Oleg A. Tretyakov

DERIVATION OF KLEIN-GORDON EQUATION FROM MAXWELL'S EQUATIONS AND STUDY OF RELATIVISTIC TIME-DOMAIN WAVEGUIDE MODES

An initial-boundary value problem for the system of Maxwells equations with time derivative is formulated and solved rigorously for transient modes in a hollow waveguide. It is supposed that the latter has perfectly conducting surface. Cross section, S; is bounded by a closed singly-connected contour of arbitrary but smooth enough shape. Hence, the TE and TM modes are under study. Every modal fleld is a product of a vector function of transverse coordinates and a scalar amplitude dependent on time, t; and axial coordinate, z: It has been established that the study comes down to, eventually, solving two autonomous problems: i) A modal basis problem. Final result of this step is deflnition of complete (in Hilbert space, L2 (S)) set of functions dependent on transverse coordinates which originates a basis. ii) A modal amplitude problem. The amplitudes are generated by the solutions to Klein-Gordon equation (KGE); derived from Maxwells equations directly, with t and z as independent variables. The solutions to KGE are invariant under relativistic Lorentz transforms and subjected to the causality principle. Special attention is paid to various ways that lead to analytical solutions to KGE. As an example, one case (among eleven others) is considered in detail. The modal amplitudes are found out explicitly and expressed via products of Airy functions with arguments dependent on t and z:

EVOLUTION EQUATIONS FOR ANALYTICAL STUDY OF DIGITAL SIGNALS IN WAVEGUIDES

excitation and propagation problem of the digital signals in a hollow waveguide is considered by an analytical timedomain method. The waveguide is geometrically regular along Oz axis, and its cross section is a closed singly connected domain. The waveguide surface is a perfect electric conductor. A complete set of TE and TM waveguide modes is obtained in Time Domain (TD) directly. Every modal field is deduced as the sum of its longitudinal and transverse vector components, where each component is the product of two factors. One factor is an element of the waveguide modal basis, which is a vector function of the transverse waveguide coordinates. The other one is a modal amplitude of appropriate field component, which is a scalar function of time t and the axial coordinate z. All the elements of the modal basis are specified via two scalar potentials. They are eigensolutions (normalized in a proper way) of Dirichlet and Neumann boundary eigenvalue problems for the Laplacian. Every element of the modal basis satisfies appropriate boundary conditions over the waveguide surface. The modal amplitudes are solutions of a system of evolution partial differential equations. The problem of Walsh function signals in the waveguide is solved explicitly in compliance with the causality principle and the special theory of relativity.

Modal basis method in radiation problems

Abslrofl - To solve radiation problems in time domain directly the modal representation of transient electromagnetic fields is considered. Using evolutionary approach the initial nonstationary three-dimensional electrodynamic problem is transformed into the problem for one-dimensional evolutionary equations. The modd basis for electromagnetic fields with arbitrary time dependence in spherical coordinate system is constructed. After elimination of the radial components of electrical and magnetic field from Maxwell equation system the four-dimensional diflerential operators are formed. It is proved that the operators are self- adjoint ones. The eigen-functions of the operators form the basis. The completeness of the basis is proved by means of Weyl Theorem about orthogonal splitting of Hilbert space. The expansion coeliicients of transient electromagnetic iield are found from the set of evolutionary equations.

Temporal Cavity Oscillations Caused by a Wide-Band Waveform

Excitation of the electromagnetic fields by a wide-band current surge, which has a beginning in time, is studied in a cavity bounded by a closed perfectly conducting surface. The cavity is filled with Debye or Lorentz dispersive medium. The fields are presented as the modal expansion in terms of the solenoidal and irrotational cavity modes with the time-dependent modal amplitudes, which should be found. Completeness of this form of solution has been proved earlier. The systems of ordinary differential equations with time derivative for the modal amplitudes are derived and solved explicitly under the initial conditions and in compliance with the causality principle. The solutions are obtained in the form of simple convolution (with respect to time variable) integrals. Numerical examples are exhibited as well.

