Evanescent waves and frustrated reflection

Place the planar interface at $z=0$, with a transparent nonmagnetic medium of index $n_1$ in $z<0$ and a lower index $n_2$ in $z>0$. Let incidence lie in the xz plane. Continuity of tangential phase requires the same real $k_x=k_0n_1\sin\theta$ on both sides. The lower-index dispersion relation then determines the squared normal wavenumber; it is not free to adjust independently to preserve a geometrical ray.

Above the critical angle, the required tangential component exceeds the entire propagating wavenumber magnitude in medium two. Its squared normal component is negative. With the $e^{-i\omega t}$ convention and a factor $e^{ik_z z}$, select $k_z=i\kappa$ with positive $\kappa$ in the semi-infinite upper medium. The alternative branch grows without bound as z increases and violates the radiation or boundedness condition for this source-free half-space.

The electric and magnetic fields must obey continuity of their tangential components when there is no free surface current. Normal displacement and magnetic flux satisfy their respective jump conditions. In particular, normal electric field need not be continuous across different permittivities. The Fresnel coefficients derived from these conditions have unit reflected magnitude above critical incidence, but generally nontrivial polarization-dependent phase. That phase can alter a reflected beam's polarization without implying energy absorption.

The evanescent field contains an exponential factor $e^{-\kappa z}$. Its amplitude penetration length is $1/\kappa$, whereas a quadratic local field magnitude falls by a factor e over distance $1/(2\kappa)$. Calling both lengths simply penetration depth invites a factor-of-two mistake. For one lossless interface and a single decaying evanescent wave, the cycle-averaged Poynting vector has no normal component but can have a tangential component. A nonzero stored field and zero outgoing normal flux are therefore compatible.

Put another high-index transparent medium beyond a gap of thickness $g$. If that medium supports the same tangential wavenumber as a propagating wave, the two interfaces can couple light across the gap. The gap field now generally includes both exponential branches because it occupies a finite interval; boundedness at infinity no longer rules out one branch inside that interval. Their relative amplitudes are fixed by matching both boundaries and can produce nonzero normal energy flux.

This frustrated total internal reflection is a classical Maxwell boundary phenomenon with a useful tunneling analogy. The factor $e^{-2\kappa g}$ suggests the large-gap attenuation scale but is not the exact transmitted-power coefficient: polarization, impedance, angle, and multiple reflections supply additional factors. For an ideal symmetric high-index pair, transmission approaches the direct-contact result as the gap vanishes. Large gaps recover isolated-interface total reflection, while absorption or a different receiving medium changes the limiting power balance.