Cross-spectral Density Matrix Research Articles

Change in coherence properties of ovally Gaussian Schell-model vortex beam in non-Kolmogorov turbulence along an uplink path

Analytical expressions are obtained for the cross-spectral density (CSD) matrix elements of an ovally Gaussian Schell-model vortex (OGSMV) beam propagating in non-Kolmogorov turbulence along uplink path based on the extended Huygens-Fresnel principle, and its coherence properties such as spectral degree of coherence (SDOC), phase distributions and coherence vortices are investigated in detail. Results indicate that the profile of the SDOC of OGSMV beam in turbulence gradually degrades into a Gaussian-like profile, and OGSMV beam with smaller ovality, larger topological charge number and initial coherence lengths will slow down this process. Interestingly, it is clearer to observe the coherence rings of the SDOC for OGSMV beam by reducing the initial auto-correlation lengths. Furthermore, one also finds that the number of elliptical edge dislocation for phase distribution of OGSMV beam is equal to topological charge number. They can provide two effectively ways for measuring topological charge number. Lastly, we used the phase screen simulation to verify our theoretical predictions. Theoretical outcomes are in good agreement with the simulations. Our results will be of important reference for optical communication.

Difference of two sinc Schell-model cross-spectral density matrices

We propose a convenient method for modeling what we believe to be novel classes of cross-spectral density (CSD), in which the difference between two CSD matrices belonging to the same class represents a supposed novel valid CSD matrix under sufficient conditions. Using such a method, we introduce what we think is a new class of CSD matrix relating to the difference of two sinc Schell-model CSD matrices, and derive the sufficient conditions for the non-negative definiteness of the resulting CSD matrix. The statistical characteristics of such electromagnetic stochastic beams propagating in free space were investigated using weighted superposition and numerical simulation methods. It is demonstrated that this new light fields with any admissible choice of source parameters possess very peculiar evolution characteristics. The results illustrate new opportunities for study new classes of electromagnetic stochastic beams.

Propagation properties of the partially coherent radially polarized twisted beam through optical system in turbulence atmosphere

In this paper, the transmission properties of the partially coherent radially polarized twisted (PCRPT) beam propagating in the turbulence atmosphere are investigated. The analytical formulas for the components of the cross-spectral density matrix for the PCRPT beam passing through the optical system in the turbulent atmosphere are developed using the Collins integral and aperture function. Research results indicate that modifying the variable parameters and dimensions of the optical system can control the near-field and far-field distributions of the beam, while providing a more flexible choice of receivers for the PCRPT beam in the receiving plane. By utilizing the Cassegrain reflector system and adjusting the optical system parameters, it is possible to achieve collimated transmission of the PCRPT beam and significantly enhance the beam transmission efficiency in turbulent atmospheric conditions. The derivation process and the research results presented in this paper can be expanded to analyze the application of optical systems to control high-dimensional beam field variations. The envisioned utilization of the results obtained from this research investigation pertains to the fields of beam shaping and optical communications.

Polarization and coherence properties of a partially coherent elegant Laguerre–Gaussian vortex beam in turbulent plasma

The analytical expressions for the cross-spectral density matrix elements of the partially coherent elegant Laguerre–Gaussian (eLG) vortex beam propagating through anisotropic turbulent plasma were derived based on the extended Huygens-Fresnel principle and used to study the changes in the polarization degree and the coherence degree of the partially coherent eLG vortex beam in the turbulent plasma. The numerical results show that the polarization degree of a partially coherent eLG vortex beam reaches a specific value (which is equal to the degree of polarization in the source plane) after a long propagation distance in the turbulent plasma. Moreover, this value is independent of the beam order, topological charge, correlation coefficient, wavelength of the source plane, anisotropy parameter, refractive index fluctuation variance, and outer and inner scales of the turbulent plasma. The results also show that with increasing distance, the coherence degree first decreases from unity and then oscillates around zero. However, this oscillation gradually disappears after traveling a long distance. Our results intuitively present the beam polarization and coherence properties through anisotropic hypersonic turbulence, which can be useful for optical communication in hypersonic turbulent environments.

