Snell-Descartes Law

Physical Law
Consider a ray of light crossing the threshold between two media.

Let its speed:
 * in medium $1$ be $v_1$
 * in medium $2$ be $v_2$.

Let it meet the threshold at:
 * an angle $\alpha_1$ from the vertical in medium 1
 * an angle $\alpha_2$ from the vertical in medium 2.

Then the Snell-Descartes law states that:
 * $\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$

Proof
The Snell-Descartes law can be derived from Fermat's Principle of Least Time as follows:

Let the ray of light travel from $A$ to $P$ in the medium $1$.

Then let it travel from $P$ to $B$ in medium $2$.


 * SnellsLaw.png

The total time $T$ required for that journey is:
 * $T = \dfrac {\sqrt{a^2 + x^2}} {v_1} + \dfrac {\sqrt{b^2 + \left({c - x}\right)^2}} {v_2}$

from the geometry of the above diagram.

From Fermat's Principle of Least Time, this time will be a minimum.

From Derivative at Maximum or Minimum, we need:
 * $\dfrac {\mathrm d T} {\mathrm d x} = 0$.

So:
 * $\dfrac x {v_1 \sqrt {a^2 + x^2} } = \dfrac {c - x} {v_2 \sqrt {b^2 + \left({c - x}\right)^2} }$

which leads directly to:
 * $\dfrac {\sin \alpha_1} {v_1} = \dfrac {\sin \alpha_2} {v_2}$

by definition of sine.