Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry

Reciprocal polarization imaging of optical activity in reflection

We present reciprocal polarization imaging for the optical activity of chiral media in reflection geometry. The method is based on the reciprocal polar decomposition of backscattering Mueller matrices accounting for the reciprocity of light waves in forward and backward scattering paths. Anisotropic depolarization is introduced to gain sensitivity to optical activity in backscattering. Experiments with glucose solutions show that while the Lu-Chipman decomposition of the backscattering Mueller matrices produces erroneous results, reciprocal polarization imaging correctly retrieves the optical activity of chiral media. The recovered optical rotation agrees with that obtained in the forward geometry and increases linearly with the concentration and thickness of the chiral media. The potential for in vivo glucose monitoring based on optical activity sensing using reciprocal polarization imaging is then discussed.

Mechanical stability of polarization signatures in biological tissue characterization

Mueller matrix imaging polarimetry (MMIP) is a promising technique for investigating structural abnormalities in pathological diagnosis. The characterization stability of polarization signatures, described by Mueller matrix parameters (MMPs), correlates with the mechanical state of the biological medium. In this study, we developed an MMIP system capable of applying quantitative forces to samples and measuring the resulting polarization signatures. Mechanical stretching experiments were conducted on a mimicking phantom and a tissue sample at different force scales. We analyzed the textural features and data distribution of MMP images and evaluated the force effect on the characterization of MMPs using the structural similarity index. The results demonstrate that changes in the mechanical microenvironment (CMM) can cause textural fluctuations in MMP images, interfering with the stability of polarization signatures. Specifically, parameters of anisotropic orientation, retardance, and optical rotation are the most sensitive to CMM, inducing a dramatic change in the overall image texture, while other parameters (e.g., polarization, diattenuation, and depolarization) exhibit locality in their response to CMM. For some MMPs, CMM can enhance regional textural contrasts. This study elucidates the mechanical stability of polarization signatures in biological tissue characterization and provides a valuable reference for further research toward minimizing CMM influence.

Optical Sensor System for 3D Jones Matrix Reconstruction of Optical Anisotropy Maps of Self-Assembled Polycrystalline Soft Matter Films

Our work uses a polarization matrix formalism to analyze and algorithmically represent optical anisotropy by open dehydration of blood plasma films. Analytical relations for Jones matrix reconstruction of optical birefringence maps of protein crystal networks of dehydrated biofluid films are found. A technique for 3D step-by-step measurement of the distributions of the elements of the Jones matrix or Jones matrix images (JMI) of the optically birefringent structure of blood plasma films (BPF) has been created. Correlation between JMI maps and corresponding birefringence images of dehydrated BPF and saliva films (SF) obtained from donors and prostate cancer patients was determined. Within the framework of statistical analysis of layer-by-layer optical birefringence maps, the parameters most sensitive to pathological changes in the structure of dehydrated films were found to be the central statistical moments of the 1st to 4th orders. We physically substantiated and experimentally determined the sensitivity of the method of 3D polarization scanning technique of BPF and SF preparations in the diagnosis of endometriosis of uterine tissue.

Discrepancy of coordinate system selection in backscattering Mueller matrix polarimetry: exploring photon coordinate system transformation invariants

In biomedical studies, Mueller matrix polarimetry is gaining increasing attention because it can comprehensively characterize polarization-related vectorial properties of the sample, which are crucial for microstructural identification and evaluation. For backscattering Mueller matrix polarimetry, there are two photon coordinate selection conventions, which can affect the following Mueller matrix parameters calculation and information acquisition quantitatively. In this study, we systematically analyze the influence of photon coordinate system selection on the backscattering Mueller matrix polarimetry. We compare the Mueller matrix elements in the right-handed-nonunitary and non-right-handed-unitary coordinate systems, and specifically deduce the changes of Mueller matrix polar decomposition, Mueller matrix Cloude decomposition and Mueller matrix transformation parameters widely used in backscattering Mueller matrix imaging as the photon coordinate system varied. Based on the theoretical analysis and phantom experiments, we provide a group of photon coordinate system transformation invariants for backscattering Mueller matrix polarimetry. The findings presented in this study give a crucial criterion of parameters selection for backscattering Mueller matrix imaging under different photon coordinate systems.

Design and analysis of chiral and achiral metasurfaces with the finite element method

The rise of metasurfaces to manipulate the polarization states of light motivates the development of versatile numerical methods able to model and analyze their polarimetric properties. Here we make use of a scattered-field formulation well suited to the Finite Element Method (FEM) to compute the Stokes-Mueller matrix of metasurfaces. The major advantage of the FEM lies in its versatility and its ability to compute the optical properties of structures with arbitrary and realistic shapes, and rounded edges and corners. We benefit from this method to design achiral, pseudo-chiral, and chiral metasurfaces with specific polarimetric properties. We compute and analyze their Mueller matrices. The accuracy of this method is assessed for both dielectric and metallic scatterers hosting Mie and plasmonic resonances.

