Anisotropic waves and reciprocity

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.

Fix a real angular frequency $\omega \in \mathbb{R}$ and a bounded open region $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with $C^1$ boundary in a material-rest frame. Here a material-rest frame is the inertial frame in which every modelled material element has fixed spatial coordinates, as stipulated by the framework's material-closure postulate. A map in $C^1(\bar{\mathrm{\Omega }})$ means the restriction of a $C^1$ map on an open neighborhood of $\bar{\mathrm{\Omega }}$. Let $\epsilon ,\mu :\bar{\mathrm{\Omega }}\to {\mathbb{C}}^{3\times 3}$ be continuous coefficient fields with units $\mathrm{F}{\mathrm{m}}^{-1}$ and $\mathrm{H}{\mathrm{m}}^{-1}$. They may be spatially varying and complex. Let $E_j,H_j\in C^1(\bar{\mathrm{\Omega }};{\mathbb{C}}^3)$ and $J_j\in C^0(\bar{\mathrm{\Omega }};{\mathbb{C}}^3)$, for $j=1,2$, be phasor amplitudes with SI units $\mathrm{V}{\mathrm{m}}^{-1}$, $\mathrm{A}{\mathrm{m}}^{-1}$, and $\mathrm{A}{\mathrm{m}}^{-2}$. The corresponding real fields use $e^{-i\omega t}$, for example $E_j^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}(x,t)=\mathrm{Re}(E_j(x)e^{-i\omega t})$. Their displacement and magnetic-induction amplitudes are $D_j=\epsilon E_j$ in $\mathrm{C}{\mathrm{m}}^{-2}$ and $B_j=\mu H_j$ in $\mathrm{T}$.

The fixed-frequency constitutive and source equations are

$D_j=\epsilon E_j,B_j=\mu H_j,\mathrm{curl}{E}_j=i\omega B_j,\mathrm{curl}{H}_j=J_j-i\omega D_j\mathrm{.}$

All operators act componentwise on the spatial variable. For $u,v\in {\mathbb{C}}^3$, define the bilinear dot product $u\cdot v={\sum }_{r=1}^3{u}_r{v}_r$ and the complex-bilinear cross product by the usual right-handed component formula. Neither product conjugates either input. The transpose $T$ is the ordinary matrix transpose, without complex conjugation. This definition calls the coefficient model reciprocal when $\epsilon (x{)}^T=\epsilon (x)$ and $\mu (x{)}^T=\mu (x)$ at every $x\in \bar{\mathrm{\Omega }}$. It places no Hermitian, positive-definite, or passivity condition on these complex matrices.

This is a fixed-frequency phasor data class. Its possibly complex coefficients are not inferred from the framework's real, instantaneous material branch; a separate response model is required to justify any such interpretation.

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

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

In the homogeneous lossless plane-wave model, let $\epsilon =\mathrm{diag}({\epsilon }_{\perp },{\epsilon }_{\perp },{\epsilon }_{\parallel })$ with ${\epsilon }_{\perp },{\epsilon }_{\parallel }>0$ in $\mathrm{F}{\mathrm{m}}^{-1}$, let $\mu >0$ be a scalar in $\mathrm{H}{\mathrm{m}}^{-1}$, and let $\omega >0$ be in ${\mathrm{s}}^{-1}$. For a nonzero source-free mode with real wavevector $k=(k_x,0,k_z)\mathrm{\ne }0$ in ${\mathrm{m}}^{-1}$, the nonzero-mode wavevectors lie on exactly the following two sheets:

$k_x^2+k_z^2=\mu {\epsilon }_{\perp }{\omega }^2,\frac{k_x^2}{{\epsilon }_{\parallel }}+\frac{k_z^2}{{\epsilon }_{\perp }}=\mu {\omega }^2\mathrm{.}$

Away from an intersection the first sheet is the ordinary sector, whose electric amplitude is parallel to $\widehat{y}$, and the second is the extraordinary sector, whose electric amplitude lies in the $xz$ plane; each then has a one-dimensional polarization space. At an intersection the two sectors span a two-dimensional polarization space. If ${\epsilon }_{\perp }={\epsilon }_{\parallel }=\epsilon$, the isotropic specialization also allows arbitrary real $k\mathrm{\ne }0$, without the $xz$-plane restriction: its nonzero-mode wavevectors lie in every direction on the common sphere $\mathrm{\mid }k{\mathrm{\mid }}^2=\mu \epsilon {\omega }^2$, and at every such wavevector the polarization space is the two-dimensional complex subspace $\{e\in {\mathbb{C}}^3:k\cdot e=0\}$. In the $xz$ section the ordinary/extraordinary labels then denote the two coordinate sectors, not distinct dispersion branches. If ${\epsilon }_{\perp }\mathrm{\ne }{\epsilon }_{\parallel }$, the sheets intersect only for $k_x=0$ and $k_z^2=\mu {\epsilon }_{\perp }{\omega }^2$; at those optic-axis directions the polarization space is again two-dimensional. The zero amplitude is not a mode in this statement.

