Analytical Propagation Research Articles

Partially coherent sources with the combined quadratic phase

We construct a class of partially coherent sources with a new phase, which is consisted of separable and inseparable quadratic phases. As examples, the Gaussian distribution and rectangular flat-topped Gaussian distributions are selected as weight functions, respectively. The analytical propagation formulas of anisotropic Gaussian Schell-model (AGSM) beams and rectangular multi-Gaussian Schell-model (RMGSM) beams with the combined quadratic phase are derived. We investigate the propagation characteristics of the AGSM beam and the RMGSM beam carrying the combined quadratic phase. The twist effects of the combined quadratic phase on the spectral density (SD) and the degree of coherence (DOC) of the AGSM beam are quantitatively analyzed, and a specific method for calculating the critical value of the phase parameter is proposed. In addition, we qualitatively analyze the twist effects of the combined quadratic phase on the SD and the DOC of the RMGSM beam. We find that the propagation of the RMGSM beam is similar to that of the AGSM beam. These results may provide new inspiration for applications of synthesizing rotating beams in optics.

A statistically coherent robust multidimensional classification scheme for water

Serious limitations of the existing water classification schemes prompted us to propose a new statistically coherent water nomenclature system. An extensive database of ionic charge-balanced concentrations of 8 elements (4 cations Ca, Mg, Na, and K; and 4 anions SO4, Cl, HCO3, and CO3), in 46,292 multivariate outlier-free simulated samples, was used for training the multidimensional classification system. The initial assignment for 16 classes was achieved from the greater molar concentration concept of each cation and anion, called the Greater molar conc model. Seven hybrid log-ratios (hlr) from 8 elemental concentrations were used for linear discriminant analysis (LDA) and canonical analysis to propose 16 multidimensional discriminant functions from the 7 hlr model. The LDA and canonical analysis were also performed on the initial molar concentrations of 7 elements, without any log-transformation, which was designated as the 7 M conc model. The robustness of these three classification systems (7 hlr, 7 M conc, and Greater molar conc) was tested against analytical uncertainty propagation and mineral-water interaction effects. The 7 hlr model, due to its higher robustness, was considered as the best option for the nomenclature of the 16 types of water. From the probability concept, it was possible to identify hybrid water types, along with the basic or primary types of water. Our water classification scheme (7 hlr model under the “basic+hybrid” option) can classify as many as 256 different classes of water. Due to the clearly high complexity of the proposed classification scheme, we developed a new online computer program WaterMClaSys_LDA (Water Molar Classification System from Linear Discriminant Analysis) available at our web portal http://tlaloc.ier.unam.mx, for use by anyone after registration and log-in. The usefulness of the new classification scheme is illustrated by applications to groundwater, lake water, and geothermal water samples from South India, Mongolia, and western Turkey, respectively.

Analytical modeling of fatigue crack propagation based on cyclic hardening and a characteristic damage length

An analytical fatigue crack growth model is developed, which combines effectively a cyclic energy dissipation approach and the strain energy density theory. The model uses cyclic material properties and relies on cyclic hardening for fatigue crack growth rate prediction. Fatigue crack propagation occurs incrementally by a characteristic damage length, which is in the length scale of striation spacing in commercial aluminum alloy, supporting the model’s physical background. Analytical crack propagation rates correlate well with existing experimental data, in material with uniform strength and strength gradient, showing the ability of the model to account for strength variation at the crack path.

Computational modeling of charge hopping dynamics along a disordered one-dimensional wire with energy gradients in quantum environments.

This computational study investigates the effects of energy gradients on charge hopping dynamics along a one-dimensional chain of discrete sites coupled to quantum bath, which is modeled at the level of Pauli master equation (PME). This study also assesses the performance of different approximations for the hopping rates. Three different methods for solving the PME, a fourth order Runge-Kutta method, numerical diagonalization of the rate matrix followed by analytic propagation, and kinetic Monte Carlo simulation method, are tested and confirmed to produce virtually identical values of time dependent mean square displacement, diffusion constant, and mobility. Five different rate expressions, exact numerical evaluation of Fermi's Golden Rule (FGR) rate, stationary phase interpolation (SPI) approximation, semiclassical approximation, classical Marcus rate, and Miller-Abrahams rate, are tested to help understand the effects of approximations in representing quantum environments in the presence of energy gradients. The results based on direct numerical evaluation of FGR rate exhibit transition from diffusive to non-diffusive behavior with the increase in the gradient and show that the charge transport in the quantum bath is more sensitive to the magnitude of the gradient and the disorder than in the classical bath. Among all the four approximations for the hopping rates, the SPI approximation is confirmed to work best overall. A comparison of two different methods to calculate the mobility identifies drift motion of the population distribution as the major source of non-diffusive behavior and provides more reliable information on the contribution of quantum bath.

