Snell refraction and total internal reflection

Assume a flat, stationary interface between homogeneous, isotropic, lossless dielectrics with positive indices. The temporal oscillation must match on both sides, so frequency is unchanged. The tangential spatial phase must also agree at every interface point. Because the wavevector magnitude is $n\omega/c$, its tangential component supplies the index-weighted sine relation. Reflection preserves the index and reverses the normal propagation component, giving equal incident and reflected angles to the normal.

Increasing index reduces wavelength at fixed frequency, and the transmitted ray bends toward the normal. Decreasing index increases wavelength and bends an existing transmitted ray away from it. These statements concern phase-index ray geometry in isotropic media. They do not calculate how much power crosses the boundary. Even normal incidence generally reflects some power because the electromagnetic impedances differ; a direction law cannot decide that fraction or its polarization dependence.

A real transmitted direction requires the computed sine to be at most one. If the incident medium has higher index, the limiting transmitted direction is tangent to the interface. Solving for its incident angle gives the critical angle. The ratio is the lower transmitted index divided by the higher incident index. Reversing that ratio produces a number greater than one and signals an incorrect assignment, rather than a mysterious angle that a calculator cannot evaluate.

For incidence strictly beyond the critical angle there is no propagating transmitted plane wave in an infinite lower-index medium. An ideal isolated lossless interface reflects all normal incident energy flux. At the critical angle itself the limiting transmitted direction is grazing, so it is useful to reserve “beyond critical” for the regime with an exponentially decaying normal field. From low index to high index no critical angle occurs; even very oblique incidence has an allowed refracted direction.

Beyond critical incidence the conserved tangential wavevector exceeds the total wavevector magnitude permitted in medium two. Its normal component becomes imaginary, producing an evanescent field that decays away from the boundary. Such a field is required by the electromagnetic boundary conditions. For the ideal single interface it carries no net time-averaged power normally away into the second half-space, although its existence matters for phase shifts, near-field coupling, and interaction with nearby matter.

Bringing a third medium within the decay distance can permit energy transfer through the thin low-index gap, often called frustrated total internal reflection. Absorption also changes the ideal unit-reflectance conclusion. A ray diagram alone cannot calculate these effects. The diagnostic habit is to ask separately whether a real transmitted direction exists, whether a field exists, and whether net normal energy flows. Those are related but distinct questions, resolved at different levels of approximation.