Stokes parameters and polarization contrasts in Young's interference experiment

We derive a spectral interference law that governs the behavior of the four Stokes parameters in Young's two-pinhole experiment with a random electromagnetic beam. In addition to the visibility of intensity fringes, we introduce three new contrast parameters that describe the interference-induced changes in the field's state of partial polarization. The polarization modulation depends on the electric field correlations at the pinholes and is closely related to the two-point Stokes parameters. The results are expected to be particularly useful in polarization interferometry and electromagnetic coherence theory. The formalism is demonstrated with specific examples.     

Electromagnetic spatial coherence wavelets and the classical laws on polarization

Malus’ law, Fresnel-Arago interference laws and Stokes parameters are analyzed in the framework of the tensor theory of the electromagnetic spatial coherence wavelets. It is shown that these laws are suitable for each individual wavelet after regarding the correlation properties of the random electromagnetic field. Stokes parameters at the wavelet scale are introduced. They deal to generalized Stokes parameters which, in contrast to the classical ones, take into account the correlation properties of the random electromagnetic field on the aperture plane, that is the contributions of radiator pairs in surroundings of that plane.     

Parameters characterizing electromagnetic wave polarization

In this paper, generalizations of the Stokes parameters and alternative characterizations of three-dimensional (3D) time-varying electromagnetic fields is introduced. One of these characteristics is the normal of the polarization plane, which, in many cases of interest, is parallel (or antiparallel) to the direction of propagation. Others are the two spectral density Stokes parameters which describe spectral intensity and circular polarization. The analysis is based on the spectral density tensor. This tensor is expanded in a base composed of the generators of the SU(3) symmetry group, as given by Gell-Mann and Y. Ne'eman [The Eight-fold Way (Benjamin, New York, 1964)] and the coefficients of this expansion are identified as generalized spectral density polarization parameters. The generators have the advantage that they obey the same algebra as the Pauli spin matrices, which is the base for expanding the 2D spectral density tensor with the Stokes parameters as coefficients. The polarization parameters introduced are formulated in the frequency domain, thereby further generalizing the theory to allow for wide-band electromagnetic waves in contrast to the traditional quasi-monochromatic formulation.     

Experimental determination of two-point Stokes parameters for a partially coherent broadband light beam

The generalized Stokes parameters, which are two-point extensions of conventional Stokes parameters, are determined for a pair of points in the cross-section of a partially coherent broadband light beam. For this purpose, using a two-mirror and two-beam-splitter assembly, unpolarized, linearly polarized and partially polarized electromagnetic beams are generated. This simple experimental method for determining generalized Stokes parameters establishes an analogy with the experimental scheme of determining conventional Stokes parameters, which in succession contributes to appreciable reduction in uncertainty in the measurements.     

Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence

We analyze the modulation of the Stokes parameters in Young's two-pinhole interference experiment with a random electromagnetic beam. We demonstrate that the electromagnetic (spectral) degree of coherence put forward in Opt. Lett.29, 328 (2004) [or its space-time analog in Opt. Express 11, 1137 (2003)] is physically related to the contrasts of modulation in the four Stokes parameters. More explicitly, the electromagnetic degree of coherence is a measure of both the visibility of the intensity fringes and the modulation contrasts of the three polarization Stokes parameters. We also show that by using suitable wave plates the modulation in any Stokes parameter can be transformed into the form of intensity variation, and hence the electromagnetic degree of coherence can be obtained experimentally by four visibility measurements.     

Generalized stokes parameters of random electromagnetic beams

A generalization of the Stokes parameters of a random electromagnetic beam is introduced. Unlike the usual Stokes parameters, which depend on one spatial variable, the generalized Stokes parameters, depend on two spatial variables. They obey precise laws of propagation, both in free space and in any linear medium, whether deterministic or random. With the help of the generalized Stokes parameters, the changes in the ordinary Stokes parameters upon propagation can be determined. Numerical examples of such changes are presented. The generalized Stokes parameters contain information not only about the polarization properties of the beam but also about its coherence properties. We illustrate this fact by expressing the degree of coherence of the electromagnetic beam in terms of one of the generalized Stokes parameters.     

