Anisotropic waves and reciprocity — Examples

Let $\epsilon =\mathrm{diag}({\epsilon }_{\perp },{\epsilon }_{\perp },{\epsilon }_{\parallel })$ with ${\epsilon }_{\perp },{\epsilon }_{\parallel }>0$ and ${\epsilon }_{\perp }\mathrm{\ne }{\epsilon }_{\parallel }$, and let $\mu >0$. Fix $\omega >0$ and real $k=(k_x,0,k_z)$ with $k_x{k}_z\mathrm{\ne }0$ on the extraordinary sheet $k_x^2\mathrm{/}{\epsilon }_{\parallel }+k_z^2\mathrm{/}{\epsilon }_{\perp }=\mu {\omega }^2$. Choose $d_0\in \mathbb{C}\setminus \{0\}$ in $\mathrm{C}{\mathrm{m}}^{-1}$ and set the displacement amplitude $d=d_0(k_z,0,-k_x)$ and electric amplitude $e={\epsilon }^{-1}d$. Then $k\cdot d=0$ but $k\cdot e=d_0{k}_x{k}_z(1\mathrm{/}{\epsilon }_{\perp }-1\mathrm{/}{\epsilon }_{\parallel })\mathrm{\ne }0$. Thus the extraordinary electric amplitude is not orthogonal to the wavevector even though the displacement amplitude is.

The algebraic example holds for every $\omega >0$. For physical interpretation, declare a finite angular-frequency 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 the supplying uniaxial classification.

Ordinary and extraordinary sheets

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Choose a one-oscillator Lorentz response with ${\epsilon }_{\mathrm{\infty }}>0$ in $\mathrm{F}{\mathrm{m}}^{-1}$, $a>0$ in $\mathrm{F}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$, ${\omega }_0>0$ and $\gamma >0$ in ${\mathrm{s}}^{-1}$, and fix $\omega >0$ in ${\mathrm{s}}^{-1}$. In the $e^{-i\omega t}$ convention its scalar absolute permittivity is $\epsilon (\omega )={\epsilon }_{\mathrm{\infty }}+a\mathrm{/}({\omega }_0^2-{\omega }^2-i\gamma \omega )$. Let $I_3$ denote the $3\times 3$ identity matrix, and let the tensor coefficients be $\mathit{\epsilon }(\omega )=\epsilon (\omega )I_3$ and $\mathit{\mu }=\mu I_3$ with real $\mu >0$ in $\mathrm{H}{\mathrm{m}}^{-1}$. These coefficients satisfy the transpose-symmetry condition of the reciprocal phasor model, while $\mathrm{Im}\epsilon (\omega )>0$. For any nonzero local electric amplitude $e\in {\mathbb{C}}^3$ in $\mathrm{V}{\mathrm{m}}^{-1}$, define the real electric field at a fixed material point by $E(t)=\mathrm{Re}(e{e}^{-i\omega t})$. Let $p_1=(\epsilon (\omega )-{\epsilon }_{\mathrm{\infty }})e$ be the oscillator polarization amplitude from the one-oscillator response, in $\mathrm{C}{\mathrm{m}}^{-2}$; put $P_1(t)=\mathrm{Re}(p_1{e}^{-i\omega t})$, and define its polarization-current contribution by $J_{\mathrm{o}\mathrm{s}\mathrm{c}}(t):={\mathrm{\partial }}_t{P}_1(t)$. Its phasor amplitude is $j_{\mathrm{o}\mathrm{s}\mathrm{c}}=-i\omega (\epsilon (\omega )-{\epsilon }_{\mathrm{\infty }})e$ in $\mathrm{A}{\mathrm{m}}^{-2}$. The cycle-averaged power density transferred from the electric field to the oscillator through this current contribution, in $\mathrm{W}{\mathrm{m}}^{-3}$, is

${\bar{q}}_{\mathrm{o}\mathrm{s}\mathrm{c}}:=\frac{1}{T}{\int }_0^T{J}_{\mathrm{o}\mathrm{s}\mathrm{c}}(t)\cdot E(t)dt=\frac{\omega }{2}\mathrm{Im}\epsilon (\omega )\mathrm{\mid }e{\mathrm{\mid }}^2>0,T=\frac{2\pi }{\omega },\mathrm{\mid }e{\mathrm{\mid }}^2:=e^{\ast }\cdot e={\sum }_{r=1}^3\mathrm{\mid }{e}_r{\mathrm{\mid }}^2\mathrm{.}$

Here $e^{\ast }$ is componentwise complex conjugation. The conjugate used in this cycle average is distinct from the bilinear, unconjugated product in the reciprocity identity.

Thus the coefficient condition that supplies the Lorentz reciprocity identity is compatible with positive cycle-averaged work on the oscillator-current contribution in this adopted model.

Reciprocity in reciprocal linear media

Analytic susceptibility and dispersion restrictions

Fixed-frequency reciprocal tensor medium

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

This calculation establishes coexistence of transpose-symmetric reciprocal coefficients and positive cycle-averaged work on the oscillator-current contribution in the stated model. It makes no claim about a time-reversal transformation.