The propagating candidate branch: cutoff, phase and group speeds, cross-section energy and flux, and energy transport at the group velocity

[Period-averaged quadratic densities] Fix a point $(x_1,x_2,x_3)$ and put $a:=\tilde{E}{e}^{i\beta x_3}$, $d:=\tilde{H}{e}^{i\beta x_3}$, so that $E=\mathrm{Re}(a{e}^{-i\omega t})$ and $H=\mathrm{Re}(d{e}^{-i\omega t})$ at that point. By [F2], $\frac{1}{T}{\int }_0^T\mathrm{\mid }E{\mathrm{\mid }}^{2}dt=\frac{1}{2}\mathrm{\mid }\tilde{E}{\mathrm{\mid }}^2$, $\frac{1}{T}{\int }_0^T\mathrm{\mid }H{\mathrm{\mid }}^{2}dt=\frac{1}{2}\mathrm{\mid }\tilde{H}{\mathrm{\mid }}^2$ and $\frac{1}{T}{\int }_0^T(E\times H)dt=\frac{1}{2}\mathrm{Re}(\tilde{E}\times \overline{\stackrel{~}{H}})$, because the phase factors $e^{\pm i\beta x_3}$ are unimodular and cancel in the inner and cross products. Hence the period averages $⟨u⟩:=\frac{1}{T}{\int }_0^{T}udt=\frac{1}{4}(\epsilon \mathrm{\mid }\tilde{E}{\mathrm{\mid }}^2+\mu \mathrm{\mid }\tilde{H}{\mathrm{\mid }}^2)$ and $⟨S⟩:=\frac{1}{T}{\int }_0^{T}Sdt=\frac{1}{2}\mathrm{Re}(\tilde{E}\times \overline{\stackrel{~}{H}})$ depend only on $(x_1,x_2)$ and not on $z$ or on the starting time, and therefore $U={\int }_{R_{ab}}⟨u⟩d{x}_{1}d{x}_2$ and $\mathrm{\Phi }={\int }_{R_{ab}}⟨S⟩\cdot e_{z}d{x}_{1}d{x}_2$ are the stated averages, independent of $z$ and of the starting time; consistency with the local balance of [F1] means ${\mathrm{\partial }}_t⟨u⟩=0$ and ${\mathrm{\partial }}_z(⟨S⟩\cdot e_z)=0$, which the displayed formulas show directly.

[Cross-section identities] For the scaled profiles $h=H_{\ast }{h}_{mn}$ and $e=E_{\ast }{e}_{mn}$ of [F1], multiplying the two gradient-integral identities of [F3] by $H_{\ast }^2$ and $E_{\ast }^2$, respectively, gives ${\int }_{R_{ab}}\mathrm{\mid }{\mathrm{\nabla }}_{t}h{\mathrm{\mid }}^2=k_c^2{\int }_{R_{ab}}{h}^2$ and ${\int }_{R_{ab}}\mathrm{\mid }{\mathrm{\nabla }}_{t}e{\mathrm{\mid }}^2=k_c^2{\int }_{R_{ab}}{e}^2$ with vanishing boundary terms (the traces ${\mathrm{\partial }}_{n}h=0$ and $e=0$ on $\mathrm{\partial }{R}_{ab}$ of The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals (a),(b)), and the pointwise identities $\mathrm{\mid }{e}_z\times {\mathrm{\nabla }}_{t}h{\mathrm{\mid }}^2=\mathrm{\mid }{\mathrm{\nabla }}_{t}h{\mathrm{\mid }}^2$ and $(e_z\times {\mathrm{\nabla }}_{t}h)\times {\mathrm{\nabla }}_{t}h=-e_z\mathrm{\mid }{\mathrm{\nabla }}_{t}h{\mathrm{\mid }}^2$ hold, with ${\mathrm{\nabla }}_{t}h\cdot (e_z\times {\mathrm{\nabla }}_{t}h)=0$; the same holds with $e$ in place of $h$.

