Wave superposition and coherence

Superposition and Interference

When two coherent light waves meet at a single point, the resulting amplitude is the sum of their amplitudes (the principle of superposition). If the two sources have the same frequency and a constant phase difference over time, this is referred to as temporal coherence; it is a necessary condition for observing stable fringes.

Path difference and conditions for interference

The path difference is defined as delta = d2 - d1, where d1 and d2 are the distances travelled by the two waves from their sources to point M. The resulting intensity is given by:

I = I1 + I2 + 2sqrt(I1I2)cos(2pi*delta/lambda)

• Constructive interference (maximum): delta = k*lambda, where k is a relative integer
• Destructive interference (minimum): delta = (k + 1/2)*lambda

Source coherence

Two independent sources (two light bulbs, for example) never produce a stable interference pattern because their phase difference varies randomly over a few nanoseconds. We therefore use either a single split source (Young’s slits, Fresnel biprism) or a laser with high temporal coherence.

Common pitfall

Be aware of the additional phase shift of pi introduced by reflection off a more refractive medium (useful for thin films; not covered here but worth knowing). Do not confuse the wavelength in a vacuum, λ, with that in a medium of refractive index n: λ_(medium) = λ/n.