A new matrix representation of the Maxwell equations based on the Riemann-Silberstein-Weber vector for a linear inhomogeneous medium (original) (raw)

Maxwell Optics: I. An exact matrix representation of the Maxwell equations in a medium

Physica Scripta, 2002

Matrix representations of the Maxwell equations are well-known. However, all these representations lack an exactness or/and are given in terms of a {\em pair} of matrix equations. We present a matrix representation of the Maxwell equation in presence of sources in a medium with varying permittivity and permeability. It is shown that such a representation necessarily requires 8times88 \times 88times8 matrices and an explicit representation for them is presented.

A new matrix formulation of the Maxwell and Dirac equations

Heliyon, 2018

Presented in this paper is a new matrix formulation of both the classical electromagnetic Maxwell equations and the relativistic quantum mechanical Dirac equation. These new matrix representations will be referred to as the Maxwell spacetime matrix equation and the Dirac spacetime matrix equation. Both are Lorentz invariant. Key to these new matrix formulations is an 8-by-8 matrix operator referred to here as the spacetime matrix operator. As it turns out, the Dirac spacetime matrix equation is equivalent to four new vector equations, which are similar in form to the four Maxwell vector equations. These new equations will be referred to as the Dirac spacetime vector equations. This allows these new vector equations to be as readily solved as solving a set of Maxwell vector equations. Based on these two new matrix approaches, two computer programs, encoded using Matlab software, have been developed and tested for determining the reflection and transmission characteristics of multilayer optical thin-film structures and multilayer quantum well-and-barrier structures. A listing of these software programs may be found in supplemental material associated with this article. Numerical results obtained based on the use of these computer programs are presented in the results section of this article.

Matrix-based approach to electrodynamics in media

2008

The Riemann -- Silberstein -- Majorana -- Oppenheimer approach to the Maxwell electrodynamics in presence of electrical sources and arbitrary media is investigated within the matrix formalism. The symmetry of the matrix Maxwell equation under transformations of the complex rotation group SO(3.C) is demonstrated explicitly. In vacuum case, the matrix form includes four real 4times44 \times 44times4 matrices alphab\alpha^{b}alphab. In

Secondary Quantization of the Electromagnetic Field in Inhomogeneous Structures Based on Formalism of Scattering Matrix

We show the procedure for the secondary quantization of the electromagnetic field in an inhomogeneous medium, based on the formalism of the scattering matrix. We demonstrate the electromagnetic field profiles, which are obtained as a result of applying the scattering matrix formalism and prove their orthogonality. Finally, we derive the Hamiltonian of the system. The obtained procedure allows us to use more complete representation of quantum theory in a layered medium to describe various effects in layered structures.

Unified matrix presentation of Maxwell's and wave equations using generalized differential matrix operators [EM engineering education]

IEEE Transactions on Education, 1998

In this paper, we introduce the concept of generalized differential matrix operators (GDMO's) that are useful for the formulation of electromagnetic boundary value problems in arbitrary orthogonal coordinate systems, e.g., Cartesian, cylindrical, and spherical. The most significant attribute of the GDMO approach is that their use helps to simplify the complicated manipulation of vector differential equations, especially in problems dealing with an anisotropic media. We show that the use of the GDMO's enable one to replace, for most problems in electromagnetics, the complicated vector differential operations with manipulation of 3 23 matrices. In addition, we demonstrate GDMO's are convenient for deriving many differentiation identities and integral theorems which find extensive applications in electromagnetics.

On the different forms of the Maxwell's electromagnetic equations in a uniform media

2006

Two known, alternative to each other, forms of the Maxwell's electromagnetic equations in a moving uniform media are investigated and discussed. Approach commonly used after Minkowski is based on the two tensors: H^{ab} = (D, H /c) and F^{ab} = (E, cB) which transform independently of each other at Lorentz transitions; relationships between fields change their form at Lorentz transformations

Solution of the Maxwell field equations in vacuum for arbitrary charge and current distributions using the methods of matrix algebra

IEEE Transactions on Education, 1993

Presented in this paper is a new matrix representation of classical electromagnetic theory. The basis 4)f this representation is a space-time, eight-by-eight difrerential matrix operator. This matrix operator is initially lormulated from the differential form of the Maxwell field equations for vacuum. The resulting matrix formulation of Maxwell's equations allows simple and direct derivation of: i he electromagnetic wave and charge continuity equations; I he Lorentz conditions and definition of the electromagnetic jotentials; the Lorentz and Coulomb gauges; the electromagnetic 2otential wave equations; and Poynting's conservation of 1:nergy theorem. A four-dimensional Fourier transform of :he matrix equations casts them into an eight-dimensional transfer theorem. The transfer function has an inverse, and this allows the equations to be inverted. This expresses the fields directly in terms of the charge and current source distributions.

The class of electromagnetic P-media and its generalization

Applying four-dimensional differential-form formalism, a novel class of electromagnetic media, labeled as that of P-media, is introduced in terms of a simple rule. It is shown that it is not possible to define the medium by expressing D and B in terms of E and H, whilst using 3D Gibbsian vectors and dyadics. Moreover, the basic properties of P-media are shown to be complementary to those of the previously known Q-media, which are defined in a somewhat similar manner. It is demonstrated that, for plane waves in a P-medium, there is no restriction to the wave one-form (corresponding to the k-vector). Importantly, the uniaxial P-medium half space also leads to another realization of the recently studied DB boundary conditions. Finally, a generalization of the class of P-media is briefly discussed. It is shown that the dispersion equation of a plane wave in the generalized Pmedium is decomposed into two conditions, each of which corresponds to a certain polarization condition. This occurrence resembles the behavior of the generalized Q-medium.