PX284 - T2 - fermat's principle

considering light travelling from point $A$ to $B$

fermat's principle

the path taken by the light ray is that which extremizes the optical path

more generally, light takes the path that minimizes the time, which neglects the other extrema
for a given path, the phase change from $A$ to $B$ is:

Δ ϕ_{A B} = \int_{A}^{B} k d l = \int_{A}^{B} \frac{ω}{v} d l = \int_{A}^{B} \frac{ω}{c} n d l

where, $k$ is the wave number, $ω$ the angular frequency, $v = c / n$ the phase velocity, and $n$ the refractive index

since $ω$ and $c$ are constants:

Δ ϕ_{A B} = k_{0} \int_{A}^{B} n d l

where, $k_{0} = ω / c$

the integral is called the 'optical path':

τ = \int_{A}^{B} n d l

the time taken for light to travel is:

Δ t = \int_{A}^{B} \frac{d l}{v} = \frac{1}{c} \int_{A}^{B} n d l

∴ Δ t = \frac{τ}{c}

by destructive interference:

\sum_{all paths} \exp (i Δ ϕ_{A B}) \to 0

except where $Δ ϕ_{A B}$ is extremized when small changes in the path do not change $Δ ϕ_{A B}$ , and therefore, interfere constructively

examples

considering $n =$ constant $⟹ τ = n \int_{A}^{B} d l$ , therefore, light takes a straight line path

now considering media with $n_{1}$ and $n_{2}$

\begin{matrix} x_{2} = x - x_{1} \\ τ = n_{1} (x_{1}^{2} + y_{1}^{2})^{1 / 2} + n_{2} ((x - x_{1})^{2} + y_{2}^{2})^{1 / 2} \end{matrix}

extremizing with respect to $x :$

\frac{\partial τ}{\partial x_{1}} = \frac{n_{1} x_{1}}{(x_{1}^{2} + y_{1}^{2})^{1 / 2}} + \frac{n_{2} (x - x_{1})}{((x - x_{1})^{2} + y_{2}^{2})^{1 / 2}}

this leads to snell's law:

n_{1} \sin i = n_{2} \sin t