Experimental verification of the electromagnetic spectral interference law using a modified version of the Young's interferometer

We present a method to verify the electromagnetic spectral interference law experimentally, which connects the combined two-beam usual Stokes parameters with the corresponding single beam usual Stokes parameters. Using polarizers and phase retarders in a modified version of the Young's interferometer, the usual Stokes parameters and the generalized Stokes parameters are determined for a laser beam. The experimentally determined single point (usual) Stokes parameters at an axial point in the observation plane for the combined beam are compared with their theoretically expected values obtained by putting the experimental data in the electromagnetic spectral interference law. A fairly good agreement is found between the two values. Not only does the proposed method provides a comparison between the two kinds of Stokes parameters but also validates the role of the generalized Stokes parameters as modulation parameters for the usual Stokes parameters.     

Shape and slope measurement by source displacement in shearography

Shearography is a full field non-contact optical technique usually used to measure the gradient of the displacement of a surface subjected to thermal or mechanical loading. This paper describes the use of shearography for surface slope and shape measurement. Interferometric speckle patterns obtained before and after displacement of the optical source are correlated to yield correlation fringes which are, in general, a mixture of slope fringes and carrier fringes. This paper contains a full treatment of the sensitivity of slope fringes to the parameters; the illumination and the imaging geometry and the magnitude and the direction of the source displacement. The slope fringes are corrected for distortion, which is due to the necessary off-axis illumination, and are scaled using parameters calculated using a mathematical model. Experimentally generated phase-stepped slope fringes are unwrapped and integrated to recover the object shape.     

Generalized Stokes parameters of random electromagnetic vortex beams propagating through atmospheric turbulence

By using the extended Huygens–Fresnel principle, the analytical expressions for the generalized Stokes parameters of random electromagnetic Gaussian Schell-model (GSM) vortex beams propagating through atmospheric turbulence are derived, and used to study the changes in spectral Stokes parameters of random electromagnetic GSM vortex beams in atmospheric turbulence and to compare the results of random electromagnetic GSM vortex-free beams. The influence of atmospheric turbulence on the spectral Stokes parameters is analyzed. The validity of our results is interpreted physically.     

A composite-structured-light 3D measurement method based on fringe parameter calibration

This paper proposes a novel method for reducing measurement error caused by spectrum overlapping in composite-structured-light 3D measurement systems. For a composite-structured-light 3D measurement system, spectrum overlapping causes parameters of each deformed phase-shifting fringe to change, and therefore leads to phase measurement errors. The proposed fringe parameter calibration method is based on the fact that variations in each deformed fringe's parameters are independent of height and reflectivity of the measured object. Three frames of composite grating are projected on the reference plane, and each carrier channel includes the information of three phase-shifting sinusoidal gratings used in Phase Measuring Profilometry (PMP). With the parameter calculation formulas of PMP, the parameters of fringes demodulated from the same carrier channel can be calculated, and therefore parameter relation coefficients between fringes demodulated from different carrier channels may be obtained. When an object is measured, these relation coefficients can be used to calibrate the parameters of the deformed phase-shifting fringes. A new 3D measurement mathematical model is established to reconstruct the shape of the object. Experimental data proved that the proposed method can effectively restrain the effect of spectrum overlapping and improve measurement accuracy by more than three times.     

Generalized (two-point) stokes-parameters representation in terms of the radial and azimuthal field components

Within the paraxial framework, a generalization of the Stokes representation is introduced in terms of the radial and azimuthal field structure of the beam cross-section. These generalized parameters depend on two spatial variables and are closely related with the cross-spectral density (CSD) matrix of the field. The propagation law of such parameters through general first-order optical systems is also given. It allows to describe, in terms of the radial and azimuthal field components, the coherence and polarization behavior of partially-polarized, partially-coherent beams.     

Generalized metric spaces for correlation fringes and applications

A novel conception and its applications, the generalized metric for correlation fringes, are presented, which can be used to develop and analyze the relevant algorithms. The relationship between the metric and the correlation fringes representation is examined and the generalized metric is defined. The theoretical analysis of traditional algorithms for correlation fringes indicates that all of them can be regarded as the generalized metrics. In addition, the generalized metric for correlation fringes is applied in developing a new algorithm. A novel generalized metric for correlation fringes based on cross entropy is presented and is proved by theoretical derivation and experimental verification. All the investigations show that it is reasonable and valid to define the generalized metric for correlation fringes in digital speckle pattern interferometry (DSPI). The fringes obtained by cross-entropy metric can be used to represent the same desired information by subtracting. There are different vector function structures among the generalized metrics, which directly influence the quality of fringes and the difficulty in post-processing.     

