Use of the van Cittert-Zernike theorem for partially polarized sources

Autocorrelation functions and power spectral densities of the Stokes parameters in a polarization speckle pattern

The concept of ensemble-average polarization and coherence has been applied to studying fluctuating Stokes parameters in a polarization speckle observed when coherent light is passed through a birefringent polarization scrambler. With the aid of the ensemble-average van Cittert-Zernike theorem for the propagation of ensemble-average polar-coherence, we invesitgate the autocorrelation functions and power spectra of the Stokes parameters to expose the dependence of the polarization-related scale-size distributions on the optical geometries in which the polarization speckle arises. A generalized concept of the Stokes ensemble-average coherence areas is introduced to deal with the polarization-related average areas associated with polarization speckle.

Spatial Coherence in Medical Ultrasound: A Review

Traditional pulse-echo ultrasound imaging heavily relies on the discernment of signals based on their relative magnitudes but is limited in its ability to mitigate sources of image degradation, the most prevalent of which is acoustic clutter. Advances in computing power and data storage have made it possible for echo data to be alternatively analyzed through the lens of spatial coherence, a measure of the similarity of these signals received across an array. Spatial coherence is not currently explicitly calculated on diagnostic ultrasound scanners but a large number of studies indicate that it can be employed to describe image quality, to adaptively select system parameters and to improve imaging and target detection. With the additional insights provided by spatial coherence, it is poised to play a significant role in the future of medical ultrasound. This review details the theory of spatial coherence in pulse-echo ultrasound and key advances made over the last few decades since its introduction in the 1980s.

Modulation of coherence-polarization property of speckles using a birefringent scatterer

The change of the spatial coherence-polarization (CP) property of a speckle pattern due to the modulation of birefringence in a scattering medium is investigated experimentally. The birefringence is introduced to the scattering medium by attaching layers of overhead projector (OHP) sheets to it. It is shown that the spatial polarization distribution can be tuned from a uniformly polarized case to a randomly polarized case by increasing the number of OHP sheets. The change of the spatial CP property with the number of OHP sheets is investigated from the study of the spatial degree of coherence and the degree of polarization, and these variations are further confirmed from the probability density function of the intensity of the speckle pattern and from the measurement of the generalized Stokes parameters. The effect of the change of the number of OHP sheets on the visibility of the intensity correlation function and the quality of the object retrieved through the scattering medium are also investigated.

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.

Bubbly flow velocity measurement in multiple scattering regime

We propose a technique to measure the velocity of a bubble cloud based on the coda correlation. The method is founded on successive recordings of multiple scattered waves from a bubble cloud. Our model predicts the dependence between the correlation coefficient of these coda waves and the velocity of the bubble cloud under diffusion approximation. The Acoustic experiments are validated by simultaneous optical measurements in a water tank, with a good agreement between the acoustical and the optical methods (relative difference smaller than 7%). This technique can be transposed to any particle flow velocity problems involving multiple scattering effects in acoustics.

Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam

We outline the propagation of an electromagnetic Gaussian Schell-model (EGSM) vortex beam through a paraxial ABCD optical system and analyze the vortex phase-induced changes of the statistical properties, such as average intensity, state of polarization, and degree of polarization (DOP), of a focused EGSM beam. It is found that one can shape the beam profile of an EGSM vortex beam in the focal plane through varying its initial topological charge, DOP, and coherence widths. Furthermore, we first report experimental generation of an EGSM vortex beam and measure its focusing properties in experiments. Our experimental results are consistent with the numerical results and may be useful in material thermal processing and particle trapping.

Van Cittert-Zernike theorem and symmetry properties of the normalized cross-spectral density matrix

The van Cittert-Zernike theorem is formulated for the normalized cross-spectral density matrix of fields emitted by spatially incoherent sources. This formulation shows that sources with homogeneous polarization characteristics at second-order lead in the far field to normalized cross-spectral density matrices that possess high-symmetry properties. These symmetry properties are lost with polarization inhomogeneities of the source. These results are illustrated with ring-shaped sources.

Generation of spatial coherence comb using Dammann grating

A new technique to generate a spatially varying coherence field, such as a coherence comb using a Dammann grating, is proposed and experimentally demonstrated. The principle of the technique lies with the vectorial van Cittert-Zernike theorem, which connects vectorial source structure with the coherence-polarization of the light. The Dammann grating is encoded into one of the polarization components of the light to shape the vectorial source structure and, consequently, the coherence-polarization of the light. Experimental results on the generation of a spatial coherence comb by the Dammann grating are presented for different orders.