Broadband Mueller ellipsometer as an all-in-one tool for spectral and temporal analysis of mutarotation kinetics

Spectroscopic Mueller matrix ellipsometry is becoming increasingly routine across physical branches of science, even outside optics. The highly sensitive tracking of the polarization-related physical properties offers a reliable and non-destructive analysis of virtually any sample at hand. If coupled with a physical model, it is impeccable in performance and irreplaceable in versatility. Nonetheless, this method is rarely adopted interdisciplinarily, and when it is, it often plays a supporting role, which does not take benefit of its full potential. To bridge this gap, we present Mueller matrix ellipsometry in the context of chiroptical spectroscopy. In this work, we utilize a commercial broadband Mueller ellipsometer to analyze the optical activity of a saccharides solution. We verify the correctness of the method in the first place by studying the well-known rotatory power of glucose, fructose, and sucrose. By employing a physically meaningful dispersion model, we obtain 2π-unwrapped absolute specific rotations. Besides that, we demonstrate the capability of tracing the glucose mutarotation kinetics from just one set of measurements. Coupling the Mueller matrix ellipsometry with the proposed dispersion model ultimately leads to the precisely determined mutarotation rate constants and spectrally and temporally resolved gyration tensor of individual glucose anomers. In this view, Mueller matrix ellipsometry may stand as an offbeat yet equal technique to those considered classical chiroptical spectroscopy techniques, which may help open new opportunities for broader polarimetric applications in biomedicine and chemistry.

Study of the spatial scale stability of Mueller matrix parameters for textural characterization of biological tissues

Mueller matrix imaging polarimetry (MMIP) is a promising technique for the textural characterization of biological tissue structures. To reveal the influence of imaging magnification on the robustness of Mueller matrix parameters (MMPs), the spatial scale stability of MMPs was studied. We established a new MMIP detector and derived the mathematical model of the spatial scale stability of MMPs. The biological tissues with well-defined structural components were imaged under different magnifications. Then, we compared and analyzed the textural features of the MMPs in the resulting images. The experimental results match the predictions of the mathematical model in these aspects: (a) magnification exhibits a strong nonlinear effect on the textural contrasts of MMPs images; (b) higher magnification does not necessarily lead to superior contrast for textural characterization; and (c) for different biological tissues, MMPs contrasts can be optimized differently, with some showing superior results. This study provides a reference for the experimental design and operation of the MMIP technique and is helpful for improving the characterization ability of MMPs.

Model for the depolarizing retarder in Mueller matrix polarimetry

We advance an analytical model describing the polarimetric response of a depolarizing retarder whose retardance varies spatially in magnitude or in orientation. The variation of the retarder parameters may be either of deterministic or of random nature. The model provides both the mean values and the uncertainties of the parameters. Its application is illustrated on two experimental examples, respectively covering the deterministic and the random cases.

Analysis and reduction of noise-induced depolarization in catheter based polarization sensitive optical coherence tomography

In catheter based polarization sensitive optical coherence tomography (PS-OCT), a optical fiber with a rapid rotation in the catheter can cause low signal-to-noise ratio (SNR), polarization state instability, phase change of PS-OCT signals and then heavy noise-induced depolarization, which has a strong impact on the phase retardation measurement of the sample. In this paper, we analyze the noise-induced depolarization and find that the effect of depolarization can be reduced by polar decomposition after incoherent averaging in the Mueller matrix averaging (MMA) method. Namely, MMA can reduce impact of noise on phase retardation mapping. We present a Monte Carlo method based on PS-OCT to numerically describe noise-induced depolarization effect and contrast phase retardation imaging results by MMA and Jones matrix averaging (JMA) methods. The peak signal to noise ratio (PSNR) of simulated images processed by MMA is higher than about 8.9 dB than that processed by JMA. We also implement experiments of multiple biological tissues using the catheter based PS-OCT system. From the simulation and experimental results, we find the polarization contrasts processed by the MMA are better than those by JMA, especially at areas with high depolarization, because the MMA can reduce effect of noise-induced depolarization on the phase retardation measurement.

Complete self-calibration compact binary magneto-optic rotator based Mueller matrix polarimetry

Magneto-optic (MO) based Mueller matrix polarimetry (MMP) has several advantages of compact size, no-mechanical movement and high speed. Inaccuracies of components in the polarization state generator (PSG) optical parameters will influence the measurement accuracy of MMP. In this paper, we present a PSG self-calibration method in the compact MMP based on binary MO polarization rotators. Since PSG can generate enough numbers of non-degenerate polarization states, the optical parameters in PSG and the Mueller matrix of the sample can totally be numerically solved, which realizes a self-calibration in the PSG. Combining the previous self-calibration method in polarization state analyzer (PSA), we realize a complete self-calibration compact MO based MMP. Based on the numerical simulation results, the errors of measured phase retardance and optical axis of the sample decrease two to three orders of magnitude after applying the PSG self-calibration method. In experimental results of a variable retarder as a sample, the Euclidean distance of retardance between the measurement and reference curves comparing PSG self-calibration with no PSG self-calibration can be reduced from 0.035 rad to 0.033 rad and the Euclidean distance of optical axis can be reduced from 3.39° to 1.51°. Compared with the experimental results, the numerical simulation results more accurately verify the performance of the presented PSG self-calibration method without being influenced by other errors because the Mueller matrix of the sample is known and the error source only comes from these components in PSG.

Mueller matrix polarimetry and polar decomposition of articular cartilage imaged in reflectance

Articular cartilage birefringence relates to zonal architecture primarily of type II collagen, which has been assessed extensively in transmission, through thin tissue sections, to evaluate cartilage repair and degeneration. Mueller matrix imaging of articular cartilage in reflection is of potential utility for non-destructive imaging in clinical and research applications. Therefore, such an imaging system was constructed to measure laser reflectance signals, calibrated, and tested with optical standards. Polar decomposition was chosen as a method to extract fundamental optical parameters from the experimental Mueller matrices, with performance confirmed by simulations. Adult bovine articular cartilage from the patellofemoral groove was found to have ∼0.93 radians retardance, low diattenuation of ∼0.2, and moderately high depolarization of 0.66. Simulations showed that variation in depolarization drives inaccuracy of depolarization and retardance maps derived by polar decomposition. These results create a basis for further investigation of the clinical utility of polarized signals from knee tissue and suggest potential approaches for improving the accuracy of polar decomposition maps.

