Cross-spectral Density Matrix Research Articles

Statistical properties of a radially polarized twisted Gaussian Schell-model beam in a uniaxial crystal

We derive the analytical formulas for the elements of the cross-spectral density matrix of a radially polarized partially coherent beam with a twist phase named radially polarized twisted Gaussian Schell-model (RPTGSM) beam propagating in a uniaxial crystal, and explore the statistical properties, such as the intensity distribution, the degree of polarization (DOP) and the state of polarization (SOP) of such beam in a uniaxial crystal with the help of the derived formulas. It is found that the statistical properties of a RPTGSM beam are closely related with the twist phase and the anisotropy of the uniaxial crystal, e.g. the twist phase leads to the rotation of the intensity, DOP and SOP distributions, and the anisotropy of the uniaxial crystal leads to the asymmetry distributions of these statistical properties. Our results will be useful for designing light field with prescribed intensity, DOP and SOP distributions, and may be useful in optical manipulations and free-space optical communications.

Generalized Stokes parameters and polarization behavior of flat-topped array laser beam propagating through oceanic turbulence

In present paper, the analytical expression for the cross-spectral density of the phase-locked Partially Coherent Flat-Topped (PCFT) array beams propagating through oceanic turbulence based on the extended Huygens–Fresnel integral is derived. Then, the polarization behavior and generalized Stokes parameters of the phase-locked PCFT array beam propagating through oceanic turbulence using the cross-spectral density matrix elements have been achieved. In this paper, the case of the clearwater oceanic turbulence has been considered, i.e. it has been assumed that the light wave is not affected by suspended organic and inorganic impurities. Due to the above-mentioned reason and the turbulence importance, the impacts of oceanic turbulence have been scrutinized, while the effects of absorption and scattering have been ignored. The changes in the Stokes parameters, the fluctuation of Stokes parameters, and consequently the state of polarization (including the spectral degree of polarization, the orientation angle and the degree of ellipticity) of the phase-locked PCFT array beams for different turbulence conditions upon propagation have been investigated. The results may find possible applications in underwater communication.

Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere

Propagation properties of a scalar twisted Gaussian Schell-model (TGSM) beam in turbulent atmosphere has been studied in detail. In this paper, we explore the propagation properties of a radially polarized TGSM beam in turbulent atmosphere. Analytical formulae for the elements of the cross-spectral density matrix of a radially polarized TGSM beam propagating in turbulent atmosphere are derived based on the extended Huygens–Fresnel integral. We find that the atmospheric turbulence induces degeneration of the intensity distribution and depolarization of a radially polarized TGSM beam on propagation, while the twist phase plays a role in anti-degradation and anti-depolarization. Furthermore, we also find that the twist phase induces not only the rotation of the beam spot but also changes of the state of polarization on propagation, which are caused by the fact that the twist phase imposes orbital angular momentum on the beam. Thus, the twist phase can be used to control the propagation properties of radially polarized partially coherent beam. Our results will be useful in some applications, such as optical trapping and free-space optical communications.

Propagation of electromagnetic multi-Gaussian Schell-model beams with astigmatic aberration in turbulent ocean.

An analytical expression for the elements of the cross-spectral density matrix of electromagnetic multi-Gaussian Schell-model beams with astigmatic aberration propagating through the turbulent ocean is derived. We investigate the statistical characteristics of the beams on propagation in great detail. It is found that, due to astigmatism, the circular symmetry of such beams suffers a certain degree of damage in the near field. In addition, astigmatism also destroys the attractive far-field flat profiles in free space. Meanwhile, we also show that astigmatism and oceanic turbulence both have certain effects on the spectral degree of polarization.

