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.     

Mueller matrices and information derived from linear polarization lidar measurements: theory

A Mueller matrix M is developed for a single-scattering process such that G(theta, phi) = T (phi(a))M T (phi(p))u, where u is the incident irradiance Stokes vector transmitted through a linear polarizer at azimuthal angle phi(p), with transmission Mueller matrix T (phi(p)), and G(theta, phi) is the polarized irradiance Stokes vector measured by a detector with a field of view F, placed after an analyzer with transmission Mueller matrix T (phi(a)) at angle phi(a). The Mueller matrix M is a function of the Mueller matrix S (theta) of the scattering medium, the scattering angle (theta, phi), and the detector field of view F. The Mueller matrixM is derived for backscattering and forward scattering, along with equations for the detector polarized irradiance measurements (e.g., cross polarization and copolarization) and the depolarization ratio. The information that can be derived from the Mueller matrix M on the scattering Mueller matrixS (theta) is limited because the detector integrates the cone of incoming radiance over a range of azimuths of 2pi for forward scattering and backscattering. However, all nine Mueller matrix elements that affect linearly polarized radiation can be derived if a spatial filter in the form of a pie-slice slit is placed in the focal plane of the detector and azimuthally dependent polarized measurements and azimuthally integrated polarized measurements are combined.     

Real-time measurement of the polarization transfer function

We present a simple method for measuring the Mueller matrix associated with a scattering medium. Without involving moving parts, four input states of polarization are generated sequentially, and for each of them all four Stokes vector parameters are simultaneously measured for the complete determination of the Mueller matrix. Two liquid-crystal variable retarders are used for controlling the input state of polarization, whereas the measurement of the state of polarization involves phase modulation with a single-pass photoelastic modulator, and Fourier analysis in two polarization channels. The setup is controlled by a computer, allowing for real-time measurement of the Mueller matrix. The method is tested on standard elements such as polarizers and quarter-wave plates, as well as on inhomogeneous particulate systems.     

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.     

Treatment of polarization in laser remote sensing of ocean water

We review existing experimental data and methods for calculating the Mueller matrix of ocean water for use as input in a simulation model applicable to laser remote sensing. Calculations of the Mueller matrix are made for scattering media of different refractive indices, shapes, and size distributions. Dependencies of the backscattering depolarization ratio as a function of the particle refractive index are presented, and we demonstrate the potential importance of polarization in bathymetric sensing.     

Degree of polarization as a criterion to obtain the nine bilinear constraints between the Mueller-Jones matrix elements

The degree of polarization is employed as a criterion to find the nine independent relations among the elements of the Mueller-Jones matrix. This procedure is applied by considering a previously determined, physically realizable Mueller matrix. On the other hand, the nine bilinear constrains are obtained by directly measuring the degree of polarization from an outgoing beam of light from an optical system by considering nine incident states of light taken from the Poincaré sphere. For practical purposes, all the incident polarization states must be scanned from the Poincaré sphere in order to satisfy the over-polarization and the over-gain conditions, respectively, for the physical realizability of the Mueller matrix.     

Mueller matrix roots depolarization parameters

The Mueller matrix roots decomposition recently proposed by Chipman in [1] and its three associated families of depolarization (amplitude depolarization, phase depolarization, and diagonal depolarization) are explored. Degree of polarization maps are used to differentiate among the three families and demonstrate the unity between phase and diagonal depolarization, while amplitude depolarization remains a distinct class. Three families of depolarization are generated via the averaging of different forms of two nondepolarizing Mueller matrices. The orientation of the resulting depolarization follows the cyclic permutations of the Pauli spin matrices. The depolarization forms of Mueller matrices from two scattering measurements are analyzed with the matrix roots decomposition-a sample of ground glass and a graphite and wood pencil tip.     

Mueller matrix measurements of algae with different shape and size distributions

The full Mueller matrix was measured to obtain the polarization state of the scattered light for a variety of algae with different shapes, wall compositions, sizes, and refractive indices. The experimental setup was a multiple laser Mueller matrix ellipsometer, by which measurements were performed for scattering angles from 16° to 160° sampled at every second degree for wavelengths of 473?nm and 532?nm. Previously, the polarization of light scattered from microalgae was investigated only for a few species, and the Mueller matrix was found to have little variation between the species. In our work a total of 11 algal species were investigated, representing diatoms, dinoflagellates, coccolithophorids, green algae, and a cryptophyte. The selection of species was made to obtain high variability in shape, size, cell wall, and refractive index. As in previous investigations, very small variations were found between species for most of the Mueller matrix elements, but noticeable variations were found for M(11), (M(12)+M(21))/2 and (M(33)+M(44))/2.     

