A glass of refractive index 1.5 is coated with a thin layer of thickness t and refractive index 1.8 . Light of wavelength `lambda` travelling in air is incident normally on the layer. It is partly reflected at the upper and the lower surfaces of the layer. It is partly reflected at the upper and the lower surfaces of the layer ant the two reflected rays interface . If `lambda = 648 nm`, obtain the least value of `t ("in" 10^(-8)m )` which the rays interface constructively.

Explanation

Here, refractive index of glass plate, `mu = 1.5`, and refractive index of coating layer, `mu' = 1.8`.
When rays of light are reflected from the glass surface and surface of coating layer, the optical path difference between the two rays.
`Deltax = 2 mu' t`, where `t` is thickness of coating layer.
As `mu' gt mu`, therefore, rays of light suffering reflection from optically denser coating layer undergo a phase change of `pi` or path diff. of `lambda//2`. Therefore, net path difference `= Deltax' = Deltax + lambda//2`
`= 2 mu' t = lambda//2`
For constructive interference
`Deltax' = n lambda`
`:. 2 mu' t + lambda//2 = n lambda`
`2 mu't = (n - (1)/(2)) lambda`
where `n = 1, 2, 3`...........
The least value of thickness of coating layer corresponding to `n = 1`
`:. 2mu' t_(min) = (1 - (1)/(2))lambda = lambda//2`
`t_(min) = (lambda)/(4mu')`
when `lambda = 648 nm, mu' = 1.8`
`t_(min) = (648)/(4 xx 1.8) = 90 nm`