Statistical modelling of the backscattered field in a one-dimensional non-stationary stochastic problem

Within the framework of an exact wave approach in the spatial-time domain, the one-dimensional stochastic problem of sound pulse scattering by a layered random medium is considered. On the basis of a unification of methods which has been developed by the authors, previously applied to the investigation of non-stationary deterministic wave problems and stochastic stationary wave problems, an analytical-numerical simulation of the behaviour of the backscattered field stochastic characteristics was carried out. Several forms of incident pulses and signals are analysed. We assume that random fluctuations of a medium are described by virtue of the Gaussian Markov process with an exponential correlation function. The most important parameters appearing in the problem are discussed; namely, the time scales of diffusion, pulse durations, the medium layer thickness or the largest observation time scale in comparison with the time scale of one correlation length for the case of a half-space. An exact pattern of the pulse backscattering processes is obtained. It is illustrated by the behaviour of the backscattered field statistical moments for all observation times which are of interest. It is shown that during the time interval when the main part of the pulse energy leaves the medium, the backscattered field is a substantially non-stationary process, having a non-zero mean value and an average intensity that decays according to a power law. There are various power indices for the different duration incident pulses, however, they are not the same as those of previous papers, which were obtained on the basis of an approximate and asymptotic analysis. We have also verified that the Gaussian law is valid for the probability density function of the backscattered field in the case of any incident pulse duration.     

The Influence of the Initial-Signal Duration on Nonstationary Reflections in a Random Medium

Using the invariant imbedding method, we study numerically the statistical characteristics of backscattering of plane-wave pulses incident normally on a random one-dimensional medium. The local reflection coefficient of the medium is simulated by a centered Gaussian process with an exponential correlation function. We consider the time dependences of the average intensity and the kurtosis coefficient of reflections for various durations and carrier frequencies of the initial signal. It is shown that the attenuation and fluctuations of reflections increase with the initial-signal duration normalized to the time scale of the medium irregularities and depend on the signal carrier. Numerical results for the average intensity are compared with previously obtained analytical solutions. We discuss the appearance conditions of a wave stochastic resonance in nonstationary reflections.     

Propagation of elliptically polarized laser pulses in isotropic gyrotropic medium with relaxation cubic nonlinearity and anomalous frequency dispersion

The self-action of elliptically polarized Gaussian laser pulses in an isotropic gyrotropic medium with an anomalous frequency dispersion and cubic Kerr nonlinearity with a finite relaxation time on the order of the pulse duration is numerically studied. It is shown that, at the output of the medium, the pulse polarization nonmonotonically varies with time. The main peak of the pulse is additionally delayed compared to the time of passing the linear medium; the value of this delay significantly depends on the polarization of the incident pulse and achieves a maximum for incident pulses whose degree of ellipticity is equal to the ratio of the material constants characterizing the local and nonlocal nonlinear optical response of the medium. It seems promising to search for possible differences in relaxation times depending on the intensity of additions to the refractive indices of the right and left circularly polarized waves by investigating the time dependence of polarization characteristics at the output of the medium.     

The transient scattering mechanism of dipole array with reflector

The transient backscattering mechanisms of a dipole array with reflector have been investigated from different aspects： time-domain, frequency-domain, and combined time-frequency domain, using 4 × 8 dipole arrays with reflector as an example. The data of scattering from the arrays under the incidence of Gaussian pulses are obtained by finite differential time domain method. The influences of the array structural parameters, incident wave parameters, and incident angles on the waveforms, spectrum, and time-frequency representations of the backscattered fields of the arrays are analysed and conclusions are drawn. From these characteristics and conclusions, it is possible to deduce the array structure inversely from the backscattered field.     

Effect of a pulse entrapping on weak localization of waves in a resonant random medium

We consider theoretically a new physical effect in coherent backscattering enhancement (CBE) of electromagnetic or acoustic non-stationary waves from a discrete random medium under condition of Mie resonant scattering. The effect manifests itself as an angle-cone broadening of a short pulsed signal CBE from the resonant random medium, compared with the case of a non-resonant random medium. The cone broadening is associated with a pulse-entrapping effect when the pulse, while propagating within the resonant random medium, spends most of the time being 'entrapped' inside scatterers. A theory for the predicted effect is based on, first, the well known relation between the contributions of the ladder and cyclical diagrams to the time spectral density of the wave electric field coherence function and, second, a recently derived radiative transfer equation with three Lorentzian kernels of delay describing a pulse entrapping in an ensemble of resonant point-like scatterers. Using the generalized Chandrasekhar H-function, we obtain an exact analytic expression for the non-stationary albedo of the semi-infinite resonant random medium, taking into account the phenomena of a pulse CBE and entrapping. A simple analytic asymptotics is found for the albedo of the later part of the scattered pulse. This asymptotics shows quantitatively how the entrapping affects the peak amplitude and peak line shape of the CBE of a short pulse.     

