# Path of light ray through varying refractive index

Suppose light ray passing through a medium with refractive index $$n=n(y)$$. In the case of an inhomogeneous medium in which $$n$$ varies continuously in the $$y$$-direction, We have curved rays that satisfy Snell's law in the form:

$$n\cos\psi=\mathrm{constant}$$ where the angle $$\psi$$ is the slope of the tangent to the path.

Fermat's principle states that

The actual path taken by a light ray between two fixed points makes the travel time of the ray stationary.

So that $$T[\mathcal{P}]=c^{-1}\int_{\mathcal{P}}nds$$ which reduce to (in present case) $$T[y]=c^{-1}\int_{x_0}^{x_1}dx \ n(1+\dot{y}^2)^{1/2}$$ with the help of Euler-Lagrange's Equation $$\frac{n}{(1+\dot{y}^2)^{1/2}}=\mathrm{constant}$$

And on writing $$\dot{y}=\tan\psi$$, this gives snell's law.

Question: If I put $$\dot{y}=0 \Rightarrow y=$$ constant that is not extremals and therefore not rays. But since such a ray would experience a constant value of $$n$$, How does the ray know that it must bend?

• Please correct your equation -- you are missing a $dx$ -- and define your notation, $\dot y = dy/dx$. I know it seams obvious to you, but for most physicists a dot means time derivative. Dec 15, 2020 at 12:49

light is a wave, which propagates in such a way that each disturbance generates a