Yongqian Wu

Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology

The accuracy of phase-shifting interferometers (PSI) is crippled by nonlinearity of the phase shifter and instability of the environment such as vibration and air turbulence. A general algorithm, utilizing Lissajous figures and ellipse fitting, of correcting the phase extraction error in the phase shifting interferometry is described in this paper. By plotting N against D, where N and D represent the numerator and denominator terms of the phase extraction function (i.e. an arctangent function) respectively, a Lissajous ellipse is created. Once the parameters of the ellipse are determined by ellipse fitting, one can transform the ellipse to a unit circle (ETC). Through this process the phase extraction error caused by random phase shift errors can be corrected successfully. Proposed method is non-iterated, adapts to all phase shifting algorithms (PSAs), and has high accuracy. Some factors that may affect the performance of proposed method are discussed in numerical simulations. Optical experiments are implemented to validate the effectiveness of proposed algorithm.

Phase shifting interferometry from two normalized interferograms with random tilt phase-shift

We propose a novel phase shifting interferometry from two normalized interferograms with random tilt phase-shift. The determination of tilt phase-shift is performed by extracting the tilted phase-shift plane from the phase difference of two normalized interferograms, and with the calculated tilt phase-shift value the phase distribution can be retrieved from the two normalized frames. By analyzing the distribution of phase difference and utilizing special points fitting method, the tilted phase-shift plane is extracted in three different cases, which relate to different magnitudes of tilts. Proposed method has been applied to simulations and experiments successfully and the satisfactory results manifest that proposed method is of high accuracy and high speed compared with the three step iterative method. Additionally, both open and closed fringe can be analyzed with proposed method. Whats more, it cannot only eliminate the small tilt-shift error caused by slight vibration in phase-shifting interferometry, but also detect the large tilt phase-shift in phase-tilting interferometry. Thus, it will relaxes the requirements on the accuracy of phase shifter, and the costly phase shifter may even be useless by applying proposed method in high amplitude vibrated circumstance to achieve high-precision analysis.

Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering

This paper presents a novel method to extract the phase shift and phase distribution from two interferograms simultaneously. By employing Hilbert-Huang transform based prefiltering, the background intensities and modulation amplitudes of the two interferograms are suppressed and normalized respectively. With the addition and subtraction operation of the two prefiltered interferograms, two parametric equations are achieved which can be regarded as the complex harmonic motion of the Lissajous figure. The phase of the Lissajous figure can be directly demodulated by the ellipse fitting algorithm. Apart from the advantages of other well-known two-step phase demodulation algorithms, i.e., high accuracy and efficiency of the Gram-Schmidt orthonormalization (GS) method and the less stringent requirement concerning the fringe number in the extreme value of interference (EVI) method, proposed Lissajous figure and ellipse fitting (LEF) approach has another bonus related to its robustness to the fluctuations of the fringe patterns noise, background intensity and modulation amplitude. Simulations demonstrate the outstanding performance of the proposed method, and experiments further corroborate its effectiveness.

Precise phase demodulation of single carrier-frequency interferogram by pixel-level Lissajous figure and ellipse fitting

Phase demodulation from a single carrier-frequency fringe pattern is becoming increasingly important particularly in areas of optical metrology such as dynamic interferometry, deflectometry and profilometry. The Fourier transform (FT) method and the spatial-carrier phase-shifting technique (SCPS) are two popular and well-established approaches to demodulation. However FT has the drawback of significant edge errors because of the Gibbs effect, whilst detuning errors for the local phase shift occur when SCPS is applied. A novel demodulation method based on pixel-level Lissajous figure and ellipse fitting (PLEF) is presented in this paper. Local demodulation in the spatial domain makes PLEF more flexible than the FT method, without spectral leakage. Based on a more adaptable approach, account is taken of variations in illumination and phase distribution over a few neighboring pixels. The mathematic demodulation model is of interest and has been demonstrated via simulation. Theoretical phase extraction error is as low as 10−4 rad. Experiments further corroborate the effectiveness of the proposed method. In conclusion, various influencing factors, e.g. variations of background/modulation, phase amplitude, carrier frequency, additive noise that may affect the precision of PLEF are discussed in detail.

Large-scale absolute surface reconstruction

In order to realize large-scale absolute surface reconstruction, a generalized iterative optimization method for solving the three-flat problem is studied. First, the idea of model-based absolute surface reconstruction is proposed, which considers the problems of absolute surface reconstruction as inverse problems. Then we take the three-flat problems as an example, we introduced two generalized iterative optimization methods for three-flat model. Finally, by both simulation and experiment, it is concluded that the block SOR method with an optimal relaxation factor converges much faster and saves more computational costs and memory space without reducing accuracy. Both simulation and experimental results indicate that the proposed iterative optimization methods are effective for solving the three-flat problem with pixel-level spatial resolution and the measuring precision of two separate measurements is 0.6 nm rms, and the cross-check test result is 0.8 nm rms. It is concluded that the proposed method can correctly reconstruct absolute figures with high efficiency and pixel-level spatial resolution.

Modeling Fizeau interferometer based on ray tracing with Zemax

A convenient method to study the influence of error sources in Fizeau is to build a ray-tracing model to simulate the error sources. In this paper an interferometer model is presented; an extension program is called to simulate the interference; and a preliminary research of several error sources is conducted. These examples demonstrate error analysis based on interferometer models is feasible and provide some guidance for optimizing our interferometer design.

Experimental Study on the Absolute Measurement of Flats

We present an absolute method of shift-rotation to calibrate the flatness of flats. Compared with the conventional three-flat test method, the method requires only two flat and no flips of the flats. And there will be no sag errors introduced by gravity deformations because of the flips of flats. The details of the method are given out in the paper. With multi-rotational measurements and translation measurements, the surface deviation of flats can be obtained with the method. Experimental results are presented to show the availability of the absolute method.