Propagation Formula Research Articles

Theoretical and computational study of a partially coherent laser beam in a marine environment

The propagation formula of a partially coherent Generalized Flattened Hermite-Cosh-Gaussian (GFHChG) beam in maritime atmospheric turbulence is derived with the help of the extended Huygens–Fresnel principle. In addition, the analytical expression for the beam width of a partially coherent GFHChG beam in the considered environment is investigated. From the numerical results based on the analytical formulae, we find that the analyzed beam can take different shapes of the profile, depending on the turbulence and beam parameters. And also, it can significantly resist turbulence with small wavelength and waist width values. On the other hand, when the medium becomes turbulent, the beam loses its characteristics and its resistance to fluctuations. Furthermore, the results reveal that the beam spreads more rapidly with the increase of the strength of turbulence, the outer scale size, and the decrease of the inner scale size. We should mention that the results gained represent a general form of numerous partially coherent laser beams such as Generalized Flattened Hermite Gaussian, Generalized Flattened Cosh-Gaussian, Hermite-Cosh-Gaussian, Cosh-Gaussian, Hermite-Gaussian and Gaussian Schell model beams.

Analyzing the spreading properties of vortex beam in turbulent biological tissues

Presenting the intensity development of a circular Laguerre-cosh-Gaussian (CLChG) beam in turbulent mouse biological tissues is the major goal of the current work. Using the power spectrum refractive index from Schmitt's model and the extended Huygens-Fresnel integral, the propagation formula of the CLChG beam is produced. In order to determine the spreading properties of the studied beam, analytical expressions of the CLChG beam's effective beam size in turbulent mouse biological tissues are constructed. Some graphical representations have to be carried out in order to discover the impacts of beam and biological turbulence parameters on this sort of beam. The findings show that the transformation of the CLChG beam into a Gaussian-like beam in the far field occurs more quickly when the beam passes through the deep dermis of the mouse. The shape of the CLChG beam can also be changed by choosing a specific value for the parameter linked to the cosh-part. Because the effective beam spot radius along the x- and y-axis are equal, we also see that the beam spot in biological tissues takes on a circular shape.

Closed-Form Error Propagation on $SE_{n}(3)$ Group for Invariant EKF With Applications to VINS

Pose estimation is important for robotic perception, path planning, etc. Robot poses can be modeled on matrix Lie groups and are usually estimated via filter-based methods. In this letter, we establish the closed-form formula for the error propagation for the Invariant extended Kalman filter (IEKF) in the presence of random noises and apply it to vision-aided inertial navigation. Moreover, we use the theoretic results to add the compensation parts for IEKF. We evaluate our algorithms via numerical simulations and experiments on the OPENVINS platform. Both simulations and the experiments performed on the public EuRoC MAV datasets demonstrate that our algorithm in particular parameters settings outperforms some state-of-art filter-based methods such as the quaternion-based EKF, first estimates Jacobian EKF, etc. The techniques of choice on the parameters are worth further investigating.

The propagation of relativistic jets in expanding media

ABSTRACT We present a comprehensive analytic model of relativistic jet propagation in expanding homologous media (ejecta). This model covers the entire jet evolution as well as a range of configurations that are relevant to binary neutron star mergers. These include low- and high-luminosity jets, unmagnetized and mildly magnetized jets, time-dependent luminosity jets, and Newtonian and relativistic head velocities. We also extend the existing solution of jets in a static medium to power-law density media with index α < 5. Our model provides simple analytic formulae (calibrated by 3D simulations) for the jet head propagation and breakout times. We find that the system evolution has two main regimes: strong and weak jets. Strong jets start their propagation immediately within the ejecta. Weak jets are unable to penetrate the ejecta at first, and breach it only after the ejecta expands significantly, thus their evolution is independent of the delay between the onset of the ejecta and the jet launching. After enough time, both strong and weak jets approach a common asymptotic phase. We find that a necessary, but insufficient, criterion for the breakout of unmagnetized (weakly magnetized) jets is $E_{j,{\rm iso,tot}} \gtrsim 3[0.4]\, {E_{ej,{\rm tot}}}\left({\, {\theta _{j,0}}}/{0.1{\rm ~rad}}\right)^2$, where Ej, iso, tot is the jet total isotropic equivalent energy, $\, {\theta _{j,0}}$ is its opening angle, and $\, {E_{ej,{\rm tot}}}$ is the ejecta energy. Applying our model to short gamma-ray bursts, we find that there is most likely a large diversity of ejecta mass, where mass ≲10−3 M⊙ (at least along the poles) is common.

