Cross-spectral Density Matrix Research Articles

Blackbody far-field coherence.

We revisit the spectral coherence properties of far-field radiation emanating from an aperture in a blackbody cavity on the basis of Kirchhoff's boundary conditions. We point out that the far zone cross-spectral density matrix derived earlier in the literature by separately propagating all three aperture-field components does not show transversality of the field at nonparaxial directions. This is not the case when Luneburg's diffraction integrals are applied on the transverse source field components to determine the entire far field. We compare the electromagnetic degrees of coherence for the two methods and show that over important angular separations their values coincide with high accuracy. The results of this work and of others concerning the far-field intensity, polarization, and paraxial angular coherence are in full agreement.

Radially polarized twisted Multi-Gaussian Schell-model beams and their statistical properties

A radially polarized twisted Multi-Gaussian Schell-model (RPT MGSM) beam based on the unified theory of coherence and polarization is introduced. The reliability conditions for the beams are also derived. We derive the analytical expressions for the elements of the cross-spectral density matrix of such beam propagating in free space and investigate the statistical properties, such as spectral density, degree of coherence, degree of polarization and state of polarization. It is found that the statistical properties of the RPT MGSM beams are closely related to the twist phase, e.g. the twist phase induces not only the rotation of the beam spot but also changes of the degree of coherence, degree of polarization and state of polarization on propagation. Specially, an interesting phenomenon is found. In the far field, when the beams without carrying twist factor, the state of polarization in the flat-top area of intensity distribution is azimuthally polarized rather than radially polarized. Our results may be useful in optical trapping and free-space optical communications.

Far-zone behaviors of scattering-induced statistical properties of partially polarized spatially and spectrally partially coherent electromagnetic pulsed beam**Project supported by the National Natural Science Foundation of China (Grant No. 11504286), the Natural Science Basic Research Program of Shaanxi Province, China (Grant No. 2019JM-470), the Fund from the International Technology Collaborative Center for Advanced Optical Manufacturing and Optoelectronic Measurement, and the Science

In this study, we explore the far-zero behaviors of a scattered partially polarized spatially and spectrally partially coherent electromagnetic pulsed beam irradiating on a deterministic medium. The analytical formula for the cross-spectral density matrix elements of this beam in the spherical coordinate system is derived. Within the framework of the first-order Born approximation, the effects of the scattering angle θ, the source parameters (i.e., the pulse duration T0 and the temporal coherence length Tcxx), and the scatterer parameter (i.e., the effective width of the medium σR) on the spectral density, the spectral shift, the spectral degree of polarization, and the degree of spectral coherence of the scattered source in the far-zero field are studied numerically and comparatively. Our work improves the scattering theory of stochastic electromagnetic beams and it may be useful for the applications involving the interaction between incident light waves and scattering media.

Data-adaptive harmonic analysis of oceanic waves and turbulent flows.

We introduce new features of data-adaptive harmonic decomposition (DAHD) that are showcased to characterize spatiotemporal variability in high-dimensional datasets of complex and mutsicale oceanic flows, offering new perspectives and novel insights. First, we present a didactic example with synthetic data for identification of coherent oceanic waves embedded in high amplitude noise. Then, DAHD is applied to analyze turbulent oceanic flows simulated by the Regional Oceanic Modeling System and an eddy-resolving three-layer quasigeostrophic ocean model, where resulting spectra exhibit a thin line capturing nearly all the energy at a given temporal frequency and showing well-defined scaling behavior across frequencies. DAHD thus permits sparse representation of complex, multiscale, and chaotic dynamics by a relatively few data-inferred spatial patterns evolving with simple temporal dynamics, namely, oscillating harmonically in time at a given single frequency. The detection of this low-rank behavior is facilitated by an eigendecomposition of the Hermitian cross-spectral matrix and resulting eigenvectors that represent an orthonormal set of global spatiotemporal modes associated with a specific temporal frequency, which in turn allows to rank these modes by their captured energy and across frequencies, and allow accurate space-time reconstruction. Furthermore, by using a correlogram estimator of the Hermitian cross-spectral density matrix, DAHD is both closely related and distinctly different from the spectral proper orthogonal decomposition that relies on Welch's periodogram as its estimator method.

Partially coherent radially polarized fractional vortex beam.