Isolating individual polarization effects from the Mueller matrix: comparison of two non-decomposition techniques

The prevailing formalisms for isolating individual polarization effects from the experimental Mueller matrix M can be broadly divided into two categories; decomposition of M to derive the individual optical effects and directly associating the individual optical effects to specific elements of M (i.e., non-decomposition techniques). Mueller matrix transformation (MMT) and direct interpretation of Mueller matrix (DIMM) are two popular techniques of the latter category. In this study, these two non-decomposition techniques (i.e., MMT and DIMM) are compared in a detailed quantitative analysis comprising of tissues (n = 53) and phantom (n = 45) samples. In particular, two commonly investigated polarimetric variables (i.e., depolarization and retardance) were calculated from the experimentally measured M using both the non-decomposition (i.e., MMT and DIMM) techniques. The comparison carried out with scatter plots (integrated with the correlation coefficients), violin plots and Bland and Altman plots revealed better agreement of depolarization-related variables (as compared to the retardance) between the two non-decomposition techniques. The comparative analyses presented here would be beneficial for the interpretation of polarimetric variables and optical characterization of turbid media.

Measuring glucose concentration in a solution based on the indices of polarimetric purity

Polarization imaging is a powerful tool, which can be applied in biomedical diagnosis and many research fields. Here, we propose a new application of the indices of polarimetric purity (IPPs) composed of P1, P2, P3, to describe the glucose concentrations (GC) changes in the scattering system. The results suggest that P1 of the IPPs is a better indicator to GC in the solution than the degree of polarization (DoP) for the forward scattering detection. Meanwhile, the fitting relation among radius of scattering particle, GCs and P1 parameter has also been calculated, in which the error of inversion is no more than 4.73%. In the backscattering detection, the fitted frequency statistical histogram of the IPPs is used to measure the GCs, and their modes can represent changing trend of GCs.

3D Mueller matrix mapping of layered distributions of depolarisation degree for analysis of prostate adenoma and carcinoma diffuse tissues

Prostate cancer is the second most common cancer globally in men, and in some countries is now the most diagnosed form of cancer. It is necessary to differentiate between benign and malignant prostate conditions to give accurate diagnoses. We aim to demonstrate the use of a 3D Mueller matrix method to allow quick and easy clinical differentiation between prostate adenoma and carcinoma tissues with different grades and Gleason scores. Histological sections of benign and malignant prostate tumours, obtained by radical prostatectomy, were investigated. We map the degree of depolarisation in the different prostate tumour tissues using a Mueller matrix polarimeter set-up, based on the superposition of a reference laser beam with the interference pattern of the sample in the image plane. The depolarisation distributions can be directly related to the morphology of the biological tissues. The dependences of the magnitude of the 1st to 4th order statistical moments of the depolarisation distribution are determined, which characterise the distributions of the depolarisation values. To determine the diagnostic potential of the method three groups of histological sections of prostate tumour biopsies were formed. The first group contained 36 adenoma tissue samples, while the second contained 36 carcinoma tissue samples of a high grade (grade 4: poorly differentiated-4 + 4 Gleason score), and the third group contained 36 carcinoma tissue samples of a low grade (grade 1: moderately differentiated-3 + 3 Gleason score). Using the calculated values of the statistical moments, tumour tissues are categorised as either adenoma or carcinoma. A high level (> 90%) accuracy of differentiation between adenoma and carcinoma samples was achieved for each group. Differentiation between the high-grade and low-grade carcinoma samples was achieved with an accuracy of 87.5%. The results demonstrate that Mueller matrix mapping of the depolarisation distribution of prostate tumour tissues can accurately differentiate between adenoma and carcinoma, and between different grades of carcinoma. This represents a first step towards the implementation of 3D Mueller matrix mapping for clinical analysis and diagnosis of prostate tumours.

Embossed topographic depolarisation maps of biological tissues with different morphological structures

Layered topographic maps of the depolarisation due to diffuse biological tissues are produced using a polarisation-holographic Mueller matrix method approach. Histological sections of myocardial tissue with a spatially structured optically anisotropic fibrillar network, and parenchymal liver tissue with a polycrystalline island structure are successfully mapped. The topography of the myocardium maps relates to the scattering multiplicity within the volume and the specific morphological structures of the biological crystallite networks. The overall depolarisation map is a convolution of the effects of these two factors. Parenchymal liver tissues behave broadly similarly, but the different biological structures present cause the degree of scattering multiplicity to increase more rapidly with increasing phase. Through statistical analysis, the dependences of the magnitudes of the first to fourth order statistical moments are determined. These moments characterise the changing distributions of the depolarisation values through the volume of biological tissues with different morphological structures. Parenchymal liver tissue depolarisation maps are characterised by larger mean and variance, and less skewness and kurtosis, compared to the distributions for the myocardium. This work demonstrates that a polarisation-holographic Mueller matrix method can be applied to the assessment of the 3D morphology of biological tissues, with applications in disease diagnosis.

