Reflected and transmitted interference complementary

Interference is the redistribution of light energy in space when two or more coherent waves superpose: the intensity is maximum at some points (constructive) and minimum at others (destructive).

Proof for a thin film

Consider a film of thickness t, refractive index μ, with refraction angle r.

Reflected light. The ray reflected from the top surface suffers a phase change of π (λ/2). Path difference between the two reflected rays:

Δ_R = 2μt cos r  ±  λ/2

• Bright: 2μt cos r = (2n+1)λ/2
• Dark: 2μt cos r = nλ

Transmitted light. The first transmitted ray goes straight through; the second is reflected twice inside the film (at the bottom and then the top surface, both from the film side, film in air). Both are reflections at a rarer medium, so there is no phase change. Path difference:

Δ_T = 2μt cos r

• Bright: 2μt cos r = nλ
• Dark: 2μt cos r = (2n+1)λ/2

Compare the conditions:

Thus the reflected and transmitted patterns are complementary: where the reflected system has a maximum, the transmitted has a minimum, and vice versa.

Energy conservation: I_reflected + I_transmitted = I_incident (no absorption). So when the reflected intensity is maximum, the transmitted intensity must be minimum. The transmitted fringes, however, have poor contrast because the two transmitted beams have very unequal amplitudes.