激光在不均匀随机媒质中的传播——多重散射的m-n阶矩方程的解

本文给出了受到多重散射的激光,在波传播方向上的物理参数的涨落是不均匀的、而垂直于传播方向上的物理参数的涨落又是均匀的随机媒质中传播时,当波受到前向小角度散射时,具有不同波数不同位置的场的矩方程的解析解.同时讨论了方程的解在激光传播研究中的一些应用.     

Simplification of an analytical solution of the fourth-moment equation

The equation for the fourth moment of a wave propagating in a multiply scattering random medium has been solved by various methods. When the analytical solutions are compared with numerical solutions of the equation it is found that the fundamental solution together with a first-order correction term agree very closely with the numerical results over a wide range of distances and scattering strengths. Unfortunately, the correction term involves multiple integrals and so is difficult to evaluate. This paper shows how some of these integrations can be carried out and the results combined in such a way that an analytical form similar to the fundamental solution is obtained involving only a single integral. This simplified combined solution also agrees very closely with the numerical results.     

Modal theory for the two-frequency mutual coherence function in random media: beam waves

Pulse propagation in a random medium is mainly determined by the two-frequency mutual coherence function which satisfies the parabolic equation. It has recently been shown that this equation can be solved by separation of variables, thereby reducing the solution for any structure function to the solution of ordinary differential equations. In this paper, the method is applied for a beam-wave excitation in a random medium. The exact solution for a quadratic medium is derived. For non-quadratic power-law media an analytical expression at equal positions is presented.     

Path integrals for wave intensity fluctuations in random media

Approximate expressions for the fourth order moment of a wave propagating in a random medium are derived by using the path integral formulation. These solutions allow the spectrum of intensity fluctuations of a multiply scattered wave to be found, and they are valid at all distances in the medium. The results obtained by path integral methods turn out to be the same as those obtained previously by solving the parabolic partial differential equation for the fourth moment. The spatial frequency spectra of intensity fluctuations are evaluated for a medium in which the irregularities have a single scale and also for one in which there is a range of scale sizes.     

Pulse propagation in randomly fluctuating media.

Pulse propagation in a weakly and randomly inhomogeneous medium is studied using a time-domain progressive wave equation. An eikonal-like approximated solution to the wave equation derived from the path integral representation is used to obtain the time-dependent statistics of pulses propagating through this random medium. This approach yields both a simple way of producing simulations of time series as well as their statistical properties.     

A uniform asymptotic integral representation for the fourth moment of the field of a wave propagating in a statistically homogeneous medium

In this paper, a uniform integral representation has been obtained for the fourth moment of the field of a wave propagating in a medium with random large-scale irregularities. The solution to the equation was obtained using a method of integral transformations and Maslov's complex WKB method. The representation obtained differs in its form from those reported thus far and in particular from those given by the method of two-scale expansions and the interference integral method. First, the paper considers the case of a plane wave incident on a layer with irregularities, followed by a treatment of the general case of an arbitrary source.     

Two-frequency mutual coherence function for electromagnetic pulse propagation over rough surfaces

The propagation of a transient electromagnetic pulse over irregular terrain is considered. We model the wave propagation using the parabolic wave equation, which is valid for near-horizontal propagation. We model the effect of scattering from the rough terrain by introducing a surface-flattening coordinate transform. This coordinate transform simplifies the boundary condition of our problem, and introduces an effective refractive index into our wave equation. As a result, the problem of propagation over an irregular surface becomes equivalent to the problem of propagation through random media. The parabolic equation is solved analytically using the path integral method. Both vertically polarized and horizontally polarized signals are treated. Cumulant expansion is introduced to obtain an approximate expression for the two-frequency mutual coherence function. From the mutual coherence function, spatial and temporal dependence of the propagating signal can be determined. It can be shown that scattering from the irregular surface can cause broadening of the transient signal. This can have a significant impact on the performance of radio communication systems.     

Reference-wave solutions for the high-frequency fields in inhomogeneous-background random media

Ray theory plays an important role in determining the propagation properties of high-frequency fields and their statistical measures in complicated random environments. For computations of the statistical measures it is therefore desirable to have a solution for the high-frequency field propagating along an isolated ray trajectory. A new reference wave is applied to obtain an analytic solution of the parabolic wave equation that describes propagation along the ray trajectory of the deterministic-background medium. The methodology is based on defining a paired-field measure as a product of an unknown field propagating in a disturbed medium and the complex-conjugate component propagating in a medium without random fluctuations. When a solution of the equation for the paired-field measure is obtained, the solution of the deterministic component can be extracted from the paired solution to determine the solution of the unknown field in an explicit form.     

