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.     

Effect of pulse entrapping on diffuse reflection from a resonant random medium: exact solution to the scalar albedo problem

A recently derived radiative transfer equation with three Lorentzian delay kernels is applied to an albedo problem of the scalar wave field produced by the diffuse reflection of a quasi-monochromatic pulse from a semi-infinite random medium consisting of resonant point-like scatterers. The albedo problem is solved exactly in terms of the Chandrasekhar H-function H(μλ), extended analytically into the complex single-scattering albedo λ plane. The resulting analytic solution for the time evolution of a diffusely reflected short pulse is used to study on the whole time axis the effect of the redistribution of the energy of the propagated pulse from the front to the rear of the pulse in cases where the pulse may for most of the propagation time through the medium be 'entrapped' inside resonant scatterers. By considering the power flux through unit area of the boundary of the medium and unit solid angle, it is shown that the relative shift of an 'energy centroid' ('centre of mass') of the pulse diffusely reflected from the resonant random medium (compared with the pulse energy centriod in the non-resonant case) is equal to the parameter describing the energy accumulation inside the scatterers. This result may be used for experimental study of resonant random media with the aid of a short-pulse technique.     

Theory of transfer with delay for trapping of nonstationary acoustic radiation in a resonant randomly inhomogeneous medium

The propagation of a quasimonochromatic wave packet of acoustic radiation in a discrete randomly-inhomogeneous medium under the condition that the carrier frequency of the packet is close to the resonance frequency of Mie scattering by an isolated scatterer is studied. The two-frequency Bethe-Salpeter equation in the form of an exact kinetic equation that takes account of the accumulation of the acoustic energy of the radiation inside the scatterers is taken as the initial equation. This kinetic equation is simplified by using the model of resonant point scatterers, the approximation of low scatterer density, and the Fraunhofer approximation in the theory of multiple scattering of waves. This leads to a new transport equation for nonstationary radiation with three Lorentzian delay kernels. In contrast to the well-known Sobolev radiative transfer equation with one Lorentzian delay kernel, the new transfer equation takes account of the accumulation of radiation energy inside the scatterers and is consistent with the Poynting theorem for nonstationary acoustic radiation. The transfer equation obtained with three Lorentzian delay kernels is used to study the Compton-Milne effect—trapping of a pulse of acoustic radiation diffusely reflected from a semi-infinite resonant randomly-inhomogeneous medium, when the pulse can spend most of its propagation time in the medium being “trapped” inside the scatterers. This specific albedo problem for the transfer equation obtained is solved by applying a generalized nonstationary invariance principle. As a result, the function describing the scattering of a diffusely reflected pulse can be expressed in terms of a generalized nonstationary Chandrasekhar H-function, satisfying a nonlinear integral equation. Simple analytical asymptotic expressions are found for the scattering function for the leading and trailing edges of a diffusely reflected δ-pulse as functions of time, the reflection angle, the mean scattering time of the radiation, the elementary delay time, and the parameter describing the accumulation of radiation energy inside the scatterers. These asymptotic expressions demonstrate quantitatively the retardation of the growth of the leading edge and the retardation of the decay of the trailing edge of a diffusely reflected δ-pulse when the conventional radiative transfer regime goes over to a regime of radiation trapping in a resonant randomly-inhomogeneous medium. Zh. éksp. Teor. Fiz. 113, 432–444 (February 1998)     

随机非球形粒子全极化散射的时间相关Mueller矩阵解

从与时间相关的矢量辐射传输方程推导一阶Mueller矩阵解,用来模拟Gauss型平面脉冲波入射下,一层随机、非均匀取向非球形粒子的全极化双站散射.数值计算了同极化和去极化脉冲响应,与入射脉冲进行了比较,说明了随机介质的物理参数,如粒子的取向和占空比、入射角、极化以及层厚等对脉冲响应的影响 关键词： 平面脉冲波 非球形粒子 Mueller矩阵     

Application of the Monte Carlo method for modeling passage of ultrashort laser pulses through an inhomogeneous medium with moving scatterers

We present the results of numerical modeling of passage of ultrashort laser pulses through an inhomogeneous layered medium with moving scatterers, based on solution of the nonsteady-state radiation transport equation by the Monte Carlo method. We consider the effects of Doppler broadening of the backscattered radiation spectrum in biological tissues. We have analyzed the dynamics of propagation of a short laser pulse within a multilayer model of human skin. We have studied the possibilities for tomography of different layers of biological tissue based on analysis of the spectrum of the scattered radiation pulse.     

