Non-singular Mueller matrices characterizing passive systems

We extended previous results and explained why it is necessary to work with SO(6,1)_e (sub-group of matrices with unit determinant and positive left upper corner element) in order to define a parametric form of an arbitrary non-singular Mueller matrix. The previously proposed solution in Devlaminck and Terrier [Definition of a parametric form of nonsingular Mueller matrices. J. Opt. Soc. Am. A 25 (2008) 2636-2643], was just a subset of this one. The issue of transmittance conditions for the matrices generated by this formalism is considered and conditions related to the elements of the polar decomposition are derived and give sufficient conditions to guarantee the transmittance conditions for the product matrix.     

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.     

Forward and reverse product decompositions of depolarizing Mueller matrices

Because of the noncommutativity of the matrix product, the three factors into which a depolarizing Mueller matrix is decomposed, i.e., the diattenuator, the retarder, and the depolarizer, form six possible products grouped into two families, as already pointed out [J. Opt. Soc. Am. A13, 1106 (1996); Opt. Lett.29, 2234 (2004)]. We show that, apart from the generalized polar decomposition generating the first family of products, there exists a dual decomposition belonging to the second family. The mathematical procedure for this dual decomposition is given, and the symmetry existing between the two decompositions is pointed out. The choice of the most appropriate decomposition for a given practical optical arrangement is likewise discussed and illustrated by simple examples.     

Differential matrix formalism for depolarizing anisotropic media

Azzam's differential matrix formalism [J. Opt. Soc. Am. 68, 1756 (1978)], originally developed for longitudinally inhomogeneous anisotropic nondepolarizing media, is extended to include depolarizing media. The generalization is physically interpreted in terms of means and uncertainties of the elementary optical properties of the medium, as well as of three anisotropy absorption parameters introduced to describe the depolarization. The formalism results in a particularly simple mathematical procedure for the retrieval of the elementary properties of a generally depolarizing anisotropic medium, assumed to be globally homogeneous, from its experimental Mueller matrix. The approach is illustrated on literature data and the conditions of its validity are identified and discussed.     

Vector parameterization of the Lorentz group transformations and polar decomposition of Mueller matrices

It is shown that the representation of the coherence matrix (the polarization density matrix) of beams of electromagnetic waves as a biquaternion corresponding to the four-vector of a pseudo-Euclidean space whose components are the intensity and the Stokes parameters provides a possibility of introducing the group transformations of these quantities isomorphic to SO(3.1) group. These transformations are a subset of the set of Mueller polarization matrices which, generally speaking, form a semigroup. The reduction of the semigroup of Mueller matrices to the group of transformations opens the possibility to use the vector parameterization of SO(3.1) group for interpretation of the polar decomposition of Mueller matrices. In particular, in this approach, the elements of the Mueller matrices corresponding to phase elements and polarizers turn out to be most simply and naturally related to their eigenpolarizations.     

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.     

Realizable differential matrices for depolarizing media

The evolution of a Stokes vector through depolarizing media is considered. A general form for the differential matrix is found that is appropriate in the presence of depolarization and it is parameterized in a manner that ensures that it yields, upon integration, a valid Mueller matrix for any choice of parameters. The form expands the more limited form for a nondepolarizing matrix given by Azzam [J. Opt. Soc. Am. 68, 1756 (1978)] and which was extended recently by others to include depolarization. A Mueller matrix decomposition is proposed that is based upon the new parameterization.     

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.     

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。     

Mueller-Stokes polarimetric characterization of transmissive liquid crystal spatial light modulator

In the twisted nematic liquid crystal spatial light modulators (TN-LCSLM), distortion of uniform twist and decrease in tilt angle of liquid crystal molecules on application of an electric field lead to amplitude and phase modulations of the transmitted or reflected wavefront, respectively. The amplitude and phase modulation characterization of TN-LCSLM using Jones calculi is simple and extensively used but does not give any information about important polarimetric parameters such as diattenuation and depolarizance. On the other hand, the characterization using Mueller calculi provides all information in terms of polarimetric properties such as diattenuation, retardance (birefringence) and depolarization. In this paper, polarimetric properties of the transmissive TN-LCSLM (HOLOEYE LC2002) are characterized measuring 17 different Mueller matrices at different addressed gray scale through Mueller Matrix Imaging Polarimeter (MMIP) at 530 nm wavelength. Lu-Chipman polar decomposition for Mueller matrix is utilized to separate out three independent Mueller matrices for diattenuation, depolarization and retardance as a function of addressed gray scale. Further, Mueller-Stokes combined formulation is used to examine the effect of depolarization present in the TN-LCSLM on six different states of polarization and evaluation of eigenpolarization states for the TN-LCSLM has been presented.     

