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Using Fermat's principle derive the laws of reflection and refraction of light on the plane interface between two media.

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Fermat’s principle : “ The actual path of propagation of light (trajectory of a light ray ) is the path which can be followed by light with in the lest time, in comparison with all other hypothetical paths between the same two points. ”

 “Above statement is the original wordings of Fermat ( A famous French scientist of 17th century)”

Deduction of the law of refraction from Fermat’s principle : 

Let the plane S be the interface between medium 1 and medium 2 with the refractive indices n1 = c/v1 and n2 = c/v2  Fig.(a). Assume, as usual, that n1 < n2 . Two points are given - one above the plane S (point A ), the other under plane S (point B ). The various distances are :

 AA1 = h1, BB1 = h2, A1B1 = l. We must find the path from A to B which can be covered by light faster than it can cover any other hypothetical path. Clearly, this path must consist of two straight lines, viz, AO in medium 1 and OB in medium 2; the point O in the plane S has to be found.

First of all, it follows from Fermat’s principle that the point O must lie on the intersection of S and a plane P, which is perpendicular to S and passes through A and B.

Indeed, let us assume that this point does not lie in the plane P; let this be point O1 in Fig. (b). Drop the perpendicular O1O2 from O1 onto P . Since AO2 < AO1 and BO2 < BO1, it is clear that the time required to traverse AO2 B is less than that needed to cover the path AO1B.

Thus, using Fermat’s principle, we see that the first law of refraction is observed : the incident and the refracted rays lie in the same plane as the perpendicular to the interface at the point where the ray is refracted. This plane is the plane P in Fig. (b); it is called the plane of incidence.

Now let us consider light rays in the plane of incidence Fig. (c). Designate A1O as x and OB1 = l - x. The time it takes a ray to travel from A to O and then from O to B is

The time depends on the value of x. According to fermat's principle, the value of x must minimize the time T. A t this value of x the derivative dT/dx equals zero :

Note : Fermat himself could not use Eqn. 2. as mathematical analysis was developed later by Newton and Leibniz. To deduce the law of the refraction of light, Fermat used his own maximum and minimum method of calculus, which, in fact, corresponded to the subsequently developed method of finding the minimum (maximum) of a function by differentiating it and equating the derivative to zero.

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