Algorithm immune to tilt phase-shifting error for phase-shifting interferometers

As a phase shifter usually suffers from both translational and tilt-shift errors during shifting, so every pixel in the same interferogram will have a different phase-shift value. Thus nonlinear phase-measurement errors cannot be avoided, but even translational-shift error has been corrected effectively. However, based on the fact that the shifted phases of all the pixels in the same interferogram remain on the phase-shift plane, by defining this plane one can eliminate a significant number of phase errors. A new algorithm that is immune to both translational- and tilt-shift errors in a phase shifter for phase-stepping interferometers is presented. A first-order Taylor series expansion replaces the nonlinear equations for defining the phase-shift plane, and iteration of the algorithm guarantees its accuracy. Results of a computer simulation show that phase-measurement errors caused by both translation- and tilt-shift error can be compensated for completely, even when the tilt-shift error is not more than ?1%.     

Phase-shifting shadow moiré based on iterative self-tuning algorithm

We proposed a general algorithm for phase-shifting shadow moiré by an iterative self-tuning algorithm. In our proposed system, the grating is translated in equal distance to introduce phase shifts across the field of view. The proposed algorithm produces accurate phase information with five interferograms and can calibrate the precise phase step during the process of the height demodulation. Compared with the traditional method, the proposed algorithm is insensitive to the height dependent effects, which is the main systematic source of error in phase-shift shadow moiré when reconstructing surfaces from fringe patterns. Numerical simulations and optical experiments show that the proposed method can eliminate the nonuniform phase-shift error and possesses a superior performance to existing typical phase-shifting algorithms.     

Generalized data reduction approach for aspheric testing in a non-null interferometer

Data reduction in non-null tests is difficult due to the presence of retrace error. We propose a simple yet effective data reduction approach for aspheric testing in a non-null interferometer. The new approach gives figure error of the aspheric by just subtracting the theoretical wavefront and first-order errors from the real wavefront obtained in the non-null interferometer. Precise prediction of the theoretical wavefront can be achieved by accurate calibration of the partial compensation system. The approach can be considered a generalization of the traditional data processing method in null tests, and errors that may affect its accuracy are discussed. A set of experiments have been carried out to demonstrate its validity and feasibility.     

Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer

Recent technological innovations have enabled the development of a new class of dynamic (vibration-insensitive) interferometer based on a CCD pixel-level phase-shifting approach. We present theoretical and experimental results for an interferometer based on this pixelated phase-shifting technique. Analyses of component errors and instrument functionality are presented. We show that the majority of error sources cause relatively small magnitude peak-to-valley errors in measurement of the order of 0.002-0.005lambda. These errors are largely mitigated by high-rate data acquisition and consequent data averaging.     

Method for designing phase-calculation algorithms for two-dimensional grating phase-shifting interferometry

We propose a design method of phase-analysis algorithms based on two-dimensional grating phase shifting for Talbot interferometry, Talbot-Lau imaging, or the Ronchi test. These algorithms are designed to separate the two orthogonal shearing wavefronts and eliminate error effects of unwanted diffraction orders, simultaneously. Taking the effect of multidiffraction into account, moving the two-dimensional grating along a certain pass leads to a series of phase-shifted interfrograms, from which two orthogonal shearing wavefronts are derived, for the tested wavefront to be retrieved. The designing process is demonstrated, and the residual errors are analyzed via simulation works and experimental comparison.     

Phase measurement in temporal speckle pattern interferometry: comparison between the phase-shifting and the Fourier transform methods

The measurement of dynamic displacements by use of speckle pattern interferometry and temporal phase unwrapping allows for the evaluation of large-object displacement fields without the propagation of spatial unwrapping errors. If a temporal carrier is introduced in one of the beams of the interferometer, phase data for whole-object displacement can be retrieved by use of a temporal phase-shifting method or a temporal Fourier transformation approach. We present a comparison between both methods of temporal phase measurement in terms of precision and execution speed. We performed the analysis by using computer-simulated speckle interferograms, an approach that allowed us to know precisely the original phase distribution and also to determine the spatial rms phase error as a function of the phase change introduced between two consecutive speckle interferograms. The performance of both methods to process experimental data is also illustrated by use of the results from a high-speed speckle interferometry study of a carbon fiber panel.     

