Continuous spectrum, characteristic modes, and leaky waves of openwaveguides

The modes of open waveguides with nonseparable cross sections are derived by means of an extension of the resonance equation for the electromagnetic field. Such modes, forming a continuous spectrum, allow us to apply to discontinuity problems in open environments the techniques originally developed for closed waveguides. In this paper, the resonance equation is generalized according to functional analysis considerations. By this approach, it is possible to derive the modal spectrum from the simultaneous diagonalization of the real and imaginary parts of the admittance of the structure. A variational interpretation of the solution of the generalized resonance equation gives additional insight into the modes of open waveguides. The generalized resonance equation, when applied to three-dimensional objects, provides the well-known characteristic modes of these structures. The relationship between continuous spectrum, characteristic modes, and leaky waves is also discussed     

Azimuthally uniform surface-plasma modes in thin metalliccylindrical shells

The pure transverse-magnetic (TM)-polarized surface-plasma bound modes of a thin metallic right circular cylindrical shell imbedded in dielectric materials are studied. Two modes are generally found to be supported by the structure. They are analogous to the two modes of the corresponding planar structure of a thin metallic film bound by dielectrics. One of the modes is shown to have sufficiently small loss to make it of potential application in fiber optics. The transcendental equation of the modes generalizes that of the planar geometry by having the cylinder radius as a parameter. The equation is solved exactly in the complex domain for all cases considered     

Leaking modes of guiding W-fibers with a large difference in the refractive index profiles

For a planar three-layer guiding fiber and a guiding W-fiber, comparative analysis of the characteristics of leaking modes is performed on the basis of numerical solution of the dispersion equation. It is shown that the complex plane distributions of solutions to the dispersion equation for the transverse wave numbers of leaking modes substantially depend on the geometrical and optical parameters of such structures. In particular, it is found out that a guiding W-fiber with a large step in the refractive index ensures the most effective filtering of leaking modes and the highest concentration of the guided mode field.     

Magnetodynamical modes in a ferrite-filled coaxial cavity

The resonance frequencies of a coaxial ferrite-filled cavity and the field distribution in the cavity are calculated. There are three different groups of modes to be distinguished: the limit-TEM mode, the EH and HE modes, and modes independent of the zco-ordinate, which are TM modes. The eigenvalue equation for all modes is derived and discussed.     

Electromagnetic resonances and Q factors of a chiral sphere

A study of electromagnetic resonances of an isotropic chiral sphere is presented. The characteristic equation of a chiral sphere, which is complex and transcendental, is obtained by forming solutions of the electromagnetic field for a chiral sphere in terms of the spherical vector wave functions. The characteristic equation is solved by using a computer. Resonant frequencies and Q factors of the first few modes for relative permittivities of two and ten are reported and discussed. These modes are hybrid modes. They are classed as either hybrid electric (HE) modes or hybrid magnetic (HM) modes. Chirality is observed to have a significant effect on the resonances of a sphere.     

Weakly guiding analysis of elliptical core step index waveguidesbased on the characteristic numbers of Mathieu's equation

A new weak guidance analysis for step index elliptical core optical fibers is developed. The method shows that weakly guiding, or linearly polarized (LP), modes are determined by and correspond uniquely to the characteristic numbers of Mathieu's equation. This is the first analysis of elliptical core weakly guiding modes which determines propagation constants and indexes modes by directly solving the underlying Mathieu's equation through the sole use of characteristic numbers. The resulting new analysis resolves an ambiguity in the standard literature concerning the classification of weakly guiding modes     

Computation of cutoff wavenumbers of TE and TM modes in waveguidesof arbitrary cross sections using a surface integral formulation

A procedure is described for obtaining the cutoff wave numbers of transverse electric (TE) and transverse magnetic (TM) modes in waveguides of arbitrary cross section. A surface integral equation approach is used in which the E-field equation has been transformed into a matrix equation using the method of moments. An iterative technique is used to pick the eigenvalues of the solution matrix which corresponds to the waveguide cutoff wave numbers. The salient features of this technique are its speed, its simplicity, and the absence of any spurious modes when waveguides of arbitrary cross section are treated. The first four modes are tabulated for various waveguides, and the results are in very good agreement with published data     

Complex TM models in a generalized Goubau line with arbitrary conductivities

The modal equation for the wavenumbers of the complex transverse magnetic (TM) modes of an open circular-cylinder coaxial waveguide with central conductor is studied for the case when the inner and outer materials have arbitrary conductivity. Analytic approximations for the wavenumhers of all of the modes are obtained for large contrast between the inner and outer materials, and simple numerical algorithms for calculating the wavenumbers are obtained for arbitrary contrast. It is shown that when the conductivities are zero, the wavenumbers group together in a set of four complex values, symmetric in the complex plane, but that the symmetry and the grouping can both be destroyed by adding conductivity to the materials. For fixed conductivities, there are principal modes (with no low-frequency cutoffs) and secondary modes (with low-frequency cutoffs) but one type mode can be converted into the other type by changing the conductivities. A numerical study of the modal equation shows how the modes of a Goubau line can be related to those of a coaxial transmission line. It shows also that the values of individual solutions of the modal equation can depend on the history of the conductivity values of the waveguide.     