New multidimensional classification scheme of altered igneous rocks from performance comparison of isometric and modified log-ratio transformations of major elements

Nomenclature in any science is of fundamental importance. From performance comparison of isometric and modified or hybrid log-ratio transformations of major elements, we present a complete multidimensional classification scheme for altered igneous rocks for High-Mg rock varieties and magma and rock types. This scheme is efficiently applicable through a computer package called Multidimensional Igneous rock Classification System (MICSys) that encapsulates four new online computer modules (HMgClaMSys_ilr, MagClaMSys_mlr, and IgRoClaMSys_mlr and a robustness evaluation module). We used a major element database of multinormally distributed 33868 samples of relatively fresh igneous rocks and compared the performance of isometric log-ratio and modified log-ratio transformations, both of which were multinormally distributed in terms of the 9 log-transformed variables as confirmed by the DOMUDaF program. The system must be trained from only fresh rocks because the prior class assignment requires an efficient scheme accepted by all, which is available only for fresh rocks. The multivariate technique of linear discriminant analysis (LDA) and canonical analysis applied to the two transformations provided the same percent success or correct classification for the respective databases. With these findings, we reinforce the recently published work on the International Union of Geological Sciences (IUGS)-consistent multidimensional classification of altered igneous rocks. The numerous tests and applications documented earlier on multidimensional classification were applied in the present work. The mathematical properties of log-ratio transformations are no longer important, provided the multidimensional technique involves the LDA and canonical analysis. MICSys available for online application at the web portal http://tlaloc.ier.unam.mx , was successfully applied to 6 new tests and 11 new applications to illustrate the usefulness of our multidimensional classification system to igneous rocks altered from post-emplacement compositional changes. Robustness of the new scheme for altered igneous rock nomenclature as compared to the conventional IUGS procedure for fresh rock nomenclature was confirmed from simulation of post-emplacement alteration of all elements in 28 different rock compositions representative of all rock types. Even greater robustness was demonstrated against analytical uncertainty propagation in 14 geochemical reference materials issued from Canada (3), France (3), Japan (5), and the USA (3). The estimation of robustness against post-emplacement changes and uncertainty propagation was facilitated by the incorporation of a fourth module in MICSys. This work, therefore, clearly shows that, for altered rocks, the new multidimensional scheme should be used, instead of the conventional classification meant for fresh rocks only.

Analytic HG-mode propagation through circular apertures with Zernike phase offset

Mode expansion methods (MEMs) have been used in both the scientific and commercial world to accurately propagate monochromatic light fields. Compared to classical methods for light propagation from scalar diffraction theory, it is often less costly and much more easily implemented into optical chains. We provide an analytic expression for the overlap between incoming and outgoing Hermite Gauss (HG) TEM modes over an aperture with potential wavefront error represented by Zernike terms for the purpose of propagating light through apertures. We show agreement between our results and those of prior studies of the topic, as well as an application of this method to an ongoing interferometry mission.

Soft x-ray spectrum unfold of K-edge filtered x-ray diode arrays using cubic splines.

