Polarimetric measurement of temporal coherence in electromagnetic light beams.

We present a method to determine the degree of temporal coherence of a quasimonochromatic vectorial light beam by polarimetric measurements. More specifically, we employ Michelson's interferometer in which the polarization Stokes parameters of the output (interference) beam are measured as a function of the time delay. Such a measurement enables us to deduce the magnitudes of the coherence (two-time) Stokes parameters, and hence the degree of coherence, of the input beam. Compared to existing methods the current technique has the benefits that neither optical elements in the arms of the interferometer nor visibility measurements are needed. The method is demonstrated with a laser diode and a filtered halogen source of various degrees of polarization.

Three-dimensional polarization effects in optical tunneling.

We consider the three-dimensional (3D) polarimetric properties of an evanescent optical field excited in the gap of a double-prism system by a random plane wave. The analysis covers the case of frustrated total internal reflection (FTIR), i.e., optical tunneling, and relies on the characteristic decomposition of the 3×3 polarization matrix. We find in particular that, for any incident partially polarized plane wave, the evanescent field inside the gap is necessarily in a nonregular, genuine 3D polarization state. We also show that the 3D polarimetric properties of the field at the second boundary are sensitive to the changes of the gap width and that the relevant effects occur for the smaller widths when the angle of incidence of the plane wave becomes larger. The results of this work uncover new aspects of the polarimetric structure of genuine 3D evanescent fields and may find applications in near-field optics and surface nanophotonics.

Dual views of the generalized degree of purity.

Several approaches and descriptors have been proposed to characterize the purity of coherency or density matrices describing physical states, including the polarimetric purity of 2D and 3D partially polarized waves. This work introduces two interpretations of the degree of purity: one derived from statistics and another from algebra. In the first one, the degree purity is expressed in terms of the mean and standard deviation of the eigenvalue spectrum of the density or coherency matrix of the corresponding state. The second one expresses the purity in terms of two specific measures obtained by decomposing the coherency matrix as a sum of traceless symmetric, antisymmetric, and scalar matrices. We believe these two approaches offer better insights into the purity measure. Furthermore, interesting relations with existing quantities in polarization optics also are described.

Information structure and general characterization of Mueller matrices.

Linear polarimetric transformations of light polarization states by the action of material media are fully characterized by corresponding Mueller matrices, which contain, in an implicit and intricate manner, all measurable information on such transformations. The general characterization of Mueller matrices relies on the positive semi-definiteness of the associated coherency matrix, which can be mathematically formulated through the nonnegativity of its eigenvalues. The enormously involved explicit algebraic form of such formulation prevents its interpretation in terms of simple physical conditions. In this work, a general and simple characterization of Mueller matrices, based on their statistical structure, is presented. The concepts associated with the retardance, enpolarization, and depolarization properties as well as the essential coupling between the latter two are straightforwardly described in the light of the new approach.

Unraveling the physical information of depolarizers.

The link between depolarization measures and physical nature and structure of material media inducing depolarization is nowadays an open question. This article shows how the joint use of two complementary sets of depolarizing metrics, namely the Indices of polarimetric purity and the Components of purity, are sufficient to completely describe the integral depolarizing properties of a sample. Based on a collection of illustrative and representative polarimetric configurations, a clear and meaningful physical interpretation of such metrics is provided, thus extending the current tools and comprehension for the study and analysis of the depolarizing properties of material media. This study could be of interest to those users dealing with depolarization or depolarizing samples.

Polarimetric imaging microscopy for advanced inspection of vegetal tissues.

Optical microscopy techniques for plant inspection benefit from the fact that at least one of the multiple properties of light (intensity, phase, wavelength, polarization) may be modified by vegetal tissues. Paradoxically, polarimetric microscopy although being a mature technique in biophotonics, is not so commonly used in botany. Importantly, only specific polarimetric observables, as birefringence or dichroism, have some presence in botany studies, and other relevant metrics, as those based on depolarization, are underused. We present a versatile method, based on a representative selection of polarimetric observables, to obtain and to analyse images of plants which bring significant information about their structure and/or the spatial organization of their constituents (cells, organelles, among other structures). We provide a thorough analysis of polarimetric microscopy images of sections of plant leaves which are compared with those obtained by other commonly used microscopy techniques in plant biology. Our results show the interest of polarimetric microscopy for plant inspection, as it is non-destructive technique, highly competitive in economical and time consumption, and providing advantages compared to standard non-polarizing techniques.

Algorithm for the numerical calculation of the serial components of the normal form of depolarizing Mueller matrices.

The normal form of a depolarizing Mueller matrix constitutes an important tool for the phenomenological interpretation of experimental polarimetric data. Due to its structure as a serial combination of three Mueller matrices, namely a canonical depolarizing Mueller matrix sandwiched between two pure (nondepolarizing) Mueller matrices, it overcomes the necessity of making a priori choices on the order of the polarimetric components, as this occurs in other serial decompositions. Because Mueller polarimetry addresses more and more applications in a wide range of areas in science, engineering, medicine, etc., the normal form decomposition has an enormous potential for the analysis of experimentally determined Mueller matrices. However, its systematic use has been limited to some extent because of the lack of numerical procedure for the calculation of each polarimetric component, in particular in the case of Type II Mueller matrices. In this work, an efficient algorithm applicable to the decomposition of both Type II and Type I Mueller matrices is presented.