Depolarization of diffusely reflecting man-made objects

The polarization properties of light scattered or diffusely reflected from seven different man-made samples are studied. For each diffusely reflecting sample an in-plane Mueller matrix bidirectional reflectance distribution function is measured at a fixed bistatic angle using a Mueller matrix imaging polarimeter. The measured profile of depolarization index with changing scattering geometry for most samples is well approximated by an inverted Gaussian function. Depolarization is minimum for specular reflection and increases asymptotically in a Gaussian fashion as the angles of incidence and scatter increase. Parameters of the Gaussian profiles fitted to the depolarization data are used to compare samples. The dependence of depolarization on the incident polarization state is compared for each Stokes basis vector: horizontal, vertical, 45 degrees, 135 degrees, and right- and left-circular polarized light. Linear states exhibit similar depolarization profiles that typically differ in value by less than 0.06 (where 1.0 indicates complete depolarization). Circular polarization states are depolarized more than linear states for all samples tested, with the output degree of polarization reduced from that of linear states by as much as 0.15. The depolarization difference between linear and circular states varies significantly between samples.     

New formalism for the analysis of polarization evolution for radiation in a weakly nonuniform, fully anisotropic medium: a magnetized plasma

The evolution equation is obtained for the state of polarization of partially polarized radiation propagating in a medium that is nonuniform and fully anisotropic, i.e., that presents both linear and circular birefringence as well as dichroism. The treatment covers the general case in which the characteristic polarizations are not necessarily orthogonal, such as occurs for propagation in a magnetized, dissipative plasma. The differential Mueller matrix that appears in the evolution equation is obtained explicitly for two particular cases. The resulting formalism is convenient for numerical integration.     

Contrast mechanisms in polarization-sensitive Mueller-matrix optical coherence tomography and application in burn imaging

We investigate the various contrast mechanisms provided by polarization-sensitive (PS) Mueller-matrix optical coherence tomography (OCT). Our PS multichannel Mueller-matrix OCT is the first, to our knowledge, to offer simultaneously comprehensive polarization-contrast mechanisms, including the amplitude of birefringence, the orientation of birefringence, and the diattenuation in addition to the polarization-independent intensity contrast, all of which can be extracted from the measured Jones or the equivalent Mueller matrix. Theoretical analysis shows that when diattenuation is negligible, the round-trip Jones matrix represents a linear retarder, which is the foundation of conventional PS-OCT, and can be calculated with a single incident polarization state, although the one-way Jones matrix generally represents an elliptical retarder; otherwise, two incident polarization states are needed. The experimental results obtained from rat skin samples, which conform well with the histology, show that Mueller OCT provides complementary structural and functional information on biological samples and reveal that polarization contrast is more sensitive to thermal degeneration of biological tissue than amplitude-based contrast. Thus, Mueller OCT has significant potential for application in the noninvasive assessment of burn depth.     

Polarization of specular reflection and near-specular scattering by a rough surface

Polarization of specular reflection and near-specular scattering (NSS) by a randomly rough surface is investigated by the use of a Mueller matrix formulation. The collective effect by a rough surface on the average specular field results in reflectance loss and polarization, which can be explained by an effective medium theory. Effects of random NSS can be represented by a scattering matrix that is partially coherent and polarized. The incoherent and unpolarized part of scattering causes depolarization, and the coherent and polarized parts of scattering change the apparent polarization properties of specular reflection. Results of a simulation and least-squares fit of ellipsometric data to the models including the NSS effect, for a black anodized aluminum sample, are presented. Simultaneous least-squares fits for both ellipsometric data and reflectance data at multiple angles of incidence at three different wavelengths gave approximately the same rms roughness, which agrees with the profilometric values reported previously.     

Normal-incidence generalized ellipsometry using the two-modulator generalized ellipsometry microscope

A new microscope is described that is capable of measuring the polarization characteristics of materials in normal-incidence reflection with a demonstrated lateral resolution of 4 microm. The instrument measures eight parameters of the sample Mueller matrix, which can be related to the diattenuation, retardation, circular diattenuation, direction of the principal axis, and the polarization factor. With proper calibration, the eight elements of the sample Mueller matrix can be determined to better than 0.001-0.002 for small values. Examples are given for aluminum, rutile (TiO2), and calcite (CaCO3).     