Cubic spline interpolation is able to recover temporally and spectrally resolved soft x-ray fluxes from an array of K-edge filtered x-ray diodes without the need for a priori assumptions about the spectrum or the geometry of the emitting volume. The mathematics of the cubic spline interpolation is discussed in detail. The analytic nature of the cubic spline solution allows for analytical error propagation, and the method of calculating the error for radiation temperature, spectral power, and confidence intervals of the unfolded spectrally resolved flux is explained. An unfold of a blackbody model demonstrates the accuracy of the cubic spline unfold. Tests of cubic spline performance using spectrally convolved detailed atomic model simulation results have been performed to measure the method's ability to conserve spectral power to within a factor of 2 or better in line-dominated regimes. The unfold is also demonstrated to work when information from the x-ray diode array is limited due to high signal-to-noise ratios or the lack of signal due to over-attenuation or over-filtration of the x-ray diode signal. The robustness of the unfold with respect to background subtraction and raw signal processing, signal alignment between diode traces, limited signal information, and initial conditions is discussed. Results from an example analysis of a halfraum drive are presented to demonstrate the capabilities of the unfold in comparison with previously established methods.

A new statistically-based methodology for variability assessment of rheological parameters in mineral processing

If variability of input data for rheological measurements is not adequately included, their associated uncertainty and subsequent modelling can be underrated. Mineral pulp rheology determination is commonly done through triplicate tests, with such variability reported as multiples of a standard deviation, with the potential for underestimation. In the present work, a novel statistically-based methodology for the estimation of uncertainty in the rheological characterization of mineral suspensions —and other parametric models— is proposed. From the variability of the experimental measurements and the analytical propagation of errors, a set of rheological profiles are generated using Monte Carlo simulations within a variability frame. The corresponding inverse problem for curve-fitting is solved individually, resulting in distributions of fitted parameters, which were statistically-analyzed to obtain representative values for both the parameter and its true variability. The methodology proposed herein has been used to explore the applicability and limitations of the Herschel-Bulkley and Bingham models under specific experimental and data analysis protocols, where the relevance of including low-shear-rate measurement points or yield stress measurements using alternative methods is exposed. Additionally, we present a case study on the effect of the concentration of NaCl on the rheological response of synthetic tailings consisting of quartz suspensions doped with kaolinite, bentonite and kaolinite-bentonite blends, using the proposed methodology with a concentric cylinder rheometer. Results show predominantly decreasing trends in yield stress as salt concentration increases, with non-monotonical behavior and strongest variability associated to the quartz-bentonite blend.

Precise line-of-sight modelling for angles-only relative navigation

This work presents a precise analytical model to reconstruct the line-of-sight vector to a target satellite over time, as required by angles-only relative navigation systems for application to rendezvous missions. The model includes the effects of the geopotential, featuring: the analytical propagation in the mean relative orbital elements (up to second-order expansion), the analytical two-way osculating/mean orbital elements’ conversion (second-order in J2 and up to a given degree and order of the geopotential), and a second-order mapping from the perturbed osculating elements’ set to the local orbital frame. Performances are assessed against the line-of-sight reconstructed out of the precise GPS-based positioning products of the PRISMA mission. The line-of-sight modelled over a far-range one day long scenario can be fitted against the true one presenting residuals of the order of ten arc-seconds, which is below the typical sensor noise at far-range.

Spiraling interaction of multi-mode solitons in self-induced harmonic potentials

We investigate the propagation and spiraling interaction of multi-mode solitons in self-induced harmonic potentials. The general analytical propagation expression for the interaction of the multi-mode solitons is derived. The interaction characteristics, such as the evolution of the intensity of the combined optical field, the spacing of each constituent soliton to the longitudinal axis, the spiraling angle and velocity, the propagation trajectory, and the phase shift are discussed with detail. We show that it is possible to control the spiraling interacting trajectories of the solitons by carefully designing the input parameters. It is proved that the motion state of each soliton satisfies the conservation of mechanical energy. The results may have potential applications in all-optical information processing and particle control.

Inertial gravity current in rectangular channels over a porous bottom: Asymptotic solutions

