Parallel film: Interference by reflected light rays

Interference in Reflected Rays

Consider a plane-parallel transparent thin film of refractive index μ and thickness t. A ray of monochromatic light is incident on the upper surface of the film at an angle of incidence i. A part of the incident ray is reflected from the upper surface, while the remaining part is refracted into the film.

The refracted ray travels inside the film and make refraction angle r. It is partially reflected from the lower surface. The two reflected rays subsequently emerge in the same direction and interfere with each other. The interference between these two reflected rays depends upon the optical path difference between them.

let draw two perpendicular $P_2A$ and $C_1M$. then effective pathdifference

$$\Delta_1 = \mu(P_1C_1 - C_1P_2)- P_1A \qquad ...(1)$$ $\because \Delta P_1MC_1 \approx \Delta P_2MC_1$ $$\therefore P_1C_1 = C_1P_2 = \frac{t}{\cos r} \qquad ...(2)$$ $$ and \qquad P_1M = MP_2 = t\tan r \qquad ...(3)$$

In $\Delta P_1AP_2$

$$\sin i = \frac{P_1A}{P_1P_2}= \frac{P_1A}{P_1N + NP_2}$$ Hence, $$P_1A = 2t \sin i \tan r \qquad..(4) \qquad \because eq.(3)$$ Thus by eq. (1), (3) and (4)

$$\Delta_1 = \mu(\frac {2t}{\cos r})- 2t \sin i \tan r $$ $$\Delta_1 = 2\mu t (\frac {1}{\cos r}- \frac {\sin i \tan r}{\mu} ) $$ $$\Delta_1 = 2\mu t (\frac {1}{\cos r}- \frac {\sin i \sin r}{\mu \cos r} ) $$ $$\Delta_1 = 2\mu t (\frac {1}{\cos r}- \frac {\sin i {\sin^2 r}}{\sin i \cos r} ) $$ $$\Delta_1 = 2\mu t (\frac {1 - \sin^2 r}{\cos r} $$ $$\Delta_1 = 2\mu t \cos r ) $$

Stokes's law

When light is reflected from a rarer medium to a denser medium, it suffers a phase change of π. This phase change is equivalent to an additional path difference of $\frac{\lambda}{2}$.

Therefore, the total optical path difference becomes:

$$\boxed{ \Delta = 2\mu t\cos r + \frac{\lambda}{2} }$$

Case (1) Constructive Interference

For constructive interference, the total optical path difference must be an integral multiple of the wavelength:

$$\Delta = n\lambda$$

Therefore:

$$2\mu t\cos r + \frac{\lambda}{2} = n\lambda$$

$$2\mu t\cos r = n\lambda - \frac{\lambda}{2}$$

Therefore, the condition for bright or constructive interference in reflected light is: $$\boxed{ 2\mu t\cos r = \frac{(2n-1)\lambda}{2}} $$ $$where, \qquad n = 1,2,3,... $$

Case (2) Destructive Interference

For destructive interference, the total path difference must be an odd multiple of half a wavelength: $$\Delta = \left(2n+1\right)\frac{\lambda}{2}$$

Therefore:

$$2\mu t\cos r + \frac{\lambda}{2} = \left(2n+1\right)\frac{\lambda}{2}$$

Therefore, the condition for dark or destructive interference in reflected light is:

$$\boxed{2\mu t\cos r = n\lambda }$$ $$where, \qquad n = 0,1,2,3,... $$