Why does light take the shortest path?

Question

There have been a lot of duplicates for this question , but my question asks a bit something else ..

1.Does the Fermat's principle have any intuitive/ mathematical explanation , If so , it would very helpful if you attach a link or simply post one .

2. Secondly, I am familiar that the Snell's Law can be derived by it and the lifeguard - water-sand situation, but none answer my doubt , which is why does light take the shortest path, where the path is between 2 points , how does the light knows those 2 points and accordingly takes the shortest path ?

Thanks

Answer 1

Snell's law of refraction is essentially equivalent to Fermat's principle. Look at the picture below, it shows two neighboring rays starting at point $S$ in a medium of refractive index $\eta_1$ and ending at point $F$ in a medium $\eta_2$, the path length from $S$ to $F$ then $$\eta_1 r +\eta_2 \rho = \eta_1 (r+dr) + \eta_2 (\rho-d\rho) \tag{1}\label{1}$$ to 1st order differentials. This can be written as $$\eta_1 dr = \eta_2 d\rho \tag{2}\label{2}.$$ Now if you divide both sides of $\eqref{2}$ by $ds$ you get Snell's law. Of course, the steps can be taken in reverse and you can get from $\eqref{2}$ to $\eqref{1}$ and get the differential from of Fermat's law of stationarity from Snell's law of refraction. The equation $\eqref{1}$ means that within an error of 2nd order, i.e, equality within a 1st order, the optical path length (time of flight) between fixed points of neighboring rays are equal.

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The argument and the picture are taken from this excellent article by Cornbleet, Geometrical Optics Reviewed: A New Light on an Old Subject, PROCEEDINGS OF THE IEEE, VOL. 71, NO. 4, APRIL 1983.