Is Snell's Law valid even for on any curved surfaces?

Proof of Snell's Law can also done by using Fermat's principle of least time on plane surfaces. Can we prove the same result for any curved surfaces ?

• It's an approximation for surfaces that have very low curvature on the scale of a wavelength. The general diffraction problem on very complex surfaces with large curvature can only be solved with the full electromagnetic field equations, which is a very hard problem. – CuriousOne Oct 31 '15 at 4:28
• @CuriousOne Is correct Snells Law is but an simplification that relies on approximations, there are therefore situations that it does not work for. – MJC Nov 28 '17 at 11:13

Consider the action $$S=\int{dt}$$ In a certain medium one can write $$S=\int{\frac{\sqrt{1+y'²}}{v}dx}$$ Here $$y'$$ represents the derivative of $$y$$ with respect to $$x$$. Solving the Euler Lagrange equations $$\frac{d}{dt} \bigg(\frac{y'}{v\sqrt{1+y'²}}\bigg)=0$$ Hence $$\frac{y'}{v\sqrt{1+y'²}}=c_1$$ Where $$c_1$$ is an integration constant. If light is moving at an angle of $$\theta$$ with the $$x$$-axis then we can write $$y'=\tan\theta$$. Substituting this in the above equation $$\boxed{\frac{\sin\theta}{v} =c_1}$$ This is Snell's law. In this derivation no surface was considered at all. For any two mediums one can write the relation (using appropriate coordinates) $$\frac{\sin\theta_1}{v_1}= \frac{\sin\theta_2}{v_2}$$ in general.