Parameters characterizing electromagnetic wave polarization

Depolarization of Vector Light Beams on Propagation in Free Space

Nonparaxial propagation of the vector vortex light beams in free space was investigated theoretically. Propagation-induced polarization changes in vector light beams with different spatial intensity distributions were analyzed. It is shown that the hybrid vector Bessel modes with polarization-OAM (orbital angular momentum) entanglement are the exact solutions of the vector Helmholtz equation. Decomposition of arbitrary vector beams in the initial plane z = 0 into these polarization-invariant beams with phase and polarization singularities was used to analyze the evolution of the polarization of light within the framework of the 2 × 2 coherency matrix formalism. It is shown that the 2D degree of polarization decreases with distance if the incident vector beam is not the modal solution. The close relationship of the degree of polarization with the quantum-mechanical purity parameter is emphasized.

Purity of 3D polarization

Measures of purity for 3D partially polarized fields, and in particular, the separation into circularly and linearly polarized contributions, are reexamined, and a new degree of total linear polarization introduced. Explicit expressions for the characteristic decomposition in terms of coherency matrix elements are presented, including the special case of an intrinsic coherency matrix. Parameterization of the coherency matrix in terms of ellipticity, and the directions of the ellipse normal and major axis are investigated. Phase consistency is discussed. A comprehensive collection of results regarding intrinsic polarization properties is presented.

Geometric Interpretation and General Classification of Three-Dimensional Polarization States through the Intrinsic Stokes Parameters

In contrast with what happens for two-dimensional polarization states, defined as those whose electric field fluctuates in a fixed plane, which can readily be represented by means of the Poincaré sphere, the complete description of general three-dimensional polarization states involves nine measurable parameters, called the generalized Stokes parameters, so that the generalized Poincaré object takes the complicated form of an eight-dimensional quadric hypersurface. In this work, the geometric representation of general polarization states, described by means of a simple polarization object constituted by the combination of an ellipsoid and a vector, is interpreted in terms of the intrinsic Stokes parameters, which allows for a complete and systematic classification of polarization states in terms of meaningful rotationally invariant descriptors.

Depolarization of Light in Optical Fibers: Effects of Diffraction and Spin-Orbit Interaction

Polarization is measured very often to study the interaction of light and matter, so the description of the polarization of light beams is of both practical and fundamental interest. This review discusses the polarization properties of structured light in multimode graded-index optical fibers, with an emphasis on the recent advances in the area of spin-orbit interactions. The basic physical principles and properties of twisted light propagating in a graded index fiber are described: rotation of the polarization plane, Laguerre–Gauss vector beams with polarization-orbital angular momentum entanglement, splitting of degenerate modes due to spin-orbit interaction, depolarization of light beams, Berry phase and 2D and 3D degrees of polarizations, etc. Special attention is paid to analytical methods for solving the Maxwell equations of a three-component field using perturbation analysis and quantum mechanical approaches. Vector and tensor polarization degrees for the description of strongly focused light beams and their geometrical interpretation are also discussed.

Eigenvectors of polarization coherency matrices

Calculation of the eigenvectors of two- and three-dimensional coherency matrices, and the four-dimensional coherency matrix associated with a Mueller matrix, is considered, especially for algebraic cases, in the light of recently published algorithms. The preferred approach is based on a combination of an evaluation of the characteristic polynomial and an adjugate matrix. The diagonal terms of the coherency matrix are given in terms of the characteristic polynomial of reduced matrices as functions of the eigenvalues of the coherency matrix. The analogous polynomial form for the off-diagonal elements of the coherency matrix is also presented. Simple expressions are given for the pure component in the characteristic decomposition.

Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects

Geometric phases are a universal concept that underpins numerous phenomena involving multi-component wave fields. These polarization-dependent phases are inherent in interference effects, spin-orbit interaction phenomena, and topological properties of vector wave fields. Geometric phases have been thoroughly studied in two-component fields, such as two-level quantum systems or paraxial optical waves. However, their description for fields with three or more components, such as generic nonparaxial optical fields routinely used in modern nano-optics, constitutes a nontrivial problem. Here we describe geometric, dynamical, and total phases calculated along a closed spatial contour in a multi-component complex field, with particular emphasis on 2D (paraxial) and 3D (nonparaxial) optical fields. We present several equivalent approaches: (i) an algebraic formalism, universal for any multi-component field; (ii) a dynamical approach using the Coriolis coupling between the spin angular momentum and reference-frame rotations; and (iii) a geometric representation, which unifies the Pancharatnam-Berry phase for the 2D polarization on the Poincaré sphere and the Majorana-sphere representation for the 3D polarized fields. Most importantly, we reveal close connections between geometric phases, angular-momentum properties of the field, and topological properties of polarization singularities in 2D and 3D fields, such as C-points and polarization Möbius strips.