Analyzing the propagation behavior of coherence and polarization degrees of a phase-locked partially coherent radial flat-topped array laser beam in underwater turbulence

In this research, based on an analytical expression for cross-spectral density (CSD) matrix elements, coherence and polarization properties of phase-locked partially coherent flat-topped (PCFT) radial array laser beams propagating through weak oceanic turbulence are analyzed. Spectral degrees of coherence and polarization are analytically calculated using CSD matrix elements. Also, the effective width of spatial degree of coherence (EWSDC) is calculated numerically. The simulation is done by considering the effects of source parameters (such as radius of the array setup's circle, effective width of the spectral degree of coherence, and wavelength) and turbulent ocean factors (such as the rate of dissipation of the turbulent kinetic energy per unit mass of fluid and relative strength of temperature and salinity fluctuations, Kolmogorov micro-scale, and rate of dissipation of the mean squared temperature) in detail. Results indicate that any change in the amount of turbulence factors that increase the turbulence power reduces the EWSDC significantly and causes the reduction in the degree of polarization, and occurs at shorter propagation distances but with smaller magnitudes. In addition, being valid for all conditions, the degradation rate of the EWSDC of Gaussian array beams are more in comparison with the PCFT ones. The simulation and calculation results are shown by graphs.

Vortex phase-induced changes of the statistical properties of a partially coherent radially polarized beam.

Partially coherent radially polarized (PCRP) beam was introduced and generated in recent years. In this paper, we investigate the statistical properties of a PCRP beam embedded with a vortex phase (i.e., PCRP vortex beam). We derive the analytical formula for the cross-spectral density matrix of a PCRP vortex beam propagating through a paraxial ABCD optical system and analyze the statistical properties of a PCRP vortex beam focused by a thin lens. It is found that the statistical properties of a PCRP vortex beam on propagation are much different from those of a PCRP beam. The vortex phase induces not only the rotation of the beam spot, but also the changes of the beam shape, the degree of polarization and the state of polarization. We also find that the vortex phase plays a role of resisting the coherence-induced degradation of the intensity distribution and the coherence-induced depolarization. Furthermore, we report experimental generation of a PCRP vortex beam for the first time. Our results will be useful for trapping and rotating particles, free-space optical communications and detection of phase object.

Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence.

Recently, we introduced a new class of radially polarized cosine-Gaussian correlated Schell-model (CGCSM) beams of rectangular symmetry based on the partially coherent electromagnetic theory [Opt. Express23, 33099 (2015)]. In this paper, we extend the work to study the second-order statistics such as the average intensity, the spectral degree of coherence, the spectral degree of polarization and the state of polarization in anisotropic turbulence based on an extended von Karman power spectrum with a non-Kolmogorov power law α and an effective anisotropic parameter. Analytical formulas for the cross-spectral density matrix elements of a radially polarized CGCSM beam in anisotropic turbulence are derived. It is found that the second-order statistics are greatly affected by the source correlation function, and the change in the turbulent statistics induces relatively small effect. The significant effect of anisotropic turbulence on the beam parameters mainly appears nearα=3.1, and decreases with the increase of the anisotropic parameter. Furthermore, the polarization state exhibits self-splitting property and each beamlet evolves into a radially polarized structure in the far field. Our work enriches the classical coherence theory and may be important for free-space optical communications.

Vector-valued Lambertian fields and their sources

The electromagnetic field within an aperture in the wall of a blackbody cavity is a known example of a Lambertian source producing a far field which is unpolarized in all directions. In this work we show that in the electromagnetic context other Lambertian sources exist whose far fields, while obeying the cosine law for the radiant intensity, differ by their polarization states and degrees. For example, the far field may be azimuthally, radially, or circularly polarized, or the polarization state may vary depending on the direction. For specific Lambertian fields generated by quasihomogeneous sources it is possible to calculate explicitly the $3\ifmmode\times\else\texttimes\fi{}3$ electric cross-spectral density matrix of the nonevanescent part of the source. This enables one to assess the source's spatial coherence and partial polarization properties. In all cases, the coherence length turns out to be roughly half a wavelength, whereas the polarization characteristics of the sources may differ significantly. Our results could find uses, for instance, in radiometry and photometry, lighting applications, and remote sensing.

Stokes parameters of phase-locked partially coherent flat-topped array laser beams propagating through turbulent atmosphere

In this study, generalized Stokes parameters of a phase-locked partially coherent flat-topped array beam based on the extended Huygens–Fresnel principle and the unified theory of coherence and polarization have been reported. Analytical formulas for 2 × 2 cross-spectral density matrix elements, and consequently Stokes parameters of a phase-locked partially coherent flat-topped array beam propagating through the turbulent atmosphere have been formulated. Effects of many physical attributes such as wavelength, turbulence strength, flatness order and other source parameters on the Stokes parameters, and therefore spectral degree of polarization upon propagation have been studied thoroughly. The behaviour of the spectral degree of coherence of a delineated beam for different source conditions has been investigated. It can be shown that four generalized Stokes parameters increase by raising the flatness order at the same propagation distance. Increasing the number of beams leads to a decrease in the Stokes parameters to zero slowly. The results are of utmost importance for optical communications.

