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

#### Solution

**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−sin^{2}r)

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**

.