Anisotropic Maxwell symbol

Use a material-rest frame: an inertial chart in which each modelled material element has fixed spatial coordinates. Use the macroscopic fields of the classical electromagnetic material model. For physical interpretation declare a finite angular-frequency band $I_{\omega }=[{\omega }_{-},{\omega }_{+}]$ with $0<{\omega }_{-}<{\omega }_{+}<\mathrm{\infty }$, in ${\mathrm{s}}^{-1}$. The constant-coefficient branch below is adopted as an effective phase-response model only for $\omega \in I_{\omega }$, as required by the material-closure postulate; it asserts no broadband material response or universal front speed. The symbol and its algebraic Maxwell equivalence remain defined for every $\omega >0$. In a homogeneous source-free region take ${\rho }_f=0$, $J_f=0$ and select the nonconducting, lossless constitutive branch

$D=\epsilon E,B=\mu H,$

where $\epsilon \in {\mathbb{R}}^{3\times 3}$ is a constant real symmetric positive-definite absolute permittivity tensor in $\mathrm{F}{\mathrm{m}}^{-1}$, and $\mu >0$ is a constant real scalar absolute permeability in $\mathrm{H}{\mathrm{m}}^{-1}$. These are declared model coefficients, not dimensionless relative coefficients. Here lossless means these coefficients are real and the conductive current density $J_c=\sigma E$ is absent, $\sigma =0$; no dispersive or absorbing coefficient is included. Fix right-handed Cartesian coordinates and the phasor convention

$E(x,t)=\mathrm{Re}\text{\hspace{0.17em}⁣}\left(e{e}^{i(k\cdot x-\omega t)}\right),B(x,t)=\mathrm{Re}\text{\hspace{0.17em}⁣}\left(b{e}^{i(k\cdot x-\omega t)}\right),$

with $k\in {\mathbb{R}}^3\setminus \{0\}$ in ${\mathrm{m}}^{-1}$, $\omega >0$ in ${\mathrm{s}}^{-1}$, and complex amplitudes $e,b\in {\mathbb{C}}^3$. Write $d=\epsilon e$ and $h={\mu }^{-1}b$ for the displacement and magnetic-intensity amplitudes, in $\mathrm{C}{\mathrm{m}}^{-2}$ and $\mathrm{A}{\mathrm{m}}^{-1}$. For complex vectors use the bilinear dot product $u\cdot v={\sum }_{r=1}^3{u}_r{v}_r$, with no conjugation. The anisotropic Maxwell symbol is the linear map

$\mathcal{M}(k,\omega )e:=k\times ({\mu }^{-1}(k\times e))+{\omega }^2\epsilon e\mathrm{.}$

A nonzero amplitude $e$ is a source-free plane-wave mode in this branch exactly when $\mathcal{M}(k,\omega )e=0$; its magnetic amplitudes are $b=(k\times e)\mathrm{/}\omega$ and $h=b\mathrm{/}\mu$. The cross product is extended complex-bilinearly. In this model $k\cdot d=0$, while $k\cdot e$ need not vanish.

Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch

The macroscopic Maxwell equations in terms of D, H and the free sources

Real plane-wave fields encoded by complex amplitudes

Framework-local SI units, the vacuum constants and the exact/measured status of their values

For the real plane-wave fields above, the source-free macroscopic equations are $\mathrm{curl}E=-{\mathrm{\partial }}_{t}B$, $\mathrm{curl}H={\mathrm{\partial }}_{t}D$, $\mathrm{div}B=0$, and $\mathrm{div}D=0$. Their phasor amplitudes are $k\times e=\omega b$, $k\times h=-\omega d$, $k\cdot b=0$, and $k\cdot d=0$: spatial differentiation multiplies the common phase by $ik$, time differentiation by $-i\omega$, and equality of the real encodings for all $(x,t)$ forces equality of their complex amplitudes.

With $d=\epsilon e$ and $h={\mu }^{-1}b$, the first curl equation gives $b=(k\times e)\mathrm{/}\omega$. Substitution into the second gives $\mathcal{M}(k,\omega )e=0$. Conversely, if this symbol equation holds, define $b=(k\times e)\mathrm{/}\omega$ and $h=b\mathrm{/}\mu$. The first curl equation then holds by definition, and the second is the symbol equation divided by $\omega$. Taking the dot product of the symbol equation with $k$ gives ${\omega }^{2}k\cdot \epsilon e=0$, hence $k\cdot d=0$; also $k\cdot b=0$ because $k\cdot (k\times e)=0$. Thus the symbol equation is equivalent to the source-free plane-wave Maxwell amplitude system under the declared coefficients and convention. It is a conditional model formulation, not evidence that any substance realizes these coefficients.