Effect of Polarization on Cross-Spectral Density Matrix

Coherence-polarization properties of different beams are experimentally measured in the far-field from the source and results are presented for incoherent sources with three different polarization features, such as unpolarized, diagonally polarized, and spatially depolarized. These results highlight the role of polarization tailoring on far-field coherence-polarization properties of the incoherent vector source. The effect of polarization on far-field coherence is analyzed using a beam cross-spectral density (CSD) matrix, and the role of polarization tailoring on the CSD matrix is demonstrated. Two-dimensional spatial distributions of all four elements of the CSD matrix are experimentally realized using a field-based interferometer with Sagnac geometry in combination with a four-step phase-shifting technique.

Polarization properties of partially coherent mixed dislocation beams transmitting in biological tissues

<sec> <b>Objective</b> The optical information change of beams acting on biological tissue can get an insight into the new optical effects of tissue, and even can provide a theoretical basis for developing biphotonic medical diagnosis and therapy technologies. Polarization technology is also widely used in the field of biological detection due to its advantages of non-contact, rich information and without staining markers. In this work, the polarization behaviors of partially coherent screw-linear edge mixed dislocation beam transmitting in biological tissue are analyzed and explored. Simultaneously, in order to more clearly and more intuitively understand a mixed dislocation beam, both the normalized intensities and phase distributions at source plane for different parameters <i>a</i> and <i>b</i> are also discussed. We hope that the obtained results will provide theoretical and experimental foundation for expanding the application of singularity beams in biological tissue imaging technology.</sec><sec> <b>Method</b> By combining the Schell term with the field distribution of the screw-linear edge mixed dislocation beam at the source plane, and based on the generalized Huygens-Fresnel principle, the analytical expressions of the cross-spectral density matrix elements of partially coherent screw-linear edge dislocation beam propagating in biological tissues are derived. Adopting the unified theory of coherence and polarization, the polarization behaviors of the beams can be investigated in detail.</sec><sec> <b>Results</b> At the source plane, the intensity has a non axisymmetric distribution, and there exists a coherent vortex with a topological charge size of 1 and a linear edge dislocation. The sign of <i>a</i> is related to the rotation direction of the phase singularity. The larger the value of <i>b</i>, the farther the linear edge dislocation is from the origin. At the source plane, the degree of polarization and ellipticity between the two identical points are independent of the four parameters: dimensionless parameter <i>a</i>, off-axis distance of edge dislocation <i>b</i>, spatial self-correlation length <i>σ</i><sub><i>yy</i></sub>, and spatial mutual-correlation length <i>σ</i><sub><i>xy</i></sub>, the orientation angle is only independent of <i>a</i> and <i>σ</i><sub><i>xy</i></sub>; the polarization of two different points is independent of <i>a</i> and <i>b</i>, but is related to <i>σ</i><sub><i>yy</i></sub> and <i>σ</i><sub><i>xy</i></sub>. In transmission, the polarization degrees and ellipticity of two different points fluctuate greatly and the orientation angle displays less fluctuation. Finally, all the polarization state parameters tend to be their corresponding values, respectively.</sec><sec> <b>Conclusions</b> The results show that when <i>b</i> is smaller, the linear edge dislocation is paraxial and plays an important role in the polarization state change; when <i>b</i> is larger, the polarization state changes of the screw-linear edge mixed dislocation beam will tend to be the pattern of spiral beams. The absolute value of the difference between <i>σ</i><sub><i>yy</i></sub> and <i>σ</i><sub><i>xy</i></sub> is also one of main factors influencing the polarization state. The sign of <i>a</i> does not affect the change in polarization state, but its magnitude can influe the change of speed. Due to more complex factors determining the correlation fluctuations between different points in the light field, the changes of two different points are more sensitive than those of the two identical points in shallow biological tissue. Beams with different parameters can be selected for different application requirements.</sec>

Tight Focusing of Circular Partially Coherent Radially Polarized Circular Airy Vortex Beam

The tight focusing properties of circular partially coherent radially polarized circular Airy vortex beams (CPCRPCAVBs) are theoretically studied in this paper. After deriving the cross-spectral density matrix of CPCRPCAVBs in the focal region of a high-NA objective, numerical calculations were performed to indicate the influence of the topological charge of the vortex phase on intensity distribution, degree of coherence and degree of polarization of the tightly focused beam. An intensity profile along the propagation axis shows that a super-length optical needle (~15 λ) can be obtained with a topological charge of 1, and a super-length dark channel (~15 λ) is observed with a topological charge of 2 or 3. In the focal plane, the rise in the number of topological charge does not distort the shapes of the coherence distribution pattern and the polarization distribution pattern, but enlarges their sizes.

