Question

Derive the conditions of maxima and minima due to interference of light reflected from thin film of uniform thickness.

Answer 1

Consider a thin film of uniform thickness (t) and R.I (μ)

On Reflected side,

The ray of light BF and DE will interfere.

The path difference between BF and DE is,
Δ = μ(BC + CD) − BG

`BC = CD = t/cosr`..........(1)

Now,

BD = (2t) tan r .......(2)

BG = BD sin i

BG = (2t) tan r sin i

`BG = 2tμsinr(sinr / cosr)`

`BG = 2μt(sin^2r/cosr)`..........(3)

Substituting (i) and (iii) in Δ :
`Δ = μ(t / cosr + t / cosr)−2μt(sin^2r / cosr)`
= 2μtcosr(1−sin²r)
Δ = 2μtcosr 

This is a geometric path difference. However, there is a phase change of π, as ray BF is reflected from a denser medium. Hence we need to add ±λ2 to path difference

Δ = 2μtcosr ± λ2

For Destructive Interference:

Δ = nλ
2μtcosr±λ2=nλ
`2μtcosr=(2n±1)λ2.....(n=0,1,2,...)`

This is the required expression for constructive Interference or Maxima.

For Destructive interference:
`Δ = (2n±1)λ/2`
`2μtcosr ± λ/2 = nλ`
2μtcosr = nλ

This is the required expression for destructive interference

.