Approximate Dispersion Equation Research Articles

Analytical Dispersion Equations of a Lossy Coaxial Waveguide in the Microwave and Visible Spectra

Analytical hybrid-mode dispersion relations of a lossy coaxial waveguide were rigorously analyzed using a mode-matching technique. In order to model a practical coaxial line with inevitable losses, we adopted an all-dielectric coaxial waveguide surrounded by the perfect electric conductor (PEC) boundary. The rigorous dispersion characteristics of the TM<sub>01</sub>, TE<sub>01</sub>, and EH<sub>11</sub> modes were investigated for lossy coaxial waveguides filled with different electrical conductivities. Based on the exact solutions, approximate but accurate dispersion equations were proposed for the TM<sub>0<i>p</i></sub>, TE<sub>0<i>p</i></sub> , EH<sub><i>mp</i></sub>, and HE<sub><i>mp</i></sub> modes in order to estimate and compare the behaviors of complex propagation constants in the microwave and visible spectra.

Smooth approximation of a varying refractive-index profile and its application in the computation of light waves

In this paper, the smooth approximation of light waves is studied for an open optical waveguide with a distinct refractive-index profile, which involves high-precision computation of the eigenmodes and corresponding eigenfunctions. During analysis, the refractive-index function is first approximated by a quadratic spline interpolation function. Since the quadratic spline function is a polynomial of degree two in every sub-interval (sub-layer), it is equivalent to a piecewise polynomial of degree two, based on which, the corresponding Sturm-Liouville eigenvalue problem of the Helmholtz operator in sub-layer can be solved analytically by the Kummer functions. Finally, the approximate dispersion equation is established to the TE case. Obviously, the approximate dispersion equations converge fast to the exact ones, as the maximum value of the sub-interval sizes tends to zero. Furthermore, eigenmodes may be obtained by Müller’s method with suitable initial values. To refine the accuracy, the equidistant partition and the non-equidistant partition are applied to divide the interval. Numerical simulations show that the eigenfunctions of the spline interpolation are much smoother than the ones with piecewise interpolation. In addition, the non-equidistant partition can help improve the accuracy and the order of convergence of general solutions reaches the third.

An Efficient Solver of Eigenmodes for a Class of Complex Optical Waveguides

In this paper, for a class of complex optical waveguide, the high-precision computation of the propagation constants β are studied. The corresponding Sturm-Liouville (S-L) problem is represented as in an open domain (open on one side), where x is a given value. Firstly, a perfectly matched layer is used to terminate the open domain. Secondly, both the equation and the complex coordinate stretching transformations are constructed. Thirdly, the S-L problem is turned to a simplified form such as in a bounded domain. Finally, the coefficient function is approximated by a piecewise polynomial of degree two. Since the simplified equation in each layer can be solved analytically by the Kummer functions, the approximate dispersion equation is established to the TE case. When the coefficient function is continuous, the approximate solutions converge fast to the exact ones, as the maximum value of the subinterval sizes tends to zero. Numerical simulations show that high-precision eigenmodes may be obtained by the Muller's method with suitable initial values.

On the Issue of Damping of a Transverse Ion Bernstein Wave in an Inhomogeneous Magnetic Field

With allowance for the centrifugal and gradient drifts of ions, a new approximate dispersion equation is obtained for the ion Bernstein (IB) wave propagating strictly transverse to the external inhomogeneous magnetic field. Using this equation, it is shown that if, in the high-temperature plasma, the drift of ions is taken into account, the resonance frequency ranges in the vicinities of the ion cyclotron frequency harmonics can strongly overlap. In this case, the ion Bernstein waves turn out to be strongly damped, despite the absence of the longitudinal component of the wave vector.

An Efficient Solver of Eigenmodes for a Class of Complex Optical Waveguides

In this paper, for a class of complex optical waveguide, the high-precision computation of the propagation constants β are studied. The corresponding Sturm-Liouville (S-L) problem is represented as in an open domain (open on one side), where x is a given value. Firstly, a perfectly matched layer is used to terminate the open domain. Secondly, both the equation and the complex coordinate stretching transformations are constructed. Thirdly, the S-L problem is turned to a simplified form such as in a bounded domain. Finally, the coefficient function is approximated by a piecewise polynomial of degree two. Since the simplified equation in each layer can be solved analytically by the Kummer functions, the approximate dispersion equation is established to the TE case. When the coefficient function is continuous, the approximate solutions converge fast to the exact ones, as the maximum value of the subinterval sizes tends to zero. Numerical simulations show that high-precision eigenmodes may be obtained by the Muller's method with suitable initial values.

