光偏振态的群论描述:转动矩阵的生成元

应用偏振光描述中的变换矩阵与群论的对应关系[1]和相应的计算理论,讨论了与偏振光学系统中的Jones矩阵、Mueller矩阵相对应的SU(2)群、SO(3)群和Lorentz群的生成元问题,给出了用单位矩阵、Pauli自旋矩阵和稀疏矩阵分别作为无耗偏振光学系统中SU(2)群元(Jones矩阵)和SO(3)群元(Mueller矩阵)生成元以及部分损耗偏振光学系统中的幺模群(Jones矩阵)和Lorentz群(Mueller矩阵)生成元的具体形式;矩阵计算理论说明这些群元的生成元表示可以简化偏振光学系统的计算。     

Transformation matrices for the Mueller-Jones formalism

The Mueller-Jones (MJ) or pure Mueller matrix formulation has been reported by using two different matrix transformations in a condensed representation. The possibility to find other transformation matrices is explored. A complete set of unitary operators (R) is found to be closely related with the MJ matrices and with the evolution of pure states on the Poincaré sphere surface. We propose an alternative deduction for the condensed representation of the MJ matrices, obtained by using the Kronecker product operation and use of R unitary matrices as a tool to combine different Mueller matrices and changes of polarized states on the Poincarè sphere surface. Finally, it is shown explicitly that the columns of the transformation matrices are the eigenvectors of the MJ matrix associated to a non-depolarizing optical system and a corollary is established as a criterion to differentiate a Mueller matrix from an MJ matrix.     

Use of Dirac matrices in polarization optics

Several matrix methods have been developed for studying polarization properties of light. Jones was the first to apply the matrix method to the study of polarization optics. In Jones matrix formalism the polarized wave field is represented by 2-element column matrix known as Jones Vector and the polarization device encountered by light is represented by a 2×2 matrix, known as the characteristic Jones matrix of the device. Mueller introduced a new matrix method where the wave field is represented by a 4-dimensional vector. The elements of the vector are the Stokes parameters of the beam. In Mueller matrix formalism the optical device is represented by a 4×4 real matrix known as ‘Mueller Matrix’ of the device. The use of coherency matrix also proves to the useful in the study of partially polarized light. Pauli spin matrices have been used to unify the different matrix treatments of polarization optical phenomena. The present article is an attempt to unify the analysis of polarization phenomena using Dirac matrices used by Dirac in quantum mechanics. We have however redefined the set of Dirac matrices in terms of the Kronecher product of Pauli spin matrices.     

Determination of the Polarization Characteristics of Objects by the Method of Three Probing Polarizations

A method of three probing polarizations for determining the polarization characteristics of objects by measuring their Mueller complete and incomplete matrices has been developed. The conditions of optimal implementation of the method from the viewpoint of minimizing the error in measurement of the Mueller matrix elements have been investigated.     

Influence of the order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex turbid media such as biological tissues

The influence of the multiplication order of the constituent basis matrices on the Mueller matrix decomposition-derived polarization parameters in complex tissue-like turbid media exhibiting simultaneous scattering and polarization effects are investigated. A polarization sensitive Monte Carlo (MC) simulation model was used to generate Mueller matrices from turbid media exhibiting simultaneous linear birefringence, optical activity and multiple scattering effects. Mueller matrix decomposition was performed with different selected multiplication orders of the constituent basis matrices, which were further analyzed to derive quantitative individual polarization medium properties. The results show that for turbid medium having weak diattenuation (differential attenuation of two orthogonal polarization states), the decomposition-derived polarization parameters are independent of the multiplication order. Importantly, the values for the extracted polarization parameters were found to be in excellent agreement with the controlled inputs, showing self-consistency in inverse decomposition analysis and successful decoupling of the individual polarization effects. These results were corroborated further by selected experimental results from phantoms having optical (scattering and polarization) properties similar to those used in the MC model. Results from tissue polarimetry confirm that the magnitude of diattenuation is generally lower compared to other polarization effects, so that the demonstrated self-consistency of the decomposition formalism with respect to the potential ambiguity of ordering of the constituent matrices should hold in biological applications.     

