scholarly journals Computing Effective Hamiltonians of Coupled Electromagnetic Systems

2021 ◽  
Author(s):  
Shaolin Liao ◽  
Lu Ou
Keyword(s):  
Boundary Condition ◽  
Dispersion Relation ◽  
Normal Modes ◽  
Computational Procedure ◽  
Hamiltonian Matrix ◽  
Effective Hamiltonian ◽  
Electromagnetic System ◽  
Wave Dynamics ◽  
Potential Applications ◽  
Free Port

In this paper, we present an efficient procedure to compute the effective Hamiltonian matrix of a coupled electromagnetic system consisting of subsystems that are coupled to a discrete number of channels through couplers. Each subsystem is described by its own effective non-Hermitian Hamiltonian and the corresponding Quasi-normal Modes (QNMs), while the coupler connecting the subsystems and the channels is described by the scattering matrix, which is equivalent to the transfer matrix, in terms of port vectors defined for the coupler. Due to the constraints imposed by the QNMs of the subsystems and the wave dynamics of the channels, as well as boundary condition constraints, constraint-free port vectors need to be chosen efficiently and they follow two rules: 1) port vectors forming loops with couplers; 2) port vectors of couplers with most constraints or with less freedom. With the constraint-free port vectors chosen, the effective Hamiltonian matrix of the coupled electromagnetic system can be obtained by imposing the boundary condition constraints. After the effective Hamiltonian is obtained, the eigenvalues, eigenvectors and dispersion relation of the coupled electromagnetic system, as well as other quantities such as the reflection and transmission, can be calculated. A 2D interstitial square coupled MRRs array is used as an example to demonstrate the computational procedure. The computation of the effective Hamiltonian matrix of a coupled electromagnetic system has many potential applications such as MRRs array, coupled Parity-Time Non-Hermitian electromagnetic system, as well as the dispersion relation of finite and infinite arrays.

scholarly journals Computing Effective Hamiltonians of Coupled Electromagnetic Systems

2021 ◽  
Author(s):  
Shaolin Liao ◽  
Lu Ou
Keyword(s):  
Boundary Condition ◽  
Dispersion Relation ◽  
Normal Modes ◽  
Computational Procedure ◽  
Hamiltonian Matrix ◽  
Effective Hamiltonian ◽  
Electromagnetic System ◽  
Wave Dynamics ◽  
Potential Applications ◽  
Free Port

In this paper, we present an efficient procedure to compute the effective Hamiltonian matrix of a coupled electromagnetic system consisting of subsystems that are coupled to a discrete number of channels through couplers. Each subsystem is described by its own effective non-Hermitian Hamiltonian and the corresponding Quasi-normal Modes (QNMs), while the coupler connecting the subsystems and the channels is described by the scattering matrix, which is equivalent to the transfer matrix, in terms of port vectors defined for the coupler. Due to the constraints imposed by the QNMs of the subsystems and the wave dynamics of the channels, as well as boundary condition constraints, constraint-free port vectors need to be chosen efficiently and they follow two rules: 1) port vectors forming loops with couplers; 2) port vectors of couplers with most constraints or with less freedom. With the constraint-free port vectors chosen, the effective Hamiltonian matrix of the coupled electromagnetic system can be obtained by imposing the boundary condition constraints. After the effective Hamiltonian is obtained, the eigenvalues, eigenvectors and dispersion relation of the coupled electromagnetic system, as well as other quantities such as the reflection and transmission, can be calculated. A 2D interstitial square coupled MRRs array is used as an example to demonstrate the computational procedure. The computation of the effective Hamiltonian matrix of a coupled electromagnetic system has many potential applications such as MRRs array, coupled Parity-Time Non-Hermitian electromagnetic system, as well as the dispersion relation of finite and infinite arrays.

Quasi-static modes of oscillation of a cold toroidal plasma

1975 ◽  
Vol 14 (1) ◽  
pp. 25-37 ◽  
Author(s):  
John D. Love
Keyword(s):  
Dispersion Relation ◽  
Cross Section ◽  
Normal Modes ◽  
Toroidal Plasma ◽  
Cylindrical Plasma ◽  
Aspect Ratios ◽  
Plasma Ring ◽  
Toroidal Geometry

The normal modes of oscillation of a cold dielectric plasma ring are analysed in the quasi-electrostatic approximation. An exact dispersion relation is derived, valid for all aspect ratios. Its solutions are shown to be extremely close to those of an infinite cylindrical plasma with cross-section equal to the minor cross-section of the ring, when the cylinder is considered as a wavelength-preserving limit of the toroidal geometry.

The effect of undulations on the electrostatic potential in a polyelectrolyte system

1994 ◽  
Vol 348 (1687) ◽  
pp. 209-228 ◽  
Keyword(s):  
Boundary Condition ◽  
Integral Equation ◽  
Electrostatic Potential ◽  
Perturbation Technique ◽  
Cylindrical Symmetry ◽  
Potential Applications ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
The Green Function ◽  
Constant Charge

I consider the effect of macromolecular undulation on the electrostatic potential around a rod-like molecule. This effort is set to demonstrate the use of a particular perturbation technique through application to a geometrical system of general colloidal interest. The Poisson—Boltzmann equation together with a constant charge boundary condition on the well defined surface of an undulating cylinder is reformulated in integral equation form by use of Green’s theorem. A perturbation solution appropriate to the deformed boundary can be extracted when the Green function is approximated by that relevant to a reference, undeformed cylinder. Numerical results demonstrate that undulation causes significant deviations (increases) in electrochemical properties from expected behaviour, assuming rigid cylindrical symmetry. By considering the total free energy of the system it is found that electrostatics tend to diminish the extent of the undulations. The predicted deviations are briefly discussed in light of measured intermolecular electrostatic forces acting in a condensed phase of close-packed DNA. The perturbation technique has potential applications to mathematically similar problems occurring in hydrodynamics.

