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.     

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-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.     

Sub-Nyquist null aspheric testing using a computer-stored compensator

A novel method to demodulate undersampled interferograms using a computer-stored undersampled compensator is presented. First, the sine and cosine of the computer-stored wave front is correlated with the interferogram that emerges from the asphere under test. Afterward, these two correlation images are used to find the phase map. The detected phase of the correlation fringes is the estimated phase difference between the software compensator and the frame-grabbed interferogram. The prior information required for this method is a good knowledge of the wave front being tested to a few wavelengths of error. Complying with this prior knowledge, the undersampled interferogram under analysis may be easily demodulated. Given that the proposed method is based on the correlation of the frame-grabbed interferogram and the computer-stored one, the method also withstands noise.     

Spatial mismatch calibration using circular carrier technique in the simultaneous phase shifting interferometry

In most simultaneous phase shifting interferometry (SPSI) systems, a group of phase shifting interferograms are captured simultaneously at the different physical locations to retrieve the phase. The data of different interferograms should be spatially matched correctly, which is hard to realize by existing methods or this spatial mismatch will lead to phase retrieving error. In this paper, a spatial mismatch calibration method is proposed, where the circular carrier is introduced in the interferograms of the SPSI system, and the modulating phases of any two interferograms can be retrieved by the demodulation technique of circular carrier interferogram. The slope of the difference between these two phases is proportional to the mismatch value, so this error can be extracted and the experiment setup calibrated. The main error sources of the proposed method are analyzed with the conclusion that its match precision can be achieved up to 0.5 pixel. In addition, the simulated interferograms and actual interferograms captured in a SPSI system are processed to validate our proposed method.     

Computation of a spectrum from a single-beam fourier-transform infrared interferogram

A new high-accuracy method has been developed to transform asymmetric single-sided interferograms into spectra. We used a fraction (short, double-sided) of the recorded interferogram and applied an iterative correction to the complete recorded interferogram for the linear part of the phase induced by the various optical elements. Iterative phase correction enhanced the symmetry in the recorded interferogram. We constructed a symmetric double-sided interferogram and followed the Mertz procedure [Infrared Phys. 7,17 (1967)] but with symmetric apodization windows and with a nonlinear phase correction deduced from this double-sided interferogram. In comparing the solution spectrum with the source spectrum we applied the Rayleigh resolution criterion with a Gaussian instrument line shape. The accuracy of the solution is excellent, ranging from better than 0.1% for a blackbody spectrum to a few percent for a complicated atmospheric radiance spectrum.     

溶液扩散系数自动测量系统

结合实时法全息干涉计量和数字图象处理技术，提出了一种溶液扩散系数的自动测量系统。在干涉图的处理中，采用了相位检测和曲线拟合技术，得到了亚象素的测量精度。     

Two-wavelength laser-diode interferometer with fractional fringe techniques

A two-wavelength interferometer with a fractional fringe technique (the method of coincidence) has been constructed by using dual frequency-ramped laser diodes. The respective wavelengths of two optical phases were measured by the heterodyne technique. The detected two phases are employed with real-time electronic processing to produce two signals that correspond to the integer and the fractional fringe numbers at a single wavelength. These summed signals can yield a synthetic phase having a single-wavelength resolution. The upper limits for the measurement accuracy are theoretically analyzed.     

Two-wavelength phase-shifting interferometry with a superimposed grating displayed on an electrically addressed spatial light modulator

A two-wavelength moire phase-shifting interferometer that uses a superimposed grating has been developed. The optical phase shifts for the two wavelengths are given by digital phase shifts of a superimposed grating displayed on a liquid-crystal spatial light modulator. A phase shift of the moire fringe is achieved by equal phase shifts with opposite signs in the two gratings. A moire phase-shifting interferometer with no moving parts and no requirement for calibration of the value of the phase shifts was obtained. Our experimental result shows measurements of the profile of a step object with a 2.65-microm synthetic wavelength.     

Phase-shifting interferometry with uncalibrated phase shifts

A computationally efficient algorithm for phase-shifting interferometry with imprecise phase shifts is developed. It permits the use of an uncalibrated phase shifter and is also insensitive to spatial intensity variations. The measurement has both spatial and temporal aspects. Comparisons are made between pixels within the same interferogram, and these comparisons are extended across a set of interferograms by a maximum-minimum procedure. A test experiment is performed and confirms the theoretical results. An additional advantage of the algorithm is that an error measure can be developed. This error measure is used to implement an error correction scheme.     

Simple multifrequency and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry for three-dimensional profilometry

We propose a simple multifrequency spatial-carrier and phase-shifting fringe-projection system based on two-wavelength lateral shearing interferometry (LSI). In this system a wedge-shaped plate lateral shearing interferometer is used and, owing to the presence of tilt, a finite number of fringes parallel to the direction of the shear appears; hence a significant spatial-carrier frequency is generated at the focus position. We further enhance the spatial-carrier frequency either by changing the wavelength of the laser light or by slight defocusing. A synthetic interferogram with low spatial-carrier frequency is obtained by use of laser light of two wavelengths simultaneously in the lateral shear interferometer. We obtain the phase-shifted fringe patterns from the same setup by simply moving the wedge plate in an in-plane parallel direction, using a linear translator. The fringe projection system was tested for measurement of the three-dimensional shape of a discontinuous object. The present system has many advantages; e.g., it is a common-path interferometry and hence is insensitive to external vibrations, is compact in size, and is relatively inexpensive.     