Backscattering response from periodic spatial patterns in random media

Abstract

In wave-based remote sensing or radio-location of distant objects in a random medium, a high-frequency electromagnetic wave is scattered by object discontinuities, and portions of the scattered radiation can traverse the same random inhomogeneities as the initial incident field. The statistical dependence of the forward–backward travelling events results in an anomaly in the backscattered intensity pattern that carries information about the scattering object. The quality of this information depends on the ability to resolve the fine-structure elements. In this work we investigate the resolving properties of periodic spatial objects by using the random propagators of the stochastic geometrical theory of diffraction.     

Peculiarities of the ionization of many-electron systems and generation of attosecond pulses in ultrashort laser fields

The ionization of a model two-electron atom in the field of a strong ultrashort laser pulse is studied by numerical integration of the nonstationary Schrödinger equation describing the dynamics of a quantum system in the field of an electromagnetic wave. Pecularities of the two-electron ionization are analyzed for pulses whose duration amounts to one to two periods of oscillation of the electric field of the wave at different frequencies of the incident radiation. For extremely short pulses, the double ionization is found to be suppressed. This effect is caused by the finiteness of the interelectron energy exchange time during the laser action. Peculiarities of the generation of high-order harmonics and single XUV attosecond pulses upon ionization of atoms by laser pulses, whose duration is within one to two optical cycles, are investigated.     

Reflection and transmission of an ultrashort electromagnetic pulse incident normal to a flat plasma layer

The process of reflection and transmission of ultrashort pulses incident normal to a flat plasma layer of finite thickness and on a step barrier is calculated within an exact mathematical approach for various parameters (barrier width and duration and carrier frequency of a corrected Gaussian pulse) of the problem.     

Measuring temporal fluctuation on femtosecond scale in a random medium

We report an observation of femtosecond optical fluctuations of transmitted light when a coherent femtosecond pulse propagates through a random medium. They are a result of random interference among scattered waves coming from different trajectories in the time domain. Temporal fluctuations are measured by using cross-correlated frequency optical gating. It is shown that a femtosecond pulse will be broadened and distorted in pulse shape while it is propagating in random medium. The real and imaginary components of transmitted electric field are also distorted severely. The average of the fluctuated transmission pulses yields a smooth profile, probability functions show good agreement with Gaussian distribution.     

Radiative transfer theory with time delay for the effect of pulse entrapping in a resonant random medium: General transfer equation and point-like scatterer model

Abstract

A pulse propagation of a vector electromagnetic wave field in a discrete random medium under the condition of Mie resonant scattering is considered on the basis of the Bethe–Salpeter equation in the two-frequency domain in the form of an exact kinetic equation which takes into account the energy accumulation inside scatterers. The kinetic equation is simplified using the transverse field and far wave zone approximations which give a new general tensor radiative transfer equation with strong time delay by resonant scattering. This new general radiative transfer equation, being specified in terms of the low-density limit and the resonant point-like scatterer model, takes the form of a new tensor radiative transfer equation with three Lorentzian time-delay kernels by resonant scattering. In contrast to the known phenomenological scalar Sobolev equation with one Lorentzian time-delay kernel, the derived radiative transfer equation does take into account effects of (i) the radiation polarization, (ii) the energy accumulation inside scatterers, (iii) the time delay in three terms, namely in terms with the Rayleigh phase tensor, the extinction coefficient and a coefficient of the energy accumulation inside scatterers, respectively (i.e. not only in a term with the Rayleigh phase tensor). It is worth noting that the derived radiative transfer equation is coordinated with Poynting's theorem for non-stationary radiation, unlike the Sobolev equation. The derived radiative transfer equation is applied to study the Compton–Milne effect of a pulse entrapping by its diffuse reflection from the semi-infinite random medium when the pulse, while propagating in the medium, spends most of its time inside scatterers. This specific albedo problem for the derived radiative transfer equation is resolved in scalar approximation using a version of the time-dependent invariance principle. In fact, the scattering function of the diffusely reflected pulse is expressed in terms of a generalized time-dependent Chandrasekhar H-function which satisfies a governing nonlinear integral equation. Simple analytic asymptotics are obtained for the scattering function of the front and the back parts of the diffusely reflected Dirac delta function incident pulse, depending on time, the angle of reflection, the mean free time, the microscopic time delay and a parameter of the energy accumulation inside scatterers. These asymptotics show quantitatively how the rate of increase of the front part and the rate of decrease of the rear part of the diffusely reflected pulse become slower with transition from the regime of conventional radiative transfer to that of pulse entrapping in the resonant random medium.     

