Waveguides and cavities — Examples

Every displayed component and parameter below is its real or complex numerical value in fixed coherent SI units: lengths in m, time in s, $\omega$ in ${\mathrm{s}}^{-1}$, $\beta$ in ${\mathrm{m}}^{-1}$, $\epsilon$ in $\mathrm{F}{\mathrm{m}}^{-1}$, $\mu$ in $\mathrm{H}{\mathrm{m}}^{-1}$, $E,\tilde{E}$ in $\mathrm{V}{\mathrm{m}}^{-1}$ and $H,\tilde{H}$ in $\mathrm{A}{\mathrm{m}}^{-1}$. The physical longitudinal normalization is $H_{\ast }=1\mathrm{A}{\mathrm{m}}^{-1}$ in The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff(a), whose numerical value is one.

Let $\epsilon ,\mu >0$, $a,b>0$, $\omega >0$, $\beta >0$ with ${\beta }^2={\omega }^2\epsilon \mu -(\pi \mathrm{/}a{)}^2>0$, and put ${\omega }_c:=\pi \mathrm{/}(a\sqrt{\epsilon \mu })$, $\theta :=\beta x_3-\omega t$. In the rectangular guide of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM with cross-section $(0,a)\times (0,b)$, take the TE candidate of The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff(a) with $(m,n)=(1,0)$, whose amplitudes are $\tilde{E}=(0,i\omega \mu \frac{a}{\pi }\sin \frac{\pi x_1}{a},0),\tilde{H}=(-i\beta \frac{a}{\pi }\sin \frac{\pi x_1}{a},0,\cos \frac{\pi x_1}{a})\mathrm{.}$ Then the real fields $E=\mathrm{Re}(\tilde{E}{e}^{i\theta })$, $B=\mu \mathrm{Re}(\tilde{H}{e}^{i\theta })$ are smooth, have $E_z=0$ (TE) with $E$ linearly polarized along $e_2$, satisfy the source-free lossless macroscopic equations and the PEC wall conditions, and carry a half-sine standing profile $\sin (\pi x_1\mathrm{/}a)$ in $x_1$ with a traveling wave in $x_3$. The cutoff of this mode is ${\omega }_c=\pi \mathrm{/}(a\sqrt{\epsilon \mu })$: real nonzero $\beta$ exists exactly for $\omega >{\omega }_c$, at $\omega ={\omega }_c$ the mode is at cutoff ($\beta =0$), and for $\omega <{\omega }_c$ it is evanescent. If $a\ge b$, then ${\omega }_c=\pi \mathrm{/}(a\sqrt{\epsilon \mu })$ is the lowest cutoff among all TE and TM candidates of the cited theorem (attained also by $(m,n)=(0,1)$ when $a=b$); the case $(m,n)=(0,0)$ is not a mode by the index restriction of the theorem. This is an explicit conditional candidate of the adopted model; it asserts no measurement and no completeness.

All displayed fields and parameters below are numerical values in fixed coherent SI units: $a,b,d,x_j$ in m, $t$ in s, $\omega$ in ${\mathrm{s}}^{-1}$, $\epsilon$ in $\mathrm{F}{\mathrm{m}}^{-1}$, $\mu$ in $\mathrm{H}{\mathrm{m}}^{-1}$, $E,E_{\mathrm{s}\mathrm{p}}$ in $\mathrm{V}{\mathrm{m}}^{-1}$ and $B$ in T. The TM longitudinal normalization is $E_{\ast }=1\mathrm{V}{\mathrm{m}}^{-1}$ in The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff(b), so the dimensionless profile below specifies the numerical electric-field components.