A new kind of partially coherent vector beam, named a partially coherent radially polarized fractional vortex (PCRPFV) beam, is introduced as a natural extension of the recently introduced scalar partially coherent fractional vortex beams [Zeng et al., Opt. Express26, 26830 (2018)10.1364/OE.26.026830]. Realizability conditions and propagation formulas for a PCRPFV beam are derived. Statistical properties of a focused PCRPFV beam, such as average intensity, degree of polarization, state of polarization and cross-spectral density matrix, are illustrated in detail and compared with that of a partially coherent radially polarized integer vortex beam and a scalar partially coherent fractional vortex beam. It is found that the statistical properties of a PCRPFV beam are qualitatively different from these simpler beam classes and are strongly determined by the vortex phase (i.e., fractional topological charge) and initial coherence width. We demonstrate experimental generation of PCRPFV beams and confirm their behavior. Our results will be useful for the rotating and trapping of particles, the detection of phase objects, and polarization lidar systems.

Anisotropic turbulence-induced statistical properties evolution of a radially polarized partially coherent pulsed beam on propagation

In this study, we introduce a new kind of partially coherent vector beam with Gaussian distribution in the spatial, temporal and correlation components named a radially polarized partially coherent Gaussian Schell-model pulsed (GSMP) beam as a natural extension of scalar Gaussian Schell-model pulsed beam. Analytical formulas for the two-frequency cross-spectral density matrix elements of this beam are derived based on the extended Huygens–Fresnel theory. Within the frame of extend anisotropic non-Kolmogorov spectrum, the effects of asymmetry anisotropic turbulence eddies on the statistical properties of a radially polarized partially coherent GSMP beam are investigated numerically. Our work improves the classical theory of stochastic electromagnetic beams on propagation and may be useful for applications involving free-space optical communications.

Synthesizing General Electromagnetic Partially Coherent Sources from Random, Correlated Complex Screens

We present a method to generate any genuine electromagnetic partially coherent source (PCS) from correlated, stochastic complex screens. The method described here can be directly implemented on existing spatial-light-modulator-based vector beam generators and can be used in any application which utilizes electromagnetic PCSs. Our method is based on the genuine cross-spectral density matrix criterion. Applying that criterion, we show that stochastic vector field realizations (corresponding to a desired electromagnetic PCS) can be generated by passing correlated Gaussian random numbers through “filters” with space-variant transfer functions. We include step-by-step instructions on how to generate the electromagnetic PCS field realizations. As an example, we simulate the synthesis of a new electromagnetic PCS. Using Monte Carlo analysis, we compute statistical moments from independent optical field realizations and compare those to the corresponding theory. We find that our method produces the desired source—the correct shape, polarization, and coherence properties—within 600 field realizations.

Spectral and coherent properties of partially polarized pulsed electromagnetic beams upon turbulent atmosphere propagation for different source conditions

The spectral and coherent properties of partially polarized pulsed electromagnetic beams propagating in a horizontal homogeneous and isotropic turbulent atmosphere channel for different source conditions are investigated in detail. Based on the extended Huygens–Fresnel principle and von Karman power spectrum, the analytical expressions for the elements of the two-frequency cross spectral density matrix, the average spectral intensity, the beam width, and the spectral degree of electromagnetic coherence are derived in turn. Finally, with the help of numerical calculations, the effects of different source conditions on the changes in the beam width, spectral and coherent properties of partially polarized pulsed Gaussian Schell-model electromagnetic beams are studied. Some theoretical results in this paper will be useful for applications involving polarized pulsed electromagnetic beams propagating through atmospheric turbulence.

Coherence and cross-spectral density matrix analysis of random wind and wave in deep water

From the collected records of simultaneously measured wind and wave data during FETCH experiment, the sample and mean power spectral density and coherence functions of the fluctuating wind velocity and wave height processes are calculated. Based on the calculated mean coherence function and interaction mechanism of wind and wave, a parametric model and simple parameter estimate method are proposed to fit the coherence of the fluctuating wind velocity and wave height processes. The model is an explicit function of the main characteristic parameters of wind and wave, hence it can be used to model other empirical coherences of the random wind and wave processes in deep water. Combining the theoretical spectrum models fitting the calculated mean power spectral density functions of the fluctuating wind velocity and wave height processes and proposed coherence model for modeling the processes, the theoretical cross-spectrum density matrix of the fluctuating wind velocity and wave height processes is established. Samples of the fluctuating wind velocity and wave height processes are simulated to illustrate the influence of coherence on the random wind and wave loads.

