Kullback relative entropy and characterization of partially polarized optical waves

Different properties of partially polarized light are discussed using the Kullback relative entropy, which provides a physically meaningful measure of proximity between probability density functions (PDFs). For optical waves with a Gaussian PDF, the standard degree of polarization is a simple function of the Kullback relative entropy between the considered optical light and a totally depolarized light of the same intensity. It is shown that the Kullback relative entropies between different PDFs allow one to define other properties such as a degree of anisotropy and a degree of non-Gaussianity. It is also demonstrated that, in dimension three, the Kullback relative entropy between a partially polarized light and a totally depolarized light can lead to natural definitions of two degrees of polarization needed to characterize the polarization state. These analyses enlighten the physical meaning of partial polarization of light waves in terms of a measure of disorder provided by the Shannon entropy.     

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.     

Numerical experiment of the Shannon entropy in partially coherent imaging by Koehler illumination to show the relationship to degree of coherence

This paper describes the Shannon entropy in a partially coherent imaging system with Koehler illumination. Numerical simulation shows that the entropy has a one-to-one relationship with the normalized mutual intensity given by the van Cittert-Zernike theorem. Analytical evaluation shows that the entropy is consistent with the definition of coherence and incoherence, which is also verified by numerical simulations. Additional numerical experiments confirm that the entropy depends on the source intensity distribution, polarization state of the source, object, and pupil. Therefore, the entropy quantitatively measures the degree of coherence of the partially coherent imaging system.     

Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields

We demonstrate monotonic, irreversible effects on the entropy, the degree of polarization, and different degrees of coherence when random, energy-preserving unitary transformations are applied to vectorial electromagnetic fields.     

Spatial partially coherent imaging

We introduce the partial coherence response function for describing the behaviour of imaging systems under spatial partially coherent illumination in the centre and the diierence coordinates notation. It involves the impulse response of the system and the spatial coherence properties of the illumination. It is shown that this function is the image cross-spectral density for a Young's pair of pinholes attached at the object plane. Furthermore, the partial coherence response function and the partial coherence transfer function constitute a Fourier pair on which the Fourier representation of partially coherent imaging can be based.     

Optical forces on small particles from partially coherent light

We put forward a theory on the optical force exerted upon a dipolar particle by a stationary and ergodic partially coherent light field. We show through a rigorous analysis that the ensemble averaged electromagnetic force is given in terms of a partial gradient of the space-variable diagonal elements of the coherence tensor. Further, by following this result we characterize the conservative and nonconservative components of this force. In addition, we establish the propagation law for the optical force in terms of the coherence function of light at a diffraction plane. This permits us to evaluate the effect of the degree of coherence on the force components by using the archetypical configuration of Young's two-aperture diffraction pattern, so often employed to characterize coherence of waves.     

Diffraction halo function of partially coherent speckle photography

We studied the diffraction halo function of partially coherent speckle photography, using the theory of partial coherence of light and speckle statistics. Numerical calculations were made on the basis of the derived expression. It was shown that the anisotropic coherence of the illumination induced by the source could lead to the inhomogeneity of the halo distribution. An experiment was performed to prove the theoretical predictions.     

Partial spatial coherence and partial polarization in random evanescent fields on lossless interfaces

We consider partial spatial coherence and partial polarization of purely evanescent optical fields generated in total internal reflection at an interface of two dielectric (lossless) media. Making use of the electromagnetic degree of coherence, we show that, in such fields, the coherence length can be notably shorter than the light's vacuum wavelength, especially at a high-index-contrast interface. Physical explanation for this behavior, analogous to the generation of incoherent light in a multimode laser, is provided. We also analyze the degree of polarization by using a recent three-dimensional formulation and show that the field may be partially polarized at a subwavelength distance from the surface even though it is fully polarized farther away. The degree of polarization can assume values unattainable by beamlike fields, indicating that electromagnetic evanescent waves generally are genuine three-dimensional fields. The results can find applications in near-field optics and nanophotonics.     

Polarization changes of partially coherent pulses propagating in optical fibers

We consider polarization changes of partially coherent pulses propagating through birefringent dispersive fibers in the linear regime. We show that the evolution of the degree of polarization across such pulses is determined not only by the coherence properties of the pulse in the source plane, but also by the spatial walk-off introduced by the group-velocity mismatch between the two polarization components. The interplay between these two factors determines the asymptotic value of the degree of polarization of an initially unpolarized statistical pulse. We compare our results with previously discussed coherence-induced polarization changes of partially coherent beams propagating in free space.     

