PX285 - B2 - fermat's law in optics

l = \int d s n (\vec{r} (s))

\begin{aligned} l & = \int n (\vec{r}) d s \\ = \int_{0}^{d_{1}} d s n_{1} + \int_{d_{1}}^{d_{1} + d_{2}} d s n_{2} \\ = n_{1} d_{1} + n_{2} d_{2} \\ = n_{1} \sqrt{h_{1}^{2} + x^{2}} + n_{2} \sqrt{h_{2}^{2} + (R - x)^{2}} \end{aligned}

\begin{aligned} \frac{d l}{d x} & = \frac{n_{1} x}{\sqrt{h_{1}^{2} + x^{2}}} + \frac{- n_{2} (R - x)}{\sqrt{h_{2}^{2} + (R - x)^{2}}} = 0 \\ 0 & = n_{1} \frac{x}{d_{1}} + n_{2} \frac{x - R}{d_{2}} \end{aligned}

$θ_{1}$ and $θ_{2}$ are angles of incidence in air and water, respectively
$d_{1} \sin θ_{1} = x$
$d_{2} \sin θ_{2} = (R - x)$

\begin{matrix} n_{1} \frac{d_{1} \sin θ_{1}}{d_{1}} - n_{2} \frac{d_{2} \sin θ_{2}}{d_{2}} = 0 \\ n_{1} \sin θ_{1} = n_{2} \sin θ_{2} \end{matrix}

l (x) |_{x = x_{m i n}} = l_{m i n} + \underset{= 0}{\underset{⏟}{\frac{d l}{d x} |_{x = x_{m i n}}}} (x - x_{m i n}) + \frac{d^{2} l}{d x^{2}} |_{x = x_{m i n}} (x - x_{m i n})^{2} + \dots

there is no difference (at lowest order) in optical path length, $l (x)$ , near $x = x_{m i n}$
the neighbouring paths have almost exactly the same optical path lengths, hence they interfere constructively
considering the phase of the light wave: $ϕ = (2 π \frac{l}{λ} - ω t)$ , so, $\frac{d ϕ}{d x} = 0 \Leftrightarrow \frac{d f}{d x} = 0$
the light 'seen' is therefore the rays passing through $x_{m i n}$
the realized path is the one where the path length, $l$ , is stationary with respect to varying paths, ie: have varying $x$