Generalized Stokes parameters of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams

The analytical expressions for the generalized (two-point) Stokes parameters of stochastic electromagnetic Gaussian Schell-model pulsed (GSMP) beams propagating in free space are obtained. The changes in the spectral Stokes parameters in free space propagation are studied. The dependence of on-axis and transverse spectral Stokes parameters of stochastic electromagnetic GSMP beams on the pulse parameters including pulse duration and temporal coherence length is stressed and illustrated numerically. The results are interpreted physically.     

Vector and tensor polarizations of light beams

Generalized Stokes vectors consisting of nine real parameters in terms of irreducible spherical and Cartesian tensor operators are considered to describe the 3D electromagnetic field. Vector and tensor polarization degrees are introduced to characterize the polarized beam. It is shown that the degree of a pure vector polarized light beam is less than 1, i.e., vector polarized light beam is only partially polarized. The geometrical interpretation of vector and tensor polarizations is presented.     

玻璃中Cr~(3+)的多声子辐射跃迁

采用多条激光窄线激发,由Cr~(3+)的荧光谱线斯托克斯位移与激发光能量之间的线性关系,分析并导出了氟磷酸盐玻璃中Cr~(3+)的一些运动晶场参数。     

On the sensitivity of coherent gradient sensing: Part II—An experimental investigation of accuracy in fracture mechanics applications

The sensitivity of the transmission ‘Coherent Gradient Sensing’ (CGS) technique is investigated experimentally in relation to the study of deformations near the tips of cracks in three-point bend specimens. Fringe data from these experiments are interpreted as gradients (geometric interpretation) and finite differences (physical interpretation) of hydrostatic stress fields. These data are used to compare the accuracy of the geometric interpretation of fringes with the physical interpretation of fringes obtained using increased sensitivity in order to confirm the theoretical findings from Part I of this investigation. Also, a least-squares fitting technique was used on the fringe data obtained from the region outside of the near tip 3D zone in order to investigate issues of K-dominance of the stress field in this region.Results from the experimental investigations reported in this paper indicated that increasing the sensitivity of the CGS technique improved the quantity and quality of fringe data. However, the apparent size of the region on the image plane dominated by 3D effects increased and the differences between the geometric and physical interpretations of CGS fringe data were increased as well. In addition, the stress field outside of the 3D zone was determined not to be strictly K-dominant, as was predicted by results from previous experiments.     

Topographic Analysis of Ocular Fundus Using the Fourier Transform Method for Projected Grating Images

In topography measurements of a local ocular fundus of the human eye using a grating-pattern projection technique, a pre-processing method to detect fringes accurately and a generalized Fourier transform method to analyze the fringe deformations are proposed. The method of analysis compensates for fringe deformations caused by instrumental characteristics. By combining the two methods, depth information can be determined with high accuracy. It is shown that the topographic parameters of the optic disc and the fundus slopes in the rim area are obtained with good reproducibility.     

Stokes parameters and degree of polarization of nonparaxial stochastic electromagnetic beams

On the basis of angular spectrum representation and the stationary-phase method, far-field expressions for generalized Stokes parameters of nonparaxial stochastic electromagnetic beams are derived, which permits us to study the changes in the ordinary Stokes parameters upon propagation, and the changes in the spectral degree of polarization of partially polarized nonparaxial stochastic electromagnetic beams. It is shown that the spectral degree of polarization changes across the section on beam propagation.     

高分辨率衍射图形的DMD并行激光干涉直写

通过将数字微反射镜（Digital Micro-mirror Device,DMD）输入阵列图形微缩干涉成像,激光干涉直写系统在光刻胶板上得到缩小的干涉光斑图形,像素特征尺寸3.5 μm,干涉条纹周期1 μm.控制刻蚀深度、干涉条纹取向和DMD输入图形的结构,系统能数字化完成2D/3D、3D光变图像、超微图形文字以及二元光学元件的制作,实现了超高分辨率图形与高效的干涉光刻.给出了实验结果.