Two-frequency mutual coherence function for electromagnetic pulse propagation over rough surfaces

The propagation of a transient electromagnetic pulse over irregular terrain is considered. We model the wave propagation using the parabolic wave equation, which is valid for near-horizontal propagation. We model the effect of scattering from the rough terrain by introducing a surface-flattening coordinate transform. This coordinate transform simplifies the boundary condition of our problem, and introduces an effective refractive index into our wave equation. As a result, the problem of propagation over an irregular surface becomes equivalent to the problem of propagation through random media. The parabolic equation is solved analytically using the path integral method. Both vertically polarized and horizontally polarized signals are treated. Cumulant expansion is introduced to obtain an approximate expression for the two-frequency mutual coherence function. From the mutual coherence function, spatial and temporal dependence of the propagating signal can be determined. It can be shown that scattering from the irregular surface can cause broadening of the transient signal. This can have a significant impact on the performance of radio communication systems.     

二维随机介质中柱面波传播及其局域性的随机泛函分析

分析了二维各向同性均匀随机介质中柱面波的传播特性及局域化现象.用随机泛函理论,在频域内将随机介电起伏展开成柱坐标系下的Wiener积分式,将波场表示为内外行柱面波的线性和,求解二维Helmholtz波动方程,得到随机介电起伏对柱面波幅度与相位调制的解析表达.由柱面波能量的空间分布验证了波的局域化现象,并求解局域化长度.二维随机介质中平面波按柱面波展开的波转换方程与非随机介质中的情形有相似的表达,但具有随机介电起伏对幅度和相位的调制,并给出数值模拟结果.     

小孔衍射和近场散射数值计算的格林函数方法

从简谐光波满足的亥姆霍兹方程出发,将由格林定理得到的介质分界面上的积分方程转化为以表面上的光波及其导数为未知量的线性方程组,并对其进行数值求解,实现了光场的数值计算。然后将这一方法应用于亚波长尺度的小孔衍射的光波以及自仿射分形表面产生的随机光场及其在近场区域范围内的传播的计算。在随机表面产生的光场计算中．提出了类比推导夫琅禾费面上散斑场自相关函数的方法产生随机表面,以及计算其导数的傅里叶变换方法。对光场的计算结果表明,在近场范围内,光场随离开表面的距离的增加而迅速变化,其传播特性完全不同于光场在远场范围内的传播特性。     

On propagation in continuous random media

The analysis of wave propagation in continuous random media typically proceeds from the parabolic wave equation with back scatter neglected. A closed hierarchy of moment equations can be obtained by using the Novikov-Furutsu theorem. When the same procedure is applied in the spatial Fourier domain, one obtains a closed hierarchy of coupled moment equations for the forward- and back-scattered wavefields that is not restricted to narrow scattering angles nor to small local perturbations. The general equations are difficult to solve, but a Markov-like approximation is suggested by the form of the scattering terms. Simple algebraic solutions can be obtained if a narrow-angle-scatter approximation is then invoked. Thus, three distinct approximations are explicit in this analysis, namely closure, Markov and narrow-angle scatter.

The results show that the extinction of the coherent wavefield has a distinctly different form from the corresponding result for propagation in a sparse distribution of discrete scatteres. Furthermore, when the scatter is constrained to narrow forwardand back-scattered cones, there is no back-scatter enhancement. These results are discussed within the context of the extension of the spectral-domain formalism to discrete random media. The general continuous-media moment equations are developed but not solved. The results correct and extend an earlier analysis that used a perturbation approach to compute the scattering functions rather than the Novikov-Furutsu theorem.     

Scattering from a rough layer of a random medium

This paper deals with scattering from a random-medium layer with rough boundaries. The fluctuations of the surface heights and medium permittivity are assumed to be small and smooth. All random quantities are assumed to be stationary and independent of each other. After the introduction of approximate boundary conditions, the system of partial differential equations is transformed into an integral equation where the fluctuations of the problem are represented as a zero-mean random operator. Employing smoothing, integral equations for the coherent fields are obtained. Use of the Helmholtz operator leads to solution for the coherent propagation constant while the boundary operators lead to coherent Fresnel coefficients. The characteristics of the results are illustrated by considering several examples.     