Nanoscale X-ray imaging of spin dynamics in yttrium iron garnet

Time-resolved scanning transmission x-ray microscopy has been used for the direct imaging of spin-wave dynamics in a thin film yttrium iron garnet (YIG) with sub-200 nm spatial resolution. Application of this x-ray transmission technique to single-crystalline garnet films was achieved by extracting a lamella (13×5×0.185 μm3) of the liquid phase epitaxy grown YIG thin film out of a gadolinium gallium garnet substrate. Spin waves in the sample were measured along the Damon-Eshbach and backward volume directions of propagation at gigahertz frequencies and with wavelengths in a range between 200 nm and 10 μm. The results were compared to theoretical models. Here, the widely used approximate dispersion equation for dipole-exchange spin waves proved to be insufficient for describing the observed Damon-Eshbach type modes. For achieving an accurate description, we made use of the full analytical theory taking mode-hybridization effects into account.

Kinetic description of the oblique propagating spin-electron acoustic waves in degenerate plasmas

An oblique propagation of the spin-electron acoustic waves in degenerate magnetized plasmas is considered in terms of quantum kinetics with the separate spin evolution, where the spin-up electrons and the spin-down electrons are considered as two different species with different equilibrium distributions. It is considered in the electrostatic limit. The corresponding dispersion equation is derived. Analysis of the dispersion equation is performed in the long-wavelength limit to find an approximate dispersion equation describing the spin-electron acoustic wave. The approximate dispersion equation is solved numerically. Real and imaginary parts of the spin-electron acoustic wave frequency are calculated for different values of the parameters describing the system. It is found that the increase in the angle between the direction of wave propagation and the external magnetic field reduces the real and imaginary parts of spin-electron acoustic wave frequency. The increase in the spin polarization decreases the real and imaginary parts of frequency either. The imaginary part of frequency has a nonmonotonic dependence on the wave vector which shows a single maximum. The imaginary part of frequency is small in comparison with the real part for all parameters in the area of applicability of the obtained dispersion equation.

Analytical Study of Hydrodynamic Instability in the Flame: 2. Account of the Viscosity of the Gas in the Cold and Hot Areas

Hydrodynamic instability is examined with consideration given to the viscosity of the fresh gas and combustion products, as well as to the dependences of the flame speed on the front curvature and of the transport coefficients on the temperature. For the perturbation frequency, an approximate second-order dispersion equation is derived. The flame is completely stable at very high viscosity or small dimensions. The greatest destabilizing role of the thermal expansion coefficient manifests itself at its relatively small values. As the expansion coefficient increases, the viscosity of the gas in the flame zone increases rapidly. In addition, the stabilizing effect according to the Markstein model is enhanced by thermal expansion.

A New Approach of Eigenmodes for Varying Refractive-Index Profile’s Waveguides

For the modal computation of an open optical waveguide with a varying refractive-index profile, two perfectly matched layers are used to terminate the waveguide, and the refractive-index profile is approximated by a piecewise polynomial of degree two. Then, the corresponding Sturm–Liouville problem (eigenvalue problem) of the Helmholtz operator in each layer can be solved analytically by the Whittaker functions, and the analytical approximate dispersion equations are established for both TE and TM cases. The approximate solutions converge fast to the exact ones for the continuous refractive-index function as the maximum value of the subinterval sizes tends to zero. Numerical simulations show that high-precision modes may be obtained by Muller’s method with suitable initial values. The validation of the proposed method is also verified by use of the finite difference method with a fine grid.

Propagation of electromagnetic waves guided by perfectly conducting model of a tape helix supported by dielectric rods

The homogeneous boundary value problem existing in the electromagnetic wave propagation in a dielectric-loaded perfectly conducting tape helix with infinitesimal tape thickness is investigated in this study. The ill-posed boundary value problem is regularised using the mollification method. The homogeneous boundary value problem is solved for the dielectric loaded perfectly conducting tape helix taking into account the exact boundary conditions for the perfectly conducting dielectric loaded tape helix. The solved approximate dispersion equation takes the form of the solvability condition for an infinite system of linear homogeneous equations namely, the determinant of the infinite order coefficient matrix is zero. For the numerical computation of the dispersion equation, all the entries of the symmetrically truncated version of the coefficient matrix are estimated by summing an adequate number of the rapidly converging series for them. The tape-current distribution is estimated from the null-space vector of the truncated coefficient matrix corresponding to a specified root of the dispersion equation. The numerical results suggest that the propagation characteristic computed by the anisotropically conducting model (that neglects the component of the tape-current density perpendicular to the winding direction) is only an abstinent approximation to consider for moderately wide tapes.