Twisted Gaussian Schell-model breathers and solitons in strongly nonlocal nonlinear media.

Based on the Snyder-Mitchell linear model and the cross-spectral density (CSD) function, the analytical propagation formula of twisted Gaussian Schell-model (TGSM) beams in strongly nonlocal nonlinear medium (SNNM) is derived. Then the propagation characteristics of TGSM beam are studied. It is found that the soliton radius is jointly determined by the initial power, coherence length, and twist factor; the degree of spatial coherence is adjusted by changing the twist factor without affecting the soliton intensity. In the case of non-soliton properties, there is a threshold of coherence length which makes partially coherent beams have the same evolution law as completely coherent beams. Furthermore, increasing the twist factor, decreasing the coherence length and initial power can improve the beam quality of the beam propagating in SNNM.

Combined effects of crab dispersion and momentum dispersion in colliders with local crab crossing scheme

In this paper, we present the effects of linear transverse-longitudinal coupling on beam size at the interaction point (IP) of a collider with a local crab crossing scheme, when time-dependent transverse deflection (crab kicks) and dispersive orbit intertwine near IP. The analytic propagation formula and the closed orbit form of the crab dispersion and momentum dispersion are derived. The nonzero momentum dispersion at crab cavities and the nonideal phase from crab cavities to IP are detailed with the derived propagation formula to predict the beam size distortion at IP with or without the beam-beam interaction. The linear results are compared with nonlinear simulation using the weak-strong beam-beam code.

Dynamic quantum-enhanced sensing without entanglement in central spin systems

We propose a dynamic quantum sensing scheme by using a quantum many-spin system composed of a central spin interacting with many surrounding spins. Starting from a generalized Ising ring model, we investigate the error propagation formula of the central spin, and it indicates that Heisenberg scaling can be reached while the probe state only needs to be a product state. Particularly, we derive an analytical form of the dynamic quantum Fisher information in a limit case, which explicitly exhibits the Heisenberg scaling. By comparing with numerical results, we demonstrate that the general case can be well approximated by the analytical result when the coupling strength among the surrounding spins is much weaker than the coupling strength between the central and surrounding spins. This analytic result guides us to find the appropriate probe state and the proper measurement time to achieve the Heisenberg scaling in realistic situations. Furthermore, we investigate various effects which are important in practical quantum systems, including the central spin Zeeman term, the anisotropy of the hyperfine interaction and the inhomogeneity of the hyperfine coupling strength. Our result indicates that the dynamic quantum-enhanced sensing scheme seems feasible in realistic quantum central spin systems, like semiconductor quantum dots.

Time domain elastic-wave full waveform inversion based on first-order approximate instantaneous frequency

The weak signals of artificial seismic records contain the subsurface medium information that is required in the inversion. But in the full waveform inversion (FWI), the weak signals contribute less to the objective functions. Therefore, how to improve the contribution of the weak signals in the objective functions of FWI is the problem that needs to be solved urgently. The research shows (Ren, D. 1980. Preliminary research on seismic record and instantaneous frequency. Oil Geophysical Prospecting 15 no. 1: 7–21) that instantaneous frequency attributes, which are very sensitive to the changes in subsurface velocity, have the potential to extract the weak signals from the seismic records. However, this frequency can only be estimated from the complex seismic signals. Empirical mode decomposition (EMD) method has been widely used in signal analysis so as to estimate the instantaneous frequency, but it is difficult to be applied in FWI due to the huge computation. In order to solve this problem, the instantaneous frequency is replaced with the first-order approximation of the exponential frequency in FWI. In this paper, the objective functions of the first-order approximate exponential frequency FWI (FRE-EFWI) in elastic waves and the source terms of its back propagation formula were derived. Besides, the FRE-EFWI method was proved to improve the contribution of the weak signals in the objective functions of FWI. In addition, the correctness and effectiveness of the method were demonstrated by the examples of FWI.