Average Intensity of a Laguerre—Gaussian Vector Vortex Beam through Inhomogeneous Atmospheric Turbulence

We investigate the evolution properties of a partially coherent Laguerre–Gaussian vector vortex (LGVV) beam through inhomogeneous atmospheric turbulence. Analytical formulae for the elements of a cross-spectral density matrix of a partially coherent LGVV beam propagating in turbulence are derived with the help of the extended Huygens–Fresnel principle. Our outcomes demonstrate that the normalized initial profile of a partially coherent beam with concentric dark rings gradually evolves into a Gaussian-like beam profile in turbulence. We also find that the beam is emitted at a large zenith angle and quickly converts to a Gaussian-like beam. Furthermore, it is also shown that a propagation beam with a large topological charge has a stronger ability to resist atmospheric turbulence. In order to confirm our numerical results, we combine the complex screen method and multi-phase screen method to simulate the propagation of a partially coherent LGVV beam in atmospheric turbulence. It is indicated that the simulation results are in good agreement with theoretical predictions. Our results will pave the way for the development of free-space optical communications and remote sensing.

Modeling sea surface noise in the presence of seamounts and ocean fronts using Nx2D and 3-D methods

This study focuses on modeling the ambient noise observed along a vertical line array in the presence of distributed surface sources and extreme bathymetric and oceanographic features. The distributed surface sources considered in this study include wind-induced noise, while the extreme bathymetric and oceanographic features comprise seamounts such as Atlantis II and the ocean front caused by the Gulf Stream. Directional vertical noise levels are modeled by computing the Green’s Function between the array elements and the distributed surface sources, estimating the average cross spectral density matrix, and calculating the plane wave response [1]. Green’s functions are computed using the Bellhop3-D ray tracing model [2] in both an Nx2D and full 3-D environment, and the resulting directional noise levels are compared. The results of this study will provide insights into the accuracy and effectiveness of the modeling methods used to predict ambient noise levels in such challenging environments, with particular emphasis on the contribution of seamounts to ambient noise. The implications of these results can be significant for thedesign and operation of underwater acoustic systems in similar environments.

Improving the quality of high-frequency surface waves retrieved from ultrashort traffic-induced noise based on eigenvalue selection

SUMMARY Stacking cross-correlations of time windows from continuous long-duration noise data is an effective solution to improve the quality of retrieved high-frequency (>1 Hz) surface waves and the accuracy of dispersion energy. The observation duration, however, is usually limited due to traffic control, making it difficult for ambient noise sources to fulfill the requirement of uniform distribution. Additionally, strong human-related noise sources exist near survey lines deployed along urban roads, which often act as interfering sources, such as local noise sources located in the non-stationary-phase zones. Local noise sources cause spurious arrivals in cross-correlations, degrade signal-to-noise ratio (SNR) of retrieved surface waves and distort their dispersion energy. To attenuate these adverse effects and improve the quality of surface waves retrieved from ultrashort noise data, we perform the eigendecomposition technique on the cross-spectral density matrix (CSDM) and apply a Wiener filter on the decomposed eigenvectors. The correct eigenvalues and the corresponding filtered eigenvectors are selected to reconstruct the CSDM related to stationary-phase sources based on the matched-field processing outputs. This procedure significantly suppresses the backpropagated signals and efficiently recovers surface waves by improving the contribution of the stationary-phase sources. We validate our scheme on a synthetic test and two practical applications and show that we obtain higher-SNR virtual shot gathers and higher-quality surface-wave dispersion images compared to seismic interferometry. Our scheme can be a new alternative technique to conduct passive seismic surveys in densely populated urban environments without being affected by local noise sources.

