Calculation and optimization of three-dimensional waveguide systems by the integral equation method

We use the well-known method of analysis of oversized electrodynamic systems, which is based on numerical solution of the integral equation for the surface current and is employed to calculate diffraction of an electromagnetic field by arbitrary conducting surfaces, for calculation of the characteristics of waveguide structures. In contrast to the finite-difference algorithms for straightforward solution of the Maxwell equations, which are dealt with the field components in the calculation space divided into cells with small sizes (compared with the wavelength), the desired quantity in this method is the current on the conducting surface bounding the structure. This reduces the required RAM capacity and speeds up calculations by several orders of magnitude in the case of oversized systems. In this work, the integral equation method is realized in the form of a computer code and is used to calculate three-dimensional multimode waveguide systems. The calculation results are compared with the results of perturbation theory and the experimental data. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 51, No. 9, pp. 742–753, September 2008.     

SiO2波导布拉格器件中声表面波及声光相互作用分析

针对SiO2光波导声光布拉格器件,计算了SiO2非对称平板波导TE模式的横向场分布;给出了SiO2/ZnO/Air层状介质结构的性能方程、运动方程和麦克斯韦方程,推导出这种层状结构的特征方程,并结合所满足的边界条件,得到了各层介质的位移及电磁场分布;计算了声表面波所引起的光学相对介质隔离率张量的变化,最后讨论了声光衍射效率和光场与声场的重叠积分、声功率、声频率、声孔径和光波导参数之间的关系。结果表明,在低频范围内光场与声表面波场重叠良好;低阶模的重叠积分始终大于高阶模重叠积分,最低阶模与声表面波相互作用最强,所需声功率最小;当声功率一定时,增加声孔径可以提高衍射效率。     

Sound scattering by a body in a planar layered waveguide

In the framework of the theory of integral equations, the problem of sound diffraction by pressure-release and acoustically stiff inhomogeneities of an oceanic waveguide is considered. The iteration method is used to obtain recurrent relations for the surface currents. The relations describe the diffraction process as a sequence of interactions between the body and the waveguide boundaries (multiple reflections). The validity condition for the zero-order approximation, which ignores multiple reflections, is formulated and physically justified.     

Directional properties of an array of sound receivers positioned in an impedance screen recess

Expressions for calculating the directional characteristics of an array of sound receivers positioned in a waveguide with impedance walls are obtained from the solution to the problem on the diffraction of a plane sound wave by the waveguide open end with impedance flanges. The waveguide can be of a finite length, and, in this case, it can be considered as an open cavity in an impedance screen. The solution of the integral equation for the sound pressure distribution over the opening area is reduced to the solution of an infinite system of algebraic equations for the coefficients of the field expansion in normal waveguide waves. Examples of calculated directional characteristics are presented for arrays with receivers positioned at different distances from the opening and for different values of the impedances of the waveguide walls and flanges.     

Diffraction of a sound wave by the open end of a flanged waveguide with impedance walls

Diffraction of a plane sound wave by the open end of an impedance-wall waveguide connected to an opening in an impedance screen is considered. The plane wave is incident on the waveguide from a free half-space. Two versions of the problem are considered: for a semi-infinite waveguide and for a finite-length waveguide with a specified bottom impedance; the impedances of the walls, screen, and waveguide bottom can be different. The finite-length waveguide can be treated as an open cavity in the impedance screen. For the cavity of zero length, the problem is reduced to the diffraction by an impedance insert in the impedance screen. The solution in the external region determines the scattered field; the solution in the internal region allows one to determine the directional pattern of an array of receivers located in the cavity. The problem is solved using the integral Helmholtz equation with a specially selected Green’s function that provides the fulfillment of the boundary conditions. Formally, the problem is reduced to an infinite system of algebraic equations. The computational results obtained for bistatic and monostatic scattering patterns are presented.     

A rigorous method for solving the problems of diffraction of monochromatic electromagnetic waves by periodic structures

We present a rigorous method for solving the problems of diffraction of monochromatic electromagnetic waves on periodic structures of many types. The method is based on transformation of the Helmholtz equation and the boundary conditions to a certain system of integral equations and a subsequent rigorous solution of this system.     

Regularization of boundary integral equations in the problems of wave field diffraction by curved boundaries

A regularization of the exact Fredholm integral equations for the field or its derivative on a scattering surface is proposed. This approach allows one to calculate the scattering or diffraction of pulsed wave fields by curved surfaces of arbitrary geometry. Mathematically, the method is based on the replacement of the exact Fredholm integral equations by their truncated analogs, in which the contributions of the geometrically shadowed regions are cancelled. This approach has a clear physical meaning and provides stable solutions even when the direct numerical solution of mathematically exact initial integral equations leads to unstable results. The method is mathematically substantiated and tested using the problem of plane-wave scattering by a cylinder as an example.     