Second-order moments of an electromagnetic Gaussian Schell-model beam in a uniaxial crystal

We derive the analytical expressions for the second-order moments of an electromagnetic Gaussian Schell-model (EGSM) beam propagating in a uniaxial crystal. With the help of the derived formulas, we study the evolution properties of the propagation factor, the effective radius of curvature and the Rayleigh range of an EGSM beam in a uniaxial crystal. It is found that the evolution properties of an EGSM beam in a uniaxial crystal are much different from its evolution properties in free space and are closely determined by the initial beam parameters and the parameters of the uniaxial crystal. The uniaxial crystal provides one way for modulating the properties of an EGSM beam.

Vectorial van Cittert-Zernike theorem based on spatial averaging: experimental demonstrations

The van Cittert-Zernike theorem is extended to the vectorial regime based on spatial averaging over the observation plane, and experimental demonstrations are presented. The theorem connects complex vectorial source structure to the degree of coherence and polarization of the spatially fluctuating vectorial field in the far field. Experimentation is carried out by making use of the space averages as a replacement of ensemble averages for the Gaussian stochastic field. For quantitative comparison with the theorem, analytical and experimental results are presented for a rectangular aperture with different vectorial source structures.

Van Cittert-Zernike theorem with Stokes parameters

We formulate the van Cittert-Zernike theorem for spatially incoherent partially polarized sources in terms of the traditional polarization (one-point) Stokes parameters and the coherence (two-point) Stokes parameters. This leads to a physically insightful connection between the source polarization and the field coherence. We show that the far-zone coherence is given by a Fourier relation of the source-plane polarization and apply the results to investigate the far-zone properties of a field emitted by an incoherent source limited by an annular aperture.

Coherence-polarization mixing in resonance gratings

We show, using rigorous diffraction theory, that resonance gratings can be used to transfer partial spatial correlation to partial polarization even if the incident light beam is fully polarized. The phenomenon is based on the fact that either of the two orthogonal polarization components can be coupled into the leaky waveguide mode, leading to a strong phase delay, while the other one is reflected without being coupled into the grating. Numerical demonstrations are based on a Gaussian Schell-model beam and a grating analysis performed by rigorous Fourier modal method.

Vectorial coherence holography

Extension of coherence holography to vectorial regime is investigated. A technique for controlling and synthesizing optical fields with desired elements of coherence-polarization matrix is proposed and experimentally demonstrated. The technique uses two separate coherence holograms, each of which is assigned to one of the orthogonal polarization components of the vectorial fields.

Propagation property of a nonuniformly polarized beam array in turbulent atmosphere

A nonuniformly polarized beam array (NUBPA) is modeled by coherent superposition of a pair of orthogonally polarized spatial modes. The propagation of a NUPBA in turbulent atmosphere is investigated based on the extended Huygens-Fresnel method. An analytical expression for the intensity profile of a NUBPA in turbulent atmosphere is presented. The influence of polarization degree, intensity of turbulence, array number, and distance between adjacent elements on the intensity profile in the receiving plane is evaluated numerically and analyzed in detail.

Jones matrix treatment for optical Fourier processors with structured polarization

We present a Jones matrix method useful to analyze coherent optical Fourier processors employing structured polarization. The proposed method is a generalization of the standard classical optical Fourier transform processor, but considering vectorial spatial functions with two complex components corresponding to two orthogonal linear polarizations. As a result we derive a Jones matrix that describes the polarization output in terms of two vectorial functions defining respectively the structured polarization input and the generalized polarization impulse response. We apply the method to show and analyze an experiment in which a regular scalar diffraction grating is converted into equivalent polarization diffraction gratings by means of an appropriate polarization filtering. The technique is further demonstrated to generate arbitrary structured polarizations. Excellent experimental results are presented.

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.

Experimental study of the relation between the degrees of coherence in space-time and space-frequency domain

We present an experimental study showing the effect of the change in the bandwidth of light on the magnitude of both the complex degree of coherence and the spectral degree of coherence at a pair of points in the cross-section of a beam. A variable bandwidth source with a Young's interferometer is utilized to produce the interference fringes. We also report for the first time that if the field is quasi-monochromatic or sufficiently narrowband, the elements of both the beam coherence polarization matrix and the cross-spectral density matrix, normalized to intensities (spectral densities) at the two points possess identical values.

Some consequences of the van Cittert-Zernike theorem for partially polarized stochastic electromagnetic fields

Coherence and polarization properties of the field generated by a partially polarized incoherent source are examined in terms of the 2x2 cross-spectral density matrix, with a view to deriving an electromagnetic counterpart of the van Cittert-Zernike theorem. Emphasis is placed on the polarization properties of the field. As some consequences of the extended theorem, we found that the spectral degree of polarization of the resultant field is generally different from that of the source and that both are the same in a special case. A different somewhat unconventional view of the extended theorem is also discussed in brief.

