Chromium-doped forsterite: dispersion measurement with white-light interferometry

Using a Michelson white-light interferometer, we measure the group-delay dispersion and third-order dispersion coefficients, d2(phi)/d(omega)2 and d3(phi)/d(omega)3, of chromium-doped forsterite (Cr:Mg2SiO4) over wavelengths of 1050-1600 nm for light polarized along both the c and b crystal axes. In this interval, the second-order dispersion for the c axis ranges from 35 fs2/mm to -14 fs2/mm, and the third-order dispersion ranges from 36 fs3/mm to 142 fs3/mm. For the b axis the second-order dispersion ranges from 35 fs2/mm to -15 fs2/mm and the third-order from 73 fs3/mm to 185 fs3/mm. Our data are relevant for the development of optimized dispersion compensation tools for Cr:Mg2SiO4 femtosecond lasers. These measurements help to clarify previously published results and show some significant discrepancies that existed, especially in the third-order dispersion. Our results should furthermore be useful to build up an analytic expression for the index of refraction of chromium forsterite.     

Thickness-profile measurement of transparent thin-film layers by white-light scanning interferometry

White-light scanning interferometry is increasingly used for precision profile metrology of engineering surfaces, but its current applications are limited primarily to opaque surfaces with relatively simple optical reflection behavior. A new attempt is made to extend the interferometric method to the thickness-profile measurement of transparent thin-film layers. An extensive frequency-domain analysis of multiple reflection is performed to allow both the top and the bottom interfaces of a thin-film layer to be measured independently at the same time by the nonlinear least-squares technique. This rigorous approach provides not only point-by-point thickness probing but also complete volumetric film profiles digitized in three dimensions.     

Dispersion measurement of tapered air-silica microstructure fiber by white-light interferometry

Dispersion properties of novel, tapered, air-silica microstructure fibers are measured between 1.3 and 1.65 microm by white-light interferometry. Dispersion values (beta2) of -181 and -152 ps2/km were obtained for 2.2- and 3-microm core sizes, respectively, at lambda = 1.55 microm.     

Direct measurement of refractive-index dispersion of transparent media by white-light interferometry

We report on a technique for measuring the refractive indices of nonabsorbing media over a broad spectral range from 0.5 to 5 microm. White-light interferometry based on a double-interferometer system consisting of a fixed Mach-Zehnder interferometer and a Fourier-transform spectrometer is used for direct measurement of the absolute rotation-dependent phase shift induced by an optical element. Refractive index n(lambda) over the whole investigated spectral range is thus obtained directly to an accuracy of 10(-4) without the need for any specific assumption about dispersion. Results for synthetic fused silica are presented and discussed.     

Sensitivity analysis of thin-film thickness measurement by vertical scanning white-light interferometry

The spectral nonlinear phase method and the Fourier amplitude method have been applied to measure the thin-film thickness profile in vertical scanning white-light interferometry (VSWLI). However, both the methods have their disadvantages, and accordingly their applications are limited. In the paper we have investigated the dependence of the sensitivities of both the methods on the thin-film thickness and refractive index, the objective numerical aperture, and the incident light spectral range of VSWLI. The relation of the Fresnel reflection coefficients on the wavelength effect is also discussed. Some important research results reveal that the combination of both Fourier amplitude and nonlinear phase methods may provide a new approach to improve the VSWLI measurement sensitivity for thin-film thickness profile.     

Theoretical measurement uncertainty of white-light interferometry on rough surfaces

A great advantage of the white-light interferometry is that it can be used for profile objects with a rough surface. A speckle pattern that arises in the image plane allows one to observethe interference; however, this pattern is also the source of the measurement uncertainty. We derive the theoretical limits of the longitudinal uncertainty by virtue of the first-order statistics of thespeckle pattern. It is shown that this uncertainty depends on the surface roughness of the measured object only; it does not depend on the setup parameters.     