[TE energy and flux] Let $h:=H_{\ast }{h}_{mn}$ and $w:=e_z\times {\mathrm{\nabla }}_{t}h$. With the TE amplitudes of [F1], $\mathrm{\mid }\tilde{E}{\mathrm{\mid }}^2=\frac{{\omega }^2{\mu }^2}{k_c^4}\mathrm{\mid }w{\mathrm{\mid }}^2=\frac{{\omega }^2{\mu }^2}{k_c^4}\mathrm{\mid }{\mathrm{\nabla }}_{t}h{\mathrm{\mid }}^2$ and $\mathrm{\mid }\tilde{H}{\mathrm{\mid }}^2=\frac{{\beta }^2}{k_c^4}\mathrm{\mid }{\mathrm{\nabla }}_{t}h{\mathrm{\mid }}^2+h^2$ (the cross term ${\mathrm{\nabla }}_{t}h\cdot e_z$ vanishes), so $⟨u⟩=\frac{1}{4}(\frac{\epsilon {\omega }^2{\mu }^2+\mu {\beta }^2}{k_c^4}\mathrm{\mid }{\mathrm{\nabla }}_{t}h{\mathrm{\mid }}^2+\mu h^2)$; integrating with step 1.2, $\int (\epsilon \mathrm{\mid }\tilde{E}{\mathrm{\mid }}^2)=\frac{\epsilon {\omega }^2{\mu }^2}{k_c^2}\int h^2$, $\int (\mu \mathrm{\mid }\tilde{H}{\mathrm{\mid }}^2)=\frac{\mu {\beta }^2}{k_c^2}\int h^2+\mu \int h^2$, and $\mu {\beta }^2+\mu k_c^2=\mu {\omega }^2\epsilon \mu$, so $\int (\epsilon \mathrm{\mid }\tilde{E}{\mathrm{\mid }}^2+\mu \mathrm{\mid }\tilde{H}{\mathrm{\mid }}^2)=\frac{2\epsilon {\mu }^2{\omega }^2}{k_c^2}\int h^2$ and $U=H_{\ast }^2\frac{{\mu }^2\epsilon {\omega }^2}{2k_c^2}{\int }_{R_{ab}}{h}_{mn}^2$ as claimed. For the flux, $\overline{\stackrel{~}{H}}=-\frac{i\beta }{k_c^2}{\mathrm{\nabla }}_{t}h+e_{z}h$ (the profile $h$ and the parameters $\beta ,\omega ,\mu$ are real), so $\tilde{E}\times \overline{\stackrel{~}{H}}=-\frac{i\omega \mu }{k_c^2}[-\frac{i\beta }{k_c^2}w\times {\mathrm{\nabla }}_{t}h+hw\times e_z]=\frac{\omega \mu \beta }{k_c^4}\mathrm{\mid }{\mathrm{\nabla }}_{t}h{\mathrm{\mid }}^2{e}_z-i\frac{\omega \mu }{k_c^2}h{\mathrm{\nabla }}_{t}h$ using $w\times {\mathrm{\nabla }}_{t}h=-e_z\mathrm{\mid }{\mathrm{\nabla }}_{t}h{\mathrm{\mid }}^2$ and $w\times e_z={\mathrm{\nabla }}_{t}h$ from [F3]; hence $\mathrm{Re}(\tilde{E}\times \overline{\stackrel{~}{H}})=\frac{\omega \mu \beta }{k_c^4}\mathrm{\mid }{\mathrm{\nabla }}_{t}h{\mathrm{\mid }}^2{e}_z$, and step 1.2 gives $\mathrm{\Phi }=\frac{1}{2}\frac{\omega \mu \beta }{k_c^4}{k}_c^2\int h^2=H_{\ast }^2\frac{\omega \mu \beta }{2k_c^2}{\int }_{R_{ab}}{h}_{mn}^2$ and the explicit norms of The separated sine and cosine rectangle profiles: eigen-identities, boundary traces, orthogonality and Dirichlet integrals (c).

[TM energy and flux] Let $e:=E_{\ast }{e}_{mn}$ and $w:=e_z\times {\mathrm{\nabla }}_{t}e$. With the TM amplitudes of [F1], $\mathrm{\mid }\tilde{E}{\mathrm{\mid }}^2=\frac{{\beta }^2}{k_c^4}\mathrm{\mid }{\mathrm{\nabla }}_{t}e{\mathrm{\mid }}^2+e^2$ and $\mathrm{\mid }\tilde{H}{\mathrm{\mid }}^2=\frac{{\omega }^2{\epsilon }^2}{k_c^4}\mathrm{\mid }{\mathrm{\nabla }}_{t}e{\mathrm{\mid }}^2$, so $⟨u⟩=\frac{1}{4}(\frac{\epsilon {\beta }^2+\mu {\omega }^2{\epsilon }^2}{k_c^4}\mathrm{\mid }{\mathrm{\nabla }}_{t}e{\mathrm{\mid }}^2+\epsilon e^2)$; integrating with step 1.2, $\int (\epsilon \mathrm{\mid }\tilde{E}{\mathrm{\mid }}^2+\mu \mathrm{\mid }\tilde{H}{\mathrm{\mid }}^2)=\frac{\epsilon {\beta }^2+\mu {\omega }^2{\epsilon }^2+\epsilon k_c^2}{k_c^2}\int e^2=\frac{2{\epsilon }^2\mu {\omega }^2}{k_c^2}\int e^2$ because $\epsilon {\beta }^2+\epsilon k_c^2=\epsilon {\omega }^2\epsilon \mu$, giving $U=E_{\ast }^2\frac{{\epsilon }^2\mu {\omega }^2}{2k_c^2}{\int }_{R_{ab}}{e}_{mn}^2$. For the flux, $\overline{\stackrel{~}{H}}=-\frac{i\omega \epsilon }{k_c^2}w$, so $\tilde{E}\times \overline{\stackrel{~}{H}}=(\frac{i\beta }{k_c^2}{\mathrm{\nabla }}_{t}e+e_{z}e)\times (-\frac{i\omega \epsilon }{k_c^2}w)=\frac{\omega \epsilon \beta }{k_c^4}\mathrm{\mid }{\mathrm{\nabla }}_{t}e{\mathrm{\mid }}^2{e}_z+i\frac{\omega \epsilon }{k_c^2}e{\mathrm{\nabla }}_{t}e$, using ${\mathrm{\nabla }}_{t}e\times w=e_z\mathrm{\mid }{\mathrm{\nabla }}_{t}e{\mathrm{\mid }}^2$ and $e_z\times w=-{\mathrm{\nabla }}_{t}e$ from [F3]; hence $\mathrm{Re}(\tilde{E}\times \overline{\stackrel{~}{H}})=\frac{\omega \epsilon \beta }{k_c^4}\mathrm{\mid }{\mathrm{\nabla }}_{t}e{\mathrm{\mid }}^2{e}_z$ and $\mathrm{\Phi }=E_{\ast }^2\frac{\omega \epsilon \beta }{2k_c^2}{\int }_{R_{ab}}{e}_{mn}^2$. This proves clause (b) in both families with the norms recorded in the profile lemma.

