The TE10 candidate of a lossless rectangular guide: explicit fields, cutoff and propagating branch

[Instantiation] With $(m,n)=(1,0)$ one has $k_c:=\pi \mathrm{/}a$, ${\mathrm{\nabla }}_t{h}_{10}=-\frac{\pi }{a}\sin \frac{\pi x_1}{a}{e}_1$ and $e_z\times {\mathrm{\nabla }}_t{h}_{10}=-\frac{\pi }{a}\sin \frac{\pi x_1}{a}{e}_2$ by [F2] and the coordinate formula for the cross product, so the amplitudes of [F1] become $\tilde{E}=-(i\omega \mu a\mathrm{/}\pi )(-\sin \frac{\pi x_1}{a})e_2=(0,i\omega \mu \frac{a}{\pi }\sin \frac{\pi x_1}{a},0)$ and $\tilde{H}=(i\beta a^2\mathrm{/}{\pi }^2)(-\frac{\pi }{a}\sin \frac{\pi x_1}{a})e_1+\cos \frac{\pi x_1}{a}{e}_z=(-i\beta \frac{a}{\pi }\sin \frac{\pi x_1}{a},0,\cos \frac{\pi x_1}{a})$, exactly as displayed. The theorem supplies smoothness, the four source-free equations, the TE property $E_z=0$ and the PEC wall conditions, and the first component ${\tilde{E}}_1={\tilde{E}}_3=0$ shows that $E$ is polarized along $e_2$; the real fields are $E=\omega \mu \frac{a}{\pi }\sin \frac{\pi x_1}{a}\mathrm{Re}(i{e}^{i\theta })e_2$ and $B=\mu \mathrm{Re}(\tilde{H}{e}^{i\theta })$, so the transverse factor $\sin (\pi x_1\mathrm{/}a)$ is a fixed half-sine standing profile and the phase $\theta =\beta x_3-\omega t$ is a traveling wave in $x_3$.

[Cutoff and propagation alternatives] Applying clause (d) of [F1] with $k_c=\pi \mathrm{/}a$ gives ${\omega }_c=\pi \mathrm{/}(a\sqrt{\epsilon \mu })$: real longitudinal wave numbers with ${\beta }^2={\omega }^2\epsilon \mu -(\pi \mathrm{/}a{)}^2$ exist exactly for $\omega \ge {\omega }_c$, with $\beta \mathrm{\ne }0$ exactly for $\omega >{\omega }_c$ and $\beta =0$ at $\omega ={\omega }_c$; for $\omega <{\omega }_c$ the two algebraic roots are $\beta =\pm i\sqrt{(\pi \mathrm{/}a{)}^2-{\omega }^2\epsilon \mu }$ and the same verified fields carry the factor $e^{-\mathrm{Im}\beta x_3}$, so the mode is evanescent.

[Lowest cutoff for $a\ge b$] Let $(m,n)\mathrm{\ne }(0,0)$ be any TE index pair and let $k_{mn}^2=(\frac{m\pi }{a}{)}^2+(\frac{n\pi }{b}{)}^2$. If $m\ge 1$ then $k_{mn}^2\ge (\pi \mathrm{/}a{)}^2$; if $m=0$ then $n\ge 1$ and $k_{mn}^2\ge (\pi \mathrm{/}b{)}^2\ge (\pi \mathrm{/}a{)}^2$ because $a\ge b$. Hence $k_{mn}^2\ge {\pi }^2\mathrm{/}{a}^2=k_{10}^2$, with equality possible only for $m=1,n=0$, or for $m=0,n=1$ when $a=b$; since the cutoff ${\omega }_c=k_c\mathrm{/}\sqrt{\epsilon \mu }$ is increasing in $k_c$, the cutoff $\pi \mathrm{/}(a\sqrt{\epsilon \mu })$ of the $(1,0)$ candidate is the lowest cutoff of all TE indices. Every TM index has $m,n\ge 1$ by [F1], so its $k_{mn}^2\ge {\pi }^2\mathrm{/}{a}^2+{\pi }^2\mathrm{/}{b}^2>{\pi }^2\mathrm{/}{a}^2$ and its cutoff is strictly larger; and $(m,n)=(0,0)$ is not a mode by the index restriction of [F1]. Thus ${\omega }_c=\pi \mathrm{/}(a\sqrt{\epsilon \mu })$ is the lowest cutoff among those TE/TM candidates, attained also by $(0,1)$ at $a=b$.

[Scope] The displayed fields are an explicit conditional candidate of the adopted lossless model, with the stated cutoff and lower-bound comparison; the quantities are declared model parameters, no measurement, resonance observation, completeness or uniqueness is asserted, and no empirical premise is used.