Second-order coherence and polarization characteristics of partially coherent radially polarized cosh-Gaussian vortex beams in a turbulent atmosphere

The second-order coherence and polarization characteristics of partially coherent radially polarized cosh-Gaussian vortex (PCRPChGV) beams in atmospheric turbulence are investigated based on the extended Huygens–Fresnel principle. The analytical propagation formulas of cross-spectral density matrix, average intensity and degree of polarization (DOP) for PCRPChGV beams in turbulence are derived. The influences of initial correlation length, cosh factor and turbulence intensity on average intensity, on-axis intensity and DOP of PCRPChGV beams are examined numerically. Results show that with increasing propagation distance, the PCRPChGV beam evolves from the initially separated double-lobe spot into overlapping double-lobe spot, and Gaussian spot. The corresponding intensities of the two Cartesian components of light field rotate markedly around the beam axis in a unique manner. Initial correlation length and cosh factor have an obvious influence on the position and size of normalized on-axis intensity peak, respectively. In addition, PCRPChGV beams in moderate turbulence have strong anti-turbulence ability.

Partially Coherent Cylindrical Vector Sources

A new class of stationary electromagnetic sources radiating outward from the surface of an infinitely long cylinder is introduced via vectorial coherent mode representation. First, two particular types of such sources are discussed: with either an electric or magnetic field aligned with the cylinder’s axis. The former case represents a scalar scenario, while the latter leads to the two-component electric field. The combination of these two types of sources is then considered by forming the three-component electric field vector. An extension to the stationary case is then made in which the electric field correlations are shown to be described by the intrinsically 3×3 cross-spectral density matrix. Several known theories of electromagnetic coherence and polarization are then invoked for the analysis of radiation, on and off the source surface. The results for the spectral density, degree of coherence, and degree of polarization are then discussed in detail. The effects of mutual correlation of modes are also outlined. The new family of sources is of importance for any application involving cylindrical sources with controllable radiation.

Recording of incoherent vector holograms using elements of the spatial cross-spectral density matrix

We present an approach to record incoherent vector holograms using a highly stable field-based interferometer employing radial shearing. The vector holograms are recorded as the elements of the 2 × 2 cross-spectral density (CSD) matrix and reconstruction of the object encoded into these holograms is demonstrated by digitally propagating the elements of the CSD matrix. The CSD matrix contains its elements in the form of a complex spatial coherence function and is connected with an incoherent vector source by the vectorial van Cittert-Zernike theorem. The experimental measurement of the two-dimensional distributions of the complex elements of the CSD matrix is realized by a Sagnac radial shearing interferometer with a phase-shifting approach. The five-step phase-shifting technique is used to measure the fringe visibility and corresponding phase which give the complex elements of the CSD matrix. Recording of incoherent vector holograms is demonstrated from the experimentally recovered elements of the CSD matrix and results are presented for two different incoherent sources, namely polarized and unpolarized. The complex nature of the CSD matrix elements helps in the digital reconstruction and focusing of the incoherent source.

Compact generation of light beams carrying robust higher-order Poincaré polarization states

We propose a protocol for compact and efficient synthesis of a random vectorial source with a higher-order Poincaré (HOP) polarization state encoded into the spatial coherence structure. The procedure is based on the complex-random-mode representation of the cross-spectral density matrix and employs a single phase-only spatial light modulator (SLM) and a common path interferometric system for the mode construction. The SLM displays a set of elaborated multiplexed holograms, which both encode the HOP polarization state and determine the statistical properties of the source. We demonstrate that the beam from the synthesized source can be highly robust against obstructions in the propagation path in the sense that the encoded HOP polarization state is well reconstructed in the far field (focal plane) even when an obstacle is introduced to largely block the source. The results are useful for the transmission of polarization-encoded information in complex media.

Three-dimensional polarization of fields radiated by partially coherent electromagnetic cylindrical sources

Partially coherent electromagnetic sources with cylindrical symmetry and infinite extent radiating outward are introduced. Their 3 × 3 cross-spectral density matrix is given through expansions of the field components in terms of basis functions related to the Hankel functions. The spectral density and the three-dimensional degree of polarization of such sources and the fields they radiate are examined. Several examples are presented and discussed. Among them, a class of cylindrical sources whose coherent vector modes coincide with the above basis functions is defined and studied.

