Electromagnetic waveguide

An electromagnetic waveguide confines an electromagnetic wave across its axis while allowing it to travel along the axis. Conducting walls or changes of refractive index select transverse normal modes: reproducible cross-sectional field patterns with their own longitudinal wavenumbers. A hollow metal pipe and an optical fiber are different realizations of this principle. Their usefulness is to deliver electromagnetic energy along a chosen route instead of letting it spread freely.

For a uniform vacuum-filled conducting guide, write a normal mode's axial and time dependence as $e^{i(\beta z-\omega t)}$. Here $\omega >0$ is angular frequency, $\beta$ is the longitudinal wavenumber, and $k_{\perp }^2$ is the transverse eigenvalue selected by the wall boundary conditions. The electromagnetic wave equation separates intoThe geometry consumes part of the available squared wavenumber: when $\omega /c<k_{\perp }$, this branch is an evanescent wave rather than a propagating normal mode. The waveguide cutoff frequency is therefore a property of a specified branch and geometry. A transverse electromagnetic mode in a suitable two-conductor geometry can have $k_{\perp }=0$ and no positive geometric cutoff. An optical fiber needs its own dielectric boundary conditions, rather than the metal-wall formulas below.

The rectangular electromagnetic waveguide makes these ideas explicit with a single sine profile. The scalar-potential conducting waveguide modes give a broader construction. At an open aperture, a confined field can radiate into the electromagnetic aperture angular spectrum; it need not remain one guided normal mode.

An optical waveguide confines an optical normal mode transversely while permitting longitudinal propagation. A higher-refractive index layer between lower-index layers provides a simple dielectric realization. In a double heterostructure laser, overlap of the guided field with the carrier-rich active layer improves useful optical gain.

A guided electromagnetic wave whose electric field has no component along the propagation axis. Its magnetic field can have a longitudinal component. In an empty perfectly conducting rectangular electromagnetic waveguide, transverse wave numbers determine a nonzero cutoff frequency. This differs from a transverse electromagnetic mode, in which both fields are transverse.

A dielectric slab surrounded by lower-index material confines waves with an oscillatory core profile and evanescent exterior tails. In an isotropic, nonmagnetic symmetric slab, an even transverse-electric field has core profile $\cos Kz$ and matches to exponential tails through $K\tan (Kd)=K^{\prime }$. The transverse constants also satisfy $K^2+K^{\prime 2}=({ϵ}_r-1){\omega }^2/c^2$ for vacuum exterior.

Set $u=Kd$, $w=K^{\prime }d$, and $\mathcal{V}=(\omega d/c)\sqrt{{ϵ}_r-1}$. The circle in the first quadrant falls from $w=\mathcal{V}$, while $u\tan u$ rises from zero to its first pole. A unique first-branch intersection exists for every $\mathcal{V}>0$. It yields a nodeless even bound field. At exactly zero index contrast there is no confined mode; as contrast tends to zero from above its tail extends without bound.

A symmetric lossless dielectric slab with arbitrarily small positive index image contrast can support its fundamental even guided mode, but confinement becomes arbitrarily weak as the image contrast vanishes. The field outside decays as $e^{-\kappa |x|}$, so its penetration distance $1/\kappa$ diverges when $\kappa \to 0$.

The existence statement for positive image contrast therefore does not imply a confined mode at exactly zero image contrast: the limiting profile ceases to be normalizable across the infinite transverse direction. This distinguishes a mode's existence from useful confinement. A finite surrounding region or nearby absorbing surface can strongly affect such a weakly confined mode even when the ideal infinite-cladding model predicts a bound solution.

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.