Propagation Properties and Application of Radially Polarized Partially Coherent Twisted Vortex Hermite‐Gaussian Schell‐Model Beams

AbstractA class of radially polarized partially coherent twisted vortex Hermite‐Gaussian Schell‐model (RPPCTVHGSM) beams is introduced. The analytical propagation formula for the cross‐spectral density matrix of this beam family is derived. It is found the component of the spectral density and the degree of polarization show twist effects upon propagation. Then, such focused beam family's feasibility to accomplish stable 3D optical capture on both low‐ and high‐index particles is theoretically and numerically analyzed. The results show that utilizing the nonzero order RPPCTVHGSM beam has no limit on the refractive index of particles. Compared with the center of the geometric focal plane, real equilibrium points will shift. Finally, the exact offset of the genuine capture point in each direction is obtained by taking the second‐order RPPCTVHGSM beam as an example. The applications of the work are envisioned in material surface processing and tracking mobile military.

On some regularity properties for the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov Equation

This work aims to study some smoothness properties concerning the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation. More precisely, we prove that the solutions to this model satisfy the so-called propagation of regularity. Roughly speaking, this principle states that if the initial data enjoys some extra smoothness prescribed on a family of half-spaces, then the regularity is propagated with infinite speed. In this sense, we prove that regardless of the scale measuring the extra regularity in such hyperplane collection, then all this regularity is also propagated by solutions of this model. Our analysis is mainly based on the deduction of propagation formulas relating homogeneous and non-homogeneous derivatives in certain regions of the plane.

Evolution properties of vortex beams through strongly nonlocal nonlinear media

The evolution properties of a vortex Hermite cosine-hyperbolic-Gaussian beam (vHChGB) propagating in a strongly nonlocal nonlinear media (SNNM) are examined theoretically based on the Snyder-Mitchell model. Mathematical and numerical calculations are performed to illustrate the propagation formula, the second-order intensity moment beam width and the curvature radius of an on-waist incident vHChGB in SNNM. The results reveal that under general conditions, the vHChGB evolves periodically upon propagation in SNNM, and the beam shape is strongly dependent on the power and the parameters of the incident beam. The beam evolution during propagation is generally breather-like, and when the incident beam power equals to the critical power value, the beam behaves like a soliton. The evolution behaviors of vHChGB in SNNM are well demonstrated, which could have promising applications in optical switch and micro-manipulations.

A tutorial introduction to quantum stochastic master equations based on the qubit/photon system

From the key composite quantum system made of a two-level system (qubit) and a harmonic oscillator (photon) with resonant or dispersive interactions, one derives the corresponding quantum Stochastic Master Equations (SME) when either the qubits or the photons are measured. Starting with an elementary discrete-time formulation based on explicit formulae for the interaction propagators, one shows how to include measurement imperfections and decoherence. This qubit/photon quantum system illustrates the Kraus-map structure of general discrete-time SME governing the dynamics of an open quantum system subject to measurement back-action and decoherence induced by the environment. Then, on the qubit/photon system, one explains the passage to a continuous-time mathematical model where the measurement signal is either a continuous real-value signal (typically homodyne or heterodyne signal) or a discontinuous and integer-value signal obtained from a counter. During this derivation, the Kraus map formulation is preserved in an infinitesimal way. Such a derivation provides also an equivalent Kraus-map formulation to the continuous-time SME usually expressed as stochastic differential equations driven either by Wiener or Poisson processes. From such Kraus-map formulation, simple linear numerical integration schemes are derived that preserve the positivity and the trace of the density operator, i.e. of the quantum state.

