Beam Coherence-polarization Matrix Research Articles

Unified matrix representation for spin and orbital angular momentum in partially coherent beams

A matrix representation is introduced for stationary beam-like light fields that allows for simultaneous treatment of the orbital angular momentum (OAM) and the second-order state of spatial coherence. This Coherence-OAM matrix is a counterpart of the 2 x 2 beam Coherence-polarization matrix in a Cartesian polarization basis or Coherence-spin angular momentum (SAM) matrix in a circular polarization basis. The general properties of the Coherence-OAM matrix are discussed and then the Coherence-SAM-OAM matrix is introduced as a unifying entity for treating beams with any coherence, spin and orbital angular momentum states.

The effects of polarization changes of stochastic electromagnetic beams on heterodyne detection in turbulence

The performance of heterodyne systems is discussed for stochastic electromagnetic beams in turbulent atmospheres by introducing a turbulence spectrum of refractive-index fluctuations. Several analytic formulae for the heterodyne detection system are presented by employing the beam coherence polarization matrix of an electromagnetic Gaussian Schell-model beam. Based on the Tatarskii spectrum model, some numerical results are given for the variation of heterodyne efficiency with the misalignment angle, detector diameter, turbulence conditions and parameters of the overlapping beams. According to the numerical results, we find that the change in heterodyne efficiency is dependent on the initial polarization of the beams, and that a turbulent atmosphere degrades the heterodyne efficiency significantly for a lager detector diameter. The change in heterodyne efficiency in a turbulent atmosphere is basically the same as that in free space for a relatively short distance zs. However, the change in heterodyne efficiency is different for a sufficiently long distance. For the deterministic received signal and the detector, the performance of the heterodyne detection can be adjusted by controlling the local oscillator signal parameters.

Propagation of focused vector laser beams in turbulent atmosphere

By adopting a combination of beam coherence-polarization matrix and extended Huygens–Fresnel integral formula, analytical formulae are derived for the average intensity, the degree of polarization and the RMS (Root-Mean-Square) beam width of focused vector laser beams propagating in a turbulent atmosphere, and the propagation of focused vector laser beams is studied in detail. The spreading properties of vector laser beams and fundamental Gaussian beams are studied comparatively. It reveals that the focused vector laser beam evolves into Gaussian profile in shorter distance compared with collimated vector laser beams, and that destroying of polarization structure of focused ones is more severe. The spreading properties of the focused vector laser beams become the same as that of fundamental Gaussian beams faster. The propagation properties of focused vector laser beams are closely related to the parameters of the beam and the structure constant of the atmospheric turbulence. The results can provide useful reference in designing fiber laser system.

Coherence–polarization properties of fields radiated from transversely periodic electromagnetic sources

Planar electromagnetic sources characterized by a periodic variation of their beam coherence–polarization matrix are investigated, as far as the polarization features of the radiated fields are concerned, within the framework of the paraxial approximation. A propagation scheme based on plane-wave decomposition leads to a longitudinal periodicity of the polarization properties of the field, thus extending the Talbot effect to the case of partially coherent electromagnetic sources. The polarization features of beams radiated from sources of this type are illustrated by means of simple examples. In particular, it is shown that completely unpolarized sources with uniform intensity profiles can be easily realized, for which the propagated field becomes perfectly polarized across some transverse planes, and vice versa.

Propagation of radially polarized beams diffracted at a circular aperture in turbulent atmosphere

Based on the extended Huygens–Fresnel integral and the beam coherence–polarization matrix, the analytical formulae for the average intensity and the degree of polarization of the radially polarized beams diffracted at a circular aperture in turbulent atmosphere are derived, which provide a convenient approach to study the propagation and polarization properties of the apertured radially polarized beams in turbulent atmosphere. The unapertured and free-space cases can be viewed as the special cases of our general result. The analyses indicate that the average intensity and the degree of polarization are closely related to the propagation distance, the structure constant of the atmospheric turbulence, and the truncation parameter. The existence of the circular aperture weakens the influence of the atmospheric turbulence on the evolution properties of the radially polarized beams.

