Multiplicative moiré two-beam phase-stepping and fourier-transform methods for the evaluation of multiple-beam fizeau patterns: a comparison

A phase-evaluation method of multiple-beam Fizeau patterns that combines two-beam phase-stepping algorithms with the moiré effect was previously reported [Appl. Opt. 34, 3639-3643(1995)]. The method is based on a multiplicative moiréimage-formation process obtained by the direct superposition of high-frequency multiple-beam Fizeau carrier fringes upon a transmission grating (working as a phase modulator). We present a comparison between this multiplicative moiré two-beam phase-stepping method and the well-known Fourier-transform method for the topographic measurement of an undoped silicon wafer. The discrepancy between the two methods yields a rms phase-difference value of the order of(~2pi/90).     

Phase retrieval errors in standard Fourier fringe analysis of digitally sampled model interferograms

We investigate the reliability of phase retrieval by use of the fringe Fourier analysis method for measuring the displacements of facets during the growth of equilibriumlike-shaped crystals. The mean phase change between two successive interferometric images contains an inherent error that emerges from the noninteger number of fringes in the image field. The magnitude of the retrieved phase error of the ideal fringe pattern is investigated as a function of spatial carrier frequency, of the initial phase setting, and of the deviation of the number of fringes from the nearest integer value. The suggested modified algorithm suppresses the error more than threefold.     

Measurement of transient deformations with dual-pulse addition electronic speckle-pattern interferometry

We describe an electronic speckle-pattern interferometry system for analyzing addition fringes generated by the transient deformation of a test object. The system is based on a frequency-doubled twin Nd:YAG laser emitting dual pulses at a TV camera field rate (50 Hz). The main advance has been the automatic, quantitative analysis of dual-pulse addition electronic speckle-pattern interferometry data by the introduction of carrier fringes and the application of Fourier methods. The carrier fringes are introduced between dual pulses by a rotating mirror that tilts the reference beam. The resulting deformation-modulated addition fringes are enhanced with a deviation filter, giving fringe visibility close to that of subtraction fringes. The phase distribution is evaluated with a Fourier-transform method with bandpass filtering. From the wrapped phase distribution, a continuous phase map is reconstructed with an iterative weighted least-squares unwrapper. Preliminary results for a thin plate excited by an acoustic shock show the suitability of the system for the quantitative evaluation of transient deformation fields.     

Assessment of a multibeam Fizeau wedge interferometer for Doppler wind lidar

The Fabry-Perot interferometer is the standard instrument for the direct detection Doppler lidar measurement of atmospheric wind speeds. The multibeam Fizeau wedge has some practical advantages over the Fabry-Perot, such as the linear fringe pattern, and is evaluated for this application. The optimal Fizeau must have a resolving power of 10(6) or more. As the multibeam Fizeau wedge is pushed to such high resolving power, the interference fringes of the device become complicated by asymmetry and secondary maxima. A simple condition for the interferometer plate reflectance, optical gap, and wedge angle reveals whether a set of parameters will yield simple, Airy-like fringes or complex Fizeau fringes. Tilting of the Fizeau wedge improves the fringe shape and permits an extension of the regime of Airy-like fringes to higher resolving power. Sufficient resolving power for the wind lidar application is shown to be possible with a large-gap, low-finesse multibeam Fizeau wedge. Liabilities of the multibeam Fizeau wedge in the wind lidar application include a smaller acceptance solid angle and calibration sensitivity to localized deviations of the plates from the ideal.     

Transient Deformation Measurement by Double-Pulsed-Subtraction TV Holography and the Fourier Transform Method

We report the measurement of transient bending waves with double-pulsed-subtraction TV holography. The correlation fringe patterns are automatically quantitatively analyzed by the application of Fourier methods. A novel optical setup with two different object-beam optical paths is demonstrated for the generation of carrier fringes. The proposed system is highly immune to environmental disturbances because the optical setup imposes no lower limit on the time separation between laser pulses. One removes the linear phase distribution due to the spatial carrier in the spatial domain by subtracting the phase of the undeformed carrier fringes from the phase of the modulated fringes. Experimental results obtained with an aluminum plate excited by the impact of a piezoelectric translator are presented.     

White-light Fizeau interferometer

A white-light Fizeau interferometer is described. Commonly, white-light fringes can be produced only by using a virtual wedge instrument such as a Michelson interferometer. By use of a series arrangement of a Fabry-Perot interferometer in front of a two-beam Fizeau interferometer, white-light fringes can be produced. For white-light fringes to be obtained, the thickness of the air gap between the Fizeau plates has to be adjusted to the same thickness as the air gap between the Fabry-Perot plates (or in more general terms to a rational multiple of this value). The contrast of the two-beam type of Fizeau fringes depends on the reflectivity of the Fabry-Perot plates.     

