Question

Derive an expression for the minimum and maximum intensity when light reflects from a thin film

Answer 1

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

On Reflected side,

The ray of light BF and DE will interfere. The path difference between BF and DE is,

?=?(BC+CD)?BG?=?(BC+CD)?BG

BC=CD=tcosr..........(1)BC=CD=tcosr..........(1)

Now,

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

BG = BD sin i

BG = (2t) tan r sin i

BG=2t?sinr(sinrcosr)BG=2t?sinr(sinrcosr) [?=sinisinr][?=sinisinr]

BG=2?tsin2rcosr......(3)BG=2?tsin2rcosr......(3)

Substituting (i) and (iii) in ? :

?=?(tcosr+tcosr)?2?tsin2rcosr?=?(tcosr+tcosr)?2?tsin2rcosr

=2?tcosr(1?sin2r)=2?tcosr(1?sin2r)

?=2?tcosr?=2?tcosr

This is a geometric path difference. However, there is a phase change of ?, as ray BF is reflected from a denser medium. Hence we need to add ±?2±?2 to path difference

?=2?tcosr±?2?=2?tcosr±?2

For Destructive Interference:

?=n??=n?

2?tcosr±?2=n?2?tcosr±?2=n?

2?tcosr=(2n±1)?2.....(n=0,1,2,...)2?tcosr=(2n±1)?2.....(n=0,1,2,...)

This is the required expression for constructive Interference or Maxima.

For Destructive interference:

?=(2n±1)?2?=(2n±1)?2

2?tcosr±?2=n?2?tcosr±?2=n?

2?tcosr=n?2?tcosr=n?

This is the required expression for destructive interference.