Focal plane filter array engineering I: rectangular lattices

Focal planes arrays (FPA) measure values proportional to an integrated irradiance with little sensitivity to wavelength or polarization in the optical wavelength range. The measurement of spectral properties is often achieved via a spatially varying color filter array. Recently spatially varying polarization filter arrays have been used to extract polarization information. Although measurement of color and polarization utilize separate physical methods, the underlying design and engineering methodology is linked. In this communication we derive a formalism which can be used to design any type of periodic filter array on a rectangular lattice. A complete system description can be obtained from the number of unit cells, the pixel shape, and the unit cell geometry. This formalism can be used to engineer the channel structure for any type of periodic tiling of a rectangular lattice for any type of optical filter array yielding irradiance measurements.

Three-dimensional polarization algebra

If light is focused or collected with a high numerical aperture lens, as may occur in imaging and optical encryption applications, polarization should be considered in three dimensions (3D). The matrix algebra of polarization behavior in 3D is discussed. It is useful to convert between the Mueller matrix and two different Hermitian matrices, representing an optical material or system, which are in the literature. Explicit transformation matrices for converting the column vector form of these different matrices are extended to the 3D case, where they are large (81×81) but can be generated using simple rules. It is found that there is some advantage in using a generalization of the Chandrasekhar phase matrix treatment, rather than that based on Gell-Mann matrices, as the resultant matrices are of simpler form and reduce to the two-dimensional case more easily. Explicit expressions are given for 3D complex field components in terms of Chandrasekhar-Stokes parameters.

Generalized Jones matrix method for homogeneous biaxial samples

The generalized Jones matrix (GJM) is a recently introduced tool to describe linear transformations of three-dimensional light fields. Based on this framework, a specific method for obtaining the GJM of uniaxial anisotropic media was recently presented. However, the GJM of biaxial media had not been tackled so far, as the previous method made use of a simplified rotation matrix that lacks a degree of freedom in the three-dimensional rotation, thus being not suitable for calculating the GJM of biaxial media. In this work we propose a general method to derive the GJM of arbitrarily-oriented homogeneous biaxial media. It is based on the differential generalized Jones matrix (dGJM), which is the three-dimensional counterpart of the conventional differential Jones matrix. We show that the dGJM provides a simple and elegant way to describe uniaxial and biaxial media, with the capacity to model multiple simultaneous optical effects. The practical usefulness of this method is illustrated by the GJM modeling of the polarimetric properties of a negative uniaxial KDP crystal and a biaxial KTP crystal for any three-dimensional sample orientation. The results show that this method constitutes an advantageous and straightforward way to model biaxial media, which show a growing relevance for many interesting applications.

Partial polarization of optical beams and near fields probed with a nanoscatterer

We consider theoretically the detection of the spectral polarization characteristics of random, partially polarized optical beams and near fields by probing them with a dipolar nanoparticle. We show that measuring the polarization state of the scattered far field with a conventional waveplate-polarizer setup, possibly in several directions, results in the full 3×3 polarization matrix at the probe site. This allows us to deduce the distributions of the degree of polarization of the field and the Stokes parameters of the polarized part of the field with a resolution limited by the probe size. Regarding random near fields we show that, in analogy with a known result on beam fields, a degree of polarization of three-component light fields put forward in recent literature can in some cases be interpreted as a ratio of the intensity in the polarized part of the light to that of the total field. We demonstrate the technique by considering the probing of a Gaussian-Schell model beam and a thermally excited near field. The method extends the current scanning-probe techniques to the detection of partial polarization of random light fields and can find applications in nanophotonics and polarization optics.

Majorization and measures of classical polarization in three dimensions

There has been much discussion in the literature about rival measures of classical polarization in three dimensions. We gather and compare the various proposed measures of polarization, creating a geometric representation of the polarization state space in the process. We use majorization, previously used in quantum information, as a criterion to establish a partial ordering on the polarization state space. Using this criterion and other considerations, the most useful polarization measure in three dimensions is found to be one dependent on the Bloch vector decomposition of the polarization matrix.

Comparison of quarter-wave retarders over finite spectral and angular bandwidths for infrared polarimetric-imaging applications

We compare three technological approaches for quarter-wave retarders within the context of polarimetric-imaging applications in the long-wave infrared (LWIR) spectrum. Performance of a commercial cadmium sulfide (CdS) crystalline waveplate, a multilayer meanderline structure, and a silicon (Si) form-birefringent retarder are evaluated under conditions of 8-12 μm broadband radiation emerging from an F/1 focusing objective. Metrics used for this comparison are the spectrally dependent axial ratio, retardance, and polarization-averaged power transmittance, which are averaged over the angular range of interest. These parameters correspond to the characteristics that would be observed at the focal-plane array (FPA) detector of an LWIR imaging polarimeter.

Entanglement and classical polarization states

We identify classical light fields as physical examples of nonquantum entanglement. A natural measure of degree of polarization emerges from this identification, and we discuss its systematic application to any optical field, whether beamlike or not.