提高Fourier变换轮廓术测量精度的新方法

在傅里叶变换轮廓术测量方法中,对离散的条想方设法进行DFT（离散傅里叶变换）运算,存在“栅栏效应”,离散频谱不能完全无误地代表原连续频谱分布,相当于在频域内发生了谱信息损失。为了减小由“栅栏效应”引入的位相误差,采用了对条纹进行空域数值加权和外插补零方法,从而提高FTP的测量精度。计算机模拟证实了所提方法的有效性。     

Multiple-beam interferometric determination of Poisson's ratio and strain distribution profiles along the cross section of bent single-mode optical fibers

Multiple-beam Fizeau fringes crossing bent single-mode optical fibers immersed in matching liquid reveal the existence of induced birefringence. Changes in the fiber cladding refractive index delta n were measured from the fringe shift to an accuracy of +/- 1 x 10(-4). Mathematical expressions were deduced to evaluate Poisson's ratio and to describe the radial strain distribution profiles of bent optical fibers from the experimental values of the fringe shifts. Experimental results were obtained from microinterferograms. Studying bent fibers by application of Fizeau fringes interferometry provides a nondestructive, accurate, sensitive, and reliable method to measure their parameters and characteristics.     

Iterative full-bandwidth wavefront reconstruction from a set of low-tilt Fizeau interferograms for high-numerical-aperture surface characterization

Full-bandwidth phase-shifting methods as well as band-limited fringe carrier techniques are both problematic when testing high-NA spherical surfaces in Fizeau interferometers. Phase stepping is usually performed by moving a sample and reference sphere relative to each other along the optical axis. At a high NA the method suffers from phase-shift inhomogeneity across the sample surface. Fringe carrier techniques rely on a minimum fringe frequency and call for an off-axis position of the sample, which in turn introduces condenser aberrations. Distortion of the imaging optics generates further apparent aberrations. We propose to combine both principles. The phase shifts are replaced by a set of very low tilts such that the sample is virtually on axis. Initial wavefront estimates are generated by a fringe carrier method. An adaptive Misell-type algorithm combines the interferometric data and iteratively improves the reconstructed wavefront until full spatial bandwidth is achieved.     

Theoretical and experimental evaluation of the modal field shift and the associated transition loss in perturbed index-profile single-mode optical fiber with application of Fizeau interferometry

Fizeau micro-interferometry is applied to evaluate some parameters of a curved single-mode optical fiber. The field shift of the fundamental mode and the associated transition loss in a perturbed index-profile fiber due to bending are determined. The preceding fiber parameters are determined as a function of the shift of multiple-beam Fizeau fringes. For a curvature range between 0.13 and 0.053 mm(-1), a range of field shift between 0.44 and 0.21 microm is determined. A fraction of the transition loss ranging between 0.0056 and 0.028 is calculated within the same curvature range. Because our method has high index resolution and spatial resolution, it shows good agreement with theory. The results and the agreement with theory indicate that the use of multiple-beam Fizeau fringes is a promising technique that is capable of determining with high accuracy some guidance parameters of the optical fibers.     

气流场ESPI载频条纹图的消噪处理与相位提取方法

为拓展ESPI方法在流动可视化测量技术中的应用,对气流场ESPI载频条纹图进行了傅立叶变换、数字全低通滤波,和傅立叶逆变换的处理以消除散斑噪声,并直接将其替换为原散斑相关条纹图,进而成为保留了流场信息的高对比度、低噪声的灰度条纹图,便于提取原散斑图中流动相位信息。根据模拟数据确定了由这两种条纹图互换所引起的空间相移量和相应流场测量中的修正量。提出了用灰度扫描法对互换后的条纹图进行一维灰度扫描,由灰度计算得出气流扰动的一维相位差分布和流场一维温度分布;同时也用FTP法对该图进行了二维相位展开和温度分布计算,亦可对多帧条纹图进行时间轴上的逐点相位提取以计算其他流场参数。结果表明,计算数据与实测符合,方法简单易行,为在气流场参数测量中应用动态ESPI方法提供了一种实用有效的方法。     

Optical fringe multiplication in moiré interferometry

An optical method for multiplication of moiré fringes is proposed to increase the sensitivity of moireé interferometry. The process involves two recording steps. In the first step, a traditional moiré interferometry setup is used. The moiré pattern containing carrier fringes and load fringes is recorded onto a glass-based holographic plate. The carrier frequency is much lower than that of the original specimen grating. The plate is then developed. In the second step, the holographic plate, regarded as a distorted specimen grating, is further examined by a similar moiré interferometry system. The frequency of the second virtual grating is arranged to be 2n times that of the carrier fringes contained in the recorded plate. As a result, the load fringes are revealed with a multiplication factor of 2n. The interpretation of the optical multiplication method from wave-front interference theory is given and an experiment is conducted.     

