Snell's law

Remarks

One proof uses Fermat's principle. A ray from $A$ in the first medium to $B$ in the second crosses the interface at a point $x$, and its optical path length is $n_1 \abs{A - x} + n_2 \abs{x - B}$. Setting the derivative in $x$ to zero gives exactly $n_1 \sin\theta_1 = n_2 \sin\theta_2$. A wave proof matches the phases of the incident and transmitted plane waves along the interface: their wave vectors must have the same component along it.

Example: light from air ($n_1 = 1$) into water ($n_2 \approx 1.33$) at $\theta_1 = 45^\circ$ has $\sin\theta_2 \approx 0.707 / 1.33 \approx 0.53$, so $\theta_2 \approx 32^\circ$; the ray bends toward the normal.

If $n_1 > n_2$ and $\sin\theta_1 > n_2 / n_1$, no angle $\theta_2$ satisfies the law. Then there is no refracted ray and the light is totally reflected.

Going deeper

For a student

Snell's law is the basic rule for tracing rays through lenses and prisms. In the paraxial approximation it becomes $n_1 \theta_1 = n_2 \theta_2$, which leads to the matrix methods of first-order optics. Its critical angle gives total internal reflection, the principle of optical fibres. In a medium with a continuously varying index stratified in layers, the quantity $n \sin\theta$ stays constant along the ray, which explains mirages.