Electromagnetic waveguide

An electromagnetic waveguide confines an electromagnetic wave transversely while allowing propagation along its axis. Conducting or dielectric boundaries select transverse modes. In a uniform guide a mode with transverse eigenvalue $k_{\perp }^2$ has longitudinal wavenumber $k_x^2=k^2-k_{\perp }^2$; below cutoff it is evanescent. At an open aperture, the confined field radiates into an angular spectrum rather than retaining a single guided mode.

For dependence $e^{i(kz-\omega t)}$, a Helmholtz equation $({\Delta }_{\perp }+{\gamma }^2)\psi =0$ generates transverse-magnetic fields $e_{\perp }={\nabla }_{\perp }\psi$, $e_z=-i{\gamma }^2\psi /k$, $b_{\perp }=\omega \hat{z}\times {\nabla }_{\perp }\psi /(k{c}^2)$, $b_z=0$, for $k\ne 0$. The conducting wall requires a Dirichlet boundary condition on $\psi$. A transverse-electric representation is $e_{\perp }=\hat{z}\times {\nabla }_{\perp }\psi$, $e_z=0$, $b_{\perp }=-k{\nabla }_{\perp }\psi /\omega$, $b_z=i{\gamma }^2\psi /\omega$, with a Neumann boundary condition. These are families of modes, not a completeness assertion for multiply connected guides and their possible TEM fields.

For an aperture normal to ${\mathbf{e}}_x$, let $F_y,F_z$ be its tangential electric field traces extended by zero outside the aperture. With the outgoing branch of $\kappa$, the Weyl plane-wave representation and the curl aperture integral give a spectrum proportional to $(-(\nu {\hat{F}}_y+\eta {\hat{F}}_z)/\kappa ,{\hat{F}}_y,{\hat{F}}_z)$. Each wave is transverse to $\mathbf{q}$. The longitudinal electric component is regenerated by oblique diffraction even when it vanishes at the input. For $G=e^{ikr}/(4\pi r)$ and a unit prefactor on the stated curl integral, the spectrum has an overall factor $1/2$; a doubled half-space Green kernel gives factor one. The normalization must follow the chosen Green kernel and equivalent-current convention.