Arrive at Snell's law of refraction, using Huygen's principle for refraction of a plane wave.

Explanation

Distance AE=`v_2 tau , tau` - time taken for optical path
Distance BC = `v_1tau, i_1` - angle of incidence in medium (1)
From the right angled `triangleABC` , sin I = `(BC)/(AC)`
and from the right angled `triangleAEC` , sin r =`(AE)/(AC)`
Hence, `(sin i)/(sin r)=(BC)/(AE)=(v_i tau)/(v_2 tau)`
or `(sin i)/(sin r) =v_1/v_2` ....(1)
By definition , absolute R.I of a medium w.r.t air/vacuum=n=`c/v`
for medium (1) = `n_1=c/v_1`
for medium (2)=`n_2=c/v_2`
So that , `v_1/v_2=n_2/n_1` ...(2)
Using (2) in (1) we get
`(sin i)/(sin r)=n_2/n_1`
where , `n_2 > n_1`.
or `n_1 sin i=n_2=sin r` ...(3)
i.e., R.I. of medium (1) times sine of angle in the medium (1)=R.I of medium (2) times sine of angle in the medium 2.
This expression (3) is known as the Snell.s law .
The ratio `n_2/n_1` can be written as `n_2` (R.I. of medium (2) w.r.t medium (1) .
Note: From (1), `n_2=(sin i_1)/(sin i_2)=v_1/v_2=lambda_1/lambda_2`
i.e., `v_1sin i_2=v_2sin i_1, lambda_2sini_1 =lambda_1 sin i_2` and `n_1 sin i_1 = n_2 sin i_2`