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Derive the laws of refraction of light using Huygens’ principle.

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Consider a plane wavefront AB of monochromatic light propagating in the direction A’A incident obliquely at an angle i on a plane refracting surface MN. This plane refracting surface MN separates two uniform and optically transparent mediums.

Let v and v be the speeds of light in medium 1 (say, a rarer medium) and medium 2 (a denser medium) respectively.

When the wavefront reaches MN at point A at t = 0, A becomes a secondary source and emits secondary waves in the second medium, while ray B’B reaches the surface MN at C at time t = T. Thus, BC = v T. During the time T, the secondary wavelet originating at A covers a distance AE in the denser medium with radius v2T.

As all the points on CE are in the same phase of wave motion, CE represents the refracted wavefront in the denser medium. CE is the tangent to the secondary wavelet starting from A. It is also a common tangent to all the secondary wavelets emitted by points between A and C. PP’ is the normal to the boundary at A.

∠A’AP = ∠BAC = the angle of incidence (i) and ∠P’AE = ∠ACE = the angle of refraction (r).

From ∆ABC and ∆AEC,

By definition, the refractive index of medium 2 with respect to medium 1,

Here, n1 and n2 are the absolute refractive indices of medium 1 and medium 2 respectively. Eq. (1) is Snell’s law of refraction. Also, it can be seen from the figure, that the incident ray and the refracted ray lie on the opposite sides of the normal and all three of them lie in the same plane.

Thus, the laws of refraction of light can be deduced by Huygens’ construction of a plane wavefront.

If v1 > v2 , i.e. n1 < n2 , then r < i (bending of the refracted ray towards the normal). 

[Notes :

1. Quite often, the terms vacuum and free space are used in the same sense. Absolute vacuum or a perfect vacuum-a region of space devoid of material, particles – does not exist. The term vacuum is also used to mean a region of space occupied by a gas at very low pressure. Free space means a region of space devoid of matter and fields. Its refractive index is 1 (by definition). Its temperature is 0 K. ε0 and μ0 are defined for free space. The refractive index of a medium with respect to air is very close to the absolute refractive index of the medium as the speed of light in air is very close to that in free space.

2. There is no lateral inversion in refraction. 

3. There is no bending of light when the angle of incidence is zero (normal incidence), r = 0 for i = 0.]

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