= 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 $\kappa\to0$.
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.
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