Two-point Stokes parameters: interpretation and properties

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.

Optimized Stokes imaging for highly resolved optical speckle fields, Part III: topological analysis of polarimetric state distributions with optimized data representations

In this last article of a three-paper series focusing on Stokes polarimetry of optical speckle fields resolved at the individual speckle grain scale, experimental results are provided on test samples of varying nature and polarization properties, and are analyzed extensively. For this purpose, a review of the classical ways of displaying Stokes polarimetric information is provided. Then, some original alternative graphical representations are introduced that ensure optimal readability and interpretability of the Stokes imaging data in the context of speckle field polarimetry, and it is shown how they can be adapted to various observation scales. Finally, these tools are implemented in order to provide a topological analysis of the distribution of the states of polarization across a speckle pattern, and in the vicinity of polarimetric singularities of the field.

Optical Sensor System for 3D Jones Matrix Reconstruction of Optical Anisotropy Maps of Self-Assembled Polycrystalline Soft Matter Films

Our work uses a polarization matrix formalism to analyze and algorithmically represent optical anisotropy by open dehydration of blood plasma films. Analytical relations for Jones matrix reconstruction of optical birefringence maps of protein crystal networks of dehydrated biofluid films are found. A technique for 3D step-by-step measurement of the distributions of the elements of the Jones matrix or Jones matrix images (JMI) of the optically birefringent structure of blood plasma films (BPF) has been created. Correlation between JMI maps and corresponding birefringence images of dehydrated BPF and saliva films (SF) obtained from donors and prostate cancer patients was determined. Within the framework of statistical analysis of layer-by-layer optical birefringence maps, the parameters most sensitive to pathological changes in the structure of dehydrated films were found to be the central statistical moments of the 1st to 4th orders. We physically substantiated and experimentally determined the sensitivity of the method of 3D polarization scanning technique of BPF and SF preparations in the diagnosis of endometriosis of uterine tissue.

Measurement of spatial coherence of light [Invited]

The most frequently used experimental techniques for measuring the spatial coherence properties of classical light fields in the space-frequency and space-time domains are reviewed and compared, with some attention to polarization effects. In addition to Young's classical two-pinhole experiment and several of its variations, we discuss methods that allow the determination of spatial coherence at higher data acquisition rates and also permit the characterization of lower-intensity light fields. These advantages are offered, in particular, by interferometric schemes that employ only beam splitters and reflective elements, and thereby also facilitate spatial coherence measurements of broadband fields.

Singular-value decomposition and electromagnetic coherence of optical beams

We investigate the implications of the singular-value decomposition of the cross-spectral density (CSD) matrix to the description of electromagnetic spectral spatial coherence of stationary light beams. We show that in a transverse plane any CSD matrix can be expressed as a mixture of two CSD matrices corresponding to beams which are fully polarized but in general spatially partially coherent. The polarization and coherence structures of these constituent beams are specified, respectively, by the singular vectors and singular values of the full CSD matrix. It follows that vector-beam coherence, including the coherence Stokes parameters and the degree of coherence, can be formulated in terms of only two correlation functions. We further establish two-point analogs of the spectral and characteristic decompositions of the polarization matrix and show that in the case of a Hermitian CSD matrix their composition is specified by the so-called degree of cross-polarization.

Probing coherence Stokes parameters of three-component light with nanoscatterers

We establish a method to determine the spectral coherence Stokes parameters of a random three-component optical field via scattering by two dipolar nanoparticles. We show that measuring the intensity and polarization-state fringes of the scattered far field in three directions allows us to construct all nine coherence Stokes parameters at the dipoles. The method extends current nanoprobe techniques to detection of the spatial coherence of random light with arbitrary three-dimensional polarization structure.

Poincaré sphere of electromagnetic spatial coherence

We introduce a Poincaré sphere construction for geometrical representation of the state of two-point spatial coherence in random electromagnetic (vectorial) beams. To this end, a novel descriptor of coherence is invoked, which shares some important mathematical properties with the polarization matrix and spans a new type of Stokes parameter space. The coherence Poincaré sphere emerges as a geometric interpretation of this novel formalism, which is developed for uniformly and nonuniformly fully polarized beams. The construction is extended to partially polarized beams as well and is demonstrated with a field having separable spatial coherence and polarization characteristics. At a single point, the coherence Poincaré sphere reduces to the conventional polarization Poincaré sphere for any state of partial polarization.

Coherence Stokes Parameters in the Description of Electromagnetic Coherence

The two-point counterparts of the traditional Stokes parameters, which are called the coherence Stokes parameters, have recently been extensively used for assessing the coherence properties of random electromagnetic light beams. In this work, we highlight their importance by emphasizing two features associated with them. First, the role of polarization in electromagnetic coherence is significantly elucidated when the coherence Stokes parameters are used. Second, the normalized coherence Stokes parameters should be regarded as the true electromagnetic counterparts of the normalized scalar-field correlation coefficient.