Bell-Plesset effects in Rayleigh-Taylor instability of finite-thickness spherical and cylindrical shells

Bell-Plesset (BP) effects account for the influence of global convergence or divergence of the fluid flow on the evolution of the interfacial perturbations embedded in the flow. The development of the Rayleigh-Taylor instability in radiation-driven spherical capsules and magnetically-driven cylindrical liners necessarily includes a significant contribution from BP effects due to the time dependence of the radius, velocity, and acceleration of the unstable surfaces or interfaces. An analytical model is presented that, for an ideal incompressible fluid and small perturbation amplitudes, exactly evaluates the BP effects in finite-thickness shells through acceleration and deceleration phases. The time-dependent dispersion equations determining the “instantaneous growth rate” are derived. It is demonstrated that by integrating this approximate growth rate over time, one can accurately evaluate the number of perturbation e-foldings during the inward acceleration phase of the implosion. In the limit of small shell thickness, exact thin-shell perturbation equations and approximate thin-shell dispersion equations are obtained, generalizing the earlier results [E. G. Harris, Phys. Fluids 5, 1057 (1962); E. Ott, Phys. Rev. Lett. 29, 1429 (1972); A. B. Bud'ko et al., Phys. Fluids B 2, 1159 (1990)].

Efficient approximations of dispersion relations in optical waveguides with varying refractive-index profiles.

In this paper we consider the problem of computing the eigen-modes for the varying refractive-index profile in an open waveguide. We first approximate the refractive-index by a piecewise polynomial of degree two, and the corresponding Sturm-Liouville problem (eigenvalue problem) of the Helmholtz operator in each layer can be solved analytically by the Kummer functions. Then, analytical approximate dispersion equations are established for both TE and TM cases. Furthermore, the approximate dispersion equations converge fast to the exact ones for the continuous refractive-index function as the maximum value of the subinterval sizes tends to zero. Suitable numerical methods, such as Müller's method or the chord secant method, may be applied to the dispersion relations to compute the eigenmodes. Numerical simulations show that our method is very practical and efficient for computing eigenmodes.

The photon magnetic moment problem revisited

The photon magnetic moment for radiation propagating in magnetized vacuum is defined as a pseudo-tensor quantity, proportional to the external electromagnetic field tensor. After expanding the eigenvalues of the polarization operator in powers of $k^2$, we obtain approximate dispersion equations (cubic in $k^2$), and analytic solutions for the photon magnetic moment, valid for low momentum and/or large magnetic field. The paramagnetic photon experiences a red shift, with opposite sign than the gravitational one, which differs for parallel and perpendicular polarizations. It is due to the drain of photon transverse momentum and energy by the external field. By defining an effective transverse momentum, the constancy of the speed of light orthogonal to the field is guaranteed. We conclude that the propagation of the photon non-parallel to the magnetic behaves as if there is a quantum compression of vacuum or warp of space-time in an amount depending on its angle with regard to the field.

TE AND TM MODES OF A VANE-LOADED CIRCULAR CYLINDRICAL WAVEGUIDE FOR GYRO-TWT APPLICATIONS

The dispersion equation governing the guided propagation of TE and TM fast wave modes of a circular cylindrical waveguide loaded by metal vanes positioned symmetrically around the wave-guide axis is derived from the exact solution of a homogeneous boundary value problem for Maxwell's equations. The dispersion equation takes the form of the solvability condition for an inflnite system of linear homogeneous algebraic equations. The approximate dispersion equation corresponding to a truncation of the inflnite-order coe-cient matrix of the inflnite system of equations to the coe-cient matrix of a flnite system of equations of su-ciently high order is solved numerically to obtain the cut-ofi wave numbers of the various propagating modes. Each cut-ofi wave number gives rise to a unique dispersion curve in the shape of a hyperbola in the !-fl plane.