Utilization of photon orbital angular momentum in the low-frequency radio domain

We show numerically that vector antenna arrays can generate radio beams that exhibit spin and orbital angular momentum characteristics similar to those of helical Laguerre-Gauss laser beams in paraxial optics. For low frequencies (< or = 1 GHz), digital techniques can be used to coherently measure the instantaneous, local field vectors and to manipulate them in software. This enables new types of experiments that go beyond what is possible in optics. It allows information-rich radio astronomy and paves the way for novel wireless communication concepts.

Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices

We assess the degree of coherence of vectorial electromagnetic fields in the space-frequency domain as the distance between the cross-spectral density matrix and the identity matrix representing completely incoherent light. This definition is compared with previous approaches. It is shown that this distance provides an upper bound for the degree of coherence and visibility for any pair of scalar waves obtained by linear combinations of the original fields. This same approach emerges when applying a previous definition of global coherence to a Young interferometer.

Prospects for lunar satellite detection of radio pulses from ultrahigh energy neutrinos interacting with the moon

The Moon provides a huge effective detector volume for ultrahigh energy cosmic neutrinos, which generate coherent radio pulses in the lunar surface layer due to the Askaryan effect. In light of presently considered lunar missions, we propose radio measurements from a Moon-orbiting satellite. First systematic Monte Carlo simulations demonstrate the detectability of Askaryan pulses from neutrinos with energies above 10(20) eV at the very low fluxes predicted in different scenarios.

Role of spatial coherence in polarization tomography

We analyze an experimental setup in which a quasi-monochromatic spatially coherent beam of light is used to probe a paraxial optical scatterer. We discuss the effect of the spatial coherence of the probe beam on the Mueller matrix representing the scatterer. We show that, according to the degree of spatial coherence of the beam, the same scattering medium can be represented by different Mueller matrices. This result should serve as a warning for experimentalists.

Correlation matrix of a completely polarized, statistically stationary electromagnetic field

It is shown that, for a 3 x 3 correlation matrix Wij(r, r, omega), (i, j = x, y, z) of the electric vector of a random, stationary electromagnetic field to represent a field that is completely polarized at a point r and frequency omega, each element of the matrix must factorize. More precisely, a necessary and sufficient condition for the correlation matrix to represent a fully polarized field at a point r is that the matrix has the form Wij(r, r, omega) = epsilon(i)*(r, omega)epsilon(j)(r, omega), where epsilon(i)(r, omega) (i = x, y, z) are deterministic functions, i.e., that all pairs of the Cartesian components of the electric field at a point r and frequency omega are completely correlated.

Minkowskian description of polarized light and polarizers

A conventional Stokes description of polarized light is considered in a four-dimensional Lorentzian space, developing a seminal idea of Paul Soleillet [Ann. Phys. (Paris) 12, 23 (1929)]. This provides a striking interpretation for the degree of polarization and the Stokes decomposition of light beams. Malus law and reciprocity theorems for polarizers are studied using this Lorentzian formalism.

Degree of polarization for optical near fields

We investigate an extension to the concept of degree of polarization that applies to arbitrary electromagnetic fields, i.e., fields whose wave fronts are not necessarily planar. The approach makes use of generalized spectral Stokes parameters that appear as coefficients, when the full 3 x 3 spectral coherence matrix is expanded in terms of the Gell-Mann matrices. By defining the degree of polarization in terms of these parameters in a manner analogous to the conventional planar-field case, we are led to a formula that consists of scalar invariants of the spectral coherence matrix only. We show that attractive physical insight is gained by expressing the three-dimensional degree of polarization explicitly with the help of the correlations between the three orthogonal spectral components of the electric field. Furthermore, we discuss the fundamental differences in characterizing the polarization state of a field by employing either the two- or the three-dimensional coherence-matrix formalism. The extension of the concept of the degree of polarization to include electromagnetic fields having structures of arbitrary form is expected to be particularly useful, for example, in near-field optics.

Degree of polarization in near fields of thermal sources: effects of surface waves

We introduce the concept of degree of polarization for electromagnetic near fields. The approach is based on the generalized Stokes parameters that appear as expansion coefficients of the 3 x 3 coherence matrix in terms of the Gell-Mann matrices. The formalism is applied to optical near fields of thermally fluctuating half-space sources with particular interest in fields that are strongly polarized owing to resonant surface plasmons or phonons. This novel method is particularly useful when assessing the full vectorial characteristics of random evanescent fields, e.g., for near-field spectroscopy and polarization microscopy.

Computer-generated space-variant polarization elements with subwavelength metal stripes

A novel method of performing two-dimensional space-variant polarization operations is presented. The method is based on determining the local direction and period of subwavelength metal-stripe gratings by use of vectorial optics to obtain any desired continuous polarization change. We demonstrate our approach with specific computer-generated space-variant polarization elements for laser radiation at 10.6mum. The polarization properties are verified with complete space-variant polarization analysis and measurement.