Carrier phase component removal: a generalized least-squares approach

In fringe projection profilometry, an object shape is evaluated through phase distribution extracted from a projected fringe pattern. For parallel illumination geometry, the carrier phase component introduced by the fringes is spatially linear, whereas nonparallel illumination would lead to a nonlinear carrier. In this study, a general approach for the removal of a nonlinear-carrier phase component is proposed. A series expansion technique is used to approximate the carrier phase function, and a least-squares method is developed to estimate the unknown coefficients of the series. The theoretical analysis is given on the basis of a divergent illumination geometry with carrier fringes in the x direction. The method is also extended to include a curved surface-fitting approach, which is applicable to various measurement system geometries.     

Simultaneous measurement of out-of-plane displacement and slope using a multiaperture DSPI system and fast Fourier transform

The simultaneous quantitative measurement of out-of-plane displacement and slope using the fast Fourier transform method with a single three-aperture digital speckle pattern interferometry (DSPI) arrangement is demonstrated. The method coherently combines two sheared object waves with a smooth reference wave at the CCD placed at the image plane of an imaging lens with a three-aperture mask placed in front of it. The apertures also introduce multiple spatial carrier fringes within the speckle. A fast Fourier transform of the image generates seven distinct diffraction halos in the spectrum. By selecting the appropriate halos, one can directly obtain two independent out-of-plane displacement phase maps and a slope phase map from the two speckle images, one before and the second after loading the object. It is also demonstrated that by subtracting the out-of-plane displacement phase maps one can generate the same slope phase map. Experimental results are presented for a circular diaphragm clamped along the edges and loaded at the center.     

Modified Fourier transform method for interferogram fringe pattern analysis

A modified Fourier transform method for interferogram fringe pattern analysis is proposed. While it retains most of the advantages of the Fourier transform method, the new method overcomes some drawbacks of the previous method. It eliminates the assumptions of slowly varying phase variation in the test section and the constant spatial carrier frequency. It also extends the frequency bandwidth and avoids phase distortion caused by discreteness of the sampling frequency. Both numerical simulation and experimental examination are performed to evaluate the performance of the method.     

Analysis of interferograms with a spatial radial carrier or closed fringes and its holographic analogy

It is well known that an interferogram can be demodulated to find the wave-front shape if a linear carrier is introduced. We show that it can also be demodulated if it has many closed fringes or a circular carrier appears. A basic assumption is that the carrier fringes are of a bandwidth adequate to contain the wave-front distortion. This phase determination, called here demodulation, is made in the space domain, as opposed to demodulation in Fourier space, but the low-pass filter characteristics must be properly chosen. For academic purposes a holographic analogy of this demodulation process is also presented, which shows that the common technique of multiplying by a sine function and a cosine function is equivalent to holographically reconstructing with a tilted-flat wave front. Alternatively, a defocused (spherical) wave front can be used as a reference to perform the reconstruction or demodulation of some closed-fringe interferograms.     

Method to obtain a clear fringe pattern with a zone-plate interferometer

When a zone-plate interferometer is used, a bright spot appears at the center of the image plane. The spot makes it difficult to analyze the interference fringes. A simple technique that is based on the principle of fringe-intensity reversal is proposed to analyze the fringes efficiently. A zone plate with a phase fraction of π/2 or 3π/2 is used in this technique to diminish the bright spot. Unlike the masking technique, no part of the data on the fringes is lost. The fringes can, therefore, be analyzed completely. The technique is described in detail, and the results of an experiment in which the shape error of a concave mirror was measured with the proposed zone plate is presented. The experimental results agree well with the results obtained with the Fizeau interferometer.     

Adaptive phase-shifting algorithm for temporal phase evaluation

Most standard temporal-phase-shifting (TPS) algorithms evaluate the phase by computing a windowed Fourier transform (WFT) of the intensity signal at the carrier frequency of the system. However, displacement of the specimen during image acquisition may cause the peak of the transform to shift away from the carrier frequency, leading to phase errors and even unwrapping failure. We present a novel TPS method that searches for the peak of the WFT and evaluates the phase at that frequency instead of at the carrier frequency. The performance of this method is compared with that of standard algorithms by using numerical simulations. Experimental results from high-speed speckle interferometry studies of carbon fiber panels are also presented.     

Speckle photography fringe analysis by the Walsh transform

Two-dimensional Walsh spectral analysis is presented as a new method of numerically processing the Young's fringes diffraction pattern from a double-exposure speckle photograph. The Walsh spectrum of the fringes is more complex than the Fourier spectrum but can be interpreted reliably by further cross correlation of the fringes with square waves. Compared with Fourier spectral analysis, the new technique gives results of almost identical accuracy but with substantially reduced computational effort. The ideas presented have relevance to the general problem of detecting and accurately determining the frequency components of a 2-D sinusoid in the presence of noise.