Mueller polarimetric imaging system with liquid crystals

We present a new polarimetric imaging system based on liquid-crystal modulators, a spectrally filtered white-light source, and a CCD camera. The whole Mueller matrix image of the sample is measured in approximately 5 s in the transmission mode. The instrument design, together with an original and easy-to-operate calibration procedure, provides high accuracy over a wide spectral range (500-700 nm). This accuracy has been assessed by measurement of a linear polarizer at different orientations and a thick wedged quartz plate as an example of a partially depolarized retarder. Polarimetric images of a stained hepatic biopsy with significant fibrosis have been taken at several wavelengths. The optical properties of Picrosirius Red stained collagen (diattenuation, retardance, and polarizance) have been measured independently from each other between 500 and 700 nm.     

Reduced form of a Mueller matrix

Abstract

Through a simple procedure based on the Lu–Chipman decomposition [S.-Y. Lu and R.C. Chipman, J. Opt. Soc. Am. A 13, 1106 (1996)], any depolarizing Mueller matrix can be transformed into a reduced form which accumulates the depolarization and polarizance properties into a set of six parameters. The simple structure of this reduced form provides straightforward ways for the general characterization of Mueller matrices as well as for the analysis of singular Mueller matrices.     

Error analysis for Mueller matrix measurement

The linear errors of Mueller matrix measurements are formulated for misalignment, depolarization, and incorrect retardation of the polarimetric components. The measured errors of a Mueller matrix depend not only on the imperfections of the measuring system but also on the Mueller matrix itself. The error matrices for different polarimetric systems are derived and also evaluated for the straight-through case. The error matrix for a polarizer-sample-analyzer system is much simpler than those for more complicated systems. The general error matrix is applied to null ellipsometry, and the obtained errors in ellipsometric parameters psi and delta are identical to the errors specifically derived for null ellipsometry with depolarization.     

Scattering by layered structures with rough surfaces: comparison of polarimetric optical scatterometer measurements with theory

A laboratory model of a layered structure with a rough upper surface (a glass microscope slide cut with a diamond saw) is used to obtain optical polarimetric data. Scatterometer measurements were made of all the Mueller matrix elements associated with light scattered in arbitrary directions. (A preliminary measurement of scattering from a smooth opaque gold film on a silicon wafer was used to validate the calculation of the Mueller matrix elements.) These measurements are compared with corresponding analytical solutions based on the full-wave approach. Physical interpretations of the analytical solutions that account for scattering upon reflection and transmission across rough interfaces are given in a companion paper. The agreement between calculations and measurements suggests that the full wave, polarimetric solutions can provide a reliable database for electromagnetic detection of rough surfaces in remote-sensing applications.     

Differential interference contrast microscopy as a polarimetric instrument

Differential interference contrast (DIC) microscopy is shown to be equivalent to an incomplete Stokes polarimeter capable of probing optical properties of materials on microscopic-length scales. The Mueller matrix for a DIC microscope is calculated for various types of samples, and the polarimetric properties for DIC component parts of a spaceflight microscope are spectrally measured. As a practical application, a measurement of the index mismatch between colloidal particles and a nearly index-matched fluid bath was performed.     

Remote sensing of penetrable objects buried beneath two-dimensional random rough surfaces by use of the Mueller matrix elements

The modified Mueller matrix elements for electromagnetic scattering from penetrable objects buried under two-dimensional random rough surfaces are investigated. This matrix relates the incident to the scattered waves, and it contains different combinations of the fully polarimetric scattering matrix elements. The statistical average of each Mueller matrix element is computed on the basis of the Monte Carlo simulations by exploiting the speed of the three-dimensional steepest-descent fast multipole method. The numerical results clearly show that relying only on the co-polarized or the cross-polarized intensities or both (i.e., vv, hh, vh, and hv) is not sufficient for sensing the buried objects. However, examining all 16 Mueller matrix elements significantly increases the possibility of detecting these objects. This technique can be used in remote sensing of scatterers buried beneath the rough ground.     

