Theoretical study on the statistical properties of phase difference between two interfering speckle fields

The statistical properties of phase difference between two interfering speckle fields are studied theoretically with the general expression of the free standard deviation of the phase difference of speckle fields. The experimental results of the standard deviations of phase difference between two interfering partially developed speckle fields in the diffractive region [Opt. Quantum Electron. 19 (1987) 59] are studied and discussed theoretically with the general expression of the free standard deviation of phase difference of speckle fields. The theoretical standard deviations of phase difference for the case of two interfering fully developed Gaussian speckle fields are compared with that which are numerically calculated from the Gaussian theory.     

Studies on the statistical properties of phase difference of speckle fields

According to the relations between the phase and the complex speckle amplitude, a phase angle is constructed to estimate the free standard deviation of phase difference of speckle fields. The relative accurate amending expression of evaluating the free standard deviation of phase difference is derived for the Gaussian speckle fields. The statistical properties of phase difference are studied for the Gaussian speckle fields of diffraction fields. The theoretical values of the free standard deviation of phase difference for the case of fully developed speckle fields are compared with the numerical values of the free standard deviation which are calculated from the Gaussian theory of the fully developed speckle fields.     

剪切散斑干涉术的统计分析

本文用统计光学方法分析了剪切散斑图的成象过程;散斑图的频谱分布以及全场滤波干涉条纹的形成.发现剪切散斑干涉条纹不仅与三维位移微分有关,而且与面内位移量有关.在此基础上又讨论了影响干涉条纹质量的有关因素,并作了实验验证.     

Real-time visualisation of deformation fields using speckle interferometry and temporal phase unwrapping

A real-time system for analysing data from speckle interferometers, and speckle shearing interferometers, has been developed. Interferograms are continuously recorded by a digital camera at a rate of 60 frames s⁻¹ with temporal phase shifting carried out at the same rate. The images are analysed using a pipeline image processor. With a standard 4-frame phase-shifting algorithm (phase steps of π/2), wrapped phase maps are calculated and displayed at 15 frames s⁻¹. These are unwrapped using a temporal phase unwrapping algorithm to provide a real-time colour-coded display of the relevant displacement component. Each camera pixel (or cluster of pixels) behaves in effect as an independent displacement sensor. The reference speckle interferogram is updated automatically at regular user-defined intervals, allowing arbitrarily large deformations to be measured and errors due to speckle decorrelation to be minimised. The system has been applied to the problem of detecting sub-surface delamination cracks in carbon fibre composite panels.     

Digital phase stepping speckle interferometry

A variation of electronic speckle pattern interferometry (ESPI) is presented whereby phase fringes are produced in contrast to speckle correlation fringes. Digital phase stepping speckle interferometry uses a phase measurement technique developed for holographic interferometry, which involves phase shifting of the reference beam in a conventional speckle interferometer. Saw-tooth phase fringes are produced by subtracting the phases of the speckle fields before and after object deformation in a digital frame store. This enables the sign of the deformation to be distinguished. After application of a special low-pass digital filter, the phase fringes are of adequate quality to be counted automatically.     

Measuring velocity of speckle fields with digital speckle pattern interferometry

We investigate a digital method for detecting the velocity of a diffusing object. The technique is based on Digital Speckle Pattern Interferometry (DSPI). A set of reference fringes is generated externally through the reference beam in a digital interferometer. As the object moves, subsequent frames are acquired and subtracted according to the normal DSPI procedure and stored. By means of the theory of first order speckle statistics applied to speckle intensity correlation, we relate the visibility variations in the reference fringes with the object velocity. Thus, by measuring the fringe visibility variation in the resulting DSPI stored frames the mean object velocity can be obtained. The theoretical results are experimentally verified.     

Statistical properties of the sum of two partially correlated speckle patterns

The probability density function and average contrast of the sum of two speckle patterns produced with various intensity ratios and correlation conditions have been experimentally investigated. The experimental results are compared with the theoretical ones obtained by Barakat and Goodman. Comparison shows excellent agreement between the experimental and theoretical results.     

Quantitative phase retrieval in dynamic laser speckle interferometry

The rapid progress of modern manufacturing and inspection technologies has posed stringent requirements on optical techniques for vibration characterization and dynamic testing. Due to its simplicity, accuracy and whole-field characters, laser speckle interferometry has served as one of the major techniques for dynamic measurement. In this paper, a two-step phase shifting method is developed for quantitative speckle phase measurement, which helps to eliminate the specklegrams needed for phase evaluation and facilitate dynamic measurement. Unlike previously reported two-step methods using fringe patterns with known phase shift of π/2, a small unknown phase shift is employed instead in the proposed method, which eliminates the need for phase shifting devices. Further investigation shows that small phase shifts are preferable over large phase shifts in this method. Shearographic experiments conducted have demonstrated the effectiveness of the proposed technique.     