Phase correction of interferograms using digital all-pass filters

A method for the phase correction of interferograms in Fourier transform infrared spectroscopy is presented. It is shown that phase error can be canceled to within an arbitrary angular precision by a low-order digital all-pass filter. Such a filter only modifies the phase of the Fourier transform of the interferogram and keeps the magnitude unchanged, like the Mertz method, for example. However, our method minimizes the asymmetric apodization that results in photometric errors when using the Mertz method alone. A practical example is provided in which phase correction over a frequency range of 800 cm(-1) to 4000 cm(-1) using a 9-pole all-pass filter resulted in a photometric error of <0.01%, much less than the 0.3% error of the Mertz method. An alternative and faster (approximately 100 ms) approach is to use an all-pass filter with lower angular precision followed by the Mertz method. Removing most of the phase error with the filter brings the interferogram to an optimal state so that the residual phase error can be completely removed with the Mertz procedure without introducing photometric error. The method can be used in most experiments, including emission spectroscopy, where conventional techniques are inadequate. A simple all-pass filter design algorithm is given.     

Effect of beat frequency on the measured phase of laser-diode heterodyne interferometry

An optical heterodyne interferometer with a frequency-ramped laser diode has been constructed. The effect of the beat frequency on the measured phase has been theoretically investigated in the frequency domain and experimentally verified. Phase errors caused by the difference between the ramp frequency and the beat frequency alter sinusoidally in accordance with the π periodicity of the interferogram. The error can be eliminated by the electronic calibration technique of the beat frequency.     

Experimental study of the phase-shift miscalibration error in phase-shifting interferometry: use of a spectrally resolved white-light interferometer

The white-light interferogram in a spectrally resolved white-light interferometer is decomposed in its constituent spectral components by a spectrometer and displayed along its chromaticity axis. A piezoelectric transducer phase shifter in such an interferometer can give a desired phase shift of pi/2 only at one wavelength. The phase shift varies continuously at all other wavelengths along the chromaticity axis. This situation is ideal for an experimental study of the phase error due to the phase-shift error in the phase-shifting technique, as it will be shown in this paper.     

Hologram Interferometry: Identification of the Sign of Surface Displacements

It is difficult to test aspheric surfaces with a Twyman-Green interferometer because the interferogram frequently contains too many fringes. A simple way of overcoming this problem is to use a lateral-shearing interferometer, in which case the number of fringes in the interferogram can be controlled by varying the shear. However, this has the drawback that two interferograms with orthogonal directions of shear are required; in addition, the accuracy with which the shape of the surface can be evaluated from measurements on photographs of the fringes is limited. In this paper it is shown how these difficulties can be overcome by using a microcomputer-controlled digital radial-shear interferometer. The values of the phase difference in the interferogram at a matrix of points covering the pupil are processed directly in the same microcomputer to give the actual shape of the surface. Typical results obtained with an off-axis paraboloid are presented.     

Data-dependent systems profilometry of two-dimensional surfaces

Fourier-transform profilometry (FTP) and data-dependent systems profilometry (DDSP) are two methods that are available for recovering one-dimensional fine surface profiles from the phase of a single interferogram. FTP has already been extended to two-dimensional surfaces; a similar extension of DDSP is introduced here. Inasmuch as this extension involves autoregressive modeling of the rows or columns of an interferogram, the feasibility of using a common model order is explored. The common order reduces not only the amount of computation but also the errors caused by the heterodyned phase-removal procedure. As autoregression requires masking the first few data values, the length of the mask is determined by means of a Green's function. A comparison shows that DDSP outperforms FTP in roughness measurements in terms of rms and center-line average. The comparison also shows that DDSP is able to recover a detailed surface, whereas FTP outlines only the global features. An interferogram regeneration procedure provides a reference surface for the verification of results.     

Phase reconstruction from three interferograms based on integral of phase gradient

A novel algorithm of phase reconstruction based on the integral of phase gradient is presented. The algorithm directly derives two real-valued partial derivatives from three phase-shifted interferograms. Through integrating the phase derivatives, the desired phase is reconstructed. During the phase reconstruction process, there is no need for an extra rewrapping manipulation to ensure values of the phase derivatives lie in the interval [?π, π] as before, thus this algorithm can prevent error or distortion brought about by the phase unwrapping operation. Additionally, this algorithm is fast and easy to implement, and insensitive to the nonuniformity of the intensity distribution of the interferogram. The feasibility of the algorithm is demonstrated by both computer simulation and experiment.     

Spatial carrier phase-shifting algorithm based on least-squares iteration

An advanced spatial carrier phase-shifting (SCPS) algorithm based on least-squares iteration is proposed to extract the phase distribution from a single spatial carrier interferogram. The proposed algorithm divides the spatial carrier interferogram into four phase-shifted interferograms. By compensating for the effects of the variations of phase shifts between pixels and the variations of background and contrast, the proposed algorithm determines the local phase shifts and phase distribution simultaneously and accurately. Numerical simulations show that the accuracy of the proposed algorithm is obviously improved by compensating for the effects of background and contrast variations. The peak to valley of the residual phase error remains less than 0.002 rad when the magnitude of spatial carrier is in the range from pi/5 to pi/2 and the direction of the spatial carrier is in the range from 25 degrees to 65 degrees. Numerical simulations and experiments demonstrate that the proposed algorithm exhibits higher precision than the existing SCPS algorithms. The proposed algorithm is sensitive to random noise, but the error can be reduced by N times if N measurements are taken and averaged.     

经验模态分解技术在SAR干涉图滤波中的应用

提出了一种基于经验模态分解技术的滤波算法。该算法可以把原始数据分解成不同尺度的信息,从初始于涉图上减去与斑点噪声所对应尺度的信息,就可以达到斑点噪声抑制的目的。与另外两种滤波算法进行比较的结果表明,该算法不仅能有效地去除斑点噪声,并且能很好地保持相位的细节信息和条纹的边缘信息,保持相位的纯洁性,而且大比例地减少了残余点的数量,更有利于相位解缠。