Measurement of the complex refractive index of isotropic materials with mueller matrix polarimetry

The complex refractive index of materials at infrared wavelengths is often determined when absorption measurements are made at selected wavelengths, and then the Kramers-Kronig relationship is used to calculate the real part of the index. Because many organic materials are highly absorbing in the infrared, absorption measurements require a short path length. We report on the use of an attenuated total internal reflection (TIR) method in combination with an infrared Mueller matrix spectropolarimeter to measure the Mueller matrix spectrum of samples from 3 to 14 mum. From the elements of the Mueller matrix the complex refractive index is determined for materials whose TIR interfaces are eigenstates of s and p polarization. The calculated index for water compares well with data found in the literature.     

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.     

Two-modulator generalized ellipsometry: experiment and calibration

A two-modulator generalized ellipsometer is described that is capable of measuring all 16 elements of a sample Mueller matrix with four measurements made at different azimuthal orientations of the polarization state generator and polarization state detector. If the sample can be described with a Mueller-Jones matrix, only a single measurement is needed. Only two calibration steps are needed to determine the fundamental operating parameters of the instrument. A reflection measurement from silicon is presented as an example, which illustrates that the elements of the Mueller-Jones matrix can be measured to an accuracy of ~0.1-0.2%.     

Use of polar decomposition for the diagnosis of oral precancer

The Mueller matrix describes all the polarizing properties of a sample and, therefore, the optical differences between noncancerous and precancerous tissue that may be present within the matrix elements. A high-speed polarimetry system that generates 16 (4x4) full Mueller matrices to characterize tissues is presented. Feature extraction is done on the Mueller matrix elements resulting in depolarization and retardance images by polar decomposition. These are used to detect and classify early oral cancers and precancerous changes in epithelium such as dysplasia. These images are compared with orthogonal polarization images and analyzed in an attempt to identity useful factors for the differentiation between cancerous lesions and their benign counterparts. Our results indicate that polarimetry has potential as a method for the in vivo early detection and diagnosis of oral premalignancy.     

Target-adaptive polarimetric synthetic aperture radar target discrimination using maximum average correlation height filters

We report the development of a technique for adaptive selection of polarization ellipse tilt and ellipticity angles such that the target separation from clutter is maximized. From the radar scattering matrix [S] and its complex components, in phase and quadrature phase, the elements of the Mueller matrix are obtained. Then, by means of polarization synthesis, the radar cross section of the radar scatters are obtained at different transmitting and receiving polarization states. By designing a maximum average correlation height filter, we derive a target versus clutter distance measure as a function of four transmit and receive polarization state angles. The results of applying this method on real synthetic aperture radar imagery indicate a set of four transmit and receive angles that lead to maximum target versus clutter discrimination. These optimum angles are different for different targets. Hence, by adaptive control of the state of polarization of polarimetric radar, one can noticeably improve the discrimination of targets from clutter.     

Producing optical vortices through forked holographic grating: study of polarization

It is important to know what happens to initial polarization during the formation of optical vortices. We use a computer-generated forked holographic grating to produce optical vortices in the laboratory and study changes in polarization, introduced by the grating, which generates optical vortices in its diffracted orders. The Mueller matrix has been used to quantify changes in the polarization in diffracted orders containing optical vortices. Decomposition of the Mueller matrices estimates the polarizing properties, such as diattenuation, retardance and depolarization of the system. We find that the system is a non-depolarizing system. The study also shows that spin and the orbital angular momentum of photons are not coupled in this process.     

Systematic effects induced by a flat isotropic dielectric slab

The instrumental polarization induced by a flat isotropic dielectric slab in microwave frequencies is discussed. We find that, in spite of its isotropic nature, such a dielectric can produce spurious polarization either by transmitting incoming anisotropic diffuse radiation or emitting when it is thermally inhomogeneous. We present evaluations of instrumental polarization generated by materials usually adopted in radio astronomy, by using the Mueller matrix formalism. As an application, results for different slabs in front of a 32 GHz receiver are discussed. Such results are based on measurements of their complex dielectric constants. We evaluate that a 0.33 cm thick Teflon slab introduces negligible spurious polarization (<2.6 x 10(-5) in transmission and <6 x 10(-7) in emission), even minimizing the leakage (<10(-8) from Q to U Stokes parameters, and vice versa) and the depolarization (approximately 1.3 x 10(-3)).