Polarimetric imaging in backscattering for the structural characterization of strongly scattering birefringent fibrous media

Interpreting the polarimetric data from fiber-like macromolecules constitutive of tissue can be difficult due to strong scattering. In this study, we probed the superficial layers of fibrous tissue models (membranes consisting of nanofibers) displaying varying degrees of alignment. To better understand the manifestation of membranes' degree of alignment in polarimetry, we analyzed the spatial variations of the backscattered light's Stokes vectors as a function of the orientation of the probing beam's linear polarization. The degree of linear polarization reflects the uniaxially birefringent behavior of the membranes. The rotational (a-)symmetry of the backscattered light's degree of linear polarization provides a measure of the membranes' degree of alignment.

Polarization in reflectance imaging

The Sinclair and Kennaugh matrices are widely used in the remote sensing discipline for signals detected in the backward direction. The connections between the Jones matrix and the Sinclair matrix, and between the Mueller matrix and the Kennaugh matrix, are explored. Different operations on the Jones matrix and their corresponding effects on the Mueller matrix, coherency matrix, and coherence vector are derived. As an example, the Sinclair matrix leads to a Mueller-Sinclair matrix, and a transformed coherence vector. The Kennaugh matrix is not, however, a Mueller matrix, but can be determined from the Mueller or Mueller-Sinclair matrices. We consider backscattering through a medium on a perfect mirror. We propose that backscattering from a uniform medium can be modeled as an effective uniform medium situated on a perfectly reflective substrate, and the elementary polarization properties derived. In this way, the concept of a uniform polarizing medium can be extended to the reflectance geometry. An experimental Mueller matrix from the literature is considered as an example.

Eigenvalues of the coherency matrix for exact backscattering

An important approach to interpretation of the Mueller matrix is based on the eigenvalues of the coherency matrix, given by the roots of a quartic characteristic equation. For the case of backscattering, one eigenvalue is zero from reciprocity arguments, and the characteristic equation reduces to a cubic. These two approaches (quartic and cubic) to calculation of the eigenvalues for exact backscattering are analytically considered and compared. As expected, the cubic approach is usually simpler, but for the special case of two zero eigenvalues, either approach reduces to the predictions of the simple quadratic characteristic equation. Either approach can be used for numerical calculation of the eigenvalues. The variation in different purity measures with the values of the Mueller matrix elements is presented. An experimental Mueller matrix for backscattering from a turbid chiral medium is investigated.

Catheter-Based Polarization Sensitive Optical Coherence Tomography Using Similar Mueller Matrix Method

OBJECTIVE

Research of catheter-based polarization sensitive optical coherence tomography (PS-OCT) is a challenging field. In this paper, we present a new polarization determination method, similar Mueller matrix (SMM) method, for a catheter-based PS-OCT system using a standard clinical catheter probe with an outer diameter of 0.9 mm.

METHODS

The SMM method can remove the diattenuation and depolarization compositions by polar decomposition. By constructing the similarity between the measured Mueller matrices and sample matrices, the phase retardance of the sample can be determined from the trace of the measured matrices.

RESULTS

In the experiments, we find that images processed by the SMM method without any averaging or phase correction have a better polarization contrast of multiple biological tissues than those by the Jones matrix based method. We also preliminarily achieve phase retardance imaging of the ex vivo porcine cardiac blood vessel.

CONCLUSION

Compared with the Jones matrix based method, the presented SMM method can provide a more robust birefringence imaging of biological tissues under low signal-to-noise ratio, depolarization, diattenuation, and phase instability.

SIGNIFICANCE

The SMM method has a potential to become a widely accepted polarization determination method for catheter-based PS-OCT.

Optical phantoms for biomedical polarimetry: a review

Calibration, quantification, and standardization of the polarimetric instrumentation, as well as interpretation and understanding of the obtained data, require the development and use of well-calibrated phantoms and standards. We reviewed the status of tissue phantoms for a variety of applications in polarimetry; more than 500 papers are considered. We divided the phantoms into five groups according to their origin (biological/nonbiological) and fundamental polarimetric properties of retardation, depolarization, and diattenuation. We found that, while biological media are generally depolarizing, retarding, and diattenuating, only one of all the phantoms reviewed incorporated all these properties, and few considered at least combined retardation and depolarization. Samples derived from biological tissue, such as tendon and muscle, remain extremely popular to quickly ascertain a polarimetric system, but do not provide quantifiable results aside from relative direction of their principal optical axis. Microspheres suspensions are the most utilized phantoms for depolarization, and combined with theoretical models can offer true quantification of depolarization or degree of polarization. There is a real paucity of birefringent phantoms despite the retardance being one of the most interesting parameters measurable with polarization techniques. Therefore, future work should be directed at generating truly reliable and repeatable phantoms for this metric determination. Diattenuating phantoms are rare and application-specific. Given that diattenuation is considered to be low in most biological tissues, the lack of such phantoms is seen as less problematic. The heterogeneity of the phantoms reviewed points to a critical need for standardization in this field. Ultimately, all research groups involved in polarimetric studies and instruments development would benefit from sharing a limited set of standardized polarimetric phantoms, as is done earlier in the round robin investigations in ellipsometry.

Method of interpreting Mueller matrix of anisotropic medium

The differential Mueller matrix is an important concept for analyzing the polarization properties of an optically homogeneous anisotropic sample, both nondepolarizing and depolarizing. In this work, we present a new method of interpreting Mueller matrix of anisotropic medium based on the relationships that exist between the components of a differential Mueller matrix and the polar components of the corresponding macroscopic Mueller matrix, and the necessary conditions are determined that guarantee the physical realizability of the generating matrices. Finally, a group of the experimental data of a sample from the literature with some known polarization properties was used to demonstrate the analysis. The work is helpful for obtaining new insights or new interpretations of the measured Mueller matrix of the medium.