Propagation properties of electromagnetic rectangular multi-Gaussian Schell-model beams in oceanic turbulence

A model of electromagnetic rectangular multi-Gaussian Schell-model (ERMGSM) beams is introduced. Its analytic expression for the elements of the cross-spectral density matrix of such beams passing through oceanic turbulence is derived. It is shown that the rectangular shape of the ERMGSM beams holds a small distance on propagation in oceanic turbulence. The spectral density, the degree of coherence and the degree of polarization of ERMGSM beams are also studied in detail. The results will be helpful for underwater communication by using ERMGSM beams.

Electromagnetic Schell-model sources generating far fields with stable and flexible concentric rings profiles.

A class of electromagnetic sources with multi-cosine Gaussian Schell-model correlation function is introduced. The realizability conditions for such sources and the beam conditions for the beams generated by them are established. Analytical formulas for the cross-spectral density matrix of such beams propagating in free space are derived and used to examine their statistical properties by numerical simulations. The results demonstrate that such beams possess invariant far-field concentric rings-like intensity patterns, uniform polarization state on all rings, and generally different degree of polarization values on different rings. By changing the summation index and other source parameters, the number of rings along with the behavior of various beam characteristics can be tuned at will.

Propagation properties of a Multi-Gaussian Schell-Model beam in a uniaxial crystal orthogonal to the optical axis

This article presents the investigation of the propagation properties of a Multi-Gaussian Schell-Model (MGSM) beam in a uniaxial crystal, orthogonal to the optical axis. The analytic expressions for the elements of the cross-spectral density matrix of an MGSM beam in such a crystal are derived. Using the derived expressions for numerical analysis, the distributions of the normalized intensity and degree of polarization for MGSM beams are studied. The results show that not only does the normalized distribution intensity of the beam form a flat shape, but also the distribution of degree of polarization reaches a plateau in the central area when propagating in a uniaxial crystals orthogonal to the optical axis. The parameters of the source and refractive indices are major factors that affect the distributions. With a suitable choice of parameters, the intensity distribution changes to a rounded rectangular plateau and the degree of polarization distribution changes to an elliptic plateau. The results also indicate that a uniaxial crystal can modulate these MGSM beam distributions, which is beneficial for the optical manipulation of microscopic particles and nonlinear optics, involving a specific profile and polarization.

Wideband multiple‐input multiple‐output radar waveform design with low peak‐to‐average ratio constraint

Multiple-input multiple-output (MIMO) radar systems allow array antennas to transmit different waveforms and enable flexible transmit beampattern synthesis. Most existing transmit beampattern synthesis methods focus on narrowband MIMO radar system configurations. In this study, the authors propose a novel technique to design transmit waveforms for wideband MIMO radar systems. This technique is based on the optimisation of the cross-spectral density matrix and achieves a low peak-to-average power ratio as desired in practical radar systems. Simulation results are provided to verify the low PAR waveform design capability corresponding to arbitrary beampatterns.

Magnetic spectral Stokes singularities of stochastic electromagnetic beams through an astigmatic lens

