Thin film interference by reflected light

Conditions for interference

1. The two sources must be coherent: a constant phase difference and the same frequency.
2. They should emit light of the same wavelength (monochromatic) and the same state of polarization.
3. The amplitudes should be nearly equal for good contrast.
4. The sources must be narrow, and the path difference small (within the coherence length).
5. The screen should be sufficiently far from the sources.

Thin film: reflected light

   air:       \ 1        / 1'
               \ i     /
   -----------A-\\---C---------  top surface
   film (μ)    r \ /  r      t
   -------------B-------------  bottom surface

A parallel beam falls at angle i on a film of thickness t and refractive index μ; the angle of refraction is r. Ray 1 is reflected at A on the top surface. Ray 2 refracts at A, reflects at B on the bottom surface, and emerges at C parallel to ray 1.

1. Optical path of ray 2 inside the film: μ(AB + BC)
2. Draw CD perpendicular to the reflected ray 1. The path of ray 1 in air up to D is AD = AC sin i; also μ = sin i / sin r
3. Geometry: AB = BC = t / cos r, and AC = 2t tan r
4. Optical path difference: μ(2t/cos r) − 2t tan r sin i = 2μt/cos r − 2μt sin²r/cos r = 2μt cos r
5. Reflection at the top surface (rarer to denser) gives an extra phase change of π, equivalent to a path of λ/2. Effective path difference: Δ = 2μt cos r − λ/2 (or +λ/2).

Conditions

• Bright (constructive): 2μt cos r − λ/2 = nλ → 2μt cos r = (2n+1) λ/2, n = 0, 1, 2 ...
• Dark (destructive): 2μt cos r − λ/2 = (2n+1)λ/2 → 2μt cos r = nλ

For normal incidence (r = 0): 2μt = (2n+1)λ/2 for bright and 2μt = nλ for dark. A very thin film (t << λ) looks dark in reflected light.