Propagating a partially coherent radially polarized beam through aligned and misaligned ABCD optical systems

In the present work the propagation of a partially coherent radially polarized beam through aligned and misaligned ABCD optical systems is discussed. The elements of the beam coherence-polarization matrix for a partially coherent radially polarized beam propagating through these systems are derived for the first time. Using the derived analytical results the intensity distribution of the focused partially coherent radially polarized beam after a low-NA lens is investigated. It is shown that the coherence of the light source affects the beam intensity profile at the focal plane significantly, while for the given coherence of the light source the focal length of the used lens does not affect the beam intensity profile at the focal plane. This is consistent with the case of a partially coherent Hermite-Gaussian beam, as expected.

Polarization properties of two Gaussian Schell-model beams in the turbulent atmosphere

General expressions are derived for the spectral degree of polarization of a beam generated by the superposition of two Gaussian Schell-model (GSM) beams propagating in a turbulent atmosphere by adopting the beam coherence-polarization matrix and Young's interference theory. We analyzed the distance of two beams, atmospheric turbulence intensity, and the location of the imaging plane affect the degree of polarization by numerical calculation. In particular, when the correlation coefficients of x and y components of the electric field are the same (i.e. δ xx = δ yy ), the degree of polarization will obtain the same value.

Polarization properties of nonparaxial partially polarized Gaussian Schell-model beams

The polarization properties of nonparaxial partially polarized Gaussian Schell-model (PGSM) beams are studied. Based on the beam coherence-polarization matrix, the propagation formula of the degree of polarization for the nonparaxial PGSM beams is derived. The paraxial approximation is dealt with as special cases of our general results. Some conditions that limit the choice of the parameters are established. The sufficiency condition of polarization invariance in propagation for the nonparaxial PGSM beams is derived. By using the derived formulae, the propagation properties of the degree of polarization for the nonparaxial PGSM beam in free space are illustrated and analyzed with numerical examples. Some detailed comparisons of the obtained results with the paraxial results are made.

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.

Determination of the Beam Coherence-Polarization Matrix of a Random Electromagnetic Beam

The beam coherence-polarization (BCP) matrix contains its elements in the form of mutual coherence functions. These are complex quantities governing the space-time correlations between the x and the y components of electric field at pair of points in the cross-section of the random electromagnetic beam. In this paper, we present mathematical expressions to determine the amplitude and the phase of the four elements of the BCP matrix which are a function of two spatial points in the optical field. We also determine experimentally the BCP matrix for an expanded laser beam introducing polarizers and rotators in widely separated beam paths of a modified version of the Young's interferometer. The real and imaginary parts of the BCP matrix elements might show the change in the polarization properties of a random electromagnetic beam during its propagation. These results might be useful for astronomical measurements also.

Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere

Analytical formulas are derived for the average irradiance and the degree of polarization of a radially or azimuthally polarized doughnut beam (PDB) propagating in a turbulent atmosphere by adopting a beam coherence-polarization matrix. It is found that the radial or azimuthal polarization structure of a radially or azimuthally PDB will be destroyed (i.e., a radially or azimuthally PDB is depolarized and becomes a partially polarized beam) and the doughnut beam spot becomes a circularly Gaussian beam spot during propagation in a turbulent atmosphere. The propagation properties are closely related to the parameters of the beam and the structure constant of the atmospheric turbulence.

Polarization properties of partially coherent flat-topped light beams diffracted through a regular polygonal aperture

Based on the beam coherence-polarization matrix and the propagation law, the polarization properties of partially coherent flat-topped light beams diffracted through a regular polygonal aperture are studied systematically. The expression for the degree of polarization of partially coherent flat-topped light beams is derived. And according to that, the expression for the degree of polarization of Gaussian Schell-model beams and the propagation of partially coherent flat-topped light beams in the free space are obtained. It is shown that the polarization properties of partially coherent flat-topped light beams diffracted through a rectangular aperture depends on the truncation parameter, spatial coherence parameters, diffraction angle, propagation distance and the order of flat-topped light beams.