CLEAN beamforming for the enhanced detection of multiple infrasonic sources

SUMMARY The detection and characterization of signals of interest in the presence of (in)coherent ambient noise is central to the analysis of infrasound array data. Microbaroms have an extended source region and a dynamical character. From the perspective of an infrasound array, these coherent noise sources appear as interfering signals that conventional beamform methods may not correctly resolve. This limits the ability of an infrasound array to dissect the incoming wavefield into individual components. In this paper, this problem will be addressed by proposing a high-resolution beamform technique in combination with the CLEAN algorithm. CLEAN iteratively selects the maximum of the f/k spectrum (i.e. following the Bartlett or the Capon method) and removes a percentage of the corresponding signal from the cross-spectral density matrix. In this procedure, the array response is deconvolved from the f/k spectral density function. The spectral peaks are retained in a ‘clean’ spectrum. A data-driven stopping criterion for CLEAN is proposed, which relies on the framework of Fisher statistics. This allows the construction of an automated algorithm that continuously extracts coherent energy until the point is reached that only incoherent noise is left in the data. CLEAN is tested on a synthetic data set and is applied to data from multiple International Monitoring System infrasound arrays. The results show that the proposed method allows for the identification of multiple microbarom source regions in the Northern Atlantic that would have remained unidentified if conventional methods had been applied.

Simulation of wind velocity time histories on long span structures modeled as non-Gaussian stochastic waves

A methodology is proposed for efficient and accurate modeling and simulation of correlated non-Gaussian wind velocity time histories along long-span structures at an arbitrarily large number of points. Currently, the most common approach is to model wind velocities as discrete components of a stochastic vector process, characterized by a Cross-Spectral Density Matrix (CSDM). To generate sample functions of the vector process, the Spectral Representation Method is one of the most commonly used, involving a Cholesky decomposition of the CSDM. However, it is a well-documented problem that as the length of the structure – and consequently the size of the vector process – increases, this Cholesky decomposition breaks down numerically. This paper extends a methodology introduced by the second and fourth authors to model wind velocities as a Gaussian stochastic wave (continuous in both space and time) by considering the stochastic wave to be non-Gaussian. The non-Gaussian wave is characterized by its frequency–wavenumber (FK) spectrum and marginal probability density function (PDF). This allows the non-Gaussian wind velocities to be modeled at a virtually infinite number of points along the length of the structure. The compatibility of the FK spectrum and marginal PDF according to translation process theory is secured using an extension of the Iterative Translation Approximation Method introduced by the second and third authors, where the underlying Gaussian FK spectrum is upgraded iteratively using the directly computed (through translation process theory) non-Gaussian FK spectrum. After a small number of computationally extremely efficient iterations, the underlying Gaussian FK spectrum is established and generation of non-Gaussian sample functions of the stochastic wave is straightforward without the need of iterations. Numerical examples are provided demonstrating that the simulated non-Gaussian wave samples exhibit the desired spectral and marginal PDF characteristics.

Optimization of Shaker Locations for Multiple Shaker Environmental Testing

For flight payloads or systems in free flight, Impedance Matched Multi-Axis Testing (IMMAT) can provide an accurate laboratory reproduction of the flight vibration environment at multiple response locations. IMMAT is performed by controlling multiple shakers attached to the system of interest, usually through slender rods so that the shakers impart negligible moments or shear forces at the attachment. The attachment usually requires that the shakers not physically support the system. Thus, IMMAT is different from other multi-degree of freedom testing where shakers for slip tables or with vertical bearings drastically change the impedance by their rigid attachment to the system or payload. Consequently, IMMAT shakers are generally smaller than used for traditional testing. In the laboratory IMMAT test, bungee cords can support the system to simulate free flight. For a system that is a flight payload, bungee cords can support a portion of the next level of assembly (such as a rack or rail) with the attached payload to greatly improve the laboratory reproduction of the payload environment with the approximate attachment impedance. Engineering judgment has historically been the basis for IMMAT test planning but provides no pre-test metrics to show whether the test setup can meet the desired requirements. For successful test planning, engineers need tools to optimize the number and location of shakers and predict the requirements for the shakers and amplifiers. Electrodynamic shakers and amplifiers have physical limitations such as maximum available amplifier current, voltage or power and shaker force or stroke. If shakers and amplifiers can barely meet required levels with a well-designed IMMAT test, improper shaker placement can cause exceedance of the limitations and failure of the test to meet required levels. We present a tool to optimize the number and locations of shakers with an objective function that performs a least square fit of the flight cross spectral density matrix while minimizing requirements on the amplifiers or shakers. In this work, an optimized IMMAT test with four shakers attached to a test article closely reproduces the vibration environment generated by a field acoustic test. The optimization is based on a model. The model consists of a modal model (derived from a finite element model) of the test article coupled to a simple calibrated electro-mechanical model of the shakers. The optimization selects shaker locations to minimize the required amplifier output voltage, but one can minimize shaker force, current, control error or some combination with appropriate physical limits.