[Bounded propagating index set] Let a candidate with indices $(m,n)$ be propagating at frequency $\omega$, that is ${\beta }^2={\omega }^2\epsilon \mu -k_{mn}^2>0$. Then $k_{mn}^2<\epsilon \mu {\omega }^2$; since $k_{mn}^2=(\frac{m\pi }{a}{)}^2+(\frac{n\pi }{b}{)}^2\ge (\frac{m\pi }{a}{)}^2$ and $\ge (\frac{n\pi }{b}{)}^2$ by [F3] and the definitions of the profiles, $(\frac{m\pi }{a}{)}^2<\epsilon \mu {\omega }^2$ and $(\frac{n\pi }{b}{)}^2<\epsilon \mu {\omega }^2$; taking positive square roots [F4] gives $m<\omega a\sqrt{\epsilon \mu }\mathrm{/}\pi$ and $n<\omega b\sqrt{\epsilon \mu }\mathrm{/}\pi$. Hence the set of propagating indices is contained in the displayed bounded box.

[Energy transport and the speed relations] From steps 2.1 and 2.2, cancelling the common positive amplitude square $H_{\ast }^2$ or $E_{\ast }^2$ gives $\mathrm{\Phi }=\frac{\beta }{\epsilon \mu \omega }U$ in both families, since $\frac{{\mu }^2\epsilon {\omega }^2}{2k_c^2}\frac{\beta }{\epsilon \mu \omega }=\frac{\omega \mu \beta }{2k_c^2}$ and $\frac{{\epsilon }^2\mu {\omega }^2}{2k_c^2}\frac{\beta }{\epsilon \mu \omega }=\frac{\omega \epsilon \beta }{2k_c^2}$. The branch function $\omega (\beta )=\sqrt{({\beta }^2+k_c^2)\mathrm{/}(\epsilon \mu )}$ of [F4] is differentiable with $v_g:={\omega }^{\mathrm{\prime }}(\beta )=\frac{\beta }{\epsilon \mu \omega (\beta )}$: indeed for real $h\mathrm{\ne }0$ the difference quotient is $\frac{2\beta +h}{\epsilon \mu (\omega (\beta +h)+\omega (\beta ))}$ (rationalizing the difference of square roots), and its limit as $h\to 0$ is $\frac{2\beta }{2\epsilon \mu \omega (\beta )}$ by [F4]; differentiating the identity $\epsilon \mu \omega (\beta {)}^2={\beta }^2+k_c^2$ with the chain rule gives the same value. Hence $\mathrm{\Phi }=U{v}_g$. Moreover $v_p:=\omega \mathrm{/}\beta$ satisfies $v_p{v}_g=\frac{\omega }{\beta }\frac{\beta }{\epsilon \mu \omega }=\frac{1}{\epsilon \mu }$; $v_p=\frac{\sqrt{{\beta }^2+k_c^2}}{\sqrt{\epsilon \mu }\beta }=\frac{1}{\sqrt{\epsilon \mu }}\sqrt{1+\frac{k_c^2}{{\beta }^2}}>\frac{1}{\sqrt{\epsilon \mu }}$ because $k_c^2>0$ and $\beta >0$; and $0<v_g=\frac{1}{\sqrt{\epsilon \mu }}\frac{\beta }{\sqrt{{\beta }^2+k_c^2}}<\frac{1}{\sqrt{\epsilon \mu }}$. This proves clause (c).

[Scope] Clauses (a) and (b) are steps 1.2, 2.1 and 2.2, with the cross-section integrals evaluated by the profile and transverse identities and the explicit norms; clause (c) is step 3.1, with the transport relation $\mathrm{\Phi }=U{v}_g$ and the speed inequalities; clause (d) is step 2.3. Every step used the lossless separated candidate of the declared model, the ideal PEC boundary model and the choice-free local energy reading of the material-energy theorem: the period average is over time at a fixed position, the flux is through one cross-section, no signal-speed interpretation of $v_p$ is asserted, no bound on information or energy speeds in general dispersive media is claimed, and no empirical premise is used. This is clause (e).