 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.