Propagation of Light in Inhomogeneous Medium

Theorem
Let $v : \R^3 \to \R$ be a real function.

Let $M$ be a 3-dimensional Euclidean space.

Let $\gamma:t \in \R \to M$ be a smooth curve embedded in $M$, where $t$ is time.

Denote its derivative time by $v$.

Suppose $M$ is filled with an optically inhomogeneous medium such that at each point speed of light is $v = \map v {x, y, z}$

Suppose $\map y x$ and $\map z x$ are real functions.

Let the light move according to Fermat's principle.

Then equations of motion have the following form:


 * $\displaystyle \dfrac {\partial v} {\partial y} \frac {\sqrt{1 + y'^2 + z'^2} } {v^2} + \dfrac \d {\d x} \frac {y'} {v\sqrt {1 + y'^2 + z'^2} } = 0$


 * $\displaystyle \dfrac {\partial v} {\partial z} \frac {\sqrt{1 + y'^2 + z'^2} } {v^2} + \dfrac \d {\d x} \frac {z'} {v \sqrt {1 + y'^2 + z'^2} } = 0$

Proof
By assumption, $\map y x$ and $\map z x$ are real functions.

This allows us to $x$ instead of $t$ to parameterize the curve.

This reduces the number of equations of motion to $2$: $\map y x$ and $\map z x$.

The time it takes to traverse the curve $\gamma$ equals:

According to Fermat's principle, light travels along the trajectory of least time.

Therefore, this integral has to be minimized $\map y x$ and $\map z x$.

It holds that:


 * $\displaystyle \dfrac {\partial} {\partial y} \frac{\sqrt{1 + x'^2 + y'^2} }{\map v {x, y, z} } = - \frac{\sqrt{1 + x'^2 + y'^2} }{\map {v^2} {x, y, z} } \dfrac {\partial v} {\partial y}$

Also:

$\displaystyle \dfrac {\d} {\d x} \dfrac{\partial }{\partial y'} \frac{\sqrt{1 + x'^2 + y'^2} }{\map v {x, y, z} } = \dfrac {\d} {\d x} \frac{y'}{v \sqrt{1 + x'^2 + y'^2} }$

Analogous relations hold for $\map z x$.

Then, by Euler's Equations and multiplication by -1 the desired result follows.