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.     

Statistical temporal behaviour of pulse wave propagation through continuous random media

Pulse waves propagating through random media suffer distortions, such as fluctuation of arrival time, temporal broadening, and alteration of skewness and kurtosis, due to both the background medium and embedded irregularities. We carry out a study on the temporal behaviour of electromagnetic pulses propagating through random media using temporal moments and an analytic solution of a two-frequency mutual coherence function recently obtained by iteration. We treat the temporal characteristics sequentially, with general expressions obtained first. Then the concise forms are given for pulse propagation in the turbulent non-dispersive atmosphere and the ionosphere, with numerical calculations for the latter. The results show that the mean arrival time is dominated by the term propagating at group velocity, and small corrections arise from higher-order dispersion of the background medium and random scattering of irregularities, but the correction from dispersion of irregularities is neglected as it is so small. As for pulse broadening in trans-ionospheric propagation, the results show that contributions are mainly from the dispersion of the background ionosphere and scattering of electron density irregularities in most cases, and the contribution of dispersion of irregularities is so small that it can be neglected. Finally, we find that the temporal skewness of a trans-ionospheric pulse is negative and its energy is shifted to the leading edge, and the contributions from scattering and dispersion of irregularities dominate over those of background, so the latter can be neglected in most cases.     

含金属散射体的中红外无序介质的光子定域化理论研究

基于Mie散射理论发现,各种金属在中红外区的各个频率点上的散射行为均极为相近,是一种在中红外区反照率高但散射效率较低的散射体.在浓度为10%时,介质的定域化参量最小只能达到6.6,并且与体系的常用基质材料无关.数值研究揭示了这种金属散射体系统与低吸收、高折射率散射体系统间的内在联系 关键词： 光子定域化 无序介质 中红外 Mie散射 金属     

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

Abstract

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.     

Single-peaked self-induced transparency pulses in a degenerate resonant medium

We obtained new analytic closed form solutions for distortionless propagation of ultra-short optical pulses through a resonant medium with overlapping Q(j) transitions for j=2. The steady-state pulse shape of this solution has only one peak. Both 2π and 0π solutions are obtained.     

Energy exchange between laser pulses in an atomic medium with a closed excitation contour

The effect the phase difference between carrier frequencies of two laser pulses has on their propagation in an optically dense medium of three-level point scatterers is investigated. The medium is exposed to constant radiation, creating a closed excitation contour. Situations are considered for pulse durations shorter and longer than the lifetime of the excited state of scatterers.     

Ray theory for a locally layered random medium

We consider acoustic pulse propagation in inhomogeneous media over relatively long propagation distances. Our main objective is to characterize the spreading of the travelling pulse due to microscale variations in the medium parameters. The pulse is generated by a point source and the medium is modelled by a smooth three-dimensional background that is modulated by stratified random fluctuations. We refer to such media as locally layered .

We show that, when the pulse is observed relative to its random arrival time, it stabilizes to a shape determined by the slowly varying background convolved with a Gaussian. The width of the Gaussian and the random travel time are determined by the medium parameters along the ray connecting the source and the point of observation. The ray is determined by high-frequency asymptotics (geometrical optics). If we observe the pulse in a deterministic frame moving with the effective slowness , it does not stabilize and its mean is broader because of the random component of the travel time. The analysis of this phenomenon involves the asymptotic solution of partial differential equations with randomly varying coefficients and is based on a new representation of the field in terms of generalized plane waves that travel in opposite directions relative to the layering.     

Energy dispersive backscattering of electrons from surface resonances of a disordered medium and 1/f noise

Anderson localised states in the bulk of a disordered medium appear as sharp resonances near the surface. The resonant backscattering leads to an energy-dependent random time delay for an incident electron. We derive an analytic expression for the delay-time probability distribution at a given energy. This is shown to give a 1/f noise for the surface currents in general.     

Interference effects in backscattering of light by a layer of a discrete random medium

Multiple backscattering of light by a layer of a discrete random medium is considered. A brief derivation of equations for describing the coherent and incoherent components of scattered light is presented. These equations are solved numerically in the approximation of doubled scattering of light by a semi-infinite medium of spherical scatterers having a size comparable with the wavelength in order to study the effect of the properties of particles on the angular dependence of interference effects. Calculations show that the half-width of the interference peak decreases upon an increase in lateral scattering by particles and that the degree of polarization has a complex angular dependence on the properties of the particles. For an optically thin layer of the medium, the relations defining the interference peak half-width and the scattering angle upon extreme linear polarization as functions of the effective refractive index are given.     