Spin-transfer torque induced domain wall ferromagnetic resonance in nanostrips

The frequency response of a Néel domain wall in a nanostrip excited by alternating spin-polarized current is studied by micromagnetic simulations. Several internal normal modes of the domain wall are excited and the corresponding spatial distributions of oscillation power are imaged. In the case of current perpendicular to the wall, the excited normal modes are mainly concentrated at the wall center and/or wall boundaries, forming edge modes, center modes, standing-wave modes and their mixed modes. The localization and spatial distribution of the modes have a close relationship with the total internal field, especially its inhomogeneity. With the decrease of saturation magnetizations Ms, the spatial inhomogeneities of the total field are gradually weakened and some domain wall normal modes disappear. In the case where the current is parallel to the wall, the wall thickness oscillation mode (or breathing mode) is excited. Furthermore, due to the geometrical confinement, high-order thickness modes such as edge- and standing wave-thickness modes are observed. The magnetization dependence of the eigen-frequency exhibits different forms for different normal modes, which can be qualitatively explained based on an approximate dispersion equation for spin-wave modes of a quasi-uniformly magnetized rectangular element.

Perfectly conducting tape-helix model for guided electromagnetic wave propagation

The homogeneous boundary value problem arising in the propagation of electromagnetic waves guided by an open tape helix modelled to be of infinitesimal tape thickness and infinite tape-material conductivity is shown to be inherently ill posed. It is demonstrated how the ill posed problem may be regularised using the mollification method. The regularised boundary value problem is then solved to yield the approximate dispersion equation which takes the form of the solvability condition for an infinite system of linear homogeneous algebraic equations viz., the determinant of the infinite-order coefficient matrix is zero. For the numerical computation of the dispersion characteristic, all the entries of the symmetrically truncated version of the coefficient matrix are estimated by summing an adequate number of the rapidly converging (after regularisation) series for them. The tape-current distribution is estimated from the null-space vector of the truncated coefficient matrix corresponding to a specified root of the dispersion equation. A comparison of the numerical results with those for the anisotropically conducting model (that neglects the component of the tape-current density perpendicular to the winding direction) of the tape helix reveals that the propagation characteristic computed on the basis of the anisotropically conducting model could be substantially in error even for moderately wide tapes.

Dispersion relation for surface plasmon polaritons in metal/nonlinear-dielectric/metal slot waveguides

We present the first (to our knowledge) exact dispersion relation for the transverse-magnetic surface plasmon polariton (SPP) modes of a plasmonic slot waveguide, which is formed by a nonlinear Kerr medium sandwiched between two metallic slabs. The obtained relation is then simplified to the case of small field intensities, while retaining nonlinear terms, to derive approximate dispersion equations for the symmetric and antisymmetric SPP modes.

Приближённое дисперсионное уравнение для атмосферных гравитационных волн

Приближённое дисперсионное уравнение для атмосферных гравитационных волн

New approach to investigation of resonant vibrations of noncircular shells based on the theory of coupled waveguides

In the present paper a new simple method of analytical description of resonant vibrations of finite noncircular cylindrical shells is developed. The method is based on the theory of coupled waveguides formed by quasiflat areas of the same noncircular shells having an infinite length (depth). The physical reason for guided wave propagation along quasiflat areas of such shells is the difference between flexural wave velocities in their quasiflat and curved areas, respectively. Using asymptotic expressions for flexural wave velocities in circular shells with different radii of curvature, approximate dispersion equations are derived for waves propagating in such waveguides and their corresponding coupling coefficients. After that, considering shells of finite length, the transition is made from the coupled guided modes to the coupled resonant vibrations of the shell. The obtained resonant frequencies and spatial distributions of the resulting vibration modes are in good agreement with the results of finite element calculations.

The fundamental electromagnetic wave of a single-wire line in a weakly absorbing medium

An approximate dispersion equation valid over a wide frequency range is obtained from the dispersion equation for the symmetric E 00 wave (the Sommerfeld wave) propagating along a finite-conductance metal wire embedded in a lossy dielectric. The boundaries of this frequency range are determined. A technique is developed for solution of the obtained approximate dispersion equation in the frequency band where the pronounced skin effect is observed. The ratio of the wire radius to the thickness of the skin layer is assumed to be no less than 10. This technique is applied to calculate the longitudinal and transverse propagation constants for 1.0-and 2.5-mm-radii copper wires over the frequency range 0.5 MHz to 1000 GHz.