 Vanishing index contrast in a dielectric slab

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.