Distributed measurements of external force induced local birefringence in spun highly birefringent optical fibers using polarimetric OFDR

We develop a local birefringence determination method of measuring the distribution of external force-induced birefringence in spun high-birefringence (HiBi) fiber (spun HiBi fiber) using polarimetric optical frequency domain reflectometry (P-OFDR). By constructing the similarity between the measured Mueller matrices and fiber under test (FUT) matrices using two input states of polarization, the total phase retardance caused by the local birefringence of FUT can be determined from the trace of the measured matrices. We measure the local birefringence of spun HiBi fibers from two different manufacturers and telecom SMF (G652.D) caused by bending, twist, and transverse stress using our presented P-OFDR system. From the experimental results, we find that bending- and twist-induced birefringences of spun HiBi fiber are much lower than those of standard SMF. More remarkably, the coating package influences the transverse stress induced birefringence of spun HiBi fibers significantly. These experimental results verify that our presented method is beneficial to evaluating and improving spun HiBi fibers' quality.

Assessment of tissue polarimetric properties using Stokes polarimetric imaging with circularly polarized illumination

Tissue-depolarization and linear-retardance are the main polarization characteristics of interest for bulk tissue characterization, and are normally interpreted from Mueller polarimetry. Stokes polarimetry can be conducted using simpler instrumentation and in a shorter time. Here, we use Stokes polarimetric imaging with circularly polarized illumination to assess the circular-depolarization and linear-retardance properties of tissue. Results obtained were compared with Mueller polarimetry in transmission and reflection geometry, respectively. It is found that circular-depolarization obtained from these 2 methods is very similar in both geometries, and that linear-retardance is highly quantitatively similar for transmission geometry and qualitatively similar for reflection geometry. The majority of tissue circular-depolarization and linear-retardance image information (represented by local image contrast features) obtained from Mueller polarimetry is well preserved from Stokes polarimetry in both geometries. These findings can be referred to for further understanding tissue Stokes polarimetric data, and for further application of Stokes polarimetry under the circumstances where short acquisition time or low optical system complexity is a priority, such as polarimetric endoscopy and microscopy.

Accurate measurements of circular and residual linear birefringences of spun fibers using binary polarization rotators

We present a method to accurately measure the birefringence properties of spun fibers using binary polarization rotators. By taking the advantages of binary polarization rotator in polarization analysis, we are able to simultaneously measure both the circular and linear birefringences in a spun fiber with high accuracy. We obtain the circular and the residual linear birefringences of the spun fiber as a function of temperature T to be 3.34 × 10^-5-5.11 × 10^-8T and 8.1 × 10^-6-1.19 × 10^-8T, respectively, with the residual linear birefringence about 4 times less than the circular birefringence. We find, for the first time with the best of authors' knowledge, that the circular and the residual linear birefringences in a spun fiber are highly linear with the temperature, with thermal coefficients of -5.11 × 10^-8 °C^-1 and -1.19 × 10^-8°C^-1, respectively, and that the relative changes per °C of the circular and residual linear birefringence are almost identical, with values of -0.152% and -0.147% respectively. We believe that the method and data presented in this paper will be beneficial for making high quality spun fibers, as well as high accuracy fiber optic current sensors.

Extended polar decomposition method of Mueller matrices for turbid media in reflection geometry

The polar decomposition method for Mueller matrices proposed by Lu-Chipman has been demonstrated and validated for many applications. However, in some situations, e.g., when analyzing the Mueller matrix of birefringent turbid media with Mie-sized scatterers acquired in reflection geometry, the method may suffer from limitations due to the assumptions required by this method. Here we extend the Lu-Chipman method and show that it can provide more reasonable results for these situations. The method has been validated experimentally with turbid phantoms. Thus, this Letter may prove useful in tissue polarimetry.

Flexible polarimetric probe for 3 × 3 Mueller matrix measurements of biological tissue

Polarimetry is a noninvasive method that uses polarised light to assess biophysical characteristics of tissues. A series of incident polarisation states illuminates a biological sample, and analysis of sample-altered polarisation states enables polarimetric tissue assessment. The resultant information can, for example, help quantitatively differentiate healthy from pathologic tissue. However, most bio-polarimetric assessments are performed using free-space optics with bulky optical components. Extension to flexible fibre-based systems is clinically desirable, but is challenging due to polarisation-altering properties of optical fibres. Here, we propose a flexible fibre-based polarimetric solution, and describe its design, fabrication, calibration, and initial feasibility demonstration in ex vivo tissue. The design is based on a flexible fibre bundle of six multimode optical fibres, each terminated with a distal polariser that ensures pre-determined output polarisation states. The resultant probe enables linear 3 × 3 Mueller matrix characterization of distal tissue. Potential in vivo Mueller matrix polarimetric tissue examinations in various directly-inaccessible body cavities are envisioned.

