Jiancheng Xu

Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method

A method combining the principal component analysis (PCA) and the least squares method (LSM) is proposed to extract the phase from interferograms with random phase shifts. The method estimates the initial phase by PCA, and then determines the correct global phase sign and reduces the residual phase error by LSM. Some factors that may influence the performance of the proposed method are analyzed and discussed, such as the number of frames used, the number of fringes in interferogram and the amplitude of random phase shifts. Numerical simulations and optical experiments are implemented to verify the effectiveness of this method. The proposed method is suitable for randomly phase-shifted interferograms.

Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts

An advanced iterative algorithm is presented to extract phase distribution from randomly and spatially nonuniform phase-shifted interferograms. The proposed algorithm divides the interferograms into small blocks and retrieves local phase shifts accurately by iterations. Therefore, the phase distribution can be calculated with high precision by eliminating the effect of tilts occurring during phase shifting. Simulated results and experiments demonstrate that the proposed algorithm exhibits high precision and converges faster than previous algorithms even when the tilt errors are up to 27.6% of the normal phase step.

Direct phase extraction from interferograms with random phase shifts

A new method that extracts phase directly from randomly phase-shifted interferograms is proposed. In this method, the background intensity and the modulation amplitude are determined by a temporal method and used as the system parameters and then the phase is extracted from one single interferogram by arc cosine function with range of (0, π). By partitioning the extracted phase distribution into watershed regions and determining the sign of phase in each region, the extracted phase is recovered to its principle phase with range of (-π, π). Numerical simulation and experiment are implemented to verify the effectiveness of this method. The method has applications in both static and dynamic phase measurement.

Phase-shift extraction for phase-shifting interferometry by histogram of phase difference

We propose a non-iterative approach to extract the unknown phase shift in phase shifting interferometry without the assumption of equal distribution of measured phase in [0,2π]. According to the histogram of the phase difference between two adjacent frames, the phase shift can be accurately extracted by finding the bin of histogram with the highest frequency. The main factors that influence the accuracy of the proposed method are analyzed and discussed, such as the random noise, the quantization bit of CCD, the number of fringe patterns used and the bin width of histogram. Numerical simulations and optical experiments are also implemented to verify the effectiveness of this method.

Tilt-shift determination and compensation in phase-shifting interferometry

A method combining spatial and time-domain processing is proposed to extract the phase from spatial carrier interferograms with tilt-shift errors. The proposed method determines the piston values and tilt-shift errors based on the Fourier transform method and then compensates for it by least-squares fitting. Therefore, phase-retrieval errors introduced by uncalibrated fluctuations (including tilt-shift errors) in the phase increments can be alleviated without the necessity of extensive iterative calculations. Simulated results and experiments demonstrate the effectiveness of the proposed method. Compared to other techniques, the proposed method extracts the phase distribution accurately without edge artifacts and waviness caused by tilt-shift errors. The ability to perform both spatial and time-domain processing in a single instrument provides the flexibility for precision metrology applications in both dynamic environments.

Principal component analysis of multiple-beam Fizeau interferograms with random phase shifts

A non-iterative method based on principal component analysis (PCA) is presented to directly extract the phase from multiple-beam Fizeau interferograms with random phase shifts. The PCA method is the approach that decomposes the multiple-beam Fizeau interferograms into many uncorrelated quadrature signals and then applies principal component analysis algorithm to extract the measured phase without any prior guess about the phase shifts. Some factors that affect the performance of the proposed method are analyzed and discussed. Numerical simulations and experiments demonstrate that the proposed method extracts phase fast and exhibits high precision. The method can be applied in high precision interferometry.

Algorithm for multiple-beam Fizeau interferograms with arbitrary phase shifts

The objective of this paper is to describe a novel method for phase extraction from multiple-beam Fizeau interferograms with arbitrary phase shifts. The approach begins with applying FFT method to estimate the phase shifts and then utilizes least-squares iterative algorithm to extract phase and phase shifts simultaneously. If the spatial carrier frequency of the fringes is high enough to separate the phase of the first-order maximum in the Fourier domain, the proposed method requires only two iterative cycles to accurately extract phase information from seven multiple-beam Fizeau interferograms with arbitrary phase shifts. Numerical simulations and experiments demonstrate the effectiveness of the proposed algorithm. A comprehensive analysis of the influences of systematic errors (spatial carrier frequency, reflectivity coefficient, and random noise) on the evaluation of phase shifts and phase is presented. The method has applications in high precision interferometry.

500-mm-aperture wavelength-tuning phase-shifting interferometer

A 500-mm-aperture wavelength-tuning phase-shifting interferometer has been developed in FOERC applied to the measurement of large optics. Also it can switches to a smaller 130-mm-aperture. We describes in detail the optical and mechanical design as well as calibration technique of phase shifter and phase-shifting algorithm design. A Zygo 4 inch standard reflective flat is used to evaluate the accuracy and repeatability of our wavelength-tuning phase-shifting system.

Interferometric measurements of mid-spatial scale surface irregularities

A simple method which can be used to map mid-spatial scale surface irregularities with high signal noise ratio is described. Two major sources of errors are analyzed and removed. One is the contributions of small-scale irregularities of the reference surface, which are subtracted by shifting the test surface laterally by a distance. The other is the spurious response of CCD, which is removed by interpolation function. The presented method is verified by simulations and experiments. It shows that it can measure mid-spatial scale surface irregularities exactly and smaller scale surface irregularities can be obtained by making measurement for a series of the lateral shifting values corresponding to one-half of the pixel space on CCD.

Design of a coherent optical interferometric system

A 100-mm-aperture high lateral resolution interferometer has been developed in Fine Optical Engineering Research Center (FOERC), which is applied to the measurement of spatial frequencies of up to 2.5 lines/mm over a 100mm field of view. The system transfer function of the interferometer is greater than 60% at near half the Nyquist frequency. To demonstrate the performance of this high lateral resolution interferometer, theoretical errors of the system are thoroughly analyzed and the design implementation is carefully studied, such as light source, wave front slope, tolerance analysis, CCD sample and so on.