Answer to Question #145518 in Optics for Srayan

Question

How can we obtain the laws of refraction using Fermat's principle?

Answer 1

Fermat’s principle states that “light travels between two points along the path

that requires the least time, as compared to other nearby paths.” From Fermat’s

principle, one can derive the law of refraction [Snell’s law]. 

Consider a light ray traveling from point A to point B in media with different indices of refraction, as shown in the figure. The time to travel between the two points is the distance in each medium divided by the speed of light in that medium.

$$t=\frac{\sqrt{x^2+h_1^2}}{\frac{c}{n_1}}+\frac{\sqrt{(1-x)^2+h_2^2}}{\frac{c}{n_2}}$$

To minimize the time we set the derivative of the time with respect to x equal to zero. We also use the definition of the sine as opposite side over hypotenuse to relate the lengths to the angles of incidence and reflection.

$$0=\frac{dt}{dx}=\frac{n_1x}{\sqrt{x^2+h_1^2}}+\frac{-n_2(1-x)}{\sqrt{(1-x)^2+h_2^2}}\\\frac{n_1x}{\sqrt{x^2+h_1^2}}=\frac{n_2(1-x)}{\sqrt{(1-x)^2+h_2^2}}\to n_1\sin{\theta_1}=n_2\sin{\theta_2}$$