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 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.     

Determination of local polarization properties of biological samples in the presence of diattenuation by use of Mueller optical coherence tomography

A unique feature of polarization-sensitive Mueller optical coherence tomography is that, by measuring Jones or Mueller matrices, it can reveal the complete polarization properties of biological samples, even in the presence of diattenuation. We map local polarization properties for the first time to our knowledge by using polar decomposition in combination with least-squares fitting to differentiate measured integrated Jones matrices with respect to depth. We also introduce the new concept of dual attenuation coefficients to characterize diattenuation per unit infinitesimal length in tissues. We experimentally verify the algorithm using measurements of a section of porcine tendon and the septum of a rat heart.     

Optimized Mueller polarimeter with liquid crystals

We demonstrate a Mueller polarimeter in which the polarization-state generator and analyzer are both composed of a linear polarizer and two liquid-crystal variable retarders. The polarimeter is designed to optimize the accuracy of the final results by minimization of the condition numbers of the modulation and analysis matrices. The polarimeter calibration, a difficult task by conventional procedures, is achieved easily by use of the eigenvalue method of Compain et al. [Appl. Opt. 38, 3490 (1999)]. The overall polarimeter performance is tested with a linear polarizer at various angles and a compensator at various retardations.     

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.     

Application of the polar decomposition to light scattering particle systems

We have applied the polar decomposition theorem (PDT) to the scattering matrix of simple systems such as an isolated sphere, either metallic or dielectric. The discrete dipole approximation (DDA) method has been used as an intermediate tool to calculate these matrices. We show how the parameters introduced by the PDT for describing the scattering by a single sphere can be easily interpreted. It is also shown that the PDT provides us with an alternative frame to conventional Mueller matrix analysis, by characterizing the systems by means of a group of independent parameters representing magnitudes of simple (virtual) elements. This step is necessary in order to translate other complex system matrices into manageable information.     

Calculations of Backscattering Mueller Matrices for Turbid Media with a Sphere Queue Model

A sphere queue model is introduced to calculate Mueller matrices of turbid media. Combined with the single scattering approximation, the backscattering Mueller matrices of turbid media can be computed rapidly by Mie theory. The numerical results agree with the azimuthal dependences of backscattering Mueller matrices＇ patterns from turbid media, which indicates that the major contribution to the Mueller matrices＇ patterns comes from the single scattering of the sphere queue, and the multiple scattering considered as a high-order correction does not change the patterns. The numerical analysis reveals that the contrast of Mueller matrices＇ patterns will decrease with increase of the concentration of media and the distance from the incident point.     

Influence of the order of diattenuator, retarder, and polarizer in polar decomposition of Mueller matrices

Polar decomposition consists of representing an arbitrary Mueller matrix with a product of three simpler matrices, but, since these matrices do not commute, the result depends on the order in which they are multiplied. We show that the six possible decompositions can be classified into two families and that one of these families always leads to physical elementary matrices, whereas the other does not.     

A simple depolarization criterion for light

A simple depolarization criterion for light is proposed. This criterion is based on the depolarization part derived from the degree of polarization formulation. Some reported Mueller matrices are employed to test its reliability and usefulness. Results prove that the criterion proposed can be employed as the first step to test the physical consistency of Mueller matrices.     

Optical diagnosis of dengue virus infected human blood using Mueller matrix polarimetry

Currently dengue fever diagnosis methods include capture ELISAs, immunofluorescence tests, and hemagglutination assays. In this study optical diagnosis of dengue virus infection in the whole blood is presented utilizing Mueller matrix polarimetry. Mueller matrices of about 50 dengue viral infected and 25 non-dengue healthy blood samples were recorded utilizing light source from 500 to 700 nm with scanning step of 10 nm. Polar decomposition of the Mueller matrices for all the blood samples was performed that yielded polarization properties including depolarization, diattenuation, degree of polarization, retardance and optical activity, out of which, depolarization index clusters up the diseased and healthy in to different separate groups. The average depolarized light in the case of dengue infection in the whole blood at 500 nm is 18%, whereas for the healthy blood samples it is 13.5%. This suggests that depolarization index of polarized light at the wavelengths of 500, 510, 520, 530 and 540 nm, we find that in case of depolarization index values are higher for dengue viral infection as compared to normal samples. This technique can effectively be used for the characterization of the dengue virus infected at an early stage of disease.