Identification and characterization of optically active materials excited by sources in the chiral medium

Biological and chemical materials that possess chiral properties can be identified and characterized through their optical activity. Optical activity (Optical Rotation and Circular Dichroism) manifests itself through the measurement of select elements of the Mueller matrix that are proportional to the cross polarized reflection and transmission coefficients for waves that are incident in free space upon the optically active medium. Thus no polar decomposition is required to identify optical activity. Thus it is not necessary to measure all sixteen elements of the Mueller Matrix. We analyze here in detail the case when the exciting sources lie below the interface in the chiral material, which could be a fluid. Explicit expressions, to within first order of the chiral measure, are obtained for the corresponding reflection and transmission matrices for circularly and linearly polarized waves. These results are compared with the results for excitations above the interface in free space. Significant differences are found between the corresponding expressions for the reflection and transmission matrices, they are related to the Mueller matrix elements and the Differential Circular Reflection. These results have numerous biological, biochemical, and medical applications.     

偏振成像激光雷达遥感中的基本变换研究

针对偏振成像激光雷达遥感的计算需求,在假设以散射体中心为主坐标系中心的情况下,研究了入射偏振光和散射偏振光坐标系的变换、散射体引起的偏振状态的变化以及散射矩阵与Mueller矩阵之间的几种最基本的变换关系,并从群论的观点给出了这些变换关系的描述。这些变换关系的进一步展开便可得到具体的激光雷达偏振遥感物理量的计算方法,这在解读偏振激光雷达遥感图像的应用中具有实际作用。     

随机介质背散射二维Mueller矩阵的数值模拟与分析

推导了随机介质背散射Mueller矩阵的直接计算公式,并运用矢量Monte Carlo方法进行了数值模拟.结果表明随机介质背散射二维Mueller矩阵方位关系随散射系数的减小而增强,而与微粒大小关系不大;Mueller矩阵元素绝对值的空间分布随径向呈近似指数规律衰减,矩阵元素的方位变化具有周期性.对称系统的二维Mueller矩阵的花样图中仅有7幅独立,其余9幅可通过对称、旋转变换得到.     

基于穆勒矩阵的模拟溢油样品荧光偏振特性研究

为了对模拟溢油样品在不同偏振态激发下诱导荧光的偏振特性进行研究,借鉴了穆勒矩阵椭偏仪的原理和结构,搭建了基于旋转波片原理的模拟溢油样品激光诱导荧光椭偏实验装置。通过特征值校准方法对该装置进行校准,获得了宽波段下偏振状态调制矩阵W(λ)和偏振状态分析矩阵A(λ)的确切调制状态,并基于荧光光谱强度矩阵Flu(λ)分别建立了轻、中、重质原油样品和柴油样品的荧光穆勒矩阵。通过极化分解方法对荧光穆勒矩阵进行分解后发现,不同样品荧光光谱的退偏振性质差异十分显著。柴油样品荧光穆勒矩阵的退偏振系数Δ(λ)没有明显的波长响应性,在荧光光谱范围内始终保持较高的退偏值,而三种原油样品的退偏系数Δ(λ)则随波长增大逐渐上升,其中,中质原油样品退偏系数随波长的变化幅度小于重质样品,超过轻质样品;就不同样品的退偏值来看,轻质原油样品最高,重质样品最低,中质原油样品介于二者之间,柴油样品的荧光退偏值略低于轻质原油样品,介于轻质和中质原油样品之间。将基于荧光穆勒矩阵极化分解后的结果与线偏振激发下样品荧光光谱的正交偏振实验结果进行对比,发现两种实验方法获得的退偏系数具有较高的吻合程度。实验还发现,四种模拟溢油样品荧光穆勒矩阵所包含的双向衰减和相位延迟性质都很微弱,不具有明显的差异。     