Forward scattering of polarized light from a turbid slab: theory and Monte Carlo simulations

It is proved that if reciprocity and mirror symmetry hold for single scattering by a particle, they also hold for multiple scattering in turbid slab media. Monte Carlo simulations generate a reduced effective Mueller matrix for forward scattering, which satisfies reciprocity and mirror symmetry, but satisfies only reciprocity if the medium contains chiral components. The scattering matrix was factorized by using the Lu-Chipman polar decomposition, which affords the polarization parameters as a function of the radial distance from the center. The depolarization coefficients decrease with increasing distance, whereas the scattering-induced linear diattenuation and retardance become larger in the middle-distance range. The optical rotation for a chiral medium increases with increasing distance.

Parameterization of the Mueller matrix

A deterministic Mueller matrix (Mueller-Jones matrix) contains seven independent parameters. By writing the so-called coherence vector in parametric form, the Mueller matrix can also be written in parametric form, where the matrix elements automatically satisfy the known relationships between each other. Three of these parameters are also related to the so-called anisotropy coefficients. The approach is generalized to express all 16 elements of a general Mueller matrix in terms of a scalar and five three-dimensional vectors. Many properties of a Mueller matrix can be written simply in terms of these vectors. Published experimental matrices are considered by this procedure.

Study of optical clearing in polarization measurements by Monte Carlo simulations with anisotropic tissue-mimicking models

We conducted Monte Carlo simulations based on anisotropic sclera-mimicking models to examine the polarization features in Mueller matrix polar decomposition (MMPD) parameters during the refractive index matching process, which is one of the major mechanisms of optical clearing. In a preliminary attempt, by changing the parameters of the models, wavelengths, and detection geometries, we demonstrate how the depolarization coefficient and retardance vary during the refractive index matching process and explain the polarization features using the average value and standard deviation of scattering numbers of the detected photons. We also study the depth-resolved polarization features during the gradual progression of the refractive index matching process. The results above indicate that the refractive index matching process increases the depth of polarization measurements and may lead to higher contrast between tissues of different anisotropies in deeper layers. MMPD-derived polarization parameters can characterize the refractive index matching process qualitatively.

Multiple scattering of polarized light in turbid birefringent media: a Monte Carlo simulation

Multiple scattering of polarized light in a birefringent turbid plane medium was studied using a Monte Carlo simulation. The reduced effective scattering Mueller matrix obtained in the simulation was factorized in two dimensions using the Lu-Chipman decomposition, yielding polarization parameters that exhibited dependences on the azimuth and the radial distance around the illumination point. We propose a double-scattering model for the propagation of polarized photons in turbid infinite plane media. When the birefringence slow axis is along the azimuth of 90° on the plane surface, the retardance becomes the largest negative along the azimuth of 0° and the largest positive along the azimuth of 90° and increases with increasing the azimuth from 0° to 90°. This azimuthal dependence may result from the overlap of the contributions from the light propagations vertical to, and lateral along, the plane surface. Thus, the dependences on the azimuth and the radial distance of the polarization parameters, such as the retardance, its orientation, optical rotation, and the depolarization coefficients, are correctly predicted.

Polarized light interaction with tissues

This tutorial-review introduces the fundamentals of polarized light interaction with biological tissues and presents some of the recent key polarization optical methods that have made possible the quantitative studies essential for biomedical diagnostics. Tissue structures and the corresponding models showing linear and circular birefringence, dichroism, and chirality are analyzed. As the basis for a quantitative description of the interaction of polarized light with tissues, the theory of polarization transfer in a random medium is used. This theory employs the modified transfer equation for Stokes parameters to predict the polarization properties of single- and multiple-scattered optical fields. The near-order of scatterers in tissues is accounted for to provide an adequate description of tissue polarization properties. Biomedical diagnostic techniques based on polarized light detection, including polarization imaging and spectroscopy, amplitude and intensity light scattering matrix measurements, and polarization-sensitive optical coherence tomography are described. Examples of biomedical applications of these techniques for early diagnostics of cataracts, detection of precancer, and prediction of skin disease are presented. The substantial reduction of light scattering multiplicity at tissue optical clearing that leads to a lesser influence of scattering on the measured intrinsic polarization properties of the tissue and allows for more precise quantification of these properties is demonstrated.

Retardance of bilayer anisotropic samples consisting of well-aligned cylindrical scatterers and birefringent media

Both cylindrical scatterers and birefringent media may contribute to the anisotropy of tissue, where anisotropy can be characterized using polarization techniques. Our previous studies have shown that a layer of well-aligned cylindrical scatterers displays anisotropic properties similar to those of a piece of birefringent media, whose equivalent extraordinary axis is along the axial direction of the cylinders. We focused on a sample consisting of two layers of anisotropic media, with each layer having a different orientation; the characteristics of this sample were representative of the properties of multilayer fibrous tissues. Using a Mueller matrix decomposition method, we examined in detail how the total retardance and the equivalent extraordinary axis of the bilayered sample varied with changes in the retardance of the two layers and the direction of the extraordinary axis. The results of this study showed that, in such bilayer samples, a layer of well-aligned cylindrical scatterers generated a retardance that behaved exactly like the retardance generated by a piece of birefringent media. The simulated results were also confirmed by the results of experiments using aligned glass fibers.

Multiple scattering of polarized light in turbid infinite planes: Monte Carlo simulations

Monte Carlo simulations were performed for infinite plane media containing spherical particles of different sizes. Most of the features of the surface plots for the elements of the effective scattering Mueller matrices are explained by the azimuthal dependence of the matrix predicted according to the theory of Raković et al. [Appl. Opt.38, 3399 (1999)10.1364/AO.38.003399APOPAI1559-128X]. The reduced effective scattering Mueller matrices calculated according to the theory have eight nonzero elements, which are only dependent on the distance from the illumination point. The reduced matrices are factorized approximately into products of a depolarizer and retarding diattenuators. The turbid infinite plane media nearly behave as a pure depolarizer at long distances and become more diattenuating and birefringent with decreasing distance.

