Total Internal Reflection

When light travels from an optically denser medium towards a rarer medium (for example, from glass or water towards air), Snell's law shows the refracted ray bends away from the normal, so the angle of refraction $r$ is always larger than the angle of incidence $i$. As $i$ is increased, $r$ increases faster, and at one particular angle of incidence -- the critical angle $C$ -- the refracted ray grazes along the boundary itself, emerging at $r=90^{\circ}$. For any angle of incidence greater than this critical angle, no refracted ray can exist at all (Snell's law would require $\sin r>1$, which is impossible), and the entire incident ray is reflected back into the denser medium, obeying the ordinary laws of reflection exactly. This phenomenon is called total internal reflection (TIR), and unlike reflection from a silvered mirror, it reflects essentially 100% of the incident light, with no absorption loss at all. Two conditions must both hold for TIR to occur: light must be travelling from a denser medium towards a rarer medium (never the other way round), and the angle of incidence inside the denser medium must exceed the critical angle for that pair of media. The critical angle itself follows directly from Snell's law by setting $r=90^{\circ}$: for a ray going from a medium of refractive index $n$ towards air ($n_{\text{air}}\approx1$), $n\sin C=1\times\sin90^{\circ}=1\quad\Rightarrow\quad\sin C=\frac{1}{n}.$ A denser medium (larger $n$) therefore has a smaller critical angle, i.e. TIR happens more readily. Familiar demonstrations and applications of TIR include the brilliant sparkle of a cut diamond (diamond's very high refractive index, about $2.42$, gives it an unusually small critical angle of about $24^{\circ}$, so light entering the stone undergoes TIR repeatedly at its many facets before emerging), the mirage seen over a hot road or desert surface (light from the sky bends progressively in layers of air that grow hotter and less dense near the ground, undergoing something close to TIR and rea …