Error in determining Mueller matrix elements and its effect on solution of the inverse problem of polarimetry

We present an analysis of the errors in measurement of the Mueller matrix elements in polarimeters that are a combination of different types of Stokes polarimeters in the detection channel and controllable polarization converters in the probing channel. As polarization converters for the probing radiation, we consider a phase plate having four different angular positions in measuring the complete Mueller matrix and a linear polarizer having different angular positions in measuring the structural parts of an incomplete 4 × 3 Mueller matrix. We have shown that the error in determining the Mueller matrix elements is distributed nonuniformly over the matrix. The nature of the error distribution over the elements and its values are different for different combinations of detection and probing channels of the polarimeter, and depend on the anisotropy of the test object. The latter dictates the choice of the optimal layout for a Mueller polarimeter for studying media with the same or different types of anisotropy and the choice of Mueller matrix elements used to solve the inverse problem of determining the anisotropy parameters.     

Mueller matrix-based optimization of reflective type twisted nematic liquid crystal SLM at oblique incidences

Mueller matrix measurements were used to characterize the polarization properties of liquid crystal-based reflective type twisted nematic (TN) special light modulator (SLM) at oblique incidence of the laser beam. The experimentally obtained Mueller matrices were used to obtain the combination of polarization optics required to optimize it for phase only modulation. The results indicate that minimum intensity modulation is obtained with the use of a polarizer followed by a quarter wave plate (QWP) in polarization state generator (PSG) arm and a QWP followed by an analyzer in polarization state analyzer arm (PSA). Polarization parameters such as retardance, rotation and depolarization were calculated from the experimentally obtained Mueller matrices using polar decomposition method at different angle of incidences of the laser beam and the results has been discussed. The similarity between retardance and depolarization curve as a function of address voltage of TNSLM indicated that depolarization is mainly associated with errors in retardance values. Further, spectral Mueller matrix measurements were used to obtain intensity modulation response in the range of wavelengths 450-700 nm for broadband applications.     

Mueller Matrix Analysis of PDL Components

A detailed Mueller-Stokes analysis of an arbitrary elliptical dichroic polarizer is presented. Explicit expressions for the Jones matrix, the Stokes parameters, and the Mueller matrix for a distributed and localized dichroic polarizer acting as a polarization-dependent loss (PDL) element are derived. Application to wavelength-dependent PDL elements is discussed. The Poincare sphere representation and the application of the Mueller matrices for the study of randomly oriented concatenated PDL elements and PDL vector measurement are addressed.     

Revisit to the symmetry relations in diffusely backscattered polarization patterns of turbid media

As there exists an inconsistency in claiming the symmetrical relations in the 16 Mueller matrix elements used to describe a turbid medium, the author restudies the symmetrical relationships between diffusely backscattered polarization patterns in isotropic turbid media and simulates all two-dimensional elements of diffusely backscattered Mueller matrix in both cases of Rayleigh and Mie scatterings using the doublescattering approximation and the Monte Carlo algorithm, respectively. The previous experimental observatious are compared with the numerically determined matrix elements, showing a good agreement in both double-scattering model and Monte Carlo simulation. The symmetrical relations between the Mueller matrix elements are clarified.     

Maximum-likelihood estimation of Mueller matrices

We show that the method of maximum-likelihood estimation, recently introduced in the context of quantum process tomography, can be applied to the determination of Mueller matrices characterizing the polarization properties of classical optical systems. Contrary to linear reconstruction algorithms, the proposed method yields physically acceptable Mueller matrices even in the presence of uncontrolled experimental errors. We illustrate our method with the case of an unphysical measured Mueller matrix taken from the literature.     