Polarization state transformation using two quarter wave plates: application to Mueller polarimetry

We show that there are a number of ways to transform an arbitrary polarization state into another with just two quarter wave plates (QWPs). We have verified this geometrically using the trajectories of the initial and final polarization states corresponding to all the fast axis orientations of a QWP on the Poincaré sphere. The exact analytical expression for the locus of polarization states has also been given, and describes the trajectory. An analytical treatment of the equations obtained through matrix operations corresponding to the transformation supports the geometrical representation. This knowledge can be used to obtain the Mueller matrix by just using QWPs, which has been shown experimentally by exploiting projections of the output states on the input states.

Singular observation of the polarization-conversion effect for a gammadion-shaped metasurface

In this article, the polarization-conversion effects of a gammadion-shaped metasurface in transmission and reflection modes are discussed. In our experiment, the polarization-conversion effect of a gammadion-shaped metasurface is investigated because of the contribution of the phase and amplitude anisotropies. According to our experimental and simulated results, the polarization property of the first-order transmitted diffraction is dominated by linear anisotropy and has weak depolarization; the first-order reflected diffraction exhibits both linear and circular anisotropies and has stronger depolarization than the transmission mode. These results are different from previously published research. The Mueller matrix ellipsometer and polar decomposition method will aid in the investigation of the polarization properties of other nanostructures.

Symmetry relationships for multiple scattering of polarized light in turbid spherical samples: theory and a Monte Carlo simulation

This paper presents a theory describing totally incoherent multiple scattering of turbid spherical samples. It is proved that if reciprocity and mirror symmetry hold for single scattering by a particle, they also hold for multiple scattering in spherical samples. Monte Carlo simulations generate a reduced effective scattering Mueller matrix, which virtually satisfies reciprocity and mirror symmetry. The scattering matrix was factorized by using the symmetric decomposition in a predefined form, as well as the Lu-Chipman polar decomposition, approximately into a product of a pure depolarizer and vertically oriented linear retarding diattenuators. The parameters of these components were calculated as a function of the polar angle. While the turbid spherical sample is a pure depolarizer at low polar angles, it obtains more functions of the retarding diattenuator with increasing polar angle.

Physically meaningful depolarization metric based on the differential Mueller matrix

We present a novel depolarization metric for Mueller matrices based on the differential Mueller formalism. The proposed metric relies on the statistical interpretation of the differential Mueller matrix. We show that the differential depolarization index successfully quantifies depolarization even when applied to specific types of Mueller matrices for which some widely used depolarization metrics yield erroneous results. Moreover, the fact that the presented metric is directly linked to the variances and covariances of the elementary anisotropic properties of the sample makes it a valuable tool to quantify depolarization on a physically meaningful basis.

Polarized light imaging in biomedicine: emerging Mueller matrix methodologies for bulk tissue assessment

Polarized light point measurements and wide-field imaging have been studied for many years in an effort to develop accurate and information-rich tissue diagnostic methods. However, the extensive depolarization of polarized light in thick biological tissues has limited the success of these investigations. Recently, advances in technology and conceptual understanding have led to a significant resurgence of research activity in the promising field of bulk tissue polarimetry. In particular, with the advent of improved measurement, analysis, and interpretation methods, including Mueller matrix decomposition, new diagnostic avenues, such as quantification of microstructural anisotropy in bulk tissues, have been enabled. Further, novel technologies have improved the speed and the accuracy of polarimetric instruments for ex vivo and in vivo diagnostics. In this paper, we review some of the recent progress in tissue polarimetry, provide illustrative application examples, and offer an outlook to the future of polarized light imaging in bulk biological tissues.

Study on the validity of 3 × 3 Mueller matrix decomposition

Using Monte Carlo simulations based on previously developed scattering models consisting of spherical and cylindrical scatterers imbedded in birefringent interstitial medium, we compare the polarization parameters extracted from the 3×3 and 4×4 Mueller matrix decomposition methods in forward and backward scattering directions. The results show that the parameters derived from the 3×3 Mueller matrix decomposition are usually not the same as those from the 4×4 Mueller matrix decomposition but display similar qualitative relations to changes in the microstructure of the sample, such as the density, size, and orientation distributions of the scatterers, and birefringence of the interstitial medium. The simulations are backed up by experiments when suitable samples are available.

Size-dependent patterns in depolarization maps from turbid medium and tissue

Mueller matrix measurements on turbid media can be used to quantify its polarization properties in terms of retardance, diattenuation, and depolarization. In particular, the depolarizing ability of such media, which is represented by the depolarization index, has been shown to be a useful diagnostic parameter. However, being a single valued metric, its dependence on a host of tissue optical parameters makes it difficult to interpret. In this paper, we show that a map of depolarization as a function of input polarization state parameters can be used to infer information about the size of scatterer and order of birefringent and depolarizing layers in turbid medium. The experiments carried out on different mice organ tissues indicate that the depolarization characteristics of tissue are closely represented by depolarization properties of intralipid. We also observed that these maps do not vary in the presence of absorption.

