Vibration-resistant phase-shifting interferometry

A method to reduce the sensitivity of phase-shifting interferometry to external vibrations is described. The returning interferogram is amplitude split to form two series of interferograms, taken simultaneously and with complementary properties, one with high temporal and low spatial resolution and the other with low temporal and high spatial resolution. The high-temporal-resolution data set is used to calculate the true phase increment between interferograms in the high-spatial-resolution data set, and a generalized phase-extraction algorithm then includes these phase increments when the topographical phases in the high-spatial-resolution data set are calculated. The measured topography thereby benefits from the best qualities of both data sets, providing increased vibration immunity without sacrificing high spatial resolution.     

Achromatic phase-shifting for white-light interferometry

The geometric (Pancharatnam) phase provides a method of introducing a variable phase shift that is almost independent of the wavelength and opens up new possibilities in broadband interferometry.     

Fourier-transform phase-shifting interferometry

Phase-shifting interferometry is a preferred technique for high-precision surface form measurements, but the difficulty in handling the intensity distortions from multiple-surface interference has limited the general use of the technique to interferometer cavities producing strict two-beam interference. I show how the capabilities of phase-shifting interferometry can be extended to address this problem using wavelength tuning techniques. The basic theory behind the technique is reviewed and applied specifically to the measurement of parallel plates, where surfaces, optical and physical thickness, and homogeneity are simultaneously obtained. Basic system requirements are derived, common error sources are discussed, and the results of the measurements are compared with theory and alternative measurement methods.     

Phase-shift calibration algorithm for phase-shifting interferometry

We propose a novel phase-shift calibration algorithm. With this technique we determine the unknown phase shift between two interferograms by examining the sums and differences of the intensities on each interferogram at the same spatial location, i.e., I1(x, y) +/- I2(x, y). These intensities are normalized so that they become sinusoidal in form. A uniformly illuminated region of the interferograms that contains at least a 2pi variation in phase is examined. The extrema of these sums and differences are found in this region and are used to find the unknown phase shift. An error analysis of the algorithm is provided. In addition, an error-correction algorithm is implemented. The method is tested by numerical simulation and implemented experimentally. The numerical tests, including digitization error, indicate that the phase step has a root-mean-square (RMS) phase error of less than 10(-6) deg. Even in the presence of added intensity noise (5% amplitude) the RMS error does not exceed 1 deg. The accuracy of the technique is not sensitive to nonlinearity in the interferogram.     

Confocal simultaneous phase-shifting interferometry

In order to implement the ultraprecise measurement with large range and long working distance in confocal microscopy, confocal simultaneous phase-shifting interferometry (C-SPSI) has been presented. Four channel interference signals, with π/2 phase shift between each other, are detected simultaneously in C-SPSI. The actual surface height is then calculated by combining the optical sectioning with the phase unwrapping in the main cycle of the interference phase response, and the main cycle is determined using the bipolar property of differential confocal microscopy. Experimental results showed that 1?nm of axial depth resolution was achieved for either low- or high-NA objective lenses. The reflectivity disturbance resistibility of C-SPSI was demonstrated by imaging a typical microcircuit specimen. C-SPSI breaks through the restriction of low NA on the axial depth resolution of confocal microscopy effectively.     

Single-step phase-shifting speckle interferometry

We propose a simplified technology of recording and processing speckle interferograms with phase shift that does not require the calibration of the phase-shifting devices and allows one to use the minimum possible number of images. We present the data on the verification of the algorithms and the results of experiments aimed at the investigation of the displacements of the surfaces of metal specimens and performed according to the proposed technology. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 43, No. 4, pp. 93–102, July–August, 2007.     

Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes

We have constructed two-wavelength phase-shifting interferometry that is insensitive to the intensity changes in interferograms associated with the current variations in two laser-diode (LD) sources by using a newly developed phase-extraction algorithm. The tested phase at a synthetic wavelength can be measured from six interferograms with different phase shifts. The algorithm becomes a simple form for seven interferograms and reduces to a minimum of five phase-shifted data in the proper conditions. We shifted the phases equally in opposite directions to one another by separately varying the stepwise currents in dual LD's on an unbalanced interferometer. The measurement accuracy has been improved compared with that of the two-wavelength four-step method. The phase error caused by the power changes in the dual LD's has been investigated theoretically and experimentally. The experimental results are shown to measure a step object with a 4.6-μm synthetic wavelength.     

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.     

The one-step phase-shifting technique for wave-front interferometry

Abstract

Hitherto, phase-shifting schemes in wave-front interferometry required a minimum of two precise phase steps in order to derive the phase. This paper outlines and demonstrates a novel technique entailing one precise phase step. Numerical analyses of phase-drift errors indicate that the proposed technique compares favourably with performances of the three- and four-bucket algorithms currently in use. Furthermore, it minimizes the period of computation by necessitating the use of three processing frames.     