Application of harmonic perturbations to the calculation of periodically corrugated waveguides

Harmonic perturbation of the profile of an infinite axisymmetric periodically-corrugated waveguide is used for the calculation of its electrodynamic characteristics. The analysis is based on the method of Fredholm-type integral equations of the second kind. The problem is solved analytically for the perturbation of a cylindrical waveguide. These analytical results serve as the basis for the solution of the inverse problem of recovery of the waveguide profile from the given dispersion curve.Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 5, pp. 467–480, May, 1995.     

Diffraction of guided waves by the end face of a dielectric slab waveguide terminated with a finite conductive strip

The diffraction of guided waves by the end face of a dielectric slab waveguide short circuited with a finite conductive strip is analyzed. An integral equation technique is employed to formulate the corresponding boundary problem. The unknown term in this integral equations is the electric field E(x) on the terminal plane of the waveguide. The homogeneous term is determined from the incident guided wave. A method of moments technique is employed to compute approximately the electric field E(x) by using Laguerre functions as describing and testing functions. The reflection coefficients of the guided waves are computed by using the approximate expression of the E(x) field. Numerical results are given for several guide and conductor plate dimensions.     

一种可稳定计算Pekeris波导中声场的波数积分方法

提出一种可稳定计算Pekeris波导中声场的波数积分方法,并在此基础上开发出一个数值模型,可用于提供Pekeris波导中声场的精确、稳定的数值解。在这个方法中,由于与深度有关的波动方程齐次解中所有的上行波与下行波均采用了合理的归一化表示,从而得到的系统方程是无条件稳定的。在简正波方法中,割线积分一般只对近场有显著影响。因此,传统的简正波模型一般都忽略割线积分对声场的贡献。但是,如果某号简正波离割线非常近,则割线积分对非常远距离的声场仍可能有显著影响。在这种情况下,传统的简正波模型由于忽略割线积分的贡献,从而得到的声场结果是不准确的。本文通过数值算例比较本文提出的波数积分模型与传统的简正波模型。数值结果表明,本文提出的模型可以提供精确、稳定的Pekeris波导中声场的数值解,而在某些情况下传统的简正波模型得到的声场结果是不准确的。因此,本文提出的模型可以作为Pekeris波导中声传播问题的标准模型使用。      

A bifurcated waveguide problem

We consider the diffraction of the dominant mode in a parallel-plate waveguide with hard boundaries which is incident on a centered soft-hard half-plane. By using the Fourier transform technique in conjunction with the Mode Matching method, the related boundary value problem is formulated as a modified Wiener-Hopf equation. The solution of the latter, which contains infinitely many constants, is found by solving numerically an infinite set of linear algebraic equations.     

Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations

For optical waveguides with high index-contrast and sharp corners, existing full-vectorial mode solvers including those based on boundary integral equations typically have only second or third order of accuracy. In this paper, a new full-vectorial waveguide mode solver is developed based on a new formulation of boundary integral equations and the so-called Neumann-to-Dirichlet operators for sub-domains of constant refractive index. The method uses the normal derivatives of the two transverse magnetic field components as the basic unknown functions, and it offers higher order of accuracy where the order depends on a parameter used in a graded mesh for handling the corners. The method relies on a standard Nyström method for discretizing integral operators and it does not require analytic properties of the electromagnetic field (which are singular) at the corners.     

Calculation of VLW fields in a waveguide with A 3-D local inhomogeneity

The problem of a point-source field in an irregular impedance waveguide is solved. The 3-D inhomogeneity of one of the walls of the waveguide is given by an area-inhomogeneous impedance. To obtain a solution within the framework of the method of integral equations, we develop a procedure for asymptotic transformation of the 2-D equation into an 1-D equation with allowance for the waves reflected from all the inhomogeneity boundaries. The obtained 1-D integral equation for points that belong to both the path line and boundary contour of the inhomogeneity is solved numerically using an original algorithm. The results of model calculations in a near-earth waveguide for the case of ionospheric perturbations that are large on the wavelength scale are given.State University, St. Petersburg. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 12, pp. 1312–1322, December, 1995.     

