Planar dielectric waveguide

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.