A comparison of transient solutions of Maxwell's equations to thatof the modified Maxwell's equations

In 1986 H.F. Harmuth introduced a modification of Maxwell's equations to study the propagation of transient electric and magnetic field strengths in lossy media. Opponents of this modification of Maxwell's equations have claimed and attempted to demonstrate that Maxwell's equations in their known forms can correctly be solved, for example by the Laplace transformation method, to obtain solutions of transient electric and associated magnetic field strengths in lossy media without encountering any difficulties. This work presents detailed computer plots of Harmuth's transient solutions of the modified Maxwell's equations and that of Maxwell's equations solved by the Laplace transformation characteristic for the two solutions, which indicate that they are not the same. It is shown that Harmuth's procedure results in physically more plausible solutions     

Signal solutions of Maxwell's equations for charge carriers withnon-negligible mass

Discusses signal solutions to Maxwell's equations for charge carriers with non-negligible mass. In order to find solutions the authors add information to Maxwell's equations by means of a physical assumption to obtain a defined solution. The authors' assumption is that magnetic dipoles and magnetic dipole currents should be represented by a magnetic (dipole) current density term just as electric dipoles and electric dipole currents-or electric polarization currents-have always been represented by an electric current density term. It is perfectly possible that other physical assumptions can be made that yield defined solutions and that will withstand public scrutiny     

Correction of Maxwell's Equations for Signals II

In the first of two companion papers it was shown that the addition of a magnetic current density to Maxwell's equations is a sufficient condition to obtain solutions in lossy propagation media for waves that are not infinitely extended periodic waves. The solutions obtained represented transients that may be used to represent signals having a beginning and an end. This second paper shows that the addition of a magnetic current density is also a necessary condition for the existence of transient solutions in lossy media. The modification of Maxwell's equations is thus necessary and sufficient for the study of the propagation of signals in lossy media.     

Reply to J.R. Waits `In defense of Stratton' [EM theory]

Additional comments on an ongoing controversy on Maxwell's equations are presented. Harmuth (see ibid., vol.EMC-30, no.1, p.90, 1988) claims that three scientists searching through Stratton's (1941) book could not find equations for the magnetic field strength associated with an electric excitation force or the electric field strength associated with a magnetic excitation force     

Local field analysis of bent graded-index planar waveguides

A local field analysis is proposed for bent planar waveguides with arbitrary refractive index profiles. Exact vector wave equations that include the gradient index, or polarization correction, term are derived for both transverse electric and transverse magnetic modes from Maxwell's equations in a local bent coordinate system. The approximate local field and correction to the propagation constants are obtained by perturbation analysis. As an example of the method, an infinitely extended parabolic index profile is studied     

Response to a letter by J.R. Wait about electromagnetic waves inseawater

In response to a series of letters by J.R. Wait, and in place of a rebuttal, the authors challenge Wait to solve a never-before-published transient problem without a magnetic current density term added to Maxwell's equations. In particular, they ask for the excited electric and the associated magnetic field strength of a planar wave due to an electric step function as excitation force in seawater, taking the ionic conductance into account by allowing the charge carriers to have a mass     

General solutions of Maxwell's equations for signals in a lossymedium. I. Electric and magnetic field strengths due to electricexponential ramp function excitation

Modification of Maxwell's equations to obtain general solutions for a lossy medium is reviewed. It is done by adding an extra term, referred to as the fictitious magnetic charge density. The solutions, which are in integral form, are solved numerically by computer for an exponential ramp function excitation. Computer plots for the electric and magnetic field strengths as functions of time at different locations in a lossy transmission medium are presented     

Low-Frequency Shielding Effectiveness of Nonuniform Enclosures

This paper presents a quasi-static approximate solution to the magnetic shielding of several nonuniform enclosures using the integral form of Maxwell's equations and insight gained from other approaches. The solution is called quasi-static as the assumptions made are from physical arguments based on low-frequency cases where the enclosure size is much less than a wavelength. The integral form of Maxwell's equations is used to obtain a first order correction to the static solution to obtain induced currents in the time-varying case. A cylindrical shell immersed in an axial magnetic field is used to illustrate the method, which is then extended to derive a formula for a similarly excited rectangular enclosure. These shields are seen to behave like a low-pass filter. Although the enclosure dimensions are small compared to the wavelength, the skin depth effects in the walls cannot be neglected even for relatively thin material as usually encountered in an enclosure. These skin effects are included in the analysis and experimental checks performed on a variety of enclosure sizes and materials, excited by a Helmholtz coil show agreement within two decibels over the 4-octave frequency range examined. No one can say whether this method offers a better solution to the shielding problem, as all solutions are approximate, but the author attempts to present an alternative formulation that aids in understanding the physical processes involved in the shielding effectiveness of an enclosure and fills some of the gaps between the plane-wave analysis and circuit approaches presently used.     