Much interest has been aroused in the polarization singularities. A new technique for metrology called singular Stokes polarimetry based on the detection of polarization singularities has been recently developed and used to detect deformations and displacements of samples on a submicron scale, to measure the topology of polarized speckle field and to study the biomedicine as well. The polarization singularities have been extensively studied theoretically, numerically and experimentally. However, most of the studiesare restricted within the frameworks of the fully coherent wave-fields. By using the spectral Stokes parameters introduced by Korotkova and Wolf[Korotkova O, Wolf E 2005 Opt. Lett. 30 198], Yan and L[Yan H, L B 2009 Opt. Lett. 34 1933] have extended the concept of the polarization singularities from fully coherent beams to partially coherent beams. On the other hand, Hajnal[Hajnal J V 1990 Proc. R. Soc. Lond. A 430 413] studied the electric and magnetic polarization singularities in free-space propagation experimentally with microwaves and confirmed that the electric and magnetic polarization singularities are not coincident in general. In this paper, taking the partially coherent edge dislocation beam for example, the explicit magnetic propagation expression for stochastic electromagnetic beam through an astigmatic lens is derived based on the representation of cross-spectral density matrix propagation. Using the spectral Stokes parameters the magnetic spectral singularities are studied in detail. It is shown that there exist magnetic spectral s12, s23 and s31 singularities of stochastic electromagnetic beams through an astigmatic lens. The magnetic spectral Stokes singularities correspond to the zero points of complex Stokes fields sij=0. s12 singularity corresponds to the circular polarization (C-point) of partially coherent beam, and s3 0(s30) means right-(left-) handedness, where the orientations of the major and minor axes of the polarization ellipse become undefined. s23 and s31 singularities must be located on L-lines, where the handedness of the polarization ellipse is undetermined (linear polarization). By suitably varying a control parameter, such as off-axis distance, slope of edge dislocation, spatial correlation length, and astigmatic coefficient or propagation distance, the motion, creation, and annihilation of magnetic spectral Stokes singularities may appear. It has been shown that a pair of C-points with equal but opposite topological charges and with similar handedness may be created or annihilated. The V point and handedness reversal of C point may take place. Compared with the electric spectral Stokes singularities of stochastic electromagnetic beams, the positions are not the same, and the left- and right-handedness spaces do not coincide. The results obtained in this paper would be useful for an in-depth understanding of polarization singularities of stochastic electromagnetic beams.

Spectral Stokes singularities of partially coherent edge dislocation beams

The polarization singularities in vector wavefields have been extensively studied analytically and experimentally. The polarization singularities can be analyzed by using electromagnetic theory or Stokes parameters, or be described in terms of complex Stokes scalar fields. In some practical applications, partially coherent beams have more advantages than fully coherent beams. Recently, the concept of the polarization singularities has been extended from fully coherent beams to partially coherent beams. In this paper, using the representation of cross-spectral density matrix propagation, the explicit propagation expressions for the partially coherent edge dislocation beams are derived in free space, and based on the spectral Stokes parameters the spectral singularities are studied in detail. It is shown that there exist spectral s12, s23 and s31 singularities of partially coherent edge dislocation beams in free-space propagation. s12 singularities correspond to circular polarization (C-points) of the partially coherent edge dislocation beams, and s30 (s30) means right-(left-) handedness, where the orientations of the major and minor axes of the polarization ellipse become undefined. s23 and s31 singularities must be located on L-lines, where the handedness of the polarization ellipse is undetermined (linear polarization). The motion, creation and annihilation of spectral Stokes singularities may appear in the variation of a controlling parameter, such as off-axis distance, slope of edge dislocation, spatial correlation length, or in the variation of the propagation distance. By suitably varying the spatial correlation length or propagation distance the V-point, the handedness reversal of C-point, creation and annihilation for a pair of oppositely charged spectral singularities take place. The creation and annihilation occur for a pair of s12 singularities with opposite topological charge but same handedness. The critical points of the controlling parameters and propagation distance, at which pairs of different spectral singularities annihilate, are not the same. The collision of the C-point and L-line results in a V-point (vector singularity), which is unstable. A small perturbation leads to the handedness reversal. At such a point the state of polarization is undetermined and the degree of polarization P=0. The results obtained in this paper would be useful for a deep understanding of polarization singularities of stochastic electromagnetic beams.

Experimental Measurement of the Generalized Stokes Parameters of a Radially Polarized Random Electromagnetic Beam

Utilizing the Young’s double slits and Mach-Zehnder interferometer, we proposed an experimental method to measure the generalized Stokes parameters of a radially polarized random electromagnetic beam. After the partially coherent beam propagating through the Young’s double slits, the interference fringe is obtained by the help of a Mach-Zehnder interferometer consisting of apertures, quarter-wave plates and polarizers. The electric cross-spectral density matrix is detected by the coherence degree of interference fringe and the density of each single slit. The generalized Stokes parameters can be obtained from the electric cross-spectral density matrix. This experiment measures the generalized Stokes parameters of the random electromagnetic beam successfully. The results show that the spectral degree of coherence for copolarized cases (xx and yy) is similar with that for cross-polaried cases (xy and yx) for the radially polarized random electromagnetic beam. This method will help us determine the change of the polarization and coherence of the light in propagation by detecting the change of the generalized Stokes parameters.