Synthesis of electromagnetic Schell-model sources

A procedure for the synthesis of the most general electromagnetic Schell-model light source is proposed. It makes use of the generalized van Cittert-Zernike theorem to produce the electromagnetic source starting from a primary spatially incoherent source, characterized by a suitable position-dependent polarization matrix. By resorting to the spectral decomposition of the polarization matrix, it is shown how such an incoherent source can be synthesized by using a Mach-Zehnder interferometer, with suitable amplitude transmittances placed in its arms, fed by two mutually uncorrelated laser beams. Examples are given for the case of electromagnetic Gaussian Schell-model sources.

Partially correlated sources with complete polarization

We show that the correlation matrix of any electromagnetic planar source endowed with complete polarization must be of a universal form, no matter how the field components are correlated across the source plane. Such form entails that both the realizability conditions and the search of vector modes of the source reduce to scalar problems. We shall also briefly dwell on how propagation affects the polarization properties of the field and, in particular, we discuss cases in which propagated light remains completely polarized.

Modal expansion for J0-correlated electromagnetic sources

We introduce a class of sources that turn out to be J0 correlated in a scalar description but exhibit varied correlation properties when examined in electromagnetic terms. We prove that a modal expansion can be built explicitly for any of these sources and give an example to illustrate the richness of their behavior.

Realizability condition for electromagnetic Schell-model sources

The set of functions that appear in the correlation matrix of an electromagnetic source must satisfy the constraint of nonnegative definiteness. Here we derive a necessary and sufficient condition for nonnegativeness for the class of electromagnetic Schell-model sources. This result also suggests a possible synthesis procedure for this type of source. As an illustration, two specific examples of electromagnetic Schell-model sources are discussed.

Electromagnetic spatial coherence wavelets

The recently introduced concept of spatial coherence wavelets is generalized to describe the propagation of electromagnetic fields in the free space. For this aim, the spatial coherence wavelet tensor is introduced as an elementary amount, in terms of which the formerly known quantities for this domain can be expressed. It allows for the analysis of the relationship between the spatial coherence properties and the polarization state of the electromagnetic wave. This approach is completely consistent with the recently introduced unified theory of coherence and polarization for random electromagnetic beams, but it provides further insight about the causal relationship between the polarization states at different planes along the propagation path.

Space-variant polarization manipulation for far-field polarimetry by use of subwavelength dielectric gratings

A method for polarimetric measurement that uses a discrete space-variant subwavelength dielectric grating is presented. One retrieves the polarization state by measuring the far-field intensity of a beam emerging from the grating followed by a polarizer. The analysis for a partially polarized, quasi-monochromatic beam is performed by use of the beam coherence polarization matrix along with an extended van Cittert-Zernike theorem. We experimentally demonstrate polarization measurements of both fully and partially polarized light.

Polarized electromagnetic radiation from spatially correlated sources

We consider the effect of spatial correlations in sources of polarized electromagnetic radiation. We evaluate the coherency matrix for a spatially extended three-dimensional source in order to obtain its dependence on the angular position of the observer in the far zone. We find a general condition on the source correlation matrix under which the polarization in the far zone shows no angular dependence. In general, however, we find that the polarization shows a rather striking dependence on the direction of observation. We illustrate this effect by considering some examples of monochromatic spatially correlated sources. The sources, assumed to be monochromatic, are constructed out of dipoles aligned along a line such that their orientation is correlated with their position. In one representative example, the dipole orientations are prescribed by a generalized form of the standard von Mises distribution for angular variables such that the azimuthal angle of dipoles is correlated with their position. In another example the tip of the dipole vector traces a helix around the symmetry axis of the source. Possible experimental observation of this effect, by considering an analog of the standard two-slit interference experiment, is also discussed. The effect may find useful applications in certain astrophysical sources.

Spectral dependence of polarized radiation due to spatial correlations

We study the polarization of light emitted by spatially correlated sources. We show that, in general, polarization acquires a nontrivial spectral dependence due to spatial correlations. The spectral dependence is found to be absent only for a special class of sources where the correlation length scales as the wavelength of light. We further study cross correlations between two spatially distinct points that are generated due to propagation. It is found that such cross correlation leads to a sufficiently strong spectral dependence of polarization which can be measured experimentally.

Gaussian Schell-model beams propagating through polarization gratings

The effects of polarization gratings on partially coherent beams are investigated by studying a Gaussian Schell-model beam impinging on a linear polarizer whose transmission axis varies periodically along one transverse direction. Analytical expressions for the beam polarization-coherence matrix after the grating are obtained. In particular, the evolution of the degree of polarization upon propagation is analyzed. Different behaviors of the output beam, depending on the beam parameters and on the period of the grating, are exhibited. In particular, it is shown that, by suitably choosing the latter quantities, it is possible to obtain not only any desirable value of the degree of polarization of the output beam but also particular distributions of such parameters across the transverse sections of the beam.