The Non-Linear Geometric Optics of the Localized Wave Fields

The new approach to the self-action theory of intensive localized pulses, based on the hydrodynamical analogy in the non-linear geometrical optics, is proposed. The complex of phenomena of amplitude-phase non-stationary evolution of the intensive localized electromagnetic wave pulses in the dispersive medium is analysed in the framework of such approach. The wide classes of exact analytical solutions of the non-linear self-action equations, connected with such pulses, are constructed. The simple form of these solutions, represented with the well-known eigen-functions of the Laplace equation in special variables, permits to divide the pulse non-linear deformation qualitatively different effects. These solutions predict the large-scale pulse self-stratification and the origin of the quick intensity increase area during the non-linear evolution of the initially smooth distribution of the wave. The characteristic points of such evolution are represented by the singularities in the exact solutions of the non-linear geometrical optics. All results, describing the dynamics of the non-linear amplitude-phase re-building of the pulse, are represented in the simple algebraic form.     

Transformation of a monochromatic plane wane by the pulse-periodic time modulation of an infinite medium

An exact solution to the problem of the transformation of a monochromatic plane wave by a finite train of equally spaced rectangular pulses of permittivity and conductivity of an infinite medium is considered. The permittivity pulse train is shifted relative to the conductivity pulse train by an arbitrary time. The problem is studied analytically in terms of the second-order Volterra integral equation describing the electromagnetic wave transformation in a medium with time-dependent parameters. The equation is solved using the resolvent technique. Expressions for the amplitude of the transformed electric field component for any time instant at any spatial point are derived and analyzed.     

基于BLT方程的微带线电磁耦合终端响应

为定量研究电磁波与微带线的耦合终端响应问题,提出一种基于BLT方程的电磁波与微带线耦合分析方法,相比传统方法计算效率高、占用内存少。将PCB板上微带线等效为有耗传输线模型,对辐照平面波进行矢量分析并求解等效激励源,运用BLT方程研究不同入射方式、脉冲波形对微带线终端的影响,并对入射波与终端响应进行了时域、频域分析。研究表明：当入射波频率与微带线长度满足一定关系式时对微带线的耦合最强,耦合电压峰值达2.4 mV;电场平行微带线入射比垂直PCB入射时的终端电压峰值大一倍;且同幅同脉宽的矩形脉冲较高斯脉冲和三角脉冲对微带线的耦合峰值电压大;不同波形的脉冲通过调节脉宽都可对微带线耦合终端电压达到mV量级。     

Correlation properties of the backscattered field in a nonstationary statistical problem

The statistical problem of the scattering of wideband pulses by a random layered medium at normal incidence is considered in the framework of the wave approach in the space—time domain. Simulated correlation functions and power spectral densities of the backscattered field are presented. They extend the earlier findings concerning the backscattered field formation and also confirm and refine a number of conclusions drawn earlier from the behavior of the field’s statistical moments. The simulation technique is free from approximations commonly used in the statistical analysis of the propagation problems and can be used to study the statistical properties of the scattered field in a wide range of time intervals, as well as to find the limits of applicability of the approximate methods.     