Correction of Maxwell's Equations for Signals I

Electromagnetic wave theory has been based on the concept of infinitely extended periodic sinusoidal waves ever since Maxwell published his theory a century ago. On the practical level this worked very well, but on the theoretical level we always had an indication that something was amiss. There was never a satisfactory concept of propagation velocity of signals within the framework of Maxwell's theory. The often-mentioned group velocity fails on two accounts, one being that it is almost always larger than the velocity of light in radio transmission through the atmosphere; the other being that its derivation implies a transmission rate of information equal to zero. A closer study shows that Maxwell's equations fail for waves with nonnegligible relative frequency bandwidth propagating in a medium with nonnegligible losses. The reason is singularities encountered in the course of calculation. The remedy is the addition of a magnetic current density which may be chosen zero after one has reached the last singularity but not before.     

Modeling of magnetizing processes

Quasi-static magnetic processes such as those found in magnetic recording are examined. The hysteresis in medium-hard magnetic materials of the type used in recording media and the magnetizability of soft magnetic materials such as those used in recording heads are discussed. Two general modeling techniques are used to describe these processes: physical modeling and phenomenological modeling. In physical modeling, the basic processes involved are simulated in order to be able to describe the basic magnetizing modes. In phenomenological models, the gross behavior of the material is described mathematically by Preisach-type models in order to couple the material properties to Maxwell's equations so as to obtain solutions of field problems. The latter models are computationally more efficient than the former, but they do not give any insight into the physical principles involved     

The Maxwell's equations in the sense of distributions

It is well known that there is no direct one-to-one correspondence between the electromagnetic theory based on the physical laws and that based on the Maxwell's differential equations. For example, in order to derive the boundary conditions from the Maxwell's differential equations, one assumes that some integral identities derived from them are valid even when the field components (or material parameters) are discontinuous. This assumption violates, in a sense, the completeness of the theory of electromagnetism based on the Maxwell's differential equations. We will prove that if one postulates that the Maxwell's equations are valid in the sense of distributions, then this incompleteness will be removed and the boundary conditions will appear implicitly in the basic differential equations.     

A generalized mass lumping technique for vector finite-element solutions of the time-dependent Maxwell equations

Time-domain finite-element solutions of Maxwell's equations require the solution of a sparse linear system involving the mass matrix at every time step. This process represents the bulk of the computational effort in time-dependent simulations. As such, mass lumping techniques in which the mass matrix is reduced to a diagonal or block-diagonal matrix are very desirable. In this paper, we present a special set of high order 1-form (also known as curl-conforming) basis functions and reduced order integration rules that, together, allow for a dramatic reduction in the number of nonzero entries in a vector finite element mass matrix. The method is derived from the Nedelec curl-conforming polynomial spaces and is valid for arbitrary order hexahedral basis functions for finite-element solutions to the second-order wave equation for the electric (or magnetic) field intensity. We present a numerical eigenvalue convergence analysis of the method and quantify its accuracy and performance via a series of computational experiments.     

General solutions of Maxwell's equations for signals in a lossymedium. III. Electric and magnetic field strengths due to electric andmagnetic sinusoidal pulse excitation

For pt.II see ibid., vol.30, no.1, p.37-40 (1988). The representation of a function with a general time variation by a series expansion of time-shifted transients is discussed. On the basis of this representation, numerical solutions of Maxwell's equations are presented for the electric and magnetic field strengths in a lossy medium due to electric and magnetic excitation functions consisting of a finite number of sinusoidal cycles. The solutions are derived by means of a time-series expansion of the available solutions for the electric and magnetic exponential ramp function excitations     

Generalized Finite-Difference Time-Domain Method Utilizing Auxiliary Differential Equations for the Full-Vectorial Analysis of Photonic Crystal Fibers

We present the generalized finite-difference time-domain full-vectorial method by reformulating the time-dependent Maxwell's curl equations with electric flux density and magnetic field intensity, with auxiliary differential equations using complex-conjugate pole-residue pairs. The model is generic and robust to treat general frequency-dependent material and nonlinear material. The Sellmeier equation is implicitly incorporated as a special case of the general formulation to account for material dispersion of fused silica. The results are in good agreement with the results from the multipole method. Kerr nonlinearity is also incorporated in the model and demonstrated. Nonlinear solutions are provided for a one ring photonic crystal fiber as an example.     

