Coherent Sources

Coherent sources have a constant phase difference and the same frequency — necessary for stable, observable interference.

Two independent ordinary light sources (e.g., two bulbs) are NOT coherent — their phase relationship fluctuates randomly on ~ns timescales, averaging the interference pattern to zero.

In practice, coherent sources are obtained from a single source by either: (a) division of wavefront (e.g., Young's slits, Fresnel's biprism) or (b) division of amplitude (e.g., thin films, Michelson interferometer).

Coherence length L_c = c·τ_c, where τ_c is the coherence time. For a laser, L_c can be metres; for sunlight ~µm.

When coherent sources superpose: I = I₁ + I₂ + 2√(I₁I₂) cos δ. For incoherent sources, the cosine term averages to zero: I = I₁ + I₂.

Lasers are highly coherent (both temporal and spatial). Sunlight has low temporal coherence — that's why thin-film colors require very thin films (within coherence length).