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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 ?

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  • $\begingroup$ 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. $\endgroup$ – CuriousOne Oct 31 '15 at 4:28
  • $\begingroup$ @CuriousOne Is correct Snells Law is but an simplification that relies on approximations, there are therefore situations that it does not work for. $\endgroup$ – MJC Nov 28 '17 at 11:13
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Yes! (I believe) Whichever "point" your light ray strikes at the curved interface (surface) just draw a tangent along that point of the surface. Then do the same thing that is done for the plane surfaces with the angential surface created, you can go ahead and draw the normal, etc. Hence, Snell's law of ray optics is valid for all the points on a curved surface (interface between the two optical media). And yes, the fermat's principle of least time on plane surfaces can now be used easily.

Assuming that the point is macroscopic and does not require us to delve into the microscopic characteristics of the interface. Since ray or geometrical optics is only valid when the obstacle is of a larger order than the wavelength of the electromagnetic wave.

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  • $\begingroup$ You rapidly run into problems when the wavelength of variations of the surface become close to the wavelength of light. See, for example, using a CD or DVD as a diffraction grating - Snell's law does not describe that at all. $\endgroup$ – Jon Custer Jan 14 at 14:20
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    $\begingroup$ @JonCuster When people are using Snell's law, they should know that the range of validity is that of geometrical optics, i.e. they should know that surfaces with variations on the scale of the wavelength are not within the scope of gemetrica optics. Still within the range of validity of geometric optics generalization of Snell law to curved surfaces is a well defined problem. This answer is the correct one, always keeping in mind the general limitations of geometric optics. Otherwise no system of lenses could work! $\endgroup$ – GiorgioP Jan 14 at 18:18
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    $\begingroup$ I would add that I find odd that people commented on the range of validity instead of starting with a plain and simple "Yes, it can be extended to smooth curved surfaces as well". Warnings on the range of validity should follow an explanation, shouldn't replace it. $\endgroup$ – GiorgioP Jan 14 at 18:24
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Yes. By drawing a tangent to the surface and considering that region of the to be flat Snell's law can be applied. A small proof is given below.

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.

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