile information to be obtained in real time.     

舵机控制延迟误差分析与补偿方法研究

舵机的性能对弹体的控制精度产生决定性影响,而受到制造工艺、安装误差等结构因素以及控制算法的影响,导致舵机控制产生延迟误差,影响弹道修正的精度。利用半实物仿真的方法对舵机延迟误差进行分析,从方法误差和结构误差两个方面对延迟误差的影响因素进行阐述,对算法误差、惯性偏差、间隙误差、摩擦等因素的影响机理进行分析;以影响因素为出发点,提出以超前控制补偿角和结构性延迟补偿角对延迟误差进行补偿。通过半实物仿真试验验证,舵机能够很好地跟随控制指令,取得良好的补偿效果,改善控制精度,为下一步的工程应用提供技术支持。     

General methods for generating phase-shifting interferometry algorithms

Two completely independent systematic approaches for designing algorithms are presented. One approach uses recursion rules to generate a new algorithm from an old one, only with an insensitivity to more error sources. The other approach uses a least-squares method to optimize the noise performance of an algorithm while constraining it to a desired set of properties. These properties might include insensitivity to detector nonlinearities as high as a certain power, insensitivity to linearly varying laser power, and insensitivity to some order to the piezoelectric transducer voltage ramp with the wrong slope. A noise figure of merit that is valid for any algorithm is also derived. This is crucial for evaluating algorithms and is what is maximized in the least-squares method. This noise figure of merit is a certain average over the phase because in general the noise sensitivity depends on it. It is valid for both quantization noise and photon noise. The equations that must be satisfied for an algorithm to be insensitive to various error sources are derived. A multivariate Taylor-series expansion in the distortions is used, and the time-varying background and signal amplitudes are expanded in Taylor series in time. Many new algorithms and families of algorithms are derived.     

Measurement error caused by speckle noise in heterodyne and quasi-heterodyne holographic interferometry

In heterodyne and quasi-heterodyne holographic interferometry, the measuring accuracy is limited by speckle noise. It is important to know the relation between measurement accuracy and speckle noise in both theory and application. This problem has been discussed in the past under the assumption that the speckle noise produces only a small measuring error. However, this assumption is not reasonable in practice. The effect of speckle noise on the measurement accuracy is analyzed by the use of a general statistical method. The results obtained reveal the general relation between the measuring accuracy and system parameters, and thus constitute important guidelines to the application of these techniques.     

Phase-stepping interferometry: methods for reducing errors caused by camera nonlinearities

Phase errors that arise in phase-stepping interferometry are discussed. Investigations were performed by use of a Twyman-Green interferometer equipped with a compensation plate with a variable and servo-controlled tilt angle. With this instrument, phase-stepping errors can be reduced to a negligible level. There are, however, phase errors that are caused by camera nonlinearities. Two methods for minimizing these errors are presented. The first method is based on the simple idea that the interference intensity at the output of a two-beam interferometer has an exact cosine shape. The camera signals were monitored as a function of the tilt angle of the compensation plate, and the deviation from the cosine form was used to produce a correction. The second method is based on the idea that, under specific conditions, errors of an average of two phase measurements may compensate for each other. Numerical calculations were performed and give evidence of this hypothesis. Each method, the signal-correction and the averaging method, drastically reduces errors in evaluation of phases. The combination of both methods is a powerful tool that allows precise phase data to be obtained with an uncertainty, in the range lambda/2000 approximately 0.3 nm, that is caused mainly by signal noise.     

High-precision shape measurement by white-light interferometry with real-time scanner error correction

White-light interferometric techniques allow high-precision shape measurement of objects with discontinuous structures by detecting the peak of the coherence envelope. These techniques assume a specific change in the optical path difference (OPD) between the interfering beams; however, the scanning device effecting that change often introduces OPD errors that are carried over to the measurements. We present a technique for measuring OPD changes from the collected interference fringes during each measurement. Information about the scan is directly fed into the algorithm, which compensates for the errors, resulting in improved measurement accuracy. The method corrects not only the scanner errors but also slowly varying vibrations. In addition, this technique can be easily adapted to any existing low-coherence interferometer because no large data storage or postprocessing is required.     

