Abstract
The fundamental macroscopic electromagnetic field equations and elementary plane wave solutions in linear, temporally dispersive absorptive media are developed in this chapter with particular emphasis on homogeneous, isotropic, locally linear (HILL), temporally absorptive dispersive media. The general frequency dependence of the dielectric permittivity, magnetic permeability, and electric conductivity is included in the analysis so that the resultant field equations rigorously apply to both perfect and imperfect dielectrics, conductors and semiconducting materials, as well as to metamaterials, over the entire frequency domain.
“We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” James Clerk Maxwell (1862).
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Notes
- 1.
Some texts refer to such low-loss materials as being imperfect dielectrics. This terminology is not used here as all dispersive dielectric media have loss over some frequency domain and consequently are imperfect.
- 2.
In sea-water, the ionic bonds in NaCl are weakened by the large relative static dielectric permittivity of H 2 O (𝜖 s ≈ 79.2), thereby effectively “dissolving” the molecule into positive and negative ions, thereby giving rise to its electrical conductivity.
- 3.
This was the driving force behind the U. S. Navy’s Project Sanguine [1, 2] which developed an extremely low frequency (ELF) 2.6 MW radio wave transmitter facility operating at 76 Hz. Because of public opposition incited by concerns of adverse health effects caused by ELF radiation, Project Sanguine was abandoned and replaced by a series of scaled-down versions, beginning with Project Seafarer in 1975, Austere ELF in 1978, and finally Project ELF in 1981 that was constructed and deployed at Clam Lake, Wisconsin and Republic, Michigan. A similar ELF system is reportedly operated by the Russian Navy at Murmansk.
- 4.
For optical wave fields, ω ∼ 1015 → 1016 r/s and k 0 ∼ 107 → 108 m−1.
- 5.
Notice that the zero frequency limit of the spatio-temporal Fourier transform of Faraday’s law \(\tilde {\tilde {\mathbf {B}}}(\mathbf {k},\omega ) = (\|c\|/\omega )\mathbf {k}\times \tilde {\tilde {\mathbf {E}}}(\mathbf {k},\omega )\) as ω → 0 exists provided that one lets k →0 simultaneously. In this idealized static limit, Maxwell’s equations are reduced into separate equations governing the electrostatic and steady-state magnetic fields.
- 6.
A square matrix \( \underline {A} = (a_{ij})\) is said to be symmetric if and only if a ij = a ji for all i, j, in which case the matrix is equal to its transpose, viz. \( \underline {A} = \underline {A}^T\). For a skew-symmetric square matrix \( \underline {B} = (b_{ij})\), b ij = −b ji for all i, j. The diagonal elements of a skew-symmetric matrix must then all be zeroes.
- 7.
A square matrix \( \underline {A} = (a_{ij})\) is said to be Hermitian if and only if \(a_{ij} = a^*_{ji}\) for all i, j, so that \( \underline {A} = \underline {A}^{\dagger }\), where the superscript † notation indicates the complex conjugate of the transpose of the matrix. The diagonal elements a jj of an Hermitian matrix must then be real-valued. For a skew-Hermitian (or anti-Hermitian) matrix \( \underline {B} = (b_{ij})\), \(b_{ij} = -b^*_{ji}\) for all i, j, so that \( \underline {B} = - \underline {B}^{\dagger }\). The diagonal elements of a skew-Hermitian matrix are then either zeroes or are pure imaginary. Finally, notice that a skew-Hermitian matrix \( \underline {B} = (b_{ij})\) may be expressed in terms of a Hermitian matrix \( \underline {C}=(c_{ij})\) as \( \underline {B} = i \underline {C}\) so that c ij = −ib ij for all i, j.
- 8.
The standard terminology of ‘principal dielectric constants’ is not adopted here because 𝜖 1, 𝜖 2, and 𝜖 3 are, in general, frequency dependent and hence, not constant.
- 9.
From the Greek for opposite (enantio) form (morphe).
- 10.
A polar vector (an ordinary vector such as the position vector r) reverses sign when the coordinate axes are reversed, whereas a pseudovector (an axial vector) does not reverse sign. The cross product of two polar vectors produces a pseudovector, whereas the dot product of a polar vector with a pseudovector produces a pseudoscalar, a scalar which changes sign under an inversion.
- 11.
This result has been criticized in an analysis [24] based upon a revised formulation of electromagnetic energy conservation [18]. However, this analysis relies, in part, on the revised constitutive relations [compare with the relations given in Eqs. (5.5) and (5.6)] D = 𝜖 ∗ ∂ E∕∂t and B = μ ∗ ∂ H∕∂t, where ∗ denotes the convolution operation. This then results in the real parts of both 𝜖(ω) and μ(ω) being odd functions of real ω and their imaginary parts are now even, in disagreement with experimental results (see Figs. 4.2 and 4.3).
- 12.
Notice that some authors choose to leave the factor \({\frac {1}{2}}\) in Eq. (5.208) when mksa units are employed, in which case \(\tilde {\mathbf {S}} = \tilde {\mathbf {E}}\times \tilde {\mathbf {H}}^*\) and \(\langle \mathbf {S}\rangle = \frac {1}{2}\Re \{\tilde {\mathbf {S}}\}\).
- 13.
There is no loss for a strictly static field, as is evident from Eq. (5.221).
- 14.
Because the electric and magnetic material properties are temporally dispersive, there may be specific angular frequency values at which one or more of the material parameters do not change across S.
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Oughstun, K.E. (2019). Fundamental Field Equations in Temporally Dispersive Media. In: Electromagnetic and Optical Pulse Propagation . Springer Series in Optical Sciences, vol 224. Springer, Cham. https://doi.org/10.1007/978-3-030-20835-6_5
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