The physics
Light reflects from both faces of a thin film. The two reflected rays meet and interfere.
The ray through the film travels an extra path 2μt. A ray that reflects off a denser medium also gains half a wave, λ/2.
1 flip, bright: 2μt = (m + ½)λ
0 or 2 flips, bright: 2μt = mλ
Light that does not reflect goes through. So a coating lets the most light through when its reflection is dark.
A film that thins by λ/2μ goes from dark to dark once.
gap = λ / (2μ × rate)
Newton's rings: a lens on a flat plate leaves a gap t = r²/2R. The ray off the plate flips, so the centre is dark.
dark ring m: r = √(mλR / μ)