Characterization of passivity in Mueller matrices.

Except for very particular and artificial experimental configurations, linear transformations of the state of polarization of an electromagnetic wave result in a reduction of the intensity of the exiting wave with respect to the incoming one. This natural passive behavior imposes certain mathematical restrictions on the Mueller matrices associated with the said transformations. Although the general conditions for passivity in Mueller matrices were presented in a previous paper [ J. Opt. Soc. Am. A17, 328 (2000)JOAOD60740-323210.1364/JOSAA.17.000328], the demonstration was incomplete. In this paper, the set of two necessary and sufficient conditions for a Mueller matrix to represent a passive medium are determined and demonstrated on the basis of its arbitrary decomposition as a convex combination of nondepolarizing and passive pure Mueller matrices. The procedure followed to solve the problem also provides an appropriate framework to identify the Mueller matrix that, among the family of proportional passive Mueller matrices, exhibits the maximal physically achievable intensity transmittance. Beyond the theoretical interest on the rigorous characterization of passivity, the results obtained, when applied to absolute Mueller polarimetry, also provide a criterion to discard those experimentally measured Mueller matrices that do not satisfy the passivity criterion.

Arbitrary decomposition of a Mueller matrix.

Mueller polarimetry involves a variety of instruments and technologies whose importance and scope of applications are rapidly increasing. The exploitation of these powerful resources depends strongly on the mathematical models that underlie the analysis and interpretation of the measured Mueller matrices and, very particularly, on the theorems for their serial and parallel decompositions. In this Letter, the most general formulation for the parallel decomposition of a Mueller matrix is presented, which overcomes certain critical limitations of the previous approaches, particularly the unnecessary exigency that the Mueller matrices of all parallel components have to be normalized in order to have equal transmittances for unpolarized light. In addition, the obtained results lead to a generalization of the polarimetric subtraction procedure and allow for a formulation of the arbitrary decomposition that integrates, in a natural way, the passivity criterion.

Intensity and spin anisotropy of three-dimensional polarization states.

Anisotropy is a natural feature of polarization states, and only fully random three-dimensional (3D) states exhibit complete isotropy. In general, differences between the strengths of the electric field components along the three orthogonal directions give rise to intensity anisotropy. Moreover, polarization states involve an average spin whose inherent vectorial nature constitutes a source of spin anisotropy. In this work, appropriate descriptors are identified to characterize quantitatively the levels of intensity anisotropy and spin anisotropy of a general 3D polarization state, leading to a novel interpretation for the degree of polarimetric purity as a measure describing the overall polarimetric anisotropy of a 3D optical field. The mathematical representation, as well as the physical features of completely intensity-isotropic 3D polarization states with a maximum spin anisotropy, are also examined. The results provide new insights into the polarimetric field structure of random 3D electromagnetic light states.

Depolarizing metrics for plant samples imaging.

Optical methods, as fluorescence microscopy or hyperspectral imaging, are commonly used for plants visualization and characterization. Another powerful collection of optical techniques is the so-called polarimetry, widely used to enhance image contrast in multiple applications. In the botanical applications framework, in spite of some works have already highlighted the depolarizing print that plant structures left on input polarized beams, the potential of polarimetric methods has not been properly exploited. In fact, among the few works dealing with polarization and plants, most of them study light scattered by plants using the Degree of Polarization (DoP) indicator. Other more powerful depolarization metrics are nowadays neglected. In this context, we highlight the potential of different depolarization metrics obtained using the Mueller matrix (MM) measurement: the Depolarization Index and the Indices of Polarimetric Purity. We perform a qualitative and quantitative comparison between DoP- and MM-based images by studying a particular plant, the Hedera maroccana. We show how Mueller-based metrics are generally more suitable in terms of contrast than DoP-based measurements. The potential of polarimetric measurements in the study of plants is highlighted in this work, suggesting they can be applied to the characterization of plants, plant taxonomy, water stress in plants, and other botanical studies.

Polarimetric nonregularity of evanescent waves.

Three-dimensional polarization states of random light can be classified into regular and nonregular according to the structure of the related 3×3 polarization matrix. Here we show that any purely evanescent wave excited in total internal reflection of a partially polarized plane-wave field is always in a nonregular polarization state. The degree of nonregularity of such evanescent waves is also studied in terms of a recently advanced measure. Nonregular evanescent waves uncover new aspects of the polarimetric structure and dimensional character of electromagnetic near fields, with potential applications in nanoscale surface optics.

Nonregularity of three-dimensional polarization states.

