Let PP. represent the surface separating medium-1 and medium-2. Let `v_(1)` and `v_(2)` be the speed of light in medium-1 and medium-2 respectively. Consider a plane wavefront AB incident in medium-1 at angle .i. on the surface PP..
According to Huygen.s principle, every point on the wavefront AB is a source of secondary wavelets.
Let the secondary wavelet from B strike the surface PP. at C in a time t. Then `BC=v_(1)t`.
The secondary wavelet from A will travel a distance `v_(2)t` as radius, draw an arc in medium-2.
The tangent from C touches the arc at E. Then `AE=v_(2)t` and `CE` is the tangential surface touching all the spheres of refracted secondary wavelets. Hence, CE is the refracted wavefront. Let r be the angle of refraction.
In the above figure, `/_BAC=i=` angle of incidence and `/_ECA=r=` angle of refraction.
`BC=v_(1)t` and `AE=v_(2)t`
From triangle `BAC`, `sini=(BC)/(AC)` and from triangle
`ECA,sinr=(AE)/(AC)`
`:.(sini)/(sinr)=(BC//AC)/(AE//AC)=(BC)/(AE)=(v_(1)t)/(v_(2)t)=(v_(2))/(v_(1))`
Since `v_(1)` is a constant in medium -1 and `v_(2)` is a constant in medium -2, `(sini)/(sinr)=(v_(2))/(v_(1))` constant.......(i)
Now , refractive index (n) of a medium : `n=(c )/(v)` or `v=(c )/(n)` where `c=` speed of light in vacuum.
For the first medium : `v_(1)=(c )/(n_(1))` and for the second
medium : `v_(2)=(c )/(n_(2))implies(v_(1))/(v_(2))=(n_(2))/(n_(1))`
Eqn. .......(i) becomes `(sini)/(sinr)=(v_(2))/(v_(1))` or `n_(1)sini=n_(2)sinr`.
This is the Snell.s law of refraction.