Turbulence-induced changes in the state of polarization of partially polarized, partially coherent pulsed electromagnetic beams propagating through anisotropic turbulence

The changes in the state of polarization of partially polarized, partially coherent pulsed electromagnetic beams propagating along a horizontal path through anisotropic turbulence are investigated. Unpolarized and partially polarized, partially coherent pulsed electromagnetic Gaussian Schell-model beams are compared in terms of the turbulence-induced changes in the spectral degree of polarization. Our analysis is focused on models pertaining to electromagnetic Gaussian Schell-model pulsed beams but can either be readily reduced to scalar or be readily generalized to other beam classes. Within the framework of the anisotropic generalized exponential spectrum, considering simultaneously the finite inner and outer scales of turbulence and the asymmetric property of turbulence eddies in the orthogonal x y -plane throughout the propagation path, we derive analytical expressions for the cross-spectral density matrix, spectral degree of polarization, orientation angle, and degree of ellipticity. Finally, we study the effects of the anisotropic turbulence parameter on the spectral degree of polarization, orientation angle, and degree of ellipticity of the beams. Our results can be useful for applications involving partially polarized, partially coherent pulsed beams propagating through atmospheric turbulence.

A modified spectral representation method to simulate non-Gaussian random vector process considering wave-passage effect

Wave-passage effect is always of great importance when performing the dynamic analysis of lager scale structures. This effect can be easily considered in the simulation of Gaussian vector process, but there is not an efficient way to consider it in simulating non-Gaussian vector process. In this paper, a method is proposed to generate stationary non-Gaussian vector process considering wave-passage effect. In the simulation of vector process, wave-passage effect leads to a complex-defined cross spectral density matrix (CSDM). To separate the phase part from the CSDM, an improved form of spectral representation method (SRM) is proposed, in which the phase part representing the wave-passage effect can be directly expressed in an explicit form in the simulation formula. Then the characteristic of phase part representing the wave-passage effect during the nonlinear translation (translating underlying Gaussian vector process to the prescribed non-Gaussian vector process) is investigated. Interestingly, it is found that, the phase part is not changed. Based on this good feature, the modified SRM and the iterative scheme can be combined to simulate non-Gaussian vector process considering wave-passage effect. Two examples, involving the simulation of seismic ground motions and wind velocity time histories, are presented to show the characteristics of the simulated vector process and to verify the capabilities of the proposed method.

A Comparison of Optimal SONAR Array Amplitude Shading Coefficients

This paper compares two different approaches to deriving shading coefficients (weights) for optimal first order and second order directional sensors (that is; sonobuoys, vectors and dyadic sensors). The first approach is an analytical or a physics-based derivation, involving computations with gradients and linearized momentum; the second is an adaptive minimum variance distortionless response (MVDR) derivation, which finds weights that minimize the cross spectral density (CSD) matrix. The two approaches are shown to be equivalent. In other words, the adaptive MVDR processing procedure does indeed converge to a physics-based solution, without any pre-existing physical knowledge of the behavior of the acoustic field. This suggests that adaptive algorithms innately seek physics-based solutions when these solutions are optimum. The intent of this short communication is not to advocate for one type of adaptive processing method over another. The observation that is presented here is important though, it confirms that at least in an idealized noise field, adaptive processing converges on an optimal set of shading coefficients, similarly derived based on well-established physical acoustics.

Generating electromagnetic Schell-model sources using complex screens with spatially varying auto- and cross-correlation functions

We present a method to generate any physically realizable electromagnetic Schell-model source. Our technique can be directly implemented on existing vector-beam generators that utilize spatial light modulators for coherence control, beam shaping, and relative phasing. This work significantly extends published research on the subject, where control over the partially coherent source’s cross-spectral density matrix was limited.We begin by presenting the statistical optics theory necessary to derive and implement our method. We then apply our technique, both analytically and in simulation, to produce two electromagnetic Schell-model sources from the literature. We demonstrate control over the full cross-spectral density matrices of both partially coherent beams. We compare the simulated results of these two sources to the corresponding theoretical or designed quantities to validate our approach. We find, through examination of the two-dimensional correlation coefficients, that both sources converge to their desired, ensemble (or by ergodicity, “long-time”) statistics within 500 random field instances.Our method and subsequent findings will be useful in any application where control over beam shape, polarization, and spatial coherence are important. These include but are not limited to free-space/underwater optical communications, directed energy, medicine, atomic optics, and optical tweezers.

