Using higher steps phase-shifting algorithms and linear least-squares fitting in white-light scanning interferometry

White-light scanning interferometry (WLSI) has been a well-established tool for measuring the profile of objects. Since the white-light source is continuous in spectrum, the phase ambiguity problem can be avoided. Specimens with discontinuous profile can therefore be possibly measured by WLSI. In this paper, using higher steps, i.e. nine and eleven steps, phase-shifting algorithms (PSAs) based on local linear conditions of envelope function were proposed. The actual zero optical path difference position was retrieved by extracting the phase value and phase compensation. Maximum intensity peak positions at five points were used to calculate the center wavelength of the light source by both phase unwrapping and linear least-squares fitting. Both simulated and experimental results showed that the proposed nine- and eleven-step PSAs have good linearity and robustness to accurately measure the profile and center wavelength of the light source.     

Micro-profile measurement based on windowed Fourier transform in white-light scanning interferometry

White-light scanning interferometry (WLSI) has been widely used in micro-profile measurement such as Micro-Electro-Mechanical Systems (MEMS) and Computer Generated Hologram (CGH) diffractive elements. It does not contain phase ambiguity problem which is often encountered in monochromatic wavelength interferometry. This paper presents an algorithm based on windowed Fourier transform (WFT) to extract the phase of a white-light interferogram and compensates for the difference in zero optical path difference (ZOPD) position in WLSI. With the WFT technique, the center wavelength of a white-light source and the phase of a white-light interferogram could be retrieved simultaneously. The effect of noise, scanning interval of a piezoelectric transducer (PZT) and the window size of WFT are also analyzed. Both simulated and experimental results show that the proposed algorithm has good noise immunity and is able to accurately measure the micro-profile of a specimen.     

Continuous wavelet transform for micro-component profile measurement using vertical scanning interferometry

White-light interferometric techniques have been widely used in three-dimensional (3D) profiling. This paper presents a new method based on vertical scanning interferometry (VSI) for the 3D profile measurement of a micro-component that contains sharp steps. The use of a white-light source in the system overcomes the phase ambiguity problem often encountered in monochromatic interferometry and also reduces speckle noises. A new algorithm based on the continuous wavelet transform (CWT) is used to retrieve the phase of an interferogram. The algorithm accurately determines local fringe peak and improves the vertical resolution of the measurement. The proposed method is highly resistant to noise and is able to achieve high accuracy. A micro-component (lamellar grating) fabricated by sacrificial etching technique is used as a test specimen to verify the proposed method. The measurement uncertainty of the experimental results is discussed.     

Shape measurement by multi-illumination method in digital holographic interferometry

In this paper, temporal phase unwrapping and short time Fourier transform (STFT) are proposed for shape measurement of an object with height steps by digital holographic interferometry (DHI). A series of digital holograms is recorded with a high-speed CCD camera using a multi-illumination method. Each pixel is processed along the time axis independently of other pixels. Two novel methods are proposed to process the wrapped phase maps: the first method is based on complex phasor (CP) and temporal phase unwrapping, and the second method is based on CP and STFT. In the STFT method, continuous phase maps are obtained by integration of the exacted instantaneous frequency. The continuous phase map can characterize the profile of the object with height steps. An experiment is conducted to verify the validity of the proposed methods.     

Determination of displacement derivative in digital holographic interferometry

Digital holography (DH) and digital shearography (DS) both play an important role in non-destructive evaluation. In this paper, a novel method based on digital holographic interferometry (DHI) and complex phasor (CP) is proposed to determine displacement derivative. An algorithm is employed to filter the imaginary and real parts of complex values without the need of direct phase manipulation. Two-dimensional short time Fourier transform (STFT) is employed subsequently to process wrapped phase maps. An experiment is conducted to demonstrate the validity of the proposed method.     

Demodulation of a single interferogram based on continuous wavelet transform and phase derivative

A novel method based on continuous wavelet transform (CWT) and guidance of phase derivative is developed to measure the phase of a single fringe pattern which contains closed fringes. Wrapped phase values are retrieved by ridge extraction algorithms based on CWT which has the capability of better noise reduction and thus increases the resolution of measurement significantly. To further reduce the noise, the scales detected by maximum ridge algorithm are filtered iteratively before retrieval of wrapped phase. The proposed method also identifies any ambiguous point in a non-monotonous fringe pattern by directly tracking an inflexion point from an unwrapped phase map without the use of a carrier. The algorithm developed is validated by computer simulation and experimental results. Based on micro interferometry the experimental results for both static and dynamic deformations of a micro structure demonstrate that the proposed method is an effective tool for the analysis of closed fringe patterns and subsequent deformation measurement. However, the proposed technique is limited to measurement of surface which is relatively smooth compared to the mean wavelength of the light source. In addition, prior knowledge of the sign of surface slope is required in cases where a spatial carrier is not available or adaptable.     