Degree of polarization for partially coherent general beams in turbulent atmosphere

The degree of polarization is found for optical excitations of cosh-Gaussian, cos-Gaussian and annular-Gaussian beams in a turbulent atmosphere. The related formulation is based on the beam coherence polarization matrix. The self and mutual coherence functions appearing in the beam coherence polarization matrix are evaluated, when the above mentioned excitations exhibit partial source coherence for self and cross fields. Plots showing the variation of the degree of polarization are provided versus the propagation length when the source size, displacement parameter, structure constant and the degree of source coherence for self and cross fields change.

Optical properties of polarization-dependent geometric phase elements with partially polarized light

The behavior of geometric phase elements illuminated with partially polarized monochromatic beams is investigated both theoretically and experimentally. The element discussed in this paper is composed of wave plates withp-retardation and a space-variant orientation angle. We found that a beam emerging from such an element comprises two polarization orders; right-and left-handed circularly polarized states with conjugate geometric phase modification. This phase equals twice the orientation angle of the space-variant wave plate comprising the element. Apart from the two polarization orders, the emerging beam coherence polarization matrix includes a ‘‘vectorial interference matrix’’ which contains information concerning the correlation between the two orthogonal, circularly polarized portions of the incident beam. In this paper we measure this correlation by a simple interference experiment. In addition, we found that the equivalent mutual intensity of the emerging beam is modulated according to the geometric phase induced by the element. Other interesting phenomena concerning propagation will be discussed theoretically and demonstrated experimentally. The experiment made use of a spherical geometric phase element that was realized by use of a space-variant subwavelength grating illuminated with CO2 laser radiation of 10.6 lm wavelength. � 2006 Elsevier B.V. All rights reserved.

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.

Propagation of partially polarized Gaussian Schell-model beams through aligned and misaligned optical systems

By use of a tensor method, the transform formulae for the beam coherence-polarization matrix of the partially polarized Gaussian Schell-model (GSM) beams through aligned and misaligned optical systems are derived. As an example, the propagation properties of the partially polarized GSM beam passing through a misaligned thin lens are illustrated numerically and discussed in detail. The derived formulae provide a convenient way to study the propagation properties of the partially polarized GSM beams through aligned and misaligned optical systems.

Parametric characterization of the spatial structure of partially coherent and partially polarized beams

On the basis of the second-order intensity moments formalism, the relationship between the spatial structure and the overall polarization characteristics of partially polarized Gaussian Schell-model beams of a certain kind has been investigated. More specifically, attention has been focused on a type of source that cannot be distinguished from ordinary Gaussian Schell-model fields when polarization measurements are disregarded. For this class of beams several general properties have been obtained that enable us to link the beam coherence polarization matrix and the beam quality parameter with certain polarization degrees recently introduced in the literature.

Far field of beams generated by quasi-homogeneous sources passing through polarization gratings

We analyze the polarization features of the beam, generated by a class of partially polarized quasi-homogeneous sources, which propagates through a polarization grating. Analytical expressions in the far zone for the beam coherence polarization matrix, the degree of polarization and the Stokes parameters are given. In particular, it is shown that, under some hypotheses, it is possible to completely and uniformly depolarize the beam in the far field. The influence of source parameters, such as the state of polarization, intensity and degree of coherence, on the degree of polarization and the Stokes parameters is also investigated.

The irradiance of partially polarized beams in a scalar treatment

It is shown that the irradiance distribution produced throughout the space by a partially polarized beam can be treated in scalar terms. To this aim it is necessary to introduce an equivalent mutual intensity across a beam section. This can be made by exploiting the properties of the beam coherence-polarization matrix.

Beam coherence-polarization matrix

We present an approach for describing the properties of a quasi-monochromatic, beam-like field that is both partially polarized and partially coherent from the spatial standpoint. It is based on the use of a single matrix, called the beam coherence-polarization matrix, whose elements have the form of mutual intensities. This approach, which can be viewed as an approximate form of Wolf's general tensorial theory of coherence, appears to be very simple, yet it is able to cover significant aspects of the beam behaviour that would not be accounted for by a scalar theory or by a local polarization matrix approach. A peculiar interference law applying to mutual intensities is derived. We show through simple examples how this approach leads to distinguish fields that would appear identical in a scalar treatment or in a local polarization matrix description. Hints for extensions are given.