The symbol classification holds algebraically for every $\omega >0$. For physical interpretation, declare a finite band $I_{\omega }=[{\omega }_{-},{\omega }_{+}]$ with $0<{\omega }_{-}<{\omega }_{+}<\mathrm{\infty }$, in ${\mathrm{s}}^{-1}$, and use the constant-coefficient model only for $\omega \in I_{\omega }$, as in its defining material closure.

Anisotropic Maxwell symbol

For the representative positive-uniaxial ratio ${\epsilon }_{\parallel }=2{\epsilon }_{\perp }$, normalize both wavevector coordinates by $\omega \sqrt{\mu {\epsilon }_{\perp }}$. The ordinary cross-section is the unit circle; the extraordinary cross-section is the ellipse $u_x^2\mathrm{/}2+u_z^2=1$. They intersect at the two optic-axis points $(0,\pm 1)$. The drawing is a normalized cross-section of the two exact equations in the Statement; the proof does not rely on it.

\begin{tikzpicture}[scale=1.25,>=stealth]
  \draw[->] (-1.8,0) -- (1.8,0) node[right] {$u_x$};
  \draw[->] (0,-1.45) -- (0,1.45) node[above] {$u_z$};
  \draw[thick,blue] (0,0) circle (1);
  \draw[thick,red,dashed] (0,0) ellipse [x radius=1.414,y radius=1];
  \fill (0,1) circle (1.4pt);
  \fill (0,-1) circle (1.4pt);
  \node[blue] at (-1.05,0.75) {ordinary};
  \node[red] at (1.15,0.78) {extraordinary};
  \node[below left] at (0,0) {$0$};
\end{tikzpicture}

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a bounded domain with $C^1$ boundary and outward unit normal $n$. Fix a real angular frequency $\omega$ in ${\mathrm{s}}^{-1}$. Let $E_j,H_j\in C^1(\bar{\mathrm{\Omega }};{\mathbb{C}}^3)$ and $J_j\in C^0(\bar{\mathrm{\Omega }};{\mathbb{C}}^3)$, for $j=1,2$, be phasor fields and electric-current densities in $\mathrm{V}{\mathrm{m}}^{-1}$, $\mathrm{A}{\mathrm{m}}^{-1}$, and $\mathrm{A}{\mathrm{m}}^{-2}$, respectively. Suppose both pairs solve the fixed-frequency Maxwell equations there with the same continuous coefficient maps $\epsilon ,\mu :\bar{\mathrm{\Omega }}\to {\mathbb{C}}^{3\times 3}$ and $\epsilon (x{)}^T=\epsilon (x)$, $\mu (x{)}^T=\mu (x)$ at every $x$. Use Euclidean volume measure $dV$ and boundary area measure $dS$. Then

${\int }_{\mathrm{\partial }\mathrm{\Omega }}n\cdot (E_1\times H_2-E_2\times H_1)dS={\int }_{\mathrm{\Omega }}(E_2\cdot J_1-E_1\cdot J_2)dV\mathrm{.}$

The dot and cross products in this identity are the complex-bilinear products defined in the reciprocal phasor model; no complex conjugation occurs. Both integrals have units W. In particular, the real symmetric positive permittivity tensor and positive scalar permeability of the lossless anisotropic plane-wave model satisfy these reciprocal-coefficient hypotheses, so the algebraic identity applies to any two such phasor solutions restricted to a bounded $C^1$ subdomain. Physical use of that anisotropic branch is restricted to its declared finite positive-frequency band $I_{\omega }=[{\omega }_{-},{\omega }_{+}]$, $0<{\omega }_{-}<{\omega }_{+}<\mathrm{\infty }$; the reciprocity identity itself asserts no broadband material response.

Fixed-frequency reciprocal tensor medium

Anisotropic Maxwell symbol

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The divergence and curl of a cross product

Divergence on a bounded C1 Euclidean domain

The symmetric conditions use transpose, not conjugate transpose. Positive absorption can therefore coexist with this reciprocity identity; the identity alone asserts neither losslessness nor time-reversal invariance.