A noise-immune method of phase unwrapping in speckle interferometry

A new phase unwrapping method that has the advantage of unwrapping of phase distribution with severe noise such as those obtained in speckle interferometry is proposed. First, computer simulations were carried out to examine the performance of the proposed method. Speckled noisy data were generated in the computer, and various types of filters having different sizes for the windows were used to carry out the phase unwrapping. It is found that the method is largely capable of phase unwrapping, and it works quite well even when the human eye cannot recognize clearly the phase jumps due to noise. Finally, the present method was applied to experimental phase distribution obtained by the speckle-shearing interferometer. The proposed method is quite simple and effective from the point of computations involved.     

Noise reduction in electronic speckle pattern interferometry fringes by merging orthogonally polarised speckle fields

The noise in sawtooth fringes generated by electronic speckle pattern interferometry (ESPI) is investigated. When deformations of depolarising objects are studied, the scattered object light can be decomposed into two orthogonal linearly polarised speckle patterns which are partially decorrelated. Their correlation coefficient decreases with increasing depolarisation coefficient of the object. By suitable merging of the phase distributions of these two speckle fields on the basis of a modulation depth analysis, the rms phase error in the ESPI sawtooth fringes can be reduced significantly.     

Statistical properties of the dynamic dichromatic laser speckle

The statistical properties of dynamic elongated speckle patterns produced in the far-field diffraction region by a moving diffuse object under illumination of the dichromatic laser beam are experimentally investigated by means of a correlation method. The experimental results show that the correlation length of the normalized autocorrelation function of time-varying dichromatic speckle intensity fluctuations depends strongly on the fibrous radial structure of elongated speckle patterns. The qualitative discussions are given for the experimental results.     

全息干涉计量中干涉计量场的统计特性

本文运用统计学方法推导了全息干涉计量中散斑计量场的统计特性.证明了:干涉计量场的复振幅为复圆高斯随机变量;干扰计量场强度的概率密度服从负指数分布;散斑干涉计量场的系综平均值代表变形信息.     

An integrated liquid crystal phase modulator for speckle shearing interferometry

A versatile phase-stepping speckle shearing interferometer is described that employed a calibrated liquid crystal as its phase modulator, operating at 514 nm. By implementation of an algorithm upon phase-stepped shearographic images it was possible to create a phase map, that provided quantitative data regarding the derivative of any out-of-plane surface displacement. The use of a liquid crystal phase modulator in an optical arrangement had advantages over alternative methods. However, it is normally limited by spatial non-uniformity of the cells birefringence. The birefringence for a nematic liquid crystal cell was determined in several cross-sectional locations, before the cell was successfully integrated into a Michelson speckle shearing interferometer. Such a scheme offers flexibility in the shearing angle, and a measurement system that may operate over an extended range of monitoring wavelengths.     

Statistical properties of speckle recorded on film

The statistical properties of speckle recorded on photographic film are studied theoretically and experimentally. When saturation and threshold effects are negligible and the gamma is equal to one, the probability distribution function of spatially integrated film transmittance has the form of a double exponential. Experimental results confirm that under those conditions, the density fluctuations caused by the speckle are independent of the signal.     

A new speckle carrier method in speckle interferometry

A new speckle carrier technique is presented, which can solve the problem in one-beam laser speckle interferometry, that the direction of the displacement vector is ambiguously determined when the point-wise method is used. The method is very easy and convenient and the results are good. Some discussion about the relationship between the carrier frequency and the displacements is given.     

Comparison of phase recovery methods in spiral speckle pattern interferometry correlation fringes

Spiral interferometry can be used as a solution to the problem of sign ambiguity presented in the conventional speckle pattern interferometric technique when the optical phase needs to be reconstructed from a single closed fringe system. Depressions and elevations of the topography corresponding to the object deformation are distinguished by the direction of rotation of the local spiral fringe pattern. In this work, we implement and compare several methods for optical phase reconstruction by analyzing a single image composed of spiral speckle pattern interferometry correlation fringes. The implemented methods are based on contour line demodulation, center line demodulation, Spiral Phase Quadrature Transform and the 2D Riesz transform with multivector structure. Contour line and center line demodulation approaches are exclusively dedicated to images containing a fringe system with spiral structure. The others are based on the 2D Riesz transform, these being well known approaches in conventional interferometry. We examine simulated experiments and analyze some of the emerging drawbacks for solving the phase reconstruction problem by using different mean values of speckle size and background noise levels. We also discuss several numerical procedures that may well improve the efficiency and robustness of the presented numerical implementations. The performance of the implemented demodulation methods is evaluated by using a universal image quality index and therefore a quantitative comparison is also presented.     

Deterministic phase retrieval from diffracted intensities speckle fields

We present a model where the phase of a wave front is obtained by processing the intensity of speckle field pattern. By using the light propagation equation and knowing the intensity of the wavefront it is possible to build a system of linear equations which can be solved to give the wanted phase information. Simulations are presented and the convergence of the solution is discussed.     

Extending the bandwidth of speckle interferometry

In speckle interferometry, the scale of the speckle array is wavelength dependent, and this limits the spectral bandwidth that can be used. A simple optical system is described to correct this effect, over a wide spectral range, by introducing a chromatic difference of magnification of opposite sign to the speckle dispersion, and variable in magnitude.