Evaluation of pressure and species concentration measurement using uncertainty propagation

This paper presents a probabilistic uncertainity evaluation method as described in the Guide to the Expression of Uncertainty in Measurements (GUM) and its application to probe measurements on pressure and fuel concentration. All sources of unceratinties are expressed as probability distributions. Consequently, the overall standard uncertainty of the quantity can be calculated using the Gaussian error propagation formula. The result of the uncertainty evaluation yields the most probable value of the measurand and describes its distribution in terms of rectangular (standard uncertainty) or gaussian (“expanded” uncertainty) distribution. A pitot-static probe and a fuel-concentration stem probe are used in order to demonstrate the principle of the probabilistic uncertainty evaluation method. The uncertainty induced by the pressure and concentration data acquisition system as well as the calibration of the fuel-concentration probe are included in the analysis. The overall “expanded” uncertainties for the measured and calculated values are presented as a function of different inlet fuel flows. In addition to this, the individual sources of uncertainty to the overall standard uncertainty are presented and discussed. Moreover, the transformation of standard uncertainty to “expanded” uncertainty will provide the deviation of the measurement in a 95% or 99% normal distributed interval instead of a 67% rectangular distributed interval.

A probabilistic uncertainty evaluation method for turbomachinery probe measurements

The paper presents a probabilistic uncertainty evaluation method described in the Guide to the Expression of Uncertainty in Measurement (GUM) [1] and its application to the field of Turbomachinery probe measurement techniques. All sources of uncertainties contributing to a measurement result are expressed in terms of probability distributions. Consequently, the overall standard uncertainty of the result can be calculated using the Gaussian error propagation formula. The result of the uncertainty evaluation yields the most probable value of the measurand and describes its distribution in terms of standard or extended uncertainties. A pneumatic five-hole-probe measurement technique has been chosen to show the principle of the probabilistic uncertainty evaluation method. The complete signal chain including the probe calibration, the modeling and the application in the turbine has been included in the analysis. The overall uncertainties of the measured flow angles and flow total and static pressures are presented as a function of the flow Mach number. In addition, the contribution of the individual sources of uncertainty to the overall standard uncertainty is shown. Based on this break down of uncertainties optimization options of the measurement chain are suggested.

An analytical framework for local and global system kinematic reliability sensitivity of robotic manipulators

This paper develops a novel analytical framework for system kinematic reliability sensitivity analysis of robotic manipulators, which can provide the analytical results of local and global reliability sensitivity defined on the pose and position errors. First, uncertainty analysis of the pose error of the end-effector is accomplished by virtue of the second-order closed-form error propagation formula on motion groups. Then, the system kinematic reliability, namely the probability of the system kinematic error located within the prescribed safe boundary, is analytically calculated. Specifically, the non-central chi-square approximation and expectation propagation technique are employed to compute the system kinematic reliability defined on the position and pose errors, respectively. On this basis, the local and global system kinematic reliability sensitivity of robotic manipulators are analytically obtained. Finally, the effectiveness of the proposed method is validated by comparison with the Monte Carlo simulation and Kriging modeling method. The results indicate the proposed method has good efficacy and efficiency in system kinematic reliability sensitivity analysis for robotic manipulators.

On the Probability and Automatic Search of Rotational-XOR Cryptanalysis on ARX Ciphers

Abstract Rotational-XOR cryptanalysis is a very recent technique for ARX ciphers. In this paper, the probability propagation formula of RX-cryptanalysis in modular addition is extended, and the calculation of RX-difference probability for any rotation parameter ($0<k<n$) can be realized. By proposing a concept of RX-offset and constructing the corresponding distribution table, the propagation of RX-difference in modular addition can be derived from the propagation of XOR-difference. Combined with the improvement of the automatic search tool for XOR-differential characteristics of ARX ciphers, we only need to add one more operation in each round, i.e. traverse the possible value of RX-offset and XOR it with the output XOR-difference of modular addition, thus it can achieve the search for RX-differential characteristics. With this method, the RX-differential distinguisher of ARX-C primitives without or with linear key schedule can be searched. For the applications, we have obtained the third-party RX-cryptanalysis results for Alzette and CHAM for the first time as far as we know.