Let $a,b,d>0$ and $\epsilon ,\mu >0$ be constants, put $E_{\mathrm{s}\mathrm{p}}(x):=e_1\sin \frac{\pi x_2}{b}\sin \frac{\pi x_3}{d}$, and for $\omega >0$ define $E(x,t):=E_{\mathrm{s}\mathrm{p}}(x)\cos (\omega t)$ and $B(x,t):=-\frac{1}{\omega }(\mathrm{curl}{E}_{\mathrm{s}\mathrm{p}})(x)\sin (\omega t)$, where the curl is the fixed-time spatial curl of Divergence and curl of a $C^1$ vector field. Then $E,B$ are smooth, $\mathrm{div}{E}_{\mathrm{s}\mathrm{p}}=0$, $\mathrm{\Delta }{E}_{\mathrm{s}\mathrm{p}}=-[(\pi \mathrm{/}b{)}^2+(\pi \mathrm{/}d{)}^2]E_{\mathrm{s}\mathrm{p}}$, and $E,B$ satisfy the four source-free macroscopic equations on the box ${\mathrm{\Omega }}_c=(0,a)\times (0,b)\times (0,d)$ in the lossless isotropic scalar branch exactly when ${\omega }^2=\frac{1}{\epsilon \mu }\left[\right(\frac{\pi }{b}{)}^2+\left(\frac{\pi }{d}{)}^2\right];$ with that resonance, $n\times E=0$ and $n\cdot B=0$ hold on all six walls, so $(E,B)$ is a PEC cavity mode. It is the cutoff instance $\beta =0$ of the TM family of The separated TE and TM candidate modes of the lossless rectangular PEC guide: exact verification, index restrictions and cutoff(b) taken with axis $e_1$, cross-section $(0,b)\times (0,d)$ and indices $(m,n)=(1,1)$: the field is independent of $x_1$, which is why it also satisfies the $x_1=0$ and $x_1=a$ end-cap conditions for every box length $a>0$. The example asserts a conditional model solution, not an observed resonance.

