Expert: Daniel Mazur - 6/28/2006

Question
Age:17
Education : 12th grade studying

When a ray of light travels from a denser medium to a rarer medium such that the angle of incidence in the denser medium is equal to the criical angle, the corresponding angle of refraction in the rarer medium if 90 degrees. According to the priciple of reversibility of light, the light ray should be able to trace back its path. Does this mean that when light is incident along the interface of two media (angle of incidence equals 90 degrees), then the angle of refraction in the denser medium is equal to the critical angle?

Answer
Hello,

I believe the answer is 'Sometimes yes, sometimes no', you have to keep in mind that you are working with models (ideal interface, wave description of light) and models always have their limits. Your question touched one such limit.

Light, as described by treory of electromagnetic waves (Maxwell), from which Snell's law and the laws of refraction and reflection are derived, is treated as a WAVE in cotinuum. This is a model, the real light is a stream of photons, quantum-mechanical particles, which individually obey certain laws of behavior, but when you have plenty of them, new laws of "group behavior" arise, and these are what we observe in refraction and reflection. It's like having molecules of water with their Brownian motion, that move together (coherently) in waves, when you sail a lake.

The real life light rays, that obey those macroscopic laws, comprise of huge ammounts of photons. The real life media (denser or rarer) comprise of atoms and molecules - something completely different from a "continuous medium", which all the idealized laws assume. A real life interface is not absolutely flat, it's atomically flat (or rough) at best.

Now, when you do the experiment with light approaching the interface from the denser medium under critical angle, yes it will bend at the interface so that it continues in the rarer medium ALONG the interface. But THIS the limit of the model, stretched to the breaking point.

I myself around your age speculated over a related issue: Up to the critical angle, the refracted ray's angle rises smoothly as a linear function of the angle of incidence phi_out = n*phi_in. BEYOND the critical angle it's not "refraction", it's "reflection" and the reflection angle has a discontinuity from the last position of the refracted ray (it jumps from 90 deg. to 90+phi_critical). And then the reflection angle moves IN THE OPPOSITE DIRECTION, i.e. phi_out = 180-phi_in. The discontinuity was fascinating, and it happened exactly at the point you asked about.

When you send a light ray parallel to the interface, just a bit above it inside the rarer medium, two basic things can happen (or a combination of both).

1) An atomical roughness (denser medium) in the way of the ray will cause another refraction, this time from the dilute medium into the denser one and the angle will be about the same as the critical angle. Here as an observer in the denser medium you will observe light approaching from one bright spot - the roughness - as a ray (quasi-onedimensional  object).

2) If there is no roughness, either the ray will not bend into the rarer one (when it is too far from the interface) at all, or it will be BRUSHING the interface all along the way. Here PARTS of the ray will continuously refract at critical angle into the denser medium. An observer from the denser medium will see the refracted light coming not from one point, but from the whole interface. There will not be one point of refraction, instead, partial refraction will happen at every interfacial atom along the light ray's path.

To sum up, you'll do best if you view the critical angle as a limit, extrapolation of a model into area, where simple laws of light propagation fail. When you shine a ray of light at critical angle from the denser medium OR shine it parallel to the interface from the rarer medium, your experiment is in between valid models. You cannot determine the ray's path based on any of the models of refraction or reflection, you cannot reliably apply any of the macroscopic laws, including the 'principle of reversibility'. You are left with quantum mechanics and a system of plenty of photons and even more atoms and electrons. This can be tackled by computers, numerically. No analytic solution is possible.

I hope my explanation was clear enough and it helps you.
Good luck!

Daniel