Study of a Time Variant Cavity System

A cavity with a time signal applied taken jointly is treated as a time variant physical system. Standard formulation of the boundary-value problem for system of Maxwells equations (with ∂t) is supplemented with the initial conditions and the causality principle. It is proved that electromagnetic field can be presented as the classical decompositions in terms of the solenoidal and irrotational modes but with time dependent modal amplitudes. For the latters, evolutionary (i.e., with time derivative) ordinary differential equations are derived and solved analytically, in quadratures. Simple explicit solutions (in elementary functions) for the modal amplitudes as functions of time are obtained in a particular case. Numerical examples are exhibited, time-domain resonances are studied.

Time-domain cavity oscillations supported by a temporally dispersive dielectric

Forced time-domain oscillations in a cavity filled with a temporally dispersive polar dielectric are studied. The cavity is bounded by a singly connected closed perfect electric conductor surface S of rather arbitrary shape. A given source pumps a signal of finite duration to the cavity. Hence, the principle of causality is involved in the formulation of the problem. The temporal cavity oscillations are obtained as a self-consistent solution to the system of Maxwells equations and Debye equation supplemented with appropriate initial conditions . Analytical solutions are obtained by using the evolutionary approach to electromagnetics proposed and implemented recently. Temporal oscillations of the cavity modes are studied. Obtained results are compared with the finite-difference time-domain solutions.

The evolution equations in study of the cavity oscillations excited by a digital signal

The problem of electromagnetic oscillations in a cavity excited by a signal of finite duration is considered. The singly connected cavity surface has arbitrary geometrical form and it is perfectly conducting physically; its volume is filled with a homogeneous lossy medium. The formulation of the problem involves the principle of causality. The problem is solved within the frames of the evolutionary approach to electromagnetics. The electromagnetic field is presented as an eigenmodal expansion with time dependent modal amplitudes. The amplitudes satisfy a system of evolution (i.e., with time derivative) ordinary differential equations, which are derived and studied. Explicit solutions are obtained satisfying the principle of causality automatically. Numerical examples for the cavity oscillations excited by the Walsh function signals are exhibited, some resonances of the digital signals are revealed.

The Real-Valued Time-Domain TE-Modes in Lossy Waveguides

The time-domain studies of the modal fields in a lossy waveguide are executed. The waveguide has a perfectly conducting surface. Its cross section domain is bounded by a singly-connected contour of rather arbitrary but enough smooth form. Possible waveguide losses are modeled by a conductive medium which fills the waveguide volume. Standard formulation of the boundary-value problem for the system of Maxwell’s equations with time derivative is given and rearranged to the transverse-longitudinal decompositions. Hilbert space of the real-valued functions of coordinates and time is chosen as a space of solutions. Complete set of the TE-time-domain modal waves is established and studied in detail. A continuity equation for the conserved energetic quantities for the time-domain modal waves propagating in the lossy waveguide is established. Instant velocity of transportation of the modal flux energy is found out as a function of time for any waveguide cross section. Fundamental solution to the problem is obtained in accordance with the causality principle. Exact explicit solutions are obtained and illustrated by graphical examples.

Radiation of arbitrary signals by plane disk

The nonsinusoidal waves radiation problem is considered. The Mode Basis Method is accommodated for free space using continuous spectrum parameters. As a result a one-dimensional Klein-Gordon equation is obtained like the evolutionary waveguide equations. The method is examined by a step function radiation problem.

Transient Electromagnetic Fields in a Cavity with Dispersive Double Negative Medium

Electromagnetic flelds in a cavity fllled with double negative dispersive medium and bounded by a closed perfectly conducting surface is studied in the Time Domain. The sought electromagnetic flelds are found in a closed form by using decomposition over cavity modes and solving in TD the difierential equations for the time varying mode amplitudes. Some features of frequency response of such an electromagnetic system are presented. Waveforms of electromagnetic flelds excited by a wideband pulse are considered.