Characterization of the electromagnetic Gaussian Schell-model beam using first-order interference

We propose a method for the characterization of electromagnetic Gaussian Schell-model (EMGSM) beams. This method utilizes the first-order interference consisting of polarization-state projections along with the two-point (generalized) Stokes parameters. The second-order field correlations employed in this method enable us to determine both the magnitude and the argument of the complex degree of electromagnetic coherence. We experimentally demonstrate this method by characterizing an EMGSM beam, which is synthesized using a laser beam passing through a rotating ground glass diffuser. This beam-characterization method is expected to be potentially useful for probing the partially coherent and partially polarized beams, and have tremendous applications in broad areas of optical communication and beam propagation.

Temporal coherence and polarization modulation of pulse trains by resonance gratings

We analyze the effects of subwavelength-period resonance gratings on temporally partially coherent optical plane-wave pulse trains. The interaction of the grating with pulses is simulated with the Fourier modal method and finite-difference time-domain method whose performances are compared. Both TE and TM linearly polarized Gaussian Schell-model pulse trains are examined, and partial temporal coherence is modeled with the identical elementary-pulse method. The polarization-dependent response of the grating is seen to lead to significant variations in the average intensity, polarization properties, and degree of temporal coherence of the reflected (and transmitted) pulse trains when the coherence time and polarization state of the incident field are altered. As an important example, we demonstrate that a fully polarized incident pulse train can become partially polarized in grating reflection. The results find use in tailoring of random electromagnetic pulse trains.

Scanning wavefront folding interferometers

We present modified scanning-type wavefront folding interferometers (WFIs), which allow spatial coherence measurements of non-uniformly correlated fields, where the degree of coherence is a function of two absolute spatial coordinates instead of coordinate separation only (Schell model). As an alternative to conventional prism-based WFI implementations, we introduce a scheme based on reflections by three mirrors. This setup allows us to avoid obstructions due to prism corners, and it is remarkably robust to polarization effects. Experimental results on measurement of fields that do not obey the Schell model are provided with the three-mirror WFI, and the results are compared to those obtained with a Young's interferometer realized using a digital micromirror device.

Young's interference experiment with electromagnetic narrowband light

We consider electromagnetic spectral spatial coherence of random stationary light beams of arbitrary spectral width. We demonstrate that the normalized spectral coherence (or two-point) Stokes parameters and the electromagnetic spectral degree of coherence can be measured by narrowband filtering the light and detecting the spectral density and the polarization-state fringes around the optical axis of Young's interferometer. It is also shown that the normalized spectral polarization (or one-point) and coherence Stokes parameters are unaffected by filtering, and that the filtered light is strictly cross-spectrally pure. We further prove theoretically and confirm experimentally that the (time-domain) electromagnetic degree of spatial coherence of the filtered light at the pinholes does not increase when the passband of the filter is reduced.

Plasmon coherence determination by nanoscattering

We present a simple and robust protocol to recover the second-order field correlations of polychromatic, statistically stationary surface plasmon polaritons (SPPs) from a spectrum measurement in the far zone of a dipolar nanoscatterer. The recovered correlations carry comprehensive information about the spectral, spatial, and temporal coherence of the SPPs. We also introduce and exemplify for the first time, to the best of our knowledge, the two-point Stokes parameters associated with partially coherent SPP fields.

Electromagnetic theory of optical coherence [Invited]

The coherence theory of random, vector-valued optical fields has been of great research interest in recent years. In this work we formulate the foundations of electromagnetic coherence theory both in the space-time and space-frequency domains, with particular emphasis on various types of optical interferometry. Analyzing statistically stationary, two-component (paraxial) electric fields in the classical and quantum-optical contexts we show fundamental connections between the conventional (polarization) Stokes parameters and the associated two-point (coherence) Stokes parameters. Measurement of the coherence and polarization properties of random vector beams by nanoparticle scattering and two-photon absorption is also addressed.

Spectrum of classes of point emitters of electromagnetic wave fields

The spectrum of classes of point emitters has been introduced as a numerical tool suitable for the design, analysis, and synthesis of non-paraxial optical fields in arbitrary states of spatial coherence. In this paper, the polarization state of planar electromagnetic wave fields is included in the spectrum of classes, thus increasing its modeling capabilities. In this context, optical processing is realized as a filtering on the spectrum of classes of point emitters, performed by the complex degree of spatial coherence and the two-point correlation of polarization, which could be implemented dynamically by using programmable optical devices.

