Invariant polarimetric contrast parameters of light with Gaussian fluctuations in three dimensions

We propose a rigorous definition of the minimal set of parameters that characterize the difference between two partially polarized states of light whose electric fields vary in three dimensions with Gaussian fluctuations. Although two such states are a priori defined by eighteen parameters, we demonstrate that the performance of processing tasks such as detection, localization, or segmentation of spatial or temporal polarization variations is uniquely determined by three scalar functions of these parameters. These functions define a "polarimetric contrast" that simplifies the analysis and the specification of processing techniques on polarimetric signals and images. This result can also be used to analyze the definition of the degree of polarization of a three-dimensional state of light with Gaussian fluctuations in comparison, with respect to its polarimetric contrast parameters, with a totally depolarized light. We show that these contrast parameters are a simple function of the degrees of polarization previously proposed by Barakat [Opt. Acta 30, 1171 (1983)] and Set?l? et al. [Phys. Rev. Lett. 88, 123902 (2002)]. Finally, we analyze the dimension of the set of contrast parameters in different particular situations.     

Entropy of partially polarized light and application to statistical processing techniques

We have analyzed entropy properties of coherent and partially polarized light in an arbitrary number of spatial dimensions. We show that for Gaussian fields, the Shannon entropy is a simple function of the intensity and of the Barakat degree of polarization. In particular, we provide a probabilistic interpretation of this definition of the degree of polarization. Using information theory results, we also deduce some physical properties of partially polarized light such as additivity of the entropy and depolarization effects induced by mixing partially polarized states of light. Finally, we demonstrate that entropy measures can play an important role in segmentation and detection tasks.     

Coherent-mode decomposition of partially polarized,partially coherent sources

It is shown that any partially polarized, partially coherent source can be expressed in terms of a suitable superposition of transverse coherent modes with orthogonal polarization states. Such modes are determined through the solution of a system of two coupled integral equations. An example, for which the modal decomposition is obtained in closed form in terms of fully linearly polarized Hermite Gaussian modes, is given.     

Quantum phase measurements and non-classical polarization states of light

Abstract

In our paper we consider the non-classical behaviour of both the Hermitian (observable) Stokes parameters of light and the phase difference of two modes that describe the quantum polarization states of optical field. To characterize the degree of polarization of light we introduce a new quantity taking into account the quantum properties of different quantum states of two orthogonally polarized modes. The problem of determination of the phase difference in two modes of optical field for the quantum polarization states of light is discussed. To describe in general such a quantum field we introduce two pairs of the phase operators: the phase angles for the Stokes parameters of light in a three-dimensional picture of the Poincaré sphere. We also consider a special type of the eight-port polarization interferometer (polarimeter) for simultaneous homodyne detection of both the Stokes parameters of light and the polarization phase operators and their fluctuations as well. Using an anisotropic (spatioperiodic) Kerr-like nonlinear medium associated with the polarization interferometer we could generate and also observe the polarization-squeezed phase states of light. The fluctuations in the phase difference between two orthogonally polarized modes for these non-classical states are less than the fluctuations for light in coherent state.     

Separation technique of a mixing of two uncorrelated and perfectly polarized lights with different coherence and polarization properties

The sum of two uncorrelated and totally polarized lights with different coherence and polarization properties usually results in a partially polarized light. It is shown in this paper that the initial totally polarized lights can be recovered from the mixed partially polarized light. The proposed technique is based on coherence analysis and does not require the knowledge of the polarization states or the coherence properties of the initial perfectly polarized beams as long as these properties are different for the two waves. Some practical optical implementations of this technique are discussed on different illustrative applications.     

Some current trends of correlation optics metrology of coherence and polarization

New feasibilities for metrology of coherence and polarization of light fields provided by correlation optics approaches are considered. This paper shows these approaches are fruitful in measuring the field parameters that are critical for optical diagnostics using the data on the degree of coherence and the state and the degree of polarization of partially coherent and inhomogeneously polarized fields.     

Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources

In studies of radiation from partially coherent sources the so-called quasi-homogeneous (QH) model sources have been very useful, for instance in elucidating the behavior of fields produced by thermal sources. The analysis of the fields generated by such sources has, however, been largely carried out in the framework of scalar wave theory. In this paper we generalize the concept of the QH source to the domain of the electromagnetic theory, and we derive expressions for the elements of the cross-spectral density matrix, for the spectral density, the spectral degree of coherence, the degree of polarization, and the Stokes parameters of the far field generated by planar QH sources of uniform states of polarization. We then derive reciprocity relations analogous to those familiar in connection with the QH scalar sources. We illustrate the results by determining the properties of the far field produced by transmission of an electromagnetic beam through a system of spatial light modulators.     