Changes in the statistical properties of electromagnetic double multi-Gaussian Schell-model beams on propagation in free space.

A class of electromagnetic random sources with a multi-Gaussian functional form in the spectral density and in the correlations part of the cross-spectral density matrix is introduced based on the genuine cross-spectral density-matrix theory. The analytic propagation formulas of the cross-spectral density matrix of such beams propagating in free space are derived by use of Collins' diffraction integral. With the help of analytic formulas, the evolution of statistical characteristics, i.e.,the spectraldensity, the spectral degree of polarization, and the spectral degree of coherence for such beams in free space are analyzed numerically. Employing the multi-Gaussian functional form in the cross-spectral density matrix introduces one more freedom in the modeling of Gaussian Schell-model sources.

Simulating random optical fields: tutorial.

Numerous applications-including optical communications, directed energy, remote sensing, and optical tweezing-utilize the principles of statistical optics and optical coherence theory. Simulation of these phenomena is, therefore, critical in the design of new technologies for these and other such applications. For this reason, this tutorial describes how to generate random electromagnetic field instances or realizations consistent with a given or desired cross-spectral density matrix for use in wave optics simulations. This tutorial assumes that the reader has knowledge of the fundamental principles of statistical optics and optical coherence theory. An extensive reference list is provided where the necessary background information can be found. We begin this tutorial with a brief summary of the coherent-mode representation and the superposition rule of stochastic electromagnetic fields as these foundational ideas form the basis of all known synthesis techniques. We then present optical field expressions that apply these concepts before discussing proper sampling and discretization. We finally compare and contrast coherent-mode- and superposition-rule-based synthesis approaches, discussing the pros and cons of each. As an example, we simulate the synthesis and propagation of an electromagnetic partially coherent field from the literature. We compare simulated or sample statistics to theory to verify that we have successfully produced the desired field and are capturing its propagation behaviors. All computer programs, including detailed explanations of the source code, are provided with this tutorial. We conclude with a brief summary.

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.

Evolution Properties of a Partially Coherent Twisted Laguerre-Gaussian Pulsed Beam Propagating through Anisotropic Atmospheric Turbulence

Analytical expressions for the cross-spectral density matrix of a partially coherent twisted Laguerre-Gaussian pulsed (PCTLGP) beam in anisotropic atmospheric turbulence are derived based on the extended Huygens–Fresnel principle. Numerical results indicate that the atmospheric turbulence induces the degeneration of the spectral intensity distribution of the PCTLGP beam, and the PCTLGP beam also shows different evolution properties on propagation in weaker turbulence and stronger turbulence. The PCTLGP beam with a negative twisted factor exhibits an advantage over the Laguerre-Gaussian pulsed beam for reducing the atmospheric turbulence-induced degeneration, and this advantage is further strengthened with increasing the topological charge, mode order and absolute value of the twisted factor. In addition, we also find that the pulse duration will affect the spectral intensity of the PCTLGP beam in turbulence. This kind of beam will show potential application value in free-space optical communications and remote sensing.

Spectral proper orthogonal decomposition using multitaper estimates

The use of multitaper estimates for spectral proper orthogonal decomposition (SPOD) is explored. Multitaper and multitaper-Welch estimators that use discrete prolate spheroidal sequences (DPSS) as orthogonal data windows are compared to the standard SPOD algorithm that exclusively relies on weighted overlapped segment averaging, or Welch’s method, to estimate the cross-spectral density matrix. Two sets of turbulent flow data, one experimental and the other numerical, are used to discuss the choice of resolution bandwidth and the bias-variance tradeoff. Multitaper-Welch estimators that combine both approaches by applying orthogonal tapers to overlapping segments allow for flexible control of resolution, variance, and bias. At additional computational cost but for the same data, multitaper-Welch estimators provide lower variance estimates at fixed frequency resolution or higher frequency resolution at similar variance compared to the standard algorithm.