Transmitted and reflected scattering matrices from an English oak leaf

The complete Mueller matrix for an English oak (Quercus robur) leaf for a fixed azimuth angle (90 degrees) was determined immediately after plucking and a day following exposure to normal room temperature and pressures. The Mueller matrices were determined for transmitted light at observation angles ranging from 0 degrees to 24 degrees and for reflected backscattering angles from 153 degrees to 170 degrees. All the measurements were taken with a He-Ne laser light source at 0.63 microm. Since positive eigenvalues were obtained for the coherence matrix, the polarimetric measurements were physically realizable. The anisotropy parameters were determined from the Jones matrices by use of the decomposition theorem. From the M33 and M44 components of the Mueller matrices, it was found that nonspherical structures within the leaf were primarily responsible for observed transmitted light scatter, and spherical structures were mostly responsible for observed backscatter. Variations in backscatter Mueller matrix elements from a fresh leaf to a second day of observation were assumed because of changes to water vapor concentration in the leaf.     

Liquid-crystal depolarizer consisting of randomly aligned hybrid orientation domains

A novel depolarization method for linearly polarized incident light that uses a liquid-crystal (LC) cell with randomly aligned hybrid orientation domains is theoretically described by use of Mueller matrix calculations. The depolarization effect of the incident linear polarization is confirmed with Stokes parameter measurements. The unique optical properties of the fabricated LC depolarizer are revealed; that is, the intensity of the transmitted light is independent of the rotation of the analyzer. The degree of polarization becomes zero when the retardation of the LC depolarizer coincides with a half-wavelength.     

Errors of Mueller matrix measurements with a partially polarized light source

The linear errors of Mueller matrix measurements, using a partially polarized light source, have been formulated for imperfections of misalignment, depolarization, and nonideal ellipsometric parameters of the polarimetric components. The error matrices for a source-polarizer system and a source-polarizer-compensator system are derived. A polarized light source, when used with an imperfect polarizer, generates extra errors in addition to those for an unpolarized source. The compensator redistributes these errors to different elements of the error matrix. The errors of the Mueller matrices for the polarizer-sample-analyzer and the polarizer-compensator-sample-analyzer systems are evaluated for a straight through case. This error analysis is applied to a Stokes method and an experiment was performed to show the errors by a polarized light source. This general analysis can be used to evaluate errors for ellipsometry and polarimetry.     

Mueller calculus of polarization change in the cube-corner retroreflector

The optical ray properties of the cube-corner retroreflector (CCR) are first recalled. The change of polarization of the radiation due to CCR reflection is then derived by use of the Mueller matrix calculus. It is found that, in general, when the faces are not ideal reflectors, the useful cross section of the CCR consists of six zones, each of which produces a different change of polarization, i.e., it gives a different Mueller matrix. All the Mueller matrices depend on wavelength. The results are quite general and can be used directly also for partially polarized radiation.     

Noise distribution of Mueller matrices retrieved with active rotating polarimeters

Two methods used to retrieve Mueller matrices from intensity measurements are revisited. It is shown that with symmetry or orthogonality considerations, numerical inversions of polarimetric equations can be avoided. With the obtained analytical formulas, noise propagation can be analyzed. If the intensity noise is a Gaussian white noise, the noise of Mueller matrices features remarkable properties. Mueller components are mutually correlated according to a scheme that involves decomposition into four blocks of 2x2 matrices. Variances are unequally distributed: the middle 2x2 block has the highest variance, the element on the bottom right has the lowest. These characteristics have been validated on experimental Mueller images of the free space.     

Structure of a general pure Mueller matrix

Changes in the radiance and state of polarization of a beam of radiation can often be described by means of a pure Mueller matrix. Such a 4 × 4 matrix transforms Stokes parameters and can be expressed in terms of the elements of a 2 × 2 Jones matrix. Relations between the two types of matrix are discussed. Explicit expressions are given for changes of a pure Mueller matrix that are caused by certain elementary changes of its Jones matrix, such as when its transpose, complex conjugate, or Hermitian conjugate are taken. It is shown that every pure Mueller matrix has a simple and elegant structure, which is embodied by interrelations that involve either only squares of the elements or only products of different elements. All possible interrelations for the elements of a general pure Mueller matrix are derived from this simple structure.     