Relation between optical Fresnel transformation and quantum tomography in two-mode entangled case

In many applications of phase-shifting interferometry (PSI) the actually recorded object wave is the diffraction field of an object surface. In this case not only the phase error but also the amplitude error in the retrieval process have effects on the object wave reconstruction in its original plane. In this paper, for the first time that we know, the amplitude errors in wave reconstruction in PSI are discussed systematically. Some general relations between the amplitude errors and the phase errors for three commonly encountered error sources, namely the phase shift error, the light source intensity fluctuation and the detector nonlinearity, are revealed. The analytical expressions of the amplitude and phase errors for six frequently used algorithms in PSI are derived. The statistical variances of the two kinds of error are introduced to describe the overall performance of these algorithms and are quantitatively calculated. A series of computer simulations are also given as a verification of our analyses. These results can be used for evaluating the complete wave errors in PSI and developing new methods for their correction.     

光热调制半导体激光波长降低干涉测量误差

本文首次将光源波长的光热调制技术用于正弦相位调制干涉仪中,消除了因直接调制激光波长引起的光强度变化对测量的影响,降低了测量误差。     

Phase-shifting interferometry with genetic algorithm-based twin image noise elimination

Phase-shifting interferometry with a genetic algorithm is proposed. The correction of unknown phase-shifting error is an important task in general phase-shifting interferometry. Since phase-shifting errors generate twin image noise in a reconstructed image, we can reduce the phase-shifting errors indirectly by trying to eliminate the twin-image noise in the reconstructed image. By Zernike polynomial expansion, the reconstructed image is represented as the evenness and oddness, where the ratio of the evenness and oddness is a measure of the amount of the twin image noise. We employ the genetic algorithm for finding the fittest phase shifts of interferograms by reducing the evenness of the reconstructed image, which leads to reduction of phase-shifting errors. This phase-shifting interferometry with a genetic algorithm is confirmed experimentally.     

Error sources and algorithms for white-light fringe estimation at low light levels

We address the problem of highly accurate phase estimation at low light levels, as required by the Space Interferometry Mission (SIM). The most stringent SIM requirement in this regard is that the average phase error over a 30-s integration time correspond to a path-length error of approximately 30 pm. Most conventional phase-estimation algorithms exhibit significant enough bias at the signal levels at which the SIM will be operating so that some correction is necessary. Several algorithms are analyzed, and methods of compensating for their bias are developed. Another source of error in phase estimation occurs because the phase is not constant over the integration period. Errors that are due to spacecraft motion, the motion of compensating optical elements, and modulation errors are analyzed and simulated. A Kalman smoothing approach for compensating for these errors is introduced.     

Measurement of air turbulence for on-machine interferometry

There is increasing demand for in situ shape measurements performed on ultraprecision processing machines. One major source of error during interferometric measurements performed on machines is fringe displacement due to external disturbances. We have developed an interferometer equipped with an electro-optic phase modulator that measures the phase of interference fringes before they are displaced by air turbulence. The frequency characteristics of air turbulence induced by a heat source are derived from successive measurements of a test surface. Experimental results show that the phase of the interference fringes can be accurately measured in the presence of air turbulence when the intensity of the fringes is sampled at a speed of several hundred hertz.     

外差干涉仪频率混叠误差分析

研究了激光外差干涉仪中存在的频率混叠现象、误差形成机理和变化规律,重点分析了在共光路外差干涉仪中由激光光源的椭圆偏振化和Ｗｏｌｌａｓｔｏｎ棱镜的安装方位角引起的频率混叠误差的大小及变化规律,对进一步提高共光路外差干涉测量精度具有现实意义。     

Wavefront measurement interferometry at the operational wavelength of extreme-ultraviolet lithography

Two basic types of interferometer, a point diffraction interferometer (PDI) and a lateral shearing interferometer (LSI) suitable for operation in the extreme-ultraviolet (EUV) wavelength region, are described. To address the challenges of wavefront measurement with an accuracy of 0.1 nm rms, we present a calibration method for the PDI that places a mask with two large windows at the image plane of the illumination point light source and a general approach to deriving the phase-shift algorithm series that eliminates the undesired zeroth-order effect in the LSI. These approaches to improving the measurement accuracy were experimentally verified by the wavefront measurements of a Schwarzschild-type EUV projection lens.     