The design of a non-cube beam-splitter used for electronic speckle pattern interferometry

A non-cube beam-splitter (NCBS) is proposed, by which an incident beam can be separated largely in a direction and then the lights from the test object and the lights from a reference surface placed adjacently to the test object can be combined to construct a simple electronic speckle pattern interferometry (ESPI) system. Two mainly useful quantitative ways, to calculate the phase change of the test object, the phase-shift and the fringe carrier method with Fourier transform, can be achieved in the ESPI system with the NCBS. Experiments with phase-shifting and fringe carrier method are completed. The experimental results show that the monolithic design of the proposed NCBS is effective in ESPI measurement and immunity to vibrations.     

非单调条纹图的相位恢复新方法

在连续小波变换和展开相位拐点识别的基础上,提出一种非单调条纹图的相位恢复新方法.包裹相位通过连续小波变换的方法提取,提出小波变换尺度步长的选择准则及对粗糙尺度作迭代均值滤波的技术.基于展开相位拐点与符号歧义点的单应性,可通过检测并修正展开相化拐点的方法恢复非单调条纹图的真实相位.给出了详细的理论推导、数值模拟及实验验证过程.数值模拟包括一维及二维的含噪声信号,实验为基于显微干涉法的微桥的静动态形貌和变形测试.模拟及实验结果表明,该法只需一幅条纹图即可准确解调非单调条纹的相位,最大误差小于4%,且具有极强的抗噪能力.这为显微干涉测量提供了有效的相位恢复新手段.     

Iterative Fourier transform method for phase map recovery in electronic speckle-shearing pattern interferometry

Increasing phase values at the edges of objects under analysis in electronic speckle shearing interferometry (ESSPI) cause errors in the actual recovering methods to obtain the spatially integrated phase. An iterative least minimum squares method is proposed here to recover the phase map from the approximated ESSPI derivatives when the displacement phase is increasing near the object edges. It is based in an iterative Fourier transform method derived from a least-squares phase map recovery algorithm. Experimental results from a rotating cylindrical bar show the validity of our approach.     

Parallel two-step phase-shifting microscopic interferometry based on a cube beamsplitter

Parallel two-step phase-shifting interferometry for microscopy is presented, and the recording condition for generalized two-step phase-shifting interferometry is discussed. A 45° tilted cube beamsplitter enables to replicate the orthogonally linear polarized object and reference waves into two parallel beams, respectively. As a consequence, two interferograms with quadrature phase shift are obtained along the two beams, and phase reconstructed with an improved algorithm. To reconstruct the phase distribution from the two-step phase-shifting interferograms, a certain recording condition should be satisfied. However, the recording condition has not ever been discussed before. In this paper, the recording condition for the two-step phase-shifting interferometry is derived and that is: the intensity of reference wave should be no less than two times object wave intensity.     

子孔径拼接法检验大口径光学镜面精度分析

介绍子孔径拼接检测大口径光学镜面的原理，即用小口径的平面干涉仪检验大口径平面的一部分，通过改变2者相对位置获得覆盖到整个被检验镜面的子孔径检测数据。提出利用最小二乘法对相邻2个子孔径重叠部分的检测数据进行计算来确定实际所有子孔径之间的位置关系，进而得到拼接而成的整体面形信息。并对子孔径拼接成的面型与实际面型的误差进行分析，建立了对子孔径拼接全口径波面恢复精度的评价指标。根据子孔径拼接原理完成了实验，并对多组子孔径数据拼接后的波面恢复精度进行了分析。实验证明，子孔径拼接检测大口径光学元件综合误差小，重复精度高。     

Fringe localization using focus measure

A method for automated evaluation of fringe localization using focus measure is demonstrated experimentally. Dual-illumination digital holographic interferometry is used to generate phase difference maps (PDMs) of a rough test object. Focus measure values of the PDMs along the axial direction (increment: 10λ=6.33 μm) yield an inverted bell-shaped curve which, in turn, facilitates the characterizations of the plane and region of fringe localization. The method can be applied in the distance measurement of rough objects and in the optimization of fringe visibility.     

利用电子散斑相移技术测量物体三维面形的方法

为了获得准确的面形测量,提出了一种相移电子散斑干涉技术测量物体面形的测量方法.利用电子散斑干涉产生载波条纹,该载波条纹受到物体表面高度的调制变得弯曲,引起载波条纹相位的变化,可运用相移技术提取物体的相位信息,最后根据高度和相位之间的关系得到物体的面形.介绍了该方法的原理,利用该方法对球冠物体进行了面形测量,证明该方法测量物体面形是可行性的.由于是采用散斑干涉的方法产生干涉条纹,因此该方法测量物体面形具有灵敏度高的优点.     