Formation of a reflected wave under propagation of a pulse with high peak intensity through a Kerr medium

Propagation through a Kerr medium of short pulses depending only on the longitudinal coordinate is investigated. High values of the peak intensity are considered for which the nonlinear part of the relative permittivity is on the order of unity. When a short pulse propagates through such a medium, the leading edge of the pulse is extended, while the trailing edge runs into the slowly propagating central part of the pulse; shock waves are generated at the trailing edge, giving rise to high spatial frequencies and backward reflected radiation. The duration of the pulse increases due to the high-frequency jet that forms at the trailing edge, and the peak intensity decreases. The spectrum of the backward reflected radiation is investigated as a function of the peak intensity of the pulse and the characteristics of the time dispersion of the medium.     

Theoretical and experimental study of blurring of a femtosecond laser pulse in a turbid medium

We propose an analytical model of the spatio-temporal structure of a short laser pulse transmitted through a layer of an optically inhomogeneous medium with high anisotropy of scattering. The light-field brightness in the medium is represented as a finite series in terms of multiplicities of the small-angle scattering, while the contribution from the higher-order scattering is allowed for as a quasi-diffuse component. The scattered-pulse structure is calculated on the basis of solving the radiative-transfer equation in the small-angle approximation with allowance for the effect of multipath light propagation. Compared with the first approximation of the multiple-scattering theory (attenuated nonscattered light plus the diffuse component), this approach makes it possible to describe more correctly the transformation of the spatio-angular distribution of light in the medium when passing from the single-scattering to multiple-scattering regime, as well as specify the temporal profile of the scattered pulse. The temporal profile of the femtosecond pulse transmitted through a layer of model scattering medium with various concentrations of scatterers is studied experimentally. The blurred-pulse structure is studied with the help of nonlinear optical gating in the case of noncollinear generation of the second harmonic. Good agreement between the theoretical and experimental time profiles of the scattered pulse is shown for the optical-thickness intervals corresponding to both the predominantly low multiplicity scattering and multiple small-angle scattering, which allows us to use the proposed analytical model for solving the inverse problem of the pulse sounding of a homogeneous turbid medium. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 4, pp. 333–348, April 2008.     

Propagation of the ultrashort laser pulses through the quantum nonlinear medium with resonant properties

The propagation of ultrashort few-cycle laser pulses through a quantum nonlinear medium with resonant properties is studied using the method of slowly varying amplitudes with allowance for dispersion effects. A self-consistent system of nonlinear wave equation for the evolution of laser field and the nonstationary Schrödinger equation that determines the material polarization response is numerically solved. Specific features of the propagation of laser pulse caused by a relatively large spectral width and the nonadiabatic character of the laser pulse are established. The rise of the slowly dumping polarization at the eigenfrequency of the medium and the consequent stretching of the laser pulse are observed. The role of the resonant absorption of the laser energy in the medium is analyzed. The nonlinear spectral broadening is demonstrated for the laser pulse propagating through the medium.     

Propagation of longitudinal waves in a random binary rod

Propagation of waves in a composite elastic rod consisting of rods with alternating properties of random length is considered. We calculate exactly the Lyapunov exponent and find its short and long wave asymptotics. Finally, we discuss conditions for propagation and localization of waves in a binary random medium.     

Propagation of longitudinal waves in a random binary rod

Propagation of waves in a composite elastic rod consisting of rods with alternating properties of random length is considered. We calculate exactly the Lyapunov exponent and find its short and long wave asymptotics. Finally, we discuss conditions for propagation and localization of waves in a binary random medium.     

Long pulse duration of the UV laser with a multi-reflected cavity: Experiment and modeling

The long pulse operation of ultraviolet (UV) laser attributed to multi-reflection is presented on an acousto-optic (AO) Q-switched ultraviolet laser system blocking green light inside a cavity. The ultraviolet laser with the highest average power of 456 mW is obtained with 20 kHz repetition-rate. The numerical simulation of UV pulse broadening was studied using Gaussian curve approximation. The conclusion suggests that the short multi-reflected cavity and high incident pump power will be beneficial to alleviate the width broadening effect. The results, for the first time, would provide the technology for controlling the pulse width of an AO Q-switched DUV laser.