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.     

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.     

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.     

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.     

Optoelectronic information encryption with phase-shifting interferometry

A technique that combines the high speed and the high security of optical encryption with the advantages of electronic transmission, storage, and decryption is introduced. Digital phase-shifting interferometry is used for efficient recording of phase and amplitude information with an intensity recording device. The encryption is performed by use of two random phase codes, one in the object plane and another in the Fresnel domain, providing high security in the encrypted image and a key with many degrees of freedom. We describe how our technique can be adapted to encrypt either the Fraunhofer or the Fresnel diffraction pattern of the input. Electronic decryption can be performed with a one-step fast Fourier transform reconstruction procedure. Experimental results for both systems including a lensless setup are shown.     

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.     

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.     

Parallel phase-shifting interferometry based on Michelson-like architecture

In this paper, we present a new scheme for parallel phase-shifting interferometry that employs a Michelson-like architecture and a simple polarization unit to generate two phase-shifting interferograms with phase shift of π/2 at a single camera exposure. The parallel phase-shifting unit is built with simple optical components, and the distance between the parallel interferograms can be adjusted conveniently. Phase reconstruction is performed by using an algorithm developed for two-step phase-shifting interferometry. The practicability of the proposed configuration and the reconstruction method is demonstrated by experiments.     

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.     

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.     

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.     

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.     

Digital image encryption and watermarking by phase-shifting interferometry

A method for both image encryption and watermarking by three-step phase-shifting interferometry is proposed. The image to be hidden is stored in three interferograms and then can be reconstructed by use of one random phase mask, several specific geometric parameters, and a certain algorithm. To further increase the security of the hidden image and confuse unauthorized receivers, images with the same or different content can be added to the interferograms, and these images will have no or only a small effect on the retrieval of the hidden image, owing to the specific property of this algorithm. All these features and the utility of this method for image retrieval from parts of interferograms are verified by computer simulations. This technique uses intensity maps as decrypted images for delivery, and both encryption and decryption can be conveniently achieved digitally. It is particularly suitable for the remote transmission of secret information via the Internet.     

Vibration-induced phase errors in high-speed phase-shifting speckle-pattern interferometry

We present results from numerical simulations of a dynamic phase-shifting speckle interferometer used in the presence of mechanical vibrations. The simulation is based on a detailed mathematical model of the system, which is used to predict the expected frequency response of the rms measurement error, in the time-varying phase difference maps, as a result of vibration. The performance of different phase-shifting algorithms is studied over a range of vibrational frequencies. Phase-difference evaluation is performed by means of temporal phase shifting and temporal phase unwrapping. It is demonstrated that longer sampling windows and higher framing rates are preferred in order to reduce the phase-change error that is due to vibration. A numerical criterion for an upper limit on the length of time window for the phase-shifting algorithm is also proposed. The numerical results are finally compared with experimental data, acquired with a phase-shifting speckle interferometer of 1000 frames/s.