Adaptive-optics system with liquid-crystal phase-shift interferometer

We develop an adaptive-optics system based on a Mach-Zehnder radial shearing interferometer with liquid-crystal-device (LCD) phase-shift interferometry (PSI). Using accurate phase calibration and transient nematic driving of the LCD, the developed three-step PSI procedure can be achieved in a time of 5 ms. The proposed Mach-Zehnder radial shearing PSI method reconstructs the phase information using a digital signal processor (DSP). The DSP then computes appropriate control signals to drive a deformable mirror in such a way as to eliminate the wavefront distortion. The current adaptive-optics system is capable of suppressing low-frequency thermal disturbances with a signal-to-noise ratio improvement of more than 20 dB and a steady-state phase error of less than 0.02pi root mean square when the control loop is operated at a frequency of 30 Hz.     

Simple, real-time method for removing the cyclic error of a homodyne interferometer with a quadrature detector system

The cyclic error of a homodyne interferometer is caused mainly by phase mixing due to the imperfection of polarizing optical components such as polarizing beam splitters. In Appl. Opt. 43, 2443 (2004), we concentrated on the relationship between these imperfect optical characteristics and the cyclic error and found the preamplifier-gains condition for removing the cyclic error. Here we demonstrate the cyclic error correction method experimentally and show that the method can be applied in real time. We obtained 0.04-nm cyclic errors, with a standard deviation above 5 microm.     

Wavelength-multiplexed phase-locked laser diode interferometer using a phase-shifting technique

We propose a new signal-processing method for eliminating measurement errors that occur in the wavelength-multiplexed phase-locked laser diode interferometer. The basic idea proposed here is a very simple but effective way to improve measurement accuracy. With our scheme, the phase in the interference signal is strictly shifted by 2pi, which enables us to eliminate measurement errors. The equivalent wavelength ? is 80 mm, and the measurement accuracy reaches ~?/600. A step-height measurement was also carried out in the experiment.     

基于波长移相的改进加权多步算法

在斐索干涉仪中测量光学平板受到多面反射光束的影响,而波长移相干涉技术是通过光源波长的改变来实现干涉信号的移相,原理上可以实现各个干涉信号的分离,在算法上要求能对多次谐波进行较好的抑制,加权多步移相算法可以实现这样的目的.笔者提出了加权多步移相算法的改进算法,可以提高算法对误差的抑制能力,最后用实验验证了改进算法的有效性.     

大长度激光测量中阿贝误差消除方法的研究

提出一种基于3路独立激光干涉仪消除大长度激光测量中的阿贝误差的方法,3路干涉仪的安装位置可布置成任意三角形。通过3路干涉仪的测量结果及被测仪器与3路干涉仪安装位置的几何关系,构造一路与被测仪器同光路的虚拟干涉仪,推导虚拟干涉仪的测长公式。该算法对干涉仪的安装位置无特殊要求,在实践中易于实现。为验证算法的有效性,依托于室内80 m大长度标准装置,通过改变被测仪器安装位置,在45 m范围内进行了3组不同的验证实验。实验结果显示消除阿贝误差后,残余的其它误差的最大值仅为1.10 μm,该算法可有效地消除阿贝误差。     

Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting

We present a new computational algorithm of phase-shifting interferometry that can effectively eliminate the uncertainty errors of reference phases encountered when we obtain multiple interferograms. The algorithm treats the reference phases as additional unknowns and we determine their exact values by analyzing interferograms using the numerical least-squares technique. A series of simulations prove that this algorithm can improve measuring accuracy because it is unaffected by the nonlinear and random errors of phase shifters.