Generation of adjustable partially coherent bottle beams by use of an axicon-lens system

We report, what is to our knowledge, the first experimental realization of partially coherent bottle beams. It is shown that partially coherent bottle beams can be achieved by the focusing of partially coherent light with an axicon-lens system. The influence of the spatial coherence of the incident partially coherent light and other parameters, such as the radius of the limiting aperture of the axicon and the distance between the axicon and the lens, on the size of the bottle beams is investigated. We find that the longer the spatial coherence length, the larger the size of the resultant bottle beams. This dependence of the size of the bottle beams on the spatial coherence of the incident light provides a facile approach for generating adjustable partially coherent bottle beams. This kind of partially coherent bottle beam may have applications in atom optics, such as in atom trapping and atom guiding, etc.     

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.     

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.     

Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere

Based on the Huygens–Fresnel principle and the unified theory of coherence and polarization of partially coherent beams, we investigate the propagation characteristics of a partially coherent radially polarized doughnut (PCRPD) beam in a turbulent atmosphere. It is found that, after propagating through a turbulent atmosphere, the doughnut beam spot is changed into a circular Gaussian beam. Moreover, the degree of coherence, the degree of polarization and the degree of cross-polarization of the beam will change on propagation, and this change is dependent upon the degree of coherence of the source and atmospheric turbulence.     

Spatial coherence effects in light scattering from metallic nanocylinders

We study the scattering of a partially coherent electromagnetic beam from metallic nanocylinders and analyze the effects of plasmon resonances on the coherence and polarization properties of the optical near field. We employ the coherent-mode representation for the incident field and solve the scattering problem independently for each mode by using a boundary-integral method. Our results show that the plasmon resonances may significantly affect the coherence and polarization characteristics of the near field and that partial coherence influences the energy flow in nanocylinder arrays.     

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.     

Exploration of the retinal nerve fiber layer thickness by measurement of the linear dichroism

An electro-optic device mounted on a slit lamp to assess the degree of polarization of a light beam that has double passed through the retina about the optic-nerve head in the living human eye is described. The asymmetric structure of the retinal nerve's fiber layer possesses a linear-form dichroism and will partially polarize an unpolarized light beam that is scattered at the fundus of the eye and has double passed the ocular media (cornea, lens, retina). This partial polarization is a function of the retinal nerve's fiber layer thickness, and its measurement may be used for exploring glaucoma and other retinal neuropathies. Experimental conditions allow us to neglect corneal dichroism. The first clinical measurements show a different degree of polarization between normal and glaucomatous eyes and a good correlation with the results obtained by optical coherence tomography.     

Influence of spatial coherence on scattering by a particle

We analyze the effect of partial spatial coherence on the scattering of light by an arbitrary particle. We extend the definition of the extinction cross section to spatially partially coherent fields. We then discuss the effect of the partial coherence on the extinction scattering cross section by introducing the Wigner transform. It is shown that for rotationally invariant scatterers, the extinction cross section does not depend on the coherence of the incident field. The effect of partial coherence on the angular behavior of the scattered intensity is also discussed in the framework of the Wigner transform. The implications for practical applications are considered.     

Reflection and transmission properties of holographic mirrors and holographic Fabry-Perot filters. II. Holographic mirrors with partially coherent light

The reflection and transmission properties of holographic mirrors (HM's) under partially coherent illumination are investigated with emphasis on the properties of reflected light. The effects of a HM on the spectrum and on the coherence properties of partially coherent incident light are studied. We show that within angular and frequency-selectivity ranges of HM's (where their reflectivities are nearly uniform), the changes in the spectrum and in the coherence properties are negligible. Yet changes are expected when the spectrum of the incident light falls beyond frequency-selectivity ranges of the HM's or when the spectrum of the reflected light is observed beyond the angular-selectivity ranges of the mirrors. We illustrate the results by considering in detail the performance of HM's when the incident light is produced by planar secondary Gaussian Schell-model sources. Some computed curves are presented.     

Propagation-induced polarization changes in partially coherent optical beams

Propagation of a partially coherent optical beam inside a linear, nondispersive, dielectric medium is studied, taking into account the vector nature of the electromagnetic field. Propagation-induced polarization changes are studied by using the Gaussian-Schell model for the cross-spectral-density tensor. The degree of polarization changes with propagation and also becomes nonuniform across the beam cross section. The extent of these changes depends on the coherence radius associated with the cross-correlation function. For optical beams with symmetric spectra, the bandwidth of the source spectra is found to play a relatively minor role.