On solution of the vector 3-D locally irregular waveguide problem

The vector 3-D problem of a point-source field in a plane waveguide with a large-scale local inhomogeneity on one of its walls is considered. The field components on the boundary surfaces comply with the Leontovich conditions, which are used as a basis for obtaining expressions for the derivatives of the field vectors normal to the boundaries; these expressions reflect the 3-D nature of the inhomogeneity. The problem is reduced to a system of 2-D integral equations allowing for overexcitation and depolarization of the field scattered by the irregularity. The system of 2-D integral equations is asymptotically transformed over the inhomogeneity region on the surface of the walls bounding the waveguide space into a system of linear integral equations, for which the integration contour is represented by the line between the source and observation point, as well as by the linear geometric contour of the irregularity.State University, St. Petersburg. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 38, No. 8, pp. 785–803, August, 1995.     

Method of integral equations in the theory of microstructured optical fibers

A method for designing microstructured optical fibers that is based on exact integral equations for the transverse components of the magnetic field of the mode is proposed. A solution to the vector waveguide problem for fibers with a finite number of circular capillaries in the round cavity of the cladding can be refined successively. Quartz fibers with hexagonal capillary rings are also studied.     

Depolarization of the electromagnetic field in a locally inhomogeneous earth-ionosphere waveguide

This paper presents a further development of the numerical-analytical method for the solution of three-dimensional problems in the theory of radio wave propagation. We consider a vector problem of the electromagnetic field of a vertical electric dipole in a plane Earth-ionosphere waveguide with a local large-scale irregularity on the anisotropic ionosphere wall. The possibility of lowering (elevating) of the local region of the upper waveguide wall with respect to the regular ionosphere level is taken into account. The field components on the boundary surfaces obey the Leontovich impedance conditions. The problem is reduced to a system of two-dimensional integral equations taking into account the overexcitation and depolarization of the field scattered by the irregularity. Using asymptotic (with respect to the parameter kr ≫1) integration along the direction perpendicular to the ray path, we transform this system to a system of one-dimensional integral equations. The system is solved numerically in the diagonal approximation, combining direct inversion of the Volterra integral operator and the subsequent iterations. The proposed method reduces the computer time required for solving the problem and is useful for the study of both small-scale and large-scale irregularities. We obtained estimates of the TE field components that are not excited by the source considered and originate entirely from field scattering by a three-dimensional irregularity disturbing the geometric regularity of the ionospheric waveguide wall. State University of St. Petersburg, Russia Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 7, pp. 617–629, July, 2000.     

Diffraction of cylindrical waves by an ideally conducting wedge in an anisotropic plasma

The problem of diffraction of cylindrical waves by an ideally conducting wedge in an anisotropic plasma is formulated and solved. The integral equations for the field are reduced to function equations, which are solved with the aid of a special function that is introduced. The properties of this function are studied. The general solution is represented as a double contour integral in the plane of a complex variable. The radiation field and surface waves for a number of special cases are analyzed: a source of cylindrical waves on an edge; at infinity; etc. Diffraction in a half-plane is studied separately.     

Light propagation in nonuniform plasmonic subwavelength waveguide arrays

We study light propagation in nanoscale periodic structures composed of dielectric and metal in the visible range. We demonstrate that diffraction curves of nonuniform waveguide arrays can be tailored by varying the geometric and dielectric features of the waveguides. The results obtained from a proper formulation of coupled mode theory for nonuniform arrays are validated through numerical solution of Maxwell equations in frequency domain.     

Modelling of finite-size grating waveguide filters on the basis of a free-space diffraction approach

The field reflection coefficientR of a high strength and short length waveguide grating section is obtained by resorting to a free-space diffraction approach whereby the waveguide grating is excited by means of a virtual high index prism. The effective index of the guided mode, the coupling coefficient between contradirectionally propagating modes of the grating waveguide as well as the modal field are first obtained in the case of an infinite grating by interpreting the results given by an exact diffraction solution by means of a three-wave coupled mode formalism. These results are then introduced in the coupled wave analysis of the structure of finite extent to provide the field at the entrance of the waveguide grating section where the usual overlap integral with the incident field delivers the reflection coefficient. We neglect only the result of radiation mode propagation. This assumption can be made when the grating section is longer than the propagating length of all the leaky modes.     

Modelling of finite-size grating waveguide filters on the basis of a free-space diffraction approach

The field reflection coefficient R of a high strength and short length waveguide grating section is obtained by resorting to a free-space diffraction approach whereby the waveguide grating is excited by means of a virtual high index prism. The effective index of the guided mode, the coupling coefficient between contradirectionally propagating modes of the grating waveguide as well as the modal field are first obtained in the case of an infinite grating by interpreting the results given by an exact diffraction solution by means of a three-wave coupled mode formalism. These results are then introduced in the coupled wave analysis of the structure of finite extent to provide the field at the entrance of the waveguide grating section where the usual overlap integral with the incident field delivers the reflection coefficient. We neglect only the result of radiation mode propagation. This assumption can be made when the grating section is longer than the propagating length of all the leaky modes.