We consider a high-Reynolds-number Boussinesq gravity current propagating in a channel above a permeable horizontal boundary. The current (of reduced gravity g′) is released from a rectangular lock (of length x0 and height h0), and after an adjustment (slumping) stage is expected to enter into a similarity stage, on which we focus here, using a thin-layer shallow-water model. The classical analytical self-similar propagation solution predicts that the length of the current is given by xN(t)=Kt2∕3 (where K is a constant and t is time from release). The height h(x,t) and velocity u(x,t) display a similarity shape of the variable y=x∕xN(t),y∈[0,1] (x is the physical coordinate measured from the backwall of the lock). This solution, which is very useful in the analysis of gravity-current problems, is invalidated by the drainage effect into the porous bottom. Here we extend the classical similarity (basic) solutions by developing a perturbation (asymptotic) expansions about the basic solution. The expansion uses the small parameter λ which represents the ratio of the typical propagation time T=x0∕(g′h0)1∕2 to the drainage time tB (a given property of the porous bottom). The perturbation terms can be calculated analytically, and we present the results of the first-order correction. This provides useful insights about the influence of the porous boundary, as compared with the classical similarity behavior: xN(t) is shorter, the profile of u(y) is deflected to lower values at the nose, and h(y) is reduced mostly at the tail. The deviation from the basic similarity solution increases like λt. In addition, we show that the drainage influence is important in reducing the transition length from the inertial (inviscid) to the viscous regimes. We compared the analytical asymptotic leading-order solution with numerical finite-difference results for various values of λ, and found excellent qualitative agreement and fair quantitative agreement. We expect that higher-order terms will improve the accuracy of the new solution, but this additional extension was not performed in this work.

Propagation properties of partially coherent anomalous hollow beams in quadratic-index media

Based on the generalized diffraction integral formula, analytical propagation expressions for a partially coherent anomalous hollow beam (AHB) propagating through a quadratic-index medium are derived. The distributions of intensity and Poynting vector for a partially coherent AHB through the quadratic-index medium are numerically and theoretically studied. It is found that the intensity and Poynting vector of the partially coherent AHB take on a periodical change. The evolution properties of the partially coherent AHB in a quadratic-index medium are closely relevant to its transverse coherence width and the quadratic-index parameter of media.

Spatially truncated Gaussian pulsed beam and its application in modeling diffraction of ultrashort pulses from hard apertures.

A new kind of pulsed beam, which we call a spatially truncated Gaussian pulsed beam, is defined to represent a Gaussian pulsed beam that is diffracted from a semi-infinite hard aperture. The analytical equations for the propagation of the spatially truncated Gaussian pulsed beam through a nonrotationally symmetric paraxial system with second-order dispersion is derived starting from the generalized spatiotemporal Huygens integral. The spatially truncated Gaussian pulsed beam is then combined with the conventional Gaussian pulsed beam decomposition method to enable the modeling of diffraction of a general ultrashort pulse from an arbitrarily shaped hard aperture. The accuracy of the analytical propagation equation derived for the propagation of the truncated Gaussian pulsed beam is evaluated by a numerical comparison with diffraction results obtained using the conventional pulse propagation method based on the Fourier transform algorithm. The application of the modified Gaussian pulsed beam decomposition method is demonstrated by propagating an ultrashort pulse after a circular aperture through a dispersive medium and a focusing aspherical lens with large chromatic aberration.

FDTD Simulation of Dispersive Metasurfaces With Lorentzian Surface Susceptibilities

A Finite-Difference Time-Domain (FDTD) simulation of broadband electromagnetic metasurfaces based on direct incorporation of Generalized Sheet Transition Conditions (GSTCs) into a conventional Yee-cell region has been proposed for arbitrary wave excitations. This is achieved by inserting a zero thickness metasurface inside bulk nodes of the Yee-cell region, giving rise to three distinct cell configurations - Symmetric Cell (SC), Asymmetric Cell (AC) and Tight Asymmetric Cell (TAC). In addition, the metasurface is modelled using electric and magnetic surface susceptibilities exhibiting a broadband Lorentzian response. As a result, the proposed model guarantees a physical and causal response from the metasurface. Several full-wave results are shown and compared with analytical Fourier propagation methods showing excellent results for both 1D and 2D field simulations. It is found that the TAC provides the fastest convergence among the three methods with minimum error.

Influence of geomagnetic field models on analytical propagator for SPIN stabilized satellites

Influence of geomagnetic field models on analytical propagator for SPIN stabilized satellites

Spin-orbital coupling of quadratic-power-exponent-phase vortex beam propagating in a uniaxial crystal.