Blind phase error suppression for color-encoded digital fringe projection profilometry

Color-encoded digital fringe projection profilometry (CDFPP) has the advantage of fast speed, non-contact and full-field testing. It is one of the most important dynamic three-dimensional (3D) profile measurement techniques. However, due to factors such as color cross-talk and gamma distortion of electro-optical devices, phase errors arise when conventional phase-shifting algorithms with fixed phase shift values are utilized to retrieve phases. In this paper, a simple and effective blind phase error suppression approach based on isotropic n-dimensional fringe pattern normalization (INFPN) and carrier squeezing interferometry (CSI) is proposed. It does not require pre-calibration for the gamma and color-coupling coefficients or the phase shift values. Simulation and experimental works show that our proposed approach is able to effectively suppress phase errors and achieve accurate measurement results in CDFPP.     

A least-squares method to cancel rigid body displacements in a hole drilling and DSPI system for measuring residual stresses

This paper presents the evaluation of a method to cancel rigid body displacements that can be introduced when a hole drilling and digital speckle pattern interferometry (DSPI) combined system is used to measure residual stresses. The proposed method is based on a least-square calculation of three correction parameters determined from two evaluation lines located near the edge of the phase map where the displacement field generated by the drilling process is supposed to be negligible. The errors introduced by the method for different residual stress levels and rigid body displacements are analysed using a numerical simulation. An application of the method to experimental data is also presented.     

Investigation of a dual-layer structure using vertical scanning interferometry

This paper proposes a method based on white light vertical scanning interferometry (VSI) to investigate a dual-layer structure. The optical arrangement is based on a modified Michelson interferometer that utilizes a reference beam and two object beams. Each object beam interferes with the reference beam and produces an interferogram. A series of interferograms are obtained on a dual-layer structure and the thickness of each layer is obtained. A continuous wavelet transform (CWT) is used to extract the envelope of each interferogram in order to determine the peak intensity that provides an indication of each layer's boundary. Tests are conducted on a semiconductor wafer and a micro-gear made of polymeric material deposited on a metal substrate. Results show that the proposed method has a good potential for investigating a dual-layer micro-structure.     

相移阴影莫尔条纹正交化解调技术

提出一种基于克莱姆正则化分析法的三帧自标定相移阴影莫尔三维轮廓技术.该技术首先采用移动光栅的方法获得相移条纹图,然后通过不同帧相移条纹图相减去除条纹图背景,进而结合克莱姆正交化法和最小二乘法,发展了一种相位解调方法,提取了测量相位.以五步Harlharan算法为参考,用不同算法对同一物体表面进行测量.结果表明,相对于典型的三步相移法和主量分析方法,提出的方法测量得到的相位误差最小(0.5rad),且简化了测量过程.     

欠采样包裹相位图的展开算法

利用四种常见算法对由球面参考光像面数字全息显微术得到的中药饮片海金沙细胞的包裹相位图进行了相位解包裹。结果表明:基于离散余弦的最小二乘算法和预条件共轭梯度算法会使欠采样区域的误差进行传递,使得解包裹相位产生较大误差;质量图导向路径跟踪算法由于未识别残差点,导致出现拉线或者孤岛区域,而且处理速度很慢;基于横向剪切干涉的最小二乘算法处理欠采样包裹相位图的效果最好,而且处理速度较快。     

Measurement of non-monotonous phase changes in temporal speckle pattern interferometry using a correlation method without a temporal carrier

Recently, a phase evaluation method was proposed to measure nanometric displacements by means of digital speckle pattern interferometry when the phase change introduced by the deformation is in the range [0,π) rad. This method is based on the evaluation of a correlation coefficient between two speckle interferograms generated by both deformation states of the object. In this paper, we present a novel technique to measure non-monotonous displacements in temporal speckle pattern interferometry using a correlation method without a temporal carrier. In this approach, the sign ambiguity is resolved automatically due to the introduction of a function that determines the correct sign of the displacement between two consecutive speckle interferograms. The rms phase errors introduced by the proposed method are determined using computer-simulated speckle interferograms. An application of the phase retrieval method to process experimental data is also presented.     

Instantaneous velocity measurement of dynamic deformation by digital holographic interferometry

A novel method for the instantaneous velocity measurement of dynamic deformation by digital holographic interferometry is proposed. During dynamic deformation, a series of digital holograms is recorded by a high-speed camera. At each pixel of the phase difference maps, phase and amplitude information are combined as complex phasor (CP). Each pixel can be then considered as an independent sensor and a sequence of complex phasors of such a sensor is analyzed by short time Fourier transform (STFT) along the time axis. A fast iterative algorithm is developed for the computation of instantaneous velocity. The displacement of each pixel can also be obtained by integration of the instantaneous velocity over time and phase unwrapping process is thus avoided. The performance of the proposed CP method is compared experimentally with the commonly used digital phase subtraction method.