Rotational symmetries of electromagnetic radiation fields

Maxwell's equations and their solutions have been examined from the point of view of the full rotation group. Two mapping theorems have been derived which give the rotational symmetries of the radiated electric and magnetic and Poynting's vector field as a function of the symmetries of the electric current density which acts as the radiating source.     

Electromagnetic Fields in Complementary and Self-Complementary Structures

Electromagnetic fields are analyzed in complementary and self-complementary structures. As is well known, Maxwell's equations exhibit a complementary (duality) symmetry between the electric and magnetic fields. Self-complementary field solutions (when these exist) on self-complementary structures have particular properties of constant impedance (resistance) associated with certain pure traveling wave fields. The contribution of this paper is that it rigorously treats the electromagnetic field in a self-complementary waveguide. It shows a self-complementary field solution is a pure traveling wave. A particular waveguide structure is selected that illustrates circumstances when a self-complementary field solution fails to exist in a self-complementary structure. A computer model was developed to analyze a family of waveguide structures that differ only in an aspect ratio parameter, an aspect ratio of 1.0 corresponding to the self-complementary case. The original structures and the corresponding complementary structures are analyzed independently. Results are presented for a wide range of frequencies and aspect ratios. At low frequencies, when only one TEM mode propagates, field solutions for the input TEM mode exhibit discontinuous behavior as the aspect ratio approaches one from above and below. In this frequency range, no time harmonic self-complementary field solution exists for the precisely self-complementary waveguide structure.     

FDTD simulation of TE and TM plane waves at nonzero incidence in arbitrary Layered media

Plane wave scattering is an important class of electromagnetic problems that is surprisingly difficult to model with the two-dimensional finite-difference time-domain (FDTD) method if the direction of propagation is not parallel to one of the grid axes. In particular, infinite plane wave interaction with dispersive half-spaces or layers must include careful modeling of the incident field. By using the plane wave solutions of Maxwell's equations to eliminate the transverse field dependence, a modified set of curl equations is derived which can model a "slice" of an oblique plane wave along grid axes. The resulting equations may be used as edge conditions on an FDTD grid. These edge conditions represent the only known way to accurately propagate plane wave pulses into a frequency dependent medium. An examination of grid dispersion between the plane wave and the modeled slice reveals good agreement. Application to arbitrary dispersive media is straightforward for the transverse magnetic (TM) case, but requires the use of an auxiliary equation for the transverse electric case, which increases complexity. In the latter case, a simplified approach, based on formulating the dual of the TM equations, is shown to be quite effective. The strength of the developed approach is illustrated with a comparison with the conventional simulation based on an analytic incident wave specification with half-space, single frequency reflection and transmission for the edges. Finally, an example of a possible biomedical application is given and the implementation of the method in the perfectly matched layer region is discussed.     

Investigation of static and quasi-static fields inherent to the pulsed FDTD method

Demonstrates that trailing dc offsets, which can affect E- or H-fields in finite-difference time-domain simulations, are physically correct static solutions of Maxwell's equations instead of being numerically induced artifacts. It is shown that they are present on the grid when sources are used, which generates nondecaying charges. Static solutions are investigated by exciting electric and magnetic dipoles models with suitable waveforms.     

Comments on `Solutions of Maxwell''s equations for generalnonperiodic waves in lossy media by M.E. El-Shandwily

The commenter notes that the intention of M.E. El-Shandwily's paper (see ibid., vol.30, no.4, p.577-82, Nov. 1988) is to demonstrate that the linear Maxwell's equations can be applied to the case of a pulse or step-function field change. If this can be demonstrated, then, it is claimed, the Harmuth Ansatz (see ibid., vol.EMC-28, no.4, p.250-8, Nov. 1986) for solving Maxwell's equations to obtain the magnetic field for impulse excitation is unnecessary. The commenter seeks to show that El-Shandwily actually implements the Harmuth Ansatz, which explains the concurrence of Harmuth's and El-Shandwily's predictions     

Solutions of Maxwell's equations for general nonperiodic waves inlossy media

Solutions of Maxwell's equations in lossy media for signals excited by a general applied source at the boundary plane are given. The excitation at the boundary plane can be through either electric or magnetic functions of any general time variation. No additional terms need be added to Maxwell's equations to obtain the solutions. Excitations by an electric step, exponential, and finite duration sinusoidal; functions of time are given as examples