Measuring the Mueller matrix of an arbitrary optical element with a universal SU(2) polarization gadget

We propose a new method for determining the Mueller matrix of an arbitrary optical element and verify it with three known optical elements. This method makes use of two universal SU(2) polarization gadgets to obtain the projection matrix directly from the experiment. It allows us to determine the Mueller matrix without precalibration of the setup, since the generated polarization states are fully determined by the azimuths of the wave plates. We calculate errors in determining the Mueller matrix and compare with other techniques.

Mueller matrix three-dimensional directional imaging of collagen fibers

A method for measuring three-dimensional (3-D) direction images of collagen fibers in biological tissue is presented. Images of the 3-D directions are derived from the measured transmission Mueller matrix images (MMIs), acquired at different incidence angles, by taking advantage of the form birefringence of the collagen fibers. The MMIs are decomposed using the recently developed differential decomposition, which is more suited to biological tissue samples than the common polar decomposition method. Validation of the 3-D direction images was performed by comparing them with images from second-harmonic generation microscopy. The comparison found a good agreement between the two methods. It is envisaged that 3-D directional imaging could become a useful tool for understanding the collagen framework for fibers smaller than the diffraction limit.

Ex vivo assessment of carbon tetrachloride (CCl(4))-induced chronic injury using polarized light spectroscopy

The liver performs various functions, such as the production and detoxification of chemicals; therefore, it is susceptible to hepatotoxins such as carbon tetrachloride (CCl4), which causes chronic injury. Thus, assessment of injury and its status of severity are of prime importance. Current work reports an ex vivo study for probing the severance of hepatic injury induced by CCl4 with polarized light over the spectral range 400-800 nm. Different concentrations of CCl4 were used to induce varying severity of hepatic injury in a rat model. Linear retardance, depolarization rates, and diagonal Mueller matrix elements (m22, m33, and m44), were successfully used as the distinguishing criterion for normal and different liver injuries. Our results show that linear retardance for injured liver samples with lower doses of CCl4 tends to increase when compared with normal liver samples, while samples injured at higher doses of CCl4 offer almost no retardance. Total, linear, and circular depolarizations follow decreasing trends with increased liver injury severity over the entire investigated wavelength range. Linear polarization states were observed to be better maintained as compared to circular polarization states for all samples. Furthermore, numerical values of diagonal elements of the experimentally measured Mueller matrix also increase with increasing doses of CCl4. Liver fibroses, change in transport albedo, and the relative refractive index of the extracellular matrix caused by CCl4 are responsible for the observed differences. These results will provide a pathway to gauge the severity of injury caused by toxic chemicals.

Singular value analysis of the Mueller matrix in comparison with eigenvalue analysis of the coherency matrix in polarization characterization of media

Singular values of the arbitrary Mueller matrix are determined to be indicators of some polarization properties of the medium such as depolarization and diattenuation. Whereas eigenvalue analysis of the coherency matrix may wrongly characterize media with simultaneous strong depolarization and diattenuation effects. The comparison between the patterns of changes in singular-value and eigenvalue trends of the coherency matrix in experimental Mueller matrices, shows that singular values of the Mueller matrix are more capable of discriminating media with close degrees of depolarization.

Physical model of differential Mueller matrix for depolarizing uniform media

In this article, we address the question of significance of the parameters of differential Mueller matrix formalism. We show how the concept of mean value and uncertainty of the optical properties recently introduced to depict this differential matrix can be related to the random fluctuations of these optical properties. From the layered-medium interpretation introduced by Jones [J. Opt. Soc. Am. 38, 671 (1948)] and extended to Mueller-Jones matrix by Azzam [J. Opt. Soc. Am. 68, 1756 (1978)], a generalization to depolarizing Mueller matrices is proposed. Based on the random Mueller-Jones matrix approach, the obtained parameterization perfectly fits the previous results from the literature. Necessary conditions of positivity on specific coefficients imposed in order to have physical Mueller matrix are introduced in a natural way and not inferred a posteriori. Interpretations of the underlying physical processes are also presented. An illustrative experimental example is provided from literature data.

Poincaré sphere mapping by Mueller matrices

By using the symmetric serial decomposition of a normalized Mueller matrix M [J. Opt. Soc. Am. A 26, 1109 (2009)] as a starting point and by considering the reciprocity property of Mueller matrices, the geometrical features of the Poincaré sphere mapping by M are analyzed in order to obtain a new parameterization of M in which the 15 representative parameters have straightforward geometrical interpretations. This approach provides a new geometry-based framework, whereby any normalized Mueller matrix M is completely described by a set of three associated ellipsoids whose geometrical and topological properties are characteristic of M. The mapping analysis considers the cases of type-I and type-II, as well as singular and nonsingular Mueller matrices. The novel parameterization is applied to several illustrative examples of experimental Mueller matrices taken from the literature.

Parametric model of the Mueller matrix of a Spectralon white reflectance standard deduced by polar decomposition techniques

Decomposition methods have been applied to in-plane Mueller matrix ellipsometric scattering data of the Spectralon reflectance standard. Data were measured at the wavelengths 532 nm and 1500 nm, using an achromatic optimal Mueller matrix scatterometer applying a photomultiplier tube and a high gain InGaAs detector for the two wavelengths. A parametric model with physical significance was deduced through analysis of the product decomposed matrices. It is found that when the data are analyzed as a function of the scattering angle, similar to particle scattering, the matrix elements are largely independent of incidence angle. To the first order, we propose that a Guassian lineshape is appropriate to describe the polarization index, while the decomposed diagonal elements of the retardance matrix have a form resembling Rayleigh single scattering. New models are proposed for the off diagonal elements of the measured Mueller matrix.