非线性介质膜波导中高阶TE模的色散特性

用微扰方法对非线性波动方程求解,导出了各阶近似下非线性介质膜波导中高阶TE模的色散方程和截止波长。研究了高阶TE模的色散特性。计算结果表明,具有功率激发阈值的高阶模的一个传播功率可能对应两个不同的传播常数,具有光学开关的特性。     

Ag-coated Long-period Fiber Grating

Based on coupled-mode theory, the eigenvalue equation of five-layered long-period fiber grating （LPFG） sensor with Ag film and gas-sensitive film overlays are firstly studied. The problem of resolving complex eigenvalue equation on five-layered LPFG is analyzed, and the method of resolution is also given. Then the eigenvalue equation of three-layered metal cladding LPFG is analyzed, and the complex transcendental equation is also discussed. The computing result shows that the coupling between the low order EH modes and the core mode is much stronger than that between the low-order HE modes and the core mode.     

分析介质光波导传播特性的Cole—Hopf变换法

采用Cole-Hopf变换法，将平板光波导导模所满足的二阶常微分方程（Helmholtz方程）变换成一阶常微分方程（Riccati方程），用较简单且有效的方法分析导模的特性。给出了多层阶跃型及渐变型平板波导导模的精确解析解和数值解法及实例；给出了非线性平板波导传播特性的数值计算法及实例。     

Modes and Losses of a Four-Mirror Ring Resonator (Letters)

An investigation is reported on modes and losses of a square ring resonator that consists of the numerical solution of the appropriate integral equation for a number of values of the mirrors' aperture and spacing. It turns out that two sets of modes can exist: circulating modes and modes between two facing mirrors. The circulating modes are of the Fabry-Perot type, and their losses can be evaluated on the basis of the losses of suitably equivalent Fabry-Perot resonators.     

Angularly propagating waves in a radially inhomogeneous, lossy dielectric cylinder and their connection with the natural modes

A physical interpretation is proposed for the natural modes of a radially inhomogeneous, lossy dielectric circular cylinder embedded in a homogeneous, lossless dielectric. These modes are either of the creeping-wave or of the whispering-gallery type. With the aid of a generalized Wentzel-Kramer-Brillouin (WKB) asymptotics, an asymptotic characteristic equation is derived for each type, which can be explained physically. Numerical results obtained from the exact characteristic equation and both approximate equations are presented and discussed.     

The effect of the ▿n2term on the modes of an optical square-law medium

Solutions of the mode problem of an optical waveguide with a square-law medium are usually based on the scalar wave equation. An exact derivation of the wave equation from Maxwell's equations results in an additional term containing the gradientnablan^{2}of the square of the refractive indexn. It is the purpose of this correspondence to evaluate the change of the propagation constants of the modes of the square-law medium that is caused by thenablan^{2}term. We find that the change is significant only for TM modes of low order, but insignificant for all other modes of a dielectric slab with parabolic index distribution.     

A simple solution for the wavenumber of off-axis modes in Y-cut Ti:LiNbO3waveguides

A general dispersion equation for off-axis modes inY-cut Ti:LiNbO3waveguides is derived. The complex wavenumber of the leaky modes is found by a simple numerical procedure.     

New computation method for characteristic modes

To compute the characteristic modes of a perfectly conducting object a new weighted eigenvalue equation is derived, using the magnetic-field integral equation. The method of moments is used to solve this eigenvalue equation, and a number of bodies of revolution are examined. The results are presented and compared with those of previous methods.     

从物理概念理解矩形波导的导波模式

本文从导体边界条件出发,解释矩形波导内不能存在TEMz模式,并分析TE_m0^z模式的场结构。根据波数和驻波概念解释导波的截止与传播以及导波的快波特性。从方向阻抗概念得到模式的波阻抗,最后由色散方程解释导波的色散特性。本文从物理概念推导出导波模式的各种特性,更有助于学生对导波模式的理解。     

Orthogonal waves on electron beams

The propagation of slow electrokinetic waves on finite electron beams which fill a conducting tunnel is discussed for modes of axial symmetry. In addition to the familiar space-charge modes, a pair of modes exists which are related to the vortex frequency (omega_{upsilon} = omega_{c} - 2theta_{0}). The finite magnetic field introduces a coupling between these modes. A set of four orthogonal modes can be derived by the use of matrix transformations, thus eliminating the necessity of solving a complicated characteristic equation. For infinitely high magnetic fields, the two additional modes disappear and the four modes reduce to the fast and slow space-charge waves.     

Generalized cable equation model for myelinated nerve fiber

Herein, the well-known cable equation for nonmyelinated axon model is extended analytically for myelinated axon formulation. The myelinated membrane conductivity is represented via the Fourier series expansion. The classical cable equation is thereby modified into a linear second order ordinary differential equation with periodic coefficients, known as Hill's equation. The general internal source response, expressed via repeated convolutions, uniformly converges provided that the entire periodic membrane is passive. The solution can be interpreted as an extended source response in an equivalent nonmyelinated axon (i.e., the response is governed by the classical cable equation). The extended source consists of the original source and a novel activation function, replacing the periodic membrane in the myelinated axon model. Hill's equation is explicitly integrated for the specific choice of piecewise constant membrane conductivity profile, thereby resulting in an explicit closed form expression for the transmembrane potential in terms of trigonometric functions. The Floquet's modes are recognized as the nerve fiber activation modes, which are conventionally associated with the nonlinear Hodgkin-Huxley formulation. They can also be incorporated in our linear model, provided that the periodic membrane point-wise passivity constraint is properly modified. Indeed, the modified condition, enforcing the periodic membrane passivity constraint on the average conductivity only leads, for the first time, to the inclusion of the nerve fiber activation modes in our novel model. The validity of the generalized transmission-line and cable equation models for a myelinated nerve fiber, is verified herein through a rigorous Green's function formulation and numerical simulations for transmembrane potential induced in three-dimensional myelinated cylindrical cell. It is shown that the dominant pole contribution of the exact modal expansion is the transmembrane potential solution of our generalized model.