Some considerations of reduction of reference phase error in phase-stepping interferometry

Positioning errors and miscalibrations of the phase-stepping device in a phase-stepping interferometer lead to systematic errors proportional to twice the measured phase distribution. We discuss the historical development of various error-compensating phase-shift algorithms from a unified mathematical point of view. Furthermore, we demonstrate experimentally that systematic errors can also be removed a posteriori. A Twyman-Green-type microlens test interferometer was used for the experiments.     

Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation

When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non-dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one-dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H¹-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h-p-version).     

Orthonormal polynomials in wavefront analysis: error analysis

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.     

Error begat error: Design error analysis and prevention in social infrastructure projects

Design errors contribute significantly to cost and schedule growth in social infrastructure projects and to engineering failures, which can result in accidents and loss of life. Despite considerable research that has addressed their error causation in construction projects they still remain prevalent. This paper identifies the underlying conditions that contribute to design errors in social infrastructure projects (e.g. hospitals, education, law and order type buildings). A systemic model of error causation is propagated and subsequently used to develop a learning framework for design error prevention. The research suggests that a multitude of strategies should be adopted in congruence to prevent design errors from occurring and so ensure that safety and project performance are ameliorated.     

Measurement and analysis technologies for magnetic pulse welding: established methods and new strategies

Magnetic pulse welding (MPW) is a fast and clean joining technique that offers the possibility to weld dissimilar metals, e.g., aluminum and steel. The high-speed collision of the joining partners is used to generate strong atomic bonded areas. Critical brittle intermetallic phases can be avoided due to the absence of external heat. These features attract the notice of industries performing large scale productions of dissimilar metal joints, like automotive and plant engineering. The most important issue is to guarantee a proper weld quality. Numerical simulations are often used to predict the welding result a priori. Nevertheless, experiments and the measurement of process parameters are needed for the validation of these data. Sensors nearby the joining zone are exposed to high pressures and intense magnetic fields which hinder the evaluation of the electrical output signals. In this paper, existing analysis tools for process development and quality assurance in MPW are reviewed. New methods for the process monitoring and weld characterization during and after MPW are introduced, which help to overcome the mentioned drawbacks of established technologies. These methods are based on optical and mechanical measuring technologies taking advantage of the hypervelocity impact flash, the impact pressure and the deformation necessary for the weld formation.     

Statistical study and experimental verification of high-resolution methods in phase-shifting interferometry

The introduction of high-resolution phase-shifting interferometry methods such as annihilation filter, state space, multiple-signal classification, minimum norm, estimation of signal parameter via rotational invariance, and maximum-likelihood estimator have enabled the estimation of phase in an interferogram in the presence of harmonics and noise. These methods are also effective in holographic moiré where incorporating two piezoelectric transducers (PZTs) yields two orthogonal displacement components simultaneously. Typically, when these methods are used, the first step involves estimating the phase steps pixelwise; then the interference phase distribution is computed by designing a Vandermonde system of equations. In this context, we present a statistical study of these methods for the case of single and dual PZTs. The performance of these methods is also compared with other conventional benchmarking algorithms involving the single PZT. The paper also discusses the significant issue of an allowable pair of phase steps in the presence of noise using a robust statistical tool such as the Cramér-Rao bound. Furthermore, experimental validations of these high-resolution methods are presented for the estimation of single phase in holographic interferometry and for the estimation of multiple phases in holographic moiré.     

Additive-subtractive phase-modulated electronic speckle interferometry: analysis of fringe visibility

Fringe-visibility issues of additive-subtractive phase-modulated (ASPM) electronic speckle pattern interferometry (ESPI) are explored. ASPM ESPI is a three-step method in which additive-speckle images are acquired rapidly in an analog fashion in every frame of a video sequence, a speckle phase modulation is intentionally introduced between frames, and a digital subtraction of consecutive pairs of additive-speckle images is performed. We show that this scheme has the good high-frequency noise immunity associated with additive-ESPI techniques as well as the good fringe visibility associated with subtractive-ESPI techniques. The method has better fringe visibility than can be obtained with purely additive ESPI and also does not suffer from the fringe distortions that can occur with subtractive ESPI in the presence of high-frequency noise. We show that even if full speckle decorrelation were to occur between the two additive speckle images that are to be subtracted, the visibility of ASPM ESPI fringes can be made to approach unity by suitable adjustment of the reference-to-object beam-intensity ratio.