Spectroscopic Mueller matrix polarimeter using four-channeled spectra

A Mueller matrix polarimeter acquired for four-channeled spectra is proposed. Both the polarizing and analyzing optics of this system consist of a linear polarizer and a high-order retarder. The polarizing elements can modulate the polarization states in the wavenumber space. By applying a Fourier transform method to a single-channeled spectrum, nine elements of the Mueller matrix can be deconvoluted without modifying the configuration of either the polarizing or analyzing optics. It is thus possible to determine the wavelength dependence of all the Mueller matrix elements from four-channeled spectra obtained using four different configurations for the polarizing and analyzing optics. The performance of this method is evaluated by measuring polarization properties, such as retardance, azimuthal angle, and linear diattenuation, from the obtained Mueller matrix in wavenumber space.     

Optical Mueller matrices in terms of geometric algebra

Connection between optical Mueller matrices and geometrical (Clifford) algebra multivectors is established. It is shown that starting from 3-dimensional (3D) Cl_3,0 algebra and using isomorphism between Cl_3,0 and even Cl_3,1⁺ subalgebra one can generate canonical Mueller matrices and their combinations that describe an optical system. It appears that representation of polarization devices in terms of geometric algebra is very compact and, in contrast to Mueller matrix approach, there is no need for speculative physical restrictions. If needed, properties of media can be logically introduced into Maxwell equation in a form of Clifford algebra via constitutive relations. Since representation of polarization by Cl_3,1 algebra is Lorentz invariant it allows to include relativistic effects of moving bodies on light polarization as well. In this paper only simple examples of connection between Mueller matrices and geometric algebra multivectors is presented.     

Differentiating characteristic microstructural features of cancerous tissues using Mueller matrix microscope

Polarized light imaging can provide rich microstructural information of samples, and has been applied to the detections of various abnormal tissues. In this paper, we report a polarized light microscope based on Mueller matrix imaging by adding the polarization state generator and analyzer (PSG and PSA) to a commercial transmission optical microscope. The maximum errors for the absolute values of Mueller matrix elements are reduced to 0.01 after calibration. This Mueller matrix microscope has been used to examine human cervical and liver cancerous tissues with fibrosis. Images of the transformed Mueller matrix parameters provide quantitative assessment on the characteristic features of the pathological tissues. Contrast mechanism of the experimental results are backed up by Monte Carlo simulations based on the sphere–cylinder birefringence model, which reveal the relationship between the pathological features in the cancerous tissues at the cellular level and the polarization parameters. Both the experimental and simulated data indicate that the microscopic transformed Mueller matrix parameters can distinguish the breaking down of birefringent normal tissues for cervical cancer, or the formation of birefringent surrounding structures accompanying the inflammatory reaction for liver cancer. With its simple structure, fast measurement and high precision, polarized light microscope based on Mueller matrix shows a good diagnosis application prospect.     

Schemes for complete compensation for polarization mode dispersion up to second order

The required structure and elements of polarization mode dispersion (PMD) compensators for complete second-order compensation are investigated by use of a general PMD vector formalism of concatenated PMD elements based on Mueller matrices and Stokes vectors. The investigation shows that two- and three-stage compensators with five independent parameters can compensate for polarization-dependent chromatic dispersion as well as the depolarization component of second-order PMD.     

Theoretical analysis of backscattered polarization patterns of turbid media containing glucose

Single scattering model and Stokes-Mueller formalism are introduced to investigate the influence of glucose on backscattered polarization patterns in turbid media. Glucose molecules rotate the polarization plane and induce changes in backscattered Mueller matrix patterns. Some Mueller matrix elements present higher optical rotation as the concentration of glucose augments. Using image subtraction and integration,linear relationship between low glucose concentration in the physiological range and optical rotation degree can be derived.     

Mueller matrix differential decomposition

We present a Mueller matrix decomposition based on the differential formulation of the Mueller calculus. The differential Mueller matrix is obtained from the macroscopic matrix through an eigenanalysis. It is subsequently resolved into the complete set of 16 differential matrices that correspond to the basic types of optical behavior for depolarizing anisotropic media. The method is successfully applied to the polarimetric analysis of several samples. The differential parameters enable one to perform an exhaustive characterization of anisotropy and depolarization. This decomposition is particularly appropriate for studying media in which several polarization effects take place simultaneously.