Application of bacteriorhodopsin film for polarization phase-shifting interferometry

Photoinduced anisotropy in bacteriorhodopsin (BR) film is based on photoanisotropic selective bleaching of BR molecules under linearly polarized excitation light. It is modulated by the polarization orientation of the linearly polarized light. The anisotropic information recorded in the BR film is read by a circularly polarized light, which is in turn converted into an elliptical polarized light by the BR film. The rotation angle and the ellipticity of the elliptical polarized light are dependent on the polarization orientation of the linearly polarized excitation light. A phase-shifting interferometer based on the photoinduced anisotropy of BR film is presented theoretically and experimentally. Phase shift is controlled by the polarization orientation of the external excitation light, thus, the phase shift can be controlled without moving parts inside the interferometer, which contributes to the mechanical stability of the system.     

Phase shift selection for two-step generalized phase-shifting interferometry

A phase shift selection method is proposed to design algorithms immune against phase shift errors in two-step generalized phase-shifting interferometry. A general formula for wavefront reconstruction error is derived, and its specific expressions for two common errors are also given. Calculation results suggest that the proper range of phase shift for general application is about from 0.5 to 2.0 rad for both the fixed and linear phase shift errors. Computer simulations have demonstrated the effectiveness of this phase shift selection method by decreasing the wave reconstruction errors to one-fifth.     

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.     

Decorrelation-induced phase errors in phase-shifting speckle interferometry

The purpose of this research is the quantitative investigation of decorrelation-induced phase errors in speckle interferometry. Measurements in speckle interferometry are inherently affected by decorrelation, i.e., by alterations of the speckle fields during measurement. Likewise, the random phases carrying the interferometric information change during decorrelation. Image plane and pupil plane decorrelation are considered for both smooth and speckle reference wave interferometers. Since the decorrelation effect depends on the aperture and the pixel size, the calculations include not only the case of speckles being well resolved by the camera but also the case of unresolved speckles. Different standard deviations of the phase error are obtained from the probability density of the pixel modulation and the phase before and after decorrelation. Most cases (apart from pupil plane decorrelation in speckle reference wave setups) appear to obey exactly the same phase error statistics. In particular, the number of speckles per pixel does not affect the phase error distribution over the whole image. The only important parameters determining the decorrelation-induced phase errors are the amount of decorrelation and the pixel modulation.     

Numerical simulations of vibration in phase-shifting interferometry

Computer simulations predict the expected rms measurement error in a phase-shifting interferometer in the presence of mechanical vibrations. The simulations involve a numerical resolution of a nonlinear mathematical model and are performed over a range of vibrational frequencies and amplitudes for three different phase-shift algorithms. Experimental research with an interference microscope and comparison with analytical solutions verify the numerical model.     

Method for designing error-compensating phase-calculation algorithms for phase-shifting interferometry

We propose a general approach to eliminating some error source effects in phase-calculation algorithms for phase-shifting interferometry. We express the actual phase shift in a convenient form that takes the errors into account and develop in series the detected phase from a generic algorithm. Setting to zero the terms of the series that involve unwanted errors leads to a set of linear equations for the algorithm coefficients, which can thus be found. By using this approach, one could develop an algorithm series for an individual interferometer based on relevant concerns about the main error sources in it and eliminate the error source effects to any desired order. Two examples of algorithm series, to eliminate distorted phase shifts caused by the geometric effect in an interferometer with a spherical Fizeau cavity and to eliminate vibration effects, are discussed.     

Blind phase shift estimation in phase-shifting interferometry

A blind phase shift estimation algorithm that allows simultaneous calculation of phases and phase shifts from three or more interferograms is presented. In phase-shifting interferometry, the phase shift errors introduce specific correlations between the calculated background intensity distribution and the fringe component. These correlations can be measured with a cross-power spectrum. By minimization of an objective function based on this cross-power spectrum, the actual phase shifts are estimated and used for phase recovery. The validity of this algorithm is verified by both the numerical simulation and the experiment results.     

Multiplicative phase-shifting interferometry using optical flow

Fringe patterns with a multiplicative phase shift among them appear in experimental techniques as photoelasticity and RGB shadow moiré, among others. These patterns cannot be processed using standard phase-shifting demodulation techniques. In this work, we propose to use a multiframe regularized optical flow algorithm to obtain the interesting modulating phase. The proposed technique has been applied to simulated and experimental interferograms obtaining satisfactory results.