Thin Film Interference: Deriving Equation

[Haftred]

My professor gave the equation:

$$2t + \frac{\lambda}{2} = (m + \frac{1}{2})\lambda$$

How did he derive this..where does it come from?

 

[marlon]

You know the formula's that express the destructive and constructive interference because of some path length-difference (see the following post)? If so, you just apply these formula's to the reflected beams on one specific side of the film. The clue is, however, that you need to know the relative magnitude of the refraction indices of the media that are separated by the film. Mostly there are three media (one on the left, one inside the film and one on the right). If a wave reflects on a medium of which the refraction index is BIGGER then that of the medium in which the wave is propagating, there will be a phase change of 0.5 times the wavelength. One can prove this, using the Fresnel relations. You will need to determine at which plane of the film, this phase change will arise. If it arises at only one of the two planes then the reflected waves will have a phase difference of 0.5 times the wavelength wtr to each other. The formula for destructive interference now expresses the actual constructive interference because of this relative phase change.

marlon

 

[marlon]

Besides, one of the two formula's should be :

$$2t = (m + \frac{1}{2}) \frac{\lambda}{n}$$

the other :

$$2t = (m ) \frac{\lambda}{n}$$


where n is the refraction index of the medium where the actual path length difference occurs.

marlon

 

[marlon]

here is more info : http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/thinfilm.html#c1

marlon