scholarly journals Hamiltonian description of wave dynamics in nonequilibrium media

1994 ◽  
Vol 1 (4) ◽  
pp. 234-248 ◽  
Author(s):  
N. N. Romanova
Keyword(s):  
Nonlinear Wave ◽  
Evolution Equations ◽  
Normal Modes ◽  
Weak Nonlinearity ◽  
Hamiltonian Description ◽  
Weakly Nonlinear ◽  
Wave Dynamics ◽  
Canonical Variables ◽  
Stable Points ◽  
Stable Area

Abstract. We consider Hamiltonian description of weakly nonlinear wave dynamics in unstable and nonequilibrium media. We construct the appropriate canonical variables in the whole wavenumber space. The essentially new element is the construction of canonical variables in a vicinity of marginally stable points where two normal modes coalesce. The commonly used normal variables are not appropriate in this domain. The mater is that the approximation of weak nonlinearity breaks down when the dynamical system is written in terms of these variables. In this case we introduce the canonical variables based on the linear combination of modes belonging to the two different branches of dispersion curve. As an example of one of the possible applications of presented results the evolution equations for weakly nonlinear wave packets in the marginally stable area are derived. These equations cannot be derived if we deal with the commonly used normal variables.

scholarly journals The Convergence of the Legendre–Galerkin Spectral Method for Constructing Atmospheric Acoustic Normal Modes

2020 ◽  
Vol 28 (03) ◽  
pp. 2050002
Author(s):  
Richard B. Evans
Keyword(s):  
Legendre Polynomials ◽  
Normal Modes ◽  
Sobolev Norm ◽  
Linear Operators ◽  
Basis Sets ◽  
Spectral Approximation ◽  
Spectral Approximations ◽  
Potential Applications ◽  
Galerkin Spectral Method ◽  
Theoretical Results

The asymptotic rate of convergence of the Legendre–Galerkin spectral approximation to an atmospheric acoustic eigenvalue problem is established, as the dimension of the approximating subspace approaches infinity. Convergence is in the [Formula: see text] Sobolev norm and is based on the existing theory [F. Chatelin, Spectral Approximations of Linear Operators (SIAM, 2011)]. The assumption is made that the eigenvalues are simple. Numerical results that help interpret the theory are presented. Eigenvalues corresponding to acoustic modes with smaller [Formula: see text] norms are especially accurately approximated, even with lower dimensioned basis sets of Legendre polynomials. The deficiencies in the potential applications of the theoretical results are noted in connection with the numerical examples.

scholarly journals Thermal instability of compressible Walters’ (Model B′) fluid in the presence of Hall currents and suspended particles

2011 ◽  
Vol 15 (2) ◽  
pp. 487-500 ◽  
Author(s):  
Urvashi Gupta ◽  
Parul Aggarwal
Keyword(s):  
Magnetic Field ◽  
Dispersion Relation ◽  
Thermal Instability ◽  
Normal Modes ◽  
Suspended Particles ◽  
Elastic Parameter ◽  
Stationary Convection ◽  
Electrically Conducting ◽  
Permissible Range ◽  
Rayleigh Numbers

Effect of Hall currents and suspended particles is considered on the hydromagnetic stability of a compressible, electrically conducting Walters? (Model B?) elastico-viscous fluid. After linearizing the relevant hydromagnetic equations, the perturbation equations are analyzed in terms of normal modes. A dispersion relation governing the effects of visco-elasticity, magnetic field, Hall currents and suspended particles is derived. It has been found that for stationary convection, the Walters? (Model B?) fluid behaves like an ordinary Newtonian fluid due to the vanishing of the visco-elastic parameter. The compressibility and magnetic field have a stabilizing effect on the system, as such their effect is to postpone the onset of thermal instability whereas Hall currents and suspended particles are found to hasten the onset of thermal instability for permissible range of values of various parameters. Also, the dispersion relation is analyzed numerically and the results shown graphically. The critical Rayleigh numbers and the wavenumbers of the associated disturbances for the onset of instability as stationary convection are obtained and the behavior of various parameters on critical thermal Rayleigh numbers has been depicted graphically. The visco-elasticity, suspended particles and Hall currents (hence magnetic field) introduce oscillatory modes in the system which were non-existent in their absence.

scholarly journals Interpretation of dispersion relations for bounded systems

1972 ◽  
Vol 7 (1) ◽  
pp. 13-48 ◽  
Author(s):  
T. D. Rognlien ◽  
S. A. Self
Keyword(s):  
Dispersion Relation ◽  
Boundary Conditions ◽  
Normal Modes ◽  
Plasma Waves ◽  
Axial Current ◽  
Linear Dispersion ◽  
Linear Dispersion Relation ◽  
End Plate ◽  
Cylindrical Form ◽  
Bounded System

A treatment is given of the problem of constructing normal modes for anarbitrarily bounded system from roots of the linear dispersion relationD( ω, k) = 0 for the corresponding infinite or periodically bounded system. For a system described by continuous macroscopic variables, and of general cylindrical form (uniform along an axisz, say), each transverse eigenmode gives rise to a set of axial normal modes constructed from a pair of dominant rootskαz(ω) ofD= 0 satisfying the boundary conditions which are characterized by complex reflexion coefficients for the dominant waves. The implications of the results for the interpretation of experiments on plasma waves and instabilities on finite cylinders are discussed, with particular reference to the effects of end-plate damping and axial current onQ-machines.

Sign in / Sign up

Export Citation Format

Share Document