Azimuthal polarization and partial coherence

Partially coherent fields with the electric field parallel to the azimuthal coordinate are analyzed by use of the exact angular spectrum representation. The known results for fully coherent fields are used to find the permitted forms of azimuthally polarized, partially coherent fields. The derived result is then used to show that this class of fields is severely restricted because the azimuthal polarization state is particularly sensitive to the correlation properties of the electric-field components. Two examples of azimuthally polarized fields are briefly examined. The first is a class of nondiffracting fields that retain the polarization state upon propagation, whereas the second is an example in which the azimuthal polarization is broken because the cross-spectral density function is not of the permitted form.     

Polarization changes in a partially coherent radially polarized doughnut beam through turbulent ocean

On the basis of the extended Huygens–Fresnel integral principle and unified theory of coherence and polarization of light, we studied the effects of oceanic turbulence on polarization properties of a partially coherent radially polarized doughnut (PCRPD) beam. The ocean-induced fluctuations in the refractive index are assumed be driven by temperature and salinity fluctuations. Numerical examples of changes in polarization properties, such as the degree of polarization, the degree of ellipticity, and the orientation angle in the oceanic turbulence for the PCRPD beam, are given. Our analysis demonstrates how polarization of the PCRPD beam is affected by statistical properties of the source and by several parameters of oceanic turbulence. We find that the propagation of the PCRPD beam is different from that of stochastic beams in oceanic turbulence. The degree of polarization for the PCRPD beam approaches a certain steady value, and the elliptical polarized state of the fully polarized portion of the beam will become fully linear in the far field.     

Size determination by use of two-dimensional Mueller matrices backscattered by optically thick random media

Here we are concerned with the systematic study of polarized light transport in thick, isotropic, homogeneous random media and of the associated inverse problem. An original spatial and intensity rescaling of the polarization transport allows one to account implicitly for the volume fraction. This parameter elimination permits a complete exploration, by means of Monte Carlo simulations of the dependence of polarized light transport on microscopic parameters. Analysis of the Mueller matrices obtained from the simulations show that additional correlations (with respect to scalar transport) are obtained between the microscopic parameters and the spatial distribution of specific elements of the Mueller matrix. As a consequence, using carefully chosen polarization states, one can determine an average particle size independently of the volume fraction of particles, with only the knowledge of the refractive-index ratio being required. This analysis is validated with experimental Mueller matrices obtained for emulsions of various size, concentration, and polydispersity.     

Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations

We propose to analyze Shannon entropy properties of partially coherent and partially polarized light with Gaussian probability distributions. It is shown that the Shannon entropy is a sum of simple functions of the intensity, of the degrees of polarization, and of the intrinsic degrees of coherence that have been recently introduced. This analysis clearly demonstrates the contribution of partial polarization and of partial coherence to the characterization of disorder of the light provided by the Shannon entropy, which is a standard measure of randomness. We illustrate these results on two simple examples.     

Radiation efficiency of partially coherent electromagnetic beams.

We present a general definition of the radiation efficiency of stationary electromagnetic fields and prove that it is bounded between zero and unity for beams of any state of coherence and polarization. The radiation efficiency may be interpreted as a measure of how directed the radiated fields are, and therefore it can be used to assess the allowed spatial coherence and intensity variations across a beam. We consider a class of partially coherent electromagnetic fields that were recently introduced in the literature and evaluate the radiation efficiencies for two particular examples, namely, the azimuthally polarized symmetric beams and the dipolar beams that are nearly linearly polarized in the central region. The results show that the radiation efficiency is fairly insensitive to the state of polarization and that it differs appreciably from unity for only small values of source and correlation widths.     

Novel Polarimetric Technique Exploring Spatiotemporal Birefringent Response of an Anti-ferroelectric Liquid Crystal Cell

Abstract

This paper describes a novel polarimetric technique for mapping the spatiotemporal birefringent parameters characterizing an anti-ferroelectric liquid crystal cell. A linearly polarized beam of light transmitted by the liquid crystal cell turns into an elliptically polarized beam. Some birefringent irregularities existing over the cell make state of polarization of the elliptically polarized beam of light be spatiotemporal. This beam works as a signal beam to interfere with a reference beam consisting of two orthogonal linearly polarized components. The resultant interference pattern is taken by a charge-coupled device video camera and recorded in a computer. The computer gives the information required for determining the spatiotemporal birefringent parameters that characterize the liquid crystal cell. The major advantage is that the two-dimensional distribution of the birefringent axes as well as the two principal refractive indices can be determined simultaneously, and no use of any optical components for polarization alignment makes it possible to follow a rapid change in birefringent parameters within the maximum frame rate of the video camera. A change in time-dependent birefringent parameter is measured at every 0·1 ms for the demonstration experiment.     