Recent studies have shown that quadratic-power-exponent-phase (QPEP) vortex and modified QPEP vortex have some novel properties and potential applications in optical manipulation, orbital angular momentum (OAM) communication, OAM multicasting and so on. In these applications, there may be potential need of processing these kinds of beams by using uniaxial crystals. In this paper, the analytical propagation equations of Gaussian QPEP vortex and modified QPEP vortex propagating in uniaxial crystals are derived and the evolution of the angular momentum via spin-orbital coupling during the propagation is investigated. This may be meaningful for guiding and promoting the applications of the QPEP vortex and modified QPEP vortex.

Application of Improved AHP-BP Neural Network in CSR Performance Evaluation Model

The evaluation of corporate social responsibility (CSR) performance may enhance companies’ willingness to undertake social responsibilities, so it is very important to improve the quality of CSR performance evaluation. Based on the three factors of economic performance, social performance and environmental performance, this paper proposed an improved analytic hierarchy process-back propagation (AHP-BP) neural network algorithm, and introduced the improved AHP-BP neural network algorithm into CSR performance evaluation model. In the stage of improved AHP, the model included the importance of the knowledge and experience of the experts by expert scoring, and reduced the subjective influence of expert judgment on the results by introducing a personality test scale. In the stage of BP neural network, trained models have been used for CSR performance evaluation. The results showed that the prediction result of improved AHP-BP neural network model was better than that of BP neural network model. Therefore, the improved AHP-BP neural network algorithm can be used as a good predictor for CSR performance evaluation.

Propagation of an Airy-Gaussian beam passing through a lens with low effective Fresnel number

We derive an analytical propagation expression for the Airy-Gaussian beam propagating through a thin lens, and calculate numerical examples to investigate its focusing properties. It is shown that the pattern formed in the focus plane can be controlled to demonstrate various geometries through tuning the beam parameters associated with the Airy function part. Moreover, we investigate the influence of the Fresnel number on the propagation properties, and find that decreasing values of the Fresnel number not only increase the size of the focusing spot, but also enhance the depth of field effectively. Our results provide a route for generating perfect squared or elliptic spot with extended depth of field through carefully adjusting the beam and optical system parameters, which may find some potential applications in various areas involving in the special beam profile.

A risk-based approach to determination of optimal inspection intervals for buried oil pipelines

Corrosion is a significant concern causing tremendous losses to all pipeline operators. To combat this operational issue, new methods and tools are needed to analyze and model degradation, to predict failure, and finally to develop strategies for prevention, control, and mitigation of corrosion in pipelines. A practical inspection and maintenance program is crucial to prevent pipeline failures due to corrosion. Risk-based inspection (RBI) is an increasingly popular and trusted method to assess and develop inspection plans. However, the determination of optimal inspection intervals is still challenging in RBI. The present study aims to develop a dynamic Bayesian network (DBN)-based approach for optimization of inspection intervals. Based on inline inspection data and analytical corrosion propagation models, DBN is applied for the estimation of both the internal and external corrosion damage as well as the probability of failure (PoF). The cost of failure (CoF) is estimated based on typical cost categories relevant to pipeline accidents. Risk is calculated as the product of PoF and CoF. A utility function to combine both the risk and the annual cost of the inspection program is also developed. The optimal interval can be found based on the curve of the utility function. The proposed approach is demonstrated through a real-world case study on an operating pipeline.

The spectral properties of a partially coherent Lommel-Gaussian beam in turbulent atmosphere

Based on the extended Huygens–Fresnel principle, the analytical propagation equations of a partially coherent Lommel-Gaussian (PCLG) beam propagating in atmospheric turbulence have been introduced. The spectral properties of the PCLG beam in turbulent atmosphere are explored. It is found by numerical simulations that after propagating in turbulent atmosphere, the intensity distribution and spectral degree of coherence (SDOC) of the PCLG beam vary on propagation. The intensity distribution of the PCLG beam depends closely on the strength of atmospheric turbulence and the parameters of the source beam, and will eventually transform into a Gaussian profile at the receiver plane. In addition, with smaller OAM quantum number and structure constant or bigger wavelength, the SDOC of the PCLG beam spreads more rapidly. The results are expected to be useful for understanding the properties of the Lommel-Gaussian beam in turbulence.