High-precision absolute distance and vibration measurement with frequency scanned interferometry

We report high-precision absolute distance and vibration measurements performed with frequency scanned interferometry using a pair of single-mode optical fibers. Absolute distance was determined by counting the interference fringes produced while scanning the laser frequency. A high-finesse Fabry-Perot interferometer was used to determine frequency changes during scanning. Two multiple-distance-measurement analysis techniques were developed to improve distance precision and to extract the amplitude and frequency of vibrations. Under laboratory conditions, measurement precision of approximately 50 nm was achieved for absolute distances ranging from 0.1 to 0.7 m by use of the first multiple-distance-measurement technique. The second analysis technique has the capability to measure vibration frequencies ranging from 0.1 to 100 Hz with an amplitude as small as a few nanometers without a priori knowledge.     

Rotation angle measurement based on white-light interferometry with a standard optical flat

We propose a simple white-light interferometric method of measuring a one-dimensional rotation angle with use of an optical plane parallel plate of standard refractive index. The phase change of the interference pattern of the interferometer during the rotation of the flat plate of known refractive index and thickness placed in one of the interferometer's arms is used for determination of the rotation angle. This method has been demonstrated for an accurate angle measurement over the angle range from 0° to 40° within a maximum uncertainty of 0.057°.     

Single-shot two-dimensional surface measurement based on spectrally resolved white-light interferometry

By analyzing the spectral domain's phase information, one can use spectrally resolved white-light interferometry (SRWLI) to obtain the profile with a single frame of an interferogram. We present here a two-dimensional (2D) SRWLI method that can be applied to measure narrow rectangle areas. A frequency comb is produced by using a Fabry-Perot (F-P) etalon to filter the broadband source. With the filtered frequency comb illumination, the interference patterns under adjacent wavelengths would be separated by a little distance, which enables us to obtain a 2D profile with a small width. The experimental details of measurement on a step sample are discussed in this paper.     

Spectrally resolved white-light interferometry for measurement of ocular dispersion.

Spectrally resolved white-light interferometry was used to measure the wavelength dependence of refractive index (i.e., dispersion) for various ocular components. Verification of the technique's efficacy was substantiated by accurate measurement of the dispersive properties of water and fused silica, which have both been well-characterized in the past by single-wavelength measurement of the refractive index. The dispersion of bovine and rabbit aqueous and vitreous humors was measured from 400 to 1100 nm. In addition, the dispersion was measured from 400 to 700 nm for aqueous and vitreous humors extracted from goat and rhesus monkey eyes. An unsuccessful attempt was also made to use the technique for dispersion measurement of bovine cornea and lens. The principles of white-light interferometry, including image analysis, measurement accuracy, and limitations of the technique, are discussed. In addition, alternate techniques and previous measurements of ocular dispersion are reviewed.     

Dispersion measurements of water with white-light interferometry

We measure the second- and third-order dispersion coefficients, d(2)k/domega(2) and d(3)k/domega(3), of water for wavelengths from 0.45 to 1.3 mum using a Michelson white-light interferometer. In this interval, the second-order dispersion ranges from 0.068 to -0.1 fs(2)/mum, and the third-order dispersion ranges from 0.048 to 1.18 fs(3)/mum. We observe an oscillation in d(2)k/domega(2) near 1.1 mum that is due to water absorption features near that wavelength. From the dispersion coefficients, derivatives of the index of refraction of water are calculated and compared with available equations. These measured values of d(2)n/dlambda(2) and d(3)n/dlambda(3) should be useful in the evaluation and improvement of existing equations for n(lambda) in water.     

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.     

Effect of wave-number error on the computation of path-length delay in white-light interferometry

We analyze the error in computed optical path-length delay when using a phase-shifting interferometry (PSI) algorithm with an error in the operating wavelength. The delay error decomposes into two terms. The first is the error in the conversion from a phase measurement to the delay because of the incorrect wavelength, and the second is the error made in the phase measurement itself that is due to the wavelength error. The most important aspect of this investigation is to ascertain this latter error. A general characterization is obtained, and a particularly simple formula is developed for the special case of least-squares estimation involving only the ratio of the wave-number error to the wave number and a multiplicative factor that is an a priori computable nonlinear function of the ratio of the modulator stroke length to the operating wavelength. Because the ultimate path-length error is a function of the two terms, a new set of PSI algorithms that compensate the computed phase error to cancel the conversion error is developed. Numerical simulations are presented to validate the analysis and establish the insensitivity of the new algorithms to wave-number error.     

