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.

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.