General state contrast imaging: an optimized polarimetric imaging modality insensitive to spatial intensity fluctuations

In active polarization imaging, one frequently needs to be insensitive to noninformative spatial intensity fluctuations. We investigate a way of solving this issue with general state contrast (GSC) imaging. It consists in acquiring two scalar polarimetric images with optimized illumination and analysis polarization states, then forming a ratio. We propose a method for maximizing the discrimination ability between a target and a background in GSC images by determining the optimal illumination and analysis states. A further advantage of this approach is to provide an objective way of quantifying the performance improvement obtained by increasing the number of degrees of freedom of a GSC imager. The efficiency of this approach is demonstrated on simulated and real-world images.     

Partial polarization in three dimensions

Various different parameters have been introduced to describe the degree of polarization of a partially polarized electromagnetic field in three dimensions. Of these, parameters based on the eigenvalues of the coherency matrix are invariant under a unitary transformation. Here, explicit expressions are presented for the eigenvalues, thus providing a geometrical interpretation of the behavior. These expressions are applied to the Huynen decomposition and allow interrelations between different parameters to be developed.     

Optical heterodyne detection of random electromagnetic beams

Abstract

In this paper we present a general analysis for the optical heterodyne detection of random electromagnetic beams. To describe the ensemble of quasimonochromatic beams which are partially polarized and partially coherent, we use a recently developed matrix treatment. We derive an expression for the signal-to-noise ratio (SNR) in terms of the beam coherence polarization matrices of the beams on the detector surface. Numerical examples are given for the SNR variation in the case of partially polarized Gaussian Schell model beams and the optimum detection is discussed in terms of beam parameters of the local oscillator.     

Sensitivity metric approach for retrieval of aerosol properties from multiangular and multispectral polarized radiances

Linearly polarized radiation is sensitive to the microphysical properties of aerosols, namely, to the particle-size distribution and refractive index. The discriminating power of polarized radiation increases strongly with the increasing range of scattering angles and the addition of multiple wavelengths. The polarization and directionality of the Earth's reflectances (POLDER) missions demonstrate that some aerosol properties can be successfully derived from spaceborne polarimetric, multiangular measurements at two visible wavelengths. We extend the concept to analyze the retrieval capabilities of a spaceborne instrument with six polarimetric channels at 412, 445, 555, 865, 1250, and 2250 nm, measuring approximately 100 scattering angles covering a range between 50 and 150 deg. Our focus is development of an analysis methodology that can help quantify the benefits of such multiangular and multispectral polarimetric measurements. To that goal we employ a sensitivity metric approach in a framework of the principal-component analysis. The radiances and noise used to construct the sensitivity metric are calculated with the realistic solar flux for representative orbital viewing geometries, accounting for surface reflection from the ground, and statistical and calibration errors of a notional instrument. Spherical aerosol particles covering a range of representative microphysical properties (effective radius, effective variance, real and imaginary parts of the refractive index, single-scattering albedo) are considered in the calculations. We find that there is a limiting threshold for the effective size (approximately 0.7 microm), below which the weak scattering intensity results in a decreased signal-to-noise ratio and minimal polarization sensitivity, precluding reliable aerosol retrievals. For such small particles, close to the Rayleigh scattering limit, the total intensity provides a much stronger aerosol signature than the linear polarization, inspiring retrieval when the combined signals of intensities and the polarization fraction are used. We also find a strong correlation between aerosol parameters, in particular between the effective size and the variance, which forces one to simultaneously retrieve at least these two parameters.     

Invariants of polarization transformations

The use of polarization-sensitive sensors is being explored in a variety of applications. Polarization diversity has been shown to improve the performance of the automatic target detection and recognition in a significant way. However, it also brings out the problems associated with processing and storing more data and the problem of polarization distortion during transmission. We present a technique for extracting attributes that are invariant under polarization transformations. The polarimetric signatures are represented in terms of the components of the Stokes vectors. Invariant algebra is then used to extract a set of signature-related attributes that are invariant under linear transformation of the Stokes vectors. Experimental results using polarimetric infrared signatures of a number of manmade and natural objects undergoing systematic linear transformations support the invariancy of these attributes.     

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.     

n-Fold Polarization Measures and Associated Thermodynamic Entropy of N Partially Coherent Pencils of Radiation

An n-fold (n=2,…,N) measure of the degree of polarization associated with an N×N coherency matrix characterizing N pencils of radiation is discussed. The measure is cast in terms of the N possible scalar invariants of the coherency matrix and automatically takes into account possible rank deficiency. When N = 2, the single polarization measure reduces to the usual expression for polarization as the fraction of the intensity contained in the polarized component. The N = 3 case is discussed as an illustrative example, with special reference to the rank deficiency of a plane wave arbitrarily oriented with respect to the given coordinate system. Finally, these results are used to redevelop von Laue's theory of the thermodynamic entropy of partially coherent pencils of radiation.