Fundamental even mode of a symmetric dielectric slab ($u\tan u=w,u^2+w^2={\mathcal{V}}^2$)

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.