基于BLT方程的微带线电磁耦合终端响应

为定量研究电磁波与微带线的耦合终端响应问题,提出一种基于BLT方程的电磁波与微带线耦合分析方法,相比传统方法计算效率高、占用内存少。将PCB板上微带线等效为有耗传输线模型,对辐照平面波进行矢量分析并求解等效激励源,运用BLT方程研究不同入射方式、脉冲波形对微带线终端的影响,并对入射波与终端响应进行了时域、频域分析。研究表明:当入射波频率与微带线长度满足一定关系式时对微带线的耦合最强,耦合电压峰值达2.4 mV;电场平行微带线入射比垂直PCB入射时的终端电压峰值大一倍;且同幅同脉宽的矩形脉冲较高斯脉冲和三角脉冲对微带线的耦合峰值电压大;不同波形的脉冲通过调节脉宽都可对微带线耦合终端电压达到mV量级。     

One-dimensional numerical simulations of random sound impulses

Abstract

We perform one-dimensional numerical simulations of small-amplitude acoustic pulses in space- and time-dependent random mass density and time-dependent velocity fields. Numerical results reveal that: (a) random fields affect the speeds, amplitudes and, consequently, shapes of sound pulses; (b) for weak random fields and short propagation times the numerical data converge with the analytical results of the mean field theory which says that a space-dependent (time-dependent) random field leads to wave attenuation (amplification) and all random fields speed up sound pulses; (c) for sufficiently strong random fields and long propagation times numerical simulations reveal pulse splitting into smaller components, parts of which propagate much slower than a wave pulse in a non-random medium. These slow waves build an initial stage of a wave localization phenomenon. However, this effect can be very weak in a real three-dimensional medium.     

Generation of an Electrostatic Pulse and a Second Harmonic EM Pulse in a Magnetoplasma

This paper presents an investigation of the generation of an electrostatic pulse by an EM pulse (Gaussian in space and time), propagating as a right/left-handed circularly polarized mode along the static magnetic field in a collisionless magnetoplasma. The generated electrostatic pulse, again interacting with an incident EM pulse, generates a second harmonic EM pulse. The effect of self-focusing of the incident EM pulse is seen to be significant on the generated pulses.     

Pulsed laser damage of proustite

Pulsed laser damage thresholds have been measured for proustite (Ag₃ As S₃) as the wavelengths 0.694, 1.065, 1.32 and 10.6 μm. The damage thresholds have been found to vary with both the wavelength and duration of the irradiating pulse. At 1.065 μm damage thresholds are 0.38 J/cm² for pulses of duration <50 ns whilst for durations >50 ns a value of 7 MW/cm² is appropriate. The results suggest that damage is initiated by absorbing inclusions approximately 0.6 μm in diameter embedded within the crystals. These inclusions are heated by an incident pulse to cause catastrophic damage of both the surface and interior of an irradiated sample. A model has been developed to enable a study of the thermal behaviour of inclusions irradiated by laser pulses with Gaussian time-dependence to be made.     

Electron energy spectrum in the field-ionized plasma

In this article, the effect of ionization on the energy spectrum of electrons within the interaction of a laser pulse with hydrogen atoms is investigated using particle-in-cell simulation codes. The results show that the behaviour of electrons' energy distribution function in the field-ionized plasma, which occurred due to the field ionization, compared with that in the pre-plasma strongly depends on the pulse shape. For short rise-time pulses (here 30 fs), due to the rapid enhancement of laser electric field, ionization occurs quickly, and as a result, there is not much difference in the electron energy in both the media. However, for pulses with rise time of 40 fs, in the pre-plasma state, the electron population reaches higher energies compared with the field-ionized plasma state. The main reason for this difference is the nonlinear wave breaking that happens earlier due to density inhomogeneity in the field-ionized plasma. On the other hand, at longer rise-time pulses (here 60 and 70 fs), electrons achieve higher energies in the field-ionized plasma than those in the case of pre-plasma. In this case, because of density fluctuations in the field-ionized plasma, the Raman backscattered radiations are seeded by a strong initial noise at the earlier times and the Mendonca condition for chaos threshold is met sooner. Therefore, the electrons gain more energy through the stochastic mechanism that is in agreement with chaotic nature of the motion.     

Splitting of an electromagnetic pulse on resonant reflection from a plasma film

It is found theoretically that the temporal profile of a quasimonochromatic electromagnetic pulse is strongly distorted on reflection from a thin (on the wavelength scale) film of a plasma-like (semiconductor, metallic) medium under plasma-resonance conditions. It is shown that an incident Gaussian pulse splits with time (completely or partially) into two reflected pulses, whose amplitudes can be controlled by varying the relationships among the parameters of the incident pulse and the film. Zh. Tekh. Fiz. 67, 65–68 (June 1997)