Stochastic functional analysis of propagation and localisation of cylindrical waves in a two-dimensional random medium

Propagation and localisation of cylindrical waves in a two-dimensional (2D) isotropic and homogeneous random medium is studied using the stochastic functional approach. By expanding the random permittivity fluctuation in the form of a Wiener integral equation, and representing the wave fields by a linear combination of outgoing and incoming waves, the scalar Helmholtz equation is solved in the cylindrical coordinates system. An analytical expression of the cylindrical wave is derived and demonstrates the localisation phenomenon, as well as the wavenumber fluctuation in the random medium. Comparing with the waves in non-random medium, the wave transfer equation between plane wave and cylindrical wave in random medium shows an additional exponential factor to indicate the modulation effect owing to the medium randomness in both the amplitude and phase. Numerical simulations are presented to illustrate the functional dependence of the localisation phenomena.     

A Green's function method for surface acoustic waves in functionally graded materials

Acoustic wave propagation in anisotropic media with one-dimensional inhomogeneity is discussed. Using a Green's function approach, the wave equation with inhomogeneous variation of elastic property and mass density is transformed into an integral equation, which is then solved numerically. The method is applied to find the dispersion relation of surface acoustic waves for a medium with continuously or discontinuously varying elastic property and mass density profiles.     

A comparison of differential and integral equations of radiative transfer

The equations of radiative transfer and of statistical equilibrium of a two-level atom are solved by means of differential and integral equations for a one-dimensional medium. The numerical solutions are compared to the analytic solution. It is found that the integral equation for piecewise quadratic source functions gives more accurate results than does the differential equation.     

Moment-screen method for wave propagation in a refractive medium with random scattering

Direct numerical solution of a parabolic equation (PE) for the second moment of the sound field in a refracting medium with random scattering is described. The method determines the mean-square sound pressure without requiring generation of random realizations of the propagation medium. The second-moment matrix is factored into components that are independently propagated with a conventional PE algorithm. A moment screen is periodically applied to attenuate the coherence of the wavefield, much as phase screens are often applied in the method of random realizations. An example involving upwind and downwind propagation in the near-ground atmosphere shows that the new direct method converges to an accurate solution faster than the method of random realizations and is particularly well suited to calculations at low frequencies.     

A Simulation of Near Field Optics by Three Dimensional Volume Integral Equation of Classical Electromagnetic Theory

A numerical simulation code for three dimensional problems of near-field optics has been developed using the volume integral equation with the moment method. The object is assumed to be continuous and macroscopic dielectric and can be treated by macroscopic Maxwell#x0027;s equations. The code can treat the large-scale moment method matrix that is obtained by the discretization of the volume integral equation. The resultant matrix equation is solved by an iteration method called the generalized minimum residual method with reasonable computational cost for simple problems of near field optics. Simulation of a simplified model of a scanning near-field optical microscope has been performed and basic polarization characteristics of the system have been investigated in detail. The code is also applied to the collection-mode of a photon scanning tunneling microscope, where the incident wave is the evanescent wave, and basic relation between near-field and far field i.e., output image, is recognized.     

Low-frequency radio wave propagation in the earth-ionosphere waveguide disturbed by a large-scale three-dimensional irregularity

In this paper, we develop further the analytical and numerical method of solving three-dimensional problems in the theory of radio wave propagation, including three-dimensional local inhomogeneities (ionospheric disturbances or Earth’s surface irregularities). To model the Earth-ionosphere waveguide, we use the surface impedance concept, by which the irregularity extending beyond one waveguide wall has an arbitrary smooth shape, and its surface can be described by the impedance. In the scalar approximation, this problem is reduced to a two-dimensional integral equation for the irregularity surface, which, by asymptotic (kr ≫ 1) integration over the coordinate transverse to the propagation path (with allowance for terms of the order of (kr)⁻¹), is reduced to a one-dimensional integral equation, in which the integration contour is the linear contour of the irregularity. The equation is solved numerically, combining the inversion of a Volterra integral operator and successive approximations. By reducing the computer times, this method enables one to study both small-scale and large-scale irregularities. The results of numerical simulation of radio wave propagation in the presence of a powerful three-dimensional ionospheric disturbance are presented as an example. State University, St. Petersburg, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 5, pp. 588–604, May, 1998.