In quantum mechanics, a particle may tunnel through a barrier it would not be able to surmount in a classical sense.
My question is this:
What are all the factors that may prevent a photon from propagating (thus it would need quantum tunneling to do so), and can they be described in a general way for any given material?
Photons have some conditions to have an evanescent wave, e.g. total internal reflection.
Suppose we have some material with index of refraction $n_1$ and a layer of another material, with smaller $n_2<n_1$. At some angle we'll see total internal reflection, i.e. when the light totally reflects, but leaves some exponentially decaying trails in layer with $n_2$. If this layer is continued with original bulk material with $n_1$, we'll intercept these evanescent waves and get normal propagating waves.
Qualitatively, this would look something like this (light goes from bottom-left side, reflected wave is not displayed):
If you necessarily want one-dimensional picture (i.e. not needing the light falling at some angle to surface), then one can use a reflecting surface, having it sufficiently thin - e.g. a thin metal foil.
In fact, any perfect reflector can be used for this (by perfect I mean 100% reflection if reflector is infinitely thick). A special case is a photonic crystal. If wavelength of the light you shine at it is in band gap of the photonic crystal, the light will totally reflect from such crystal because it'll have imaginary wave vector inside, which also means evanescent wave. If the photonic crystal is thin enough, you can have the light tunneled through it and it'll partially go to the other side and then propagate as usual.
The potential barrier problem and solution in quantum mechanics is discussed within the solutions of Schrodineger's equation in which there exist potentials, and the solutions of the equations with the boundary conditions give the wave function of a particle, i.e an entity with a mass. In addition it is a non relativistic equation.Thus in this framework:
relativistic velocities are not allowed as a particle cannot be described by a wave function that is a solution of the Schrodinger equation.
The photon enters as an interpretation of the energy conservation between transitions of bound state energy levels, a hypothesis that has been amply experimentally observed and validated the use of the Schrodinger equation in first quantization.
The answers to the question What equation describes the wavefunction of a single photon?, asked here a while ago, cover the way the photon is described in first (Dirac equation) and second quantization.
In this preprint a view is suggested of using Maxwell's equations for the wave function of the photon. One could use these and define a barrier and solve for it to get the behavior of the photon's probability to pass the barrier, but it is not a simple problem I could tackle. Generally when one calculates behaviors of photons on reaching a barrier it is wise to use classical solutions of Maxwell's equations. Transmission, reflectance etc describe the behavior of light at barrier. Going down to the individual photon is complicated mathematically and not worth the effort since it can be shown that the classical and the quantum description for photons merges naturally.
