Derivation for optical path length and the origins of the formula

Question

So I've learnt that the formula for optical path length is $OPL = ns$, where $n$ is refractive index of the medium and $s$ is its geometrical length, the problem is i cant really get around this formula like where does it come from? I tried deriving it but couldnt get anywhere.

I tried adjusting the last equation using snell's law, got nowhere.

Brain Stroke Patient · Answer

The quantity $ns$ has a physical meaning. I think that's what you're trying to figure out. If light travels a distance $s$ through a medium with refractive index $n$ in a time $T$, then the distance light would travel in the same time $T$ in a vacuum would be $ns$.
$$T = frac{s}{v} = frac{ns}{c}$$
and so the distance travelled in vacuum in this time is
$$cT = ns$$

Derivation for optical path length and the origins of the formula

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