Study on early warnings of strategic risk during the process of firms’ sustainable innovation based on an optimized genetic BP neural networks model: Evidence from Chinese manufacturing firms

Strategic risk is an inevitable question of reality, which leads to a significant impact that negatively affects firms’ overall development, even threatening their survival. Most of extant studies on early warning models merely take the financial factors into account, whereas strategic risk pertaining to firms involves a diverse group of non-financial factors. In addition, the growing attention attached to the sustainable development of ecological and social environment has raised specific concerns to manufacturing firms, which need especially manage the risks generated from the process of sustainable innovation. However, scholars have paid far less attention to early warning models that comprehensively address the above problems. It is crucial to facilitate manufacturing firms to timely detect strategic risk during the process of sustainable innovation, whereas the existing early warning models are not reliable enough to warn the potential risks. Therefore, this study sets out to introduce the sustainable risk into the early warning indicators system of strategic risk. Then, Chinese manufacturing firms are applied to construct an early warning model which based on the back-propagation (BP) neural network optimized by genetic algorithm (GA). The results provide evidence that the proposed model shows a better fitting effect, higher prediction precision and higher convergence speed when compared with conventional models. In addition, the weights of the early warning model obtained from the training are substituted into the forward propagation formula, which enables to determine the relative importance of risk factors. It is of great significance for the firms to prioritize risk management actions. Furthermore, the new model enables manufacturing firms to warn the strategic risk during the process of sustainable innovation while striking a balance among economic, environmental, and social performances.

Analytical investigation into error propagation of power spectral density transmissibility (PSDT) based on coherence function

Power spectral density transmissibility (PSDT) has attracted widespread interests because it is insensitive to the nature of excitation and has no demand for measuring the input. The classical spectral estimation procedure is usually subject to measurement and spectral leakage errors, which might cause distortion in the estimation of PSDT. This study investigates how the error of PSD estimation propagates into PSDT estimation based on analytical approximation formulas. Based on perturbation theory in tandem with statistical moments of the ratio of random variables, both first- and second-order asymptotic expressions of the mean and variance of PSDT estimation are derived in terms of coherence functions, PSDT measurements, and the number of averages. Given these theoretical findings, it is further revealed that the variance of PSDT estimation around the system poles approaches zero, and the variances of the magnitude and phase of the mode shape decrease to local minima. The performance of the analytical error propagation formulas is validated through numerical and field test data. Parametric studies into the effects of recording durations, window types, and reference outputs on the statistics of PSDT estimation and mode shape estimation are also conducted.

Influence of inhomogeneous stochasticity on the falsifiability of mean-field theories and examples from accretion disc modelling

ABSTRACT Despite spatial and temporal fluctuations in turbulent astrophysical systems, mean-field theories can be used to describe their secular evolution. However, observations taken over time scales much shorter than dynamical time scales capture a system in a single state of its turbulence ensemble. Comparing with mean-field theory can falsify the latter only if the theory is additionally supplied with a quantified precision. The central limit theorem provides appropriate estimates to the precision only when fluctuations contribute linearly to an observable and with constant coherent scales. Here, we introduce an error propagation formula that relaxes both limitations, allowing for non-linear functional forms of observables and inhomogeneous coherent scales and amplitudes of fluctuations. The method is exemplified in the context of accretion disc theories, where inhomogeneous fluctuations in the surface temperature are propagated to the disc emission spectrum – the latter being a non-linear and non-local function of the former. The derived precision depends non-monotonically on emission frequency. Using the same method, we investigate how binned spectral fluctuations in telescope data change with the spectral resolving power. We discuss the broader implications for falsifiability of a mean-field theory.

Random error analysis of normalized Fourier coefficient in dual-rotating compensator Mueller matrix ellipsometer

Ever since the dual-rotating compensator Mueller matrix ellipsometer (DRC-MME) was developed, improving its measurement accuracy has been an important research topic. Studies indicate that the measurement errors in the Mueller matrix of a DRC-MME, including systematic and random errors, significantly influence the measurement accuracy. Accordingly, this study focuses on random errors. A numerical formula for the random errors of normalized Fourier coefficients was derived and applied to estimate the random errors of the measured Mueller matrix. The formula was validated through simulations and experiments. Furthermore, the use of a random error propagation formula to minimize effect of random error on the final measurement results by setting appropriate configurations, including those of software and hardware, was demonstrated. Finally, for real-time measurements, an improved merit function for nonlinear regression algorithms (such as by Levenberg–Marquardt) is proposed.