Simulation of wind velocities on long span structures: A novel stochastic wave based model

A methodology is presented for efficient and accurate modeling of correlated wind velocities along long span structures at a virtually infinite number of points. Currently, the standard approach is to model wind velocities as discrete components of a multivariate stochastic vector process, characterized by a Cross-Spectral Density Matrix. To simulate sample realizations of the vector process, the Spectral Representation Method is one of the most commonly used, which involves a Cholesky decomposition of the Cross-Spectral Density Matrix. However, it is a well-known issue that as the length of the structure, and consequently the size of the vector process, increases, this Cholesky decomposition breaks down numerically. Alternatively, this paper introduces the use of the frequency–wavenumber spectrum to model the wind velocities as a stochastic “wave”, continuous in both space and time. This allows the wind velocities to be modeled at a virtually infinite number of points along the length of the structure. In this paper, the relationship between the Cross Spectral Density Matrix and the frequency–wavenumber spectrum is first examined. The frequency–wavenumber spectrum is then derived for wind velocities. Numerical examples are carried out demonstrating that the simulated wave samples exhibit the desired spectral and coherence characteristics. The efficiency of this method, specifically through the use of the Fast Fourier Transform, is also demonstrated.

Eigenanalysis-Based Adaptive Interference Suppression and Its Application in Acoustic Source Range Estimation

Passive localization of weak sources in the presence of strong interferences is generally a difficult task. In this paper, an eigenanalysis-based adaptive interference suppression (EAAIS) method is proposed for estimating the bearing and subsequently the weak source range. Assuming the target of interest (TOI) is in a known bearing sector, first we define a contribution ratio for each eigenvector of the cross-spectral density matrix (CSDM). The eigenvectors not dominated by the TOI are adaptively identified and removed for interference suppression. The remaining eigenvectors are then used to reconstruct the CSDM for TOI bearing estimation. Finally, the acoustic source range estimation is achieved with the reconstructed CSDM, by matching the beam intensity striation pattern, where it is not required to estimate the waveguide-invariant parameter. The effectiveness of the proposed TOI localization is demonstrated using several experimental data sets.

Stochastic-basis matched-field processing

Matched-field processing (MFP) suffers serious degradation due to environmental mismatch between received acoustic-field vectors and modeled replica vectors. Physical reasons for degradation include uncertainty caused by incomplete descriptions of the parameters and fields necessary for correct specification of the acoustic waveguide (i.e., environmental uncertainty), and system uncertainty associated with incomplete knowledge of the array configuration, source depth, etc. Recent research in the theory of stochastic-basis expansions (polynomial chaos) provides a mathematically consistent way of incorporating both types of uncertainty into MFP. Such expansions are used efficiently to construct replica matrices that steer high-rank subspaces capable of capturing signals with uncertain wavefronts. When combined with cross-spectral density matrices, stochastic-basis steering matrices enable the design of new processors with properties not previously evaluated in a MFP context. A maximum likelihood processor is developed which incorporates environmental uncertainty through polynomial chaos expansions. The processor can be written as a Frobenius product between an estimated cross-spectral density matrix and the inverse of a stochastic-basis replica matrix. This talk will outline the theoretical foundation of stochastic-basis MFP, and illustrate the method with some simulations. [Work supported by the Office of Naval Research.]

Nonparaxial propagation of a multi-Gaussian Schell-model beam

Based on the vectorial Rayleigh–Sommerfeld diffraction integral formulas, analytical expressions for the elements of 3 × 3 cross-spectral density matrix of polarized multi-Gaussian Schell-model (MGSM) beam’s nonparaxial propagation in free space are derived, and used to investigate its propagation properties. As a special case of nonparaxial propagation, the paraxial case has also been considered by the reduced paraxial approximation expressions. The distributions of spectral density, spectral degree of coherence, and spectral degree of polarization of nonparaxial MGSM beam are illustrated by numerical examples. Results show that the statistical properties of MGSM beam’s nonparaxial propagation in free space are closely related to the initial beam’s correlation width related parameter and transverse width related parameter These characteristics of MGSM beam might find applications in laser scanning, optical trapping, and free space optical communication.