An algorithm for profilometry by white-light phase-shifting interferometry

Abstract

In the case of white-light interferometry, the usual phase shifting methods cannot be used because the modulation in the interference intensity is due to both the phase variation and to the coherence envelope. We propose a practical approach in which the coherence envelope is considered as locally linear. The classical four-point calculation algorithm is adapted to a set of seven intensities which are weighted with adapted coefficients in order to compensate for the effect of the coherence envelope. The capabilities of this method are described theoretically.     

Surface profile measurement in white-light scanning interferometry using a three-chip color CCD

White-light scanning interferometry (WLSI) is a useful technique to measure surface profile when a test object contains discontinuous structures or microstructures. A black and white CCD camera is usually utilized to capture interferograms, and a series of corresponding algorithms is used to achieve the profile measurement. However, the color information in the interferograms is lost. A novel profile measurement method that uses phase information in different color channels (red-green-blue) of an interferogram obtained using a three-chip color CCD in WLSI is proposed. The phase values are extracted by a windowed Fourier transform algorithm. Simulation and experimental results are presented to demonstrate the validity of the proposed method.     

A new p-control chart with measurement error correction

Control charts are important tools to monitor quality of products. One of useful applications is to monitor the proportion of non-conforming products. However, in practical applications, measurement error is ubiquitous and may occur due to false records or misclassification, which makes the observed proportion different from the underlying true proportion. It is also well-known that ignoring measurement error effects provides biases, and is expected that the resulting control charts may incur wrong detection. In this paper, we study this important problem and propose a valid method to correct for measurement error effects and obtain error-eliminated control chart for the proportion of non-conforming products. In addition, unlike traditional approaches, the corrected EWMA p-control chart provides asymmetric control limits and is flexible to handle the data with small sample size. Numerical results are conducted to justify the validity of the corrected EWMA p-control chart and verify the necessity of measurement error correction.     

Standardizing measurement error with periodic static characteristic correction

Error regulation is considered for means of measurement with periodically corrected static characteristics used as components of data-acquisition systems. Basic principles are considered for metrological support to such measuring instruments during checking.Translated from Izmeritelnaya Tekhnika, No. 9, pp. 8–10, September, 2004.     

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.     

Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector

A structured light system using a digital video projector is widely used for 3D shape measurement. However, the nonlinear gamma of the projector causes the projected fringe patterns to be nonsinusoidal, which results in phase error and therefore measurement error. It has been shown that, by using a small look-up table (LUT), this type of phase error can be reduced significantly for a three-step phase-shifting algorithm. We prove that this algorithm is generic for any phase-shifting algorithm. Moreover, we propose a new LUT generation method by analyzing the captured fringe image of a flat board directly. Experiments show that this error compensation algorithm can reduce the phase error to at least 13 times smaller.     

孔径径向扫描白光散斑干涉法的研究

提出了一种用于动态研究的新方法-孔径径向扫描白光散斑干涉法,介绍了两孔和直三孔径向扫描器.该方法是一种非接触式的用于动态测量的白光散斑法.分析了孔径径向扫描白光散斑干涉法的基本原理,给出了白光散斑全场滤波分析的平均光强解析式和实验结果.该方法的优点是利用散斑的调频技术能在一张白光散斑图上记录物体的连续动态变形的信息,在全场滤波分时,能将物体动态变形的信息方便地提取出来,得到清晰的全场白光散斑条纹图.直三孔径向扫描法能得到物体在x方向、y方向和45°方向上的变形信息.在实时全场滤波分析观察时,将滤波孔连续地沿径向扫描,可观察到物体连续动态变形的全过程,从而为物体的动态研究提供了一种新方法.