Regular states of polarization are defined as those that can be decomposed into a pure state (fully polarized), a two-dimensional (2D) unpolarized state (a state whose polarization ellipse evolves fully randomly in a fixed plane), and a three-dimensional (3D) unpolarized state (a state whose polarization ellipse evolves fully randomly in the 3D space) [Phys. Rev. A95, 053856 (2017)PLRAAN1050-294710.1103/PhysRevA.95.053856]. For nonregular states, the middle component can be considered as an equiprobable mixture of two pure states, whose polarization ellipses lie in different planes. In this work, we identify a perfect nonregular state and introduce the degree of nonregularity as a measure of the proximity of a nonregular state to regularity. We also analyze and interpret the notion of polarization-state regularity in terms of polarimetric parameters. Our results bring new insights into the polarimetric structure of 3D light fields.

Polarimetric imaging of biological tissues based on the indices of polarimetric purity.

We highlight the interest of using the indices of polarimetric purity (IPPs) to the inspection of biological tissues. The IPPs were recently proposed in the literature and they result in a further synthetization of the depolarizing properties of samples. Compared with standard polarimetric images of biological samples, IPP-based images lead to larger image contrast of some biological structures and to a further physical interpretation of the depolarizing mechanisms inherent to the samples. In addition, unlike other methods, their calculation do not require advanced algebraic operations (as is the case of polar decompositions), and they result in 3 indicators of easy implementation. We also propose a pseudo-colored encoding of the IPP information that leads to an improved visualization of samples. This last technique opens the possibility of tailored adjustment of tissues contrast by using customized pseudo-colored images. The potential of the IPP approach is experimentally highlighted along the manuscript by studying 3 different ex-vivo samples. A significant image contrast enhancement is obtained by using the IPP-based methods, compared to standard polarimetric images.

Basic properties and classification of Mueller matrices derived from their statistical definition.

Starting from the statistical definition of the Mueller matrix, we derive the relationships between basic properties, such as the number of contact points of the characteristic ellipsoid with the Poincaré sphere and the rank of the covariance matrix. This approach enables the comprehensive classification of any experimental depolarizing Mueller matrix into one of six possible classes, thus making possible phenomenological interpretation in terms of a specific fluctuating Jones generator or of a finite sum of nondepolarizing Mueller matrices.

Synthesis and characterization of depolarizing samples based on the indices of polarimetric purity.

In this work, we discuss the interest of using the indices of polarimetric purity (IPPs) as a criterion for the characterization and classification of depolarizing samples. We prove how differences in the depolarizing capability of samples, not seen by the commonly used depolarization index PΔ, are identified by the IPPs. The above-stated result is analyzed from a theoretical point of view and experimentally verified through a set of polarimetric measurements. We show how the approach presented here can be useful in easily synthetizing depolarizing samples with controlled depolarizing features, just by properly combining low-cost fully polarizing elements (such as linear retarders or polarizers).

On optimal filtering of measured Mueller matrices.

While any 2D mixed state of polarization of light can be represented by a combination of a pure state and a fully random state, any Mueller matrix can be represented by a convex combination of a pure component and three additional components whose randomness is scaled in a proper and objective way. Such characteristic decomposition constitutes the appropriate framework for the characterization of the polarimetric randomness of the system represented by a given Mueller matrix and provides criteria for the optimal filtering of noise in experimental polarimetry.

Invariant quantities of a nondepolarizing Mueller matrix.

Orthogonal Mueller matrices can be considered as corresponding either to retarders or to generalized transformations of the polarization basis for the representation of Stokes vectors, so that they constitute the only type of Mueller matrices that preserve the degree of polarization and the intensity of any partially polarized input Stokes vector. The physical quantities that remain invariant when a nondepolarizing Mueller matrix is transformed through its product by different types of orthogonal Mueller matrices are identified and interpreted, providing a better knowledge of the information contained in a nondepolarizing Mueller matrix.

Singular Mueller matrices.

Singular Mueller matrices play an important role in polarization algebra and have peculiar properties that stem from the fact that either the medium exhibits maximum diattenuation and/or polarizance or because its associated canonical depolarizer has the property of fully randomizing the circular component (at least) of the states of polarization of light incident on it. The formal reasons for which the Mueller matrix M of a given medium is singular are systematically investigated, analyzed, and interpreted in the framework of the serial decompositions and the characteristic ellipsoids of M. The analysis allows for a general classification and geometric representation of singular Mueller matrices, which are of potential usefulness to experimentalists dealing with such media.

Components of purity of a three-dimensional polarization state.

The degree of polarimetric purity of a three-dimensional (3D) polarization state is a measure of the closeness to a pure state and can be expressed as a weighted quadratic average of two indices of polarimetric purity, invariant with respect to unitary transformations and defined in terms of the relative weights of certain incoherent components of the state. An alternative view of the polarimetric purity is formulated in terms of three contributions, namely, the degree of directionality, the degree of linear polarization, and the degree of circular polarization. While the indices of polarimetric purity give complete information on the structure of randomness but are insensitive to other attributes of the state of polarization, the three components of purity are invariant under orthogonal transformations (rotations in the real space) and provide a meaningful framework for the representation of 3D polarization states in terms of quantities that are intrinsic for each given state.