Detection of electromagnetic degree of coherence with nanoscatterers: comparison with Young's interferometer

We show theoretically that the (spectral) electromagnetic degree of spatial coherence of a random, stationary light beam can be measured by using two dipolar nanoscatterers instead of aperture diffraction as in traditional Young's interferometer. The method is based on considering individually the correlation functions associated with the six polarization states that make up the coherence (two-point) Stokes parameters and observing separately the visibilities and the locations of the intensity fringes created by the interfering dipole fields, leading to a complete characterization of the beam's second-order spatial coherence. The novel technique, although introduced in this work for beams, paves the way toward the detection of spatial coherence in nonparaxial optical near-fields for which the use of nanoscatterers is necessary.

Experimental determination of generalized Stokes parameters

A new technique to determine generalized Stokes parameters by making use of the speckle holographic technique and the two-point intensity correlation is proposed and experimentally demonstrated. Assuming Gaussian statistics and spatial stationarity, a speckle hologram is generated prior to the fourth-order correlation, i.e., the two-point intensity correlation. This measurement technique offers a complete retrieval of complex generalized Stokes parameters. The application of the proposed technique is experimentally demonstrated for two different, random source structures.

Complete characterization of partially coherent and partially polarized optical fields

We suggest a method to access the second-order, or two-point, Stokes parameters of a partially coherent and partially polarized Gaussian model optical field from an intensity interferometry experiment. Through a remarkably simple experimental arrangement, it is possible to measure the two-point and one-point Stokes parameters simultaneously, allowing the reconstruction of the coherence matrix and the polarization matrix, thus completely characterizing the optical field both statistically and locally on the observation plane. Developments, automation, and applications are pointed out.

Purity of partial polarization in the frequency and time domains

We discuss the properties of fields whose polarization states remain invariant in diffractive devices; in particular, in Young's interferometer. We investigate the conditions for such fields, called pure partially polarized fields, in both the space-time and space-frequency domains and show that the frequency-domain condition is a more general one. We also discuss the relationships of polarization purity and nonquantum entanglement between polarization and spatial (or temporal) correlation.

Polarimetric imaging beyond the speckle grain scale

We address an experimental Stokes imaging setup allowing one to explore the polarimetric properties of a speckle light field with spatial resolution well beyond the speckle grain scale. We detail how the various experimental difficulties inherent to such measurements can be overcome with a dedicated measurement protocol involving a careful speckle registration step. The setup and protocol are then validated on a metallic reference sample, and used to measure the state of polarization (SOP) of light in each pixel of highly resolved speckle patterns (>2000 pixels per speckle grain) resulting from the scattering of an incident coherent beam on samples exhibiting different polarimetric properties. Evolution of the SOP with spatial averaging and across adjacent speckle grains is eventually addressed.

On experimental characterization of polarization fluctuation dynamics in random optical beams

Polarization correlation functions that characterize the rate of change of the instantaneous polarization state of an arbitrary fluctuating electromagnetic beam were recently introduced. In this work, we describe techniques that enable the measurement of these functions leading to the determination of the so-called polarization time of a random field.

Hanbury Brown-Twiss effect with electromagnetic waves

The classic Hanbury Brown-Twiss experiment is analyzed in the space-frequency domain by taking into account the vectorial nature of the radiation. We show that as in scalar theory, the degree of electromagnetic coherence fully characterizes the fluctuations of the photoelectron currents when a random vector field with Gaussian statistics is incident onto the detectors. Interpretation of this result in terms of the modulations of optical intensity and polarization state in two-beam interference is discussed. We demonstrate that the degree of cross-polarization may generally diverge. We also evaluate the effects of the state of polarization on the correlations of intensity fluctuations in various circumstances.

Generalized Stokes parameters of rectangular hard-edge diffracted random electromagnetic beams

Based on the complex Gaussian expansion method for two dimensional rectangular hard-edged aperture, the analytical formulas for the generalized Stokes parameters of random electromagnetic beams through a paraxial ABCD optical system with rectangular hard-edged aperture are derived. With the help of the analytical formulae, the changes in statistical properties of rectangular hard-edge diffracted random electromagnetic beams, such as in the spectral density, in the spectral degree of coherence, in the polarization properties and so on, can be determined. Numerical examples of such changes are presented and discussed.

Partially correlated thin annular sources: the vectorial case

We develop an electromagnetic analysis for partially correlated thin annular sources. The elements of the correlation matrix are assumed to depend only on the angular distance between two typical points. For any such source, we show how the modal expansion can be found. Correlation changes upon free propagation are discussed. Further, examples and possible synthesis schemes are presented.