Propagation properties of radially polarized multi-Gaussian Schell-model beams in oceanic turbulence.

By utilizing the extended Huygens-Fresnel principle, we derive the analytical formulas for the cross-spectral density matrix elements of a radially polarized multi-Gaussian Schell-model (RPMGSM) beam propagating in oceanic turbulence. Effects of beam parameters and oceanic turbulence parameters on the propagation properties of RPMGSM beams are investigated in detail by numerical simulation. Our results show that the RPMGSM beam with larger beam order M has an advantage over the radially polarized Gaussian Schell-model beam for reducing turbulence-induced degradation. Compared with temperature-induced turbulence fluctuation, the salinity-induced turbulence fluctuation makes a greater contribution to the influence on propagation properties of RPMGSM beams.

Condition for keeping polarization invariant on propagation in space-to-ground optical communication downlink

We consider a Gaussian Schell-model (GSM) laser beam applied in space-to-ground optical communication downlink. Based on the derived analytic expression of cross-spectral density matrix of GSM laser beam, the sufficient conditions of the polarization properties remaining invariant through the space-to-ground optical communication downlink are obtained. The conditions reveal when the source light is linear polarization beam or the three coherence lengths ( δxx, δxy and δyy) are equal to each other, the polarization properties unchanged upon propagation. With full consideration of the coherence lengths and the zenith angle, the conditions of keeping polarization properties invariance is verified by numerical calculation.

Acoustic Classification of Surface and Underwater Vessels in the Ocean Using Supervised Machine Learning.

Four data-driven methods—random forest (RF), support vector machine (SVM), feed-forward neural network (FNN), and convolutional neural network (CNN)—are applied to discriminate surface and underwater vessels in the ocean using low-frequency acoustic pressure data. Acoustic data are modeled considering a vertical line array by a Monte Carlo simulation using the underwater acoustic propagation model, KRAKEN, in the ocean environment of East Sea in Korea. The raw data are preprocessed and reorganized into the phone-space cross-spectral density matrix (pCSDM) and mode-space cross-spectral density matrix (mCSDM). Two additional matrices are generated using the absolute values of matrix elements in each CSDM. Each of these four matrices is used as input data for supervised machine learning. Binary classification is performed by using RF, SVM, FNN, and CNN, and the obtained results are compared. All machine-learning algorithms show an accuracy of >95% for three types of input data—the pCSDM, mCSDM, and mCSDM with the absolute matrix elements. The CNN is the best in terms of low percent error. In particular, the result using the complex pCSDM is encouraging because these data-driven methods inherently do not require environmental information. This work demonstrates the potential of machine learning to discriminate between surface and underwater vessels in the ocean.

Spectral Empirical Orthogonal Function Analysis of Weather and Climate Data

Abstract We apply spectral empirical orthogonal function (SEOF) analysis to educe climate patterns as dominant spatiotemporal modes of variability from reanalysis data. SEOF is a frequency-domain variant of standard empirical orthogonal function (EOF) analysis, and computes modes that represent the statistically most relevant and persistent patterns from an eigendecomposition of the estimated cross-spectral density matrix (CSD). The spectral estimation step distinguishes the approach from other frequency-domain EOF methods based on a single realization of the Fourier transform, and results in a number of desirable mathematical properties: at each frequency, SEOF yields a set of orthogonal modes that are optimally ranked in terms of variance in the L2 sense, and that are coherent in both space and time by construction. We discuss the differences between SEOF and other competing approaches, as well as its relation to dynamical modes of stochastically forced, nonnormal linear dynamical systems. The method is applied to ERA-Interim and ERA-20C reanalysis data, demonstrating its ability to identify a number of well-known spatiotemporal coherent meteorological patterns and teleconnections, including the Madden–Julian oscillation (MJO), the quasi-biennial oscillation (QBO), and the El Niño–Southern Oscillation (ENSO) (i.e., a range of phenomena reoccurring with average periods ranging from months to many years). In addition to two-dimensional univariate analyses of surface data, we give examples of multivariate and three-dimensional meteorological patterns that illustrate how this technique can systematically identify coherent structures from different sets of data. The MATLAB code used to compute the results presented in this study, including the download scripts for the reanalysis data, is freely available online.