Assume the lossless isotropic scalar constant branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch, the macroscopic equations of The macroscopic Maxwell equations in terms of D, H and the free sources with ${\rho }_f=0$, $J_f=0$, and the straight-guide, TEM and separated-phasor conventions of Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM; put $Z:=\sqrt{\mu \mathrm{/}\epsilon }$ and $v:=(\epsilon \mu {)}^{-1\mathrm{/}2}$. (a) [Potential form] Let $A\subseteq {\mathbb{R}}^2$ be a bounded connected $C^1$ domain and let $\phi \in C^2(\bar{A};\mathbb{C})$ (with real-valued potentials included) satisfy $\mathrm{\Delta }\phi =0$ on $A$ (the plane Laplacian of The Laplacian of a $C^2$ function and of a $C^2$ vector field; a $C^2$ function with $\mathrm{\Delta }\phi =0$ is called harmonic). For either sign $\zeta \in \{+1,-1\}$ define $\tilde{E}:=-\mathrm{\nabla }\phi$, $\beta :=\zeta \omega \mathrm{/}v$ and $\tilde{H}:=\zeta e_z\times \tilde{E}\mathrm{/}Z$; then the separated real fields $E=\mathrm{Re}(\tilde{E}{e}^{i(\beta x_3-\omega t)})$, $B=\mu \mathrm{Re}(\tilde{H}{e}^{i(\beta x_3-\omega t)})$ satisfy the source-free lossless macroscopic equations with $E_z=B_z=0$ (TEM), and the PEC condition on a smooth wall with unit normal $n$ is equivalent to ${\mathrm{\partial }}_{\tau }\phi =0$ along that boundary curve, so that $\phi$ is constant on each connected component of $\mathrm{\partial }A$. Moreover, if $A$ is star-shaped (Star-shaped open subsets of Euclidean space), every separated TEM phasor with $C^1$ transverse amplitudes has on the interior a complex harmonic potential $\phi \in C^2(A;\mathbb{C})$, obtained by applying On a star-shaped open domain, closed, exact, conservative, path-independent, and zero-loop are equivalent to the real and imaginary electric amplitudes separately. If the field is nonzero, the equations force ${\beta }^2=\epsilon \mu {\omega }^2$, hence $\beta =\pm \omega \mathrm{/}v$, and $\tilde{H}=\beta e_z\times \tilde{E}\mathrm{/}(\omega \mu )$, giving the two displayed signs. If the field is zero, $\tilde{E}=\tilde{H}=0$ solves the equations for every longitudinal label $\beta$, with no dispersion conclusion. The boundary potential description requires the stated continuous differentiable boundary trace (in particular the initial $C^2(\bar{A})$ potential has it); the interior exactness supplier alone is not asserted to produce a real potential or automatic boundary extension. (b) [Single-conductor obstruction] If in the potential-form setting $\mathrm{\partial }A$ is connected, then $\phi$ is constant on $\mathrm{\partial }A$, hence $\phi$ is constant on $\bar{A}$ by the two-dimensional classical Dirichlet uniqueness Uniqueness for the classical dirichlet problem, applied to its real and imaginary parts with their respective constants as second solutions, and therefore $\tilde{E}=\tilde{H}=0$: a single-conductor cross-section with connected boundary admits no nonzero TEM field in the potential form. The same potential-form obstruction applies to a rectangular cross-section with a potential in $C^2(A)\cap C(\bar{A})$ having constant boundary trace, because this uniqueness supplier does not require a $C^1$ boundary. For rectangular separated TEM amplitudes that are $C^1$ on the interior and continuous on the closure with the classical PEC traces on open edges, the star-centre radial primitive extends continuously to the rectangle closure. The edgewise tangential derivative is the electric trace, so PEC makes the potential constant on each open edge and continuity joins those constants at the corners. Thus this stated rectangular class has no nonzero TEM field. (c) [Coaxial example] For the following Lebesgue polar-coordinate energy calculation assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $0<a<b$, $A:=\{x\in {\mathbb{R}}^2:a<\mathrm{\mid }x\mathrm{\mid }<b\}$ (two boundary components), fix $V\in \mathbb{R}$, choose the forward label $k:=\omega \mathrm{/}v$, and define $\phi (x):=V\ln (b\mathrm{/}\mathrm{\mid }x\mathrm{\mid })\mathrm{/}\ln (b\mathrm{/}a)$, so that $\tilde{E}=V{e}_r\mathrm{/}(\mathrm{\mid }x\mathrm{\mid }\ln (b\mathrm{/}a))$ with $e_r:=x\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ and $\tilde{H}=(V\mathrm{/}(Z\mathrm{\mid }x\mathrm{\mid }\ln (b\mathrm{/}a)))e_{\phi }$ with $e_{\phi }:=e_z\times e_r$. Then $\mathrm{\Delta }\phi =0$ on $A$: indeed ${\mathrm{\partial }}_i\ln \mathrm{\mid }x\mathrm{\mid }=x_i\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2$ by The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t and the chain rule (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$), and the second partial derivatives sum to zero; hence $\phi$ is constant on each boundary circle (values $V$ at $\mathrm{\mid }x\mathrm{\mid }=a$ and $0$ at $\mathrm{\mid }x\mathrm{\mid }=b$), the fields satisfy the source-free lossless macroscopic equations and the PEC conditions, and the relation $k=\omega \mathrm{/}v$ is nondispersive with no cutoff. The amplitude integrals computed in polar coordinates with Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma (the Jacobian factor $r$ is read from that formula) are $W_e:=\frac{1}{2}\epsilon {\int }_A\mathrm{\mid }\tilde{E}{\mathrm{\mid }}^{2}dA=\frac{1}{2}C{V}^2,W_m:=\frac{1}{2}\mu {\int }_A\mathrm{\mid }\tilde{H}{\mathrm{\mid }}^{2}dA=\frac{1}{2}C{V}^2,C:=\frac{2\pi \epsilon }{\ln (b\mathrm{/}a)};$ the instantaneous per-length energies are $\frac{1}{2}C{V}^2{\cos }^2\theta$ and $\frac{1}{2}L{I}^2{\cos }^2\theta$ with $\theta =k{x}_3-\omega t$; equivalently, with the per-length inductance $L:=\mu \ln (b\mathrm{/}a)\mathrm{/}(2\pi )$ in $\mathrm{H}{\mathrm{m}}^{-1}$, per-length capacitance $C$ in $\mathrm{F}{\mathrm{m}}^{-1}$, and wall-circulation amplitude $I:=2\pi V\mathrm{/}(Z\ln (b\mathrm{/}a))$ in $\mathrm{A}$ (identified with the inner-conductor sheet-current amplitude when its conductor-side magnetic intensity trace is zero) one has $W_m=\frac{1}{2}L{I}^2$ and $LC=\epsilon \mu$. For $V\mathrm{\ne }0$ the two boundary components carry distinct constants, allowing a nonzero field; $V=0$ gives equal constants and the zero field. (d) [Scope] The obstruction clause (b) is proved for the continuous-boundary harmonic-potential form. The star-shaped interior classification in (a) covers both signs and the zero field, with boundary conclusions under the stated potential-trace hypotheses; the general reduction of an arbitrary TEM field to the potential form on a merely simply connected non-star-shaped cross-section is not asserted here. The coaxial clause is an explicit conditional computation in the declared model with no empirical content; no lossy, radiation or signal-speed claim is made.