Fully tunable active polarization imager for contrast enhancement and partial polarimetry

We present the design and the practical implementation of a polarimetric imaging system based on liquid-crystal modulators that allows generation and analysis of any polarization state on the Poincaré sphere. This system is more versatile than standard Mueller imagers that are based on optimized, but limited, sets of illumination and analysis states. Examples of benefits brought by these extra degrees of freedom are illustrated on two different applications: contrast enhancement and extraction of partial polarimetric properties of a scene.     

Interpretation of nondepolarizing Mueller matrices based on singular-value decomposition

A product decomposition of a nondepolarizing Mueller matrix consisting of the sequence of three factors--a first linear retarder, a horizontal or vertical "retarding diattenuator," and a second linear retarder--is proposed. Each matrix factor can be readily identified with one or two basic polarization devices such as partial polarizers and retardation waveplates. The decomposition allows for a straightforward interpretation and parameterization of an experimentally determined Mueller matrix in terms of an arrangement of polarization devices and their characteristic parameters: diattenuations, retardances, and axis azimuths. Its application is illustrated on an experimentally determined Mueller matrix.     

Optimization of a dual-rotating-retarder Mueller matrix polarimeter

The dual-rotating-retarder configuration is one of the most common forms of the Mueller matrix polarimeter. I perform an optimization of this polarimeter configuration by minimizing the condition number of the system data reduction matrix. I find the optimum retardance for the rotating retarders to be 127 degrees. If exactly 16 intensity measurements are used for a Mueller matrix calculation, a complex relationship exists between the condition number and the sizes of the angular increments of the two retarders. If many intensity measurements are made, thus overspecifying the calculation, I find broad optimal ranges of angular increments of the two retarders that yield essentially equal performance. Experimental results are given.     

Characteristic properties of Mueller matrices

A complete and minimum set of necessary and sufficient conditions for a real 4 x 4 matrix to be a physical Mueller matrix is obtained. An additional condition is presented to complete the set of known conditions, namely, the four conditions obtained from the nonnegativity of the eigenvalues of the Hermitian matrix H associated with a Mueller matrix M and the transmittance condition. Using the properties of H, a demonstration is also presented of Tr(M(T)M) = 4m(2)00 as being a necessary and sufficient condition for a physical Mueller matrix to be a pure Mueller matrix.     

Polarization characterization of biological tissues using Stokes vector decomposition

A method is proposed for characterizing the polarization properties of a depolarizing anisotropic medium based on decomposition of the Stokes vector representing the light emerging from the medium. The exiting Stokes vector can be considered as being decomposed into two parts, namely a completely unpolarized and a completely polarized part. Then, the Mueller matrix representing the sample can be extracted as a superposition of two distinguishable parts. The differential Mueller formalism is applied to one part of the Muller matrix. Here, explicit expressions are presented for the extraction of polarization properties of the medium. Results indicate that the proposed method is a promising method for characterization of complicated media such as biological tissues.     

Mueller Matrices and Depolarization Criteria

Abstract

The question of whether a given Mueller matrix represents a deterministic or a non-deterministic system is analysed by means of a matrix condition. The possibility of replacing this matrix condition by a scalar condition is examined. It is shown that this is permissible only for those cases where a Hermitian matrix constructed from the Mueller matrix is positive semidefinite.     

Multifusion, Multispectral, Optical Polarimetric Imaging Sensing Principles

The goal of this study is to present novel, multifusion optical imaging sensing principles, based on active-multispectral polarimetric imaging of targets surrounded by scattered media. Specifically, the novelty of this study consists in the fusion of multispectral images, with polarimetric imaging principles, forming image differences. The experimental results indicate clearly that, high-contrast multispectral Mueller polarimetric image differences, as well as Degree of Linear Polarization (DOLP) images can be obtained from transmitted or backscattered photons, from targets embedded in turbid media.     

Clustering of polarization-encoded images

Polarization-encoded imaging consists of the distributed measurements of polarization parameters for each pixel of an image. We address clustering of multidimensional polarization-encoded images. The spatial coherence of polarization information is considered. Two methods of analysis are proposed: polarization contrast enhancement and a more-sophisticated image-processing algorithm based on a Markovian model. The proposed algorithms are applied and validated with two different Mueller images acquired by a fully polarimetric imaging system.