Interference and coherence

The main symbols used in this lesson:

Drop two stones into a pond at the same moment. Where the two sets of ripples cross, some places are unusually calm and others are unusually lively.

The waves are doing what waves always do: superposing. Where a crest meets a crest the displacements add and the water moves more; where a crest meets a trough they cancel and the water barely moves.

This pattern of strong and weak regions is interference. It is the same superposition you met with two pulses, applied to two continuous waves.

Take a point P and two sources. The wave from each source travels a different distance to reach P. The difference between those distances is the path difference.

• If the path difference is a whole number of wavelengths, $0, \lambda, 2\lambda, \ldots$, the waves arrive in phase: crest meets crest, and P is a maximum.
• If the path difference is an odd number of half wavelengths, $\dfrac{\lambda}{2}, \dfrac{3\lambda}{2}, \ldots$, they arrive in antiphase: crest meets trough, and P is a minimum.

The same rule in terms of phase: a phase difference of $0, 2\pi, 4\pi, \ldots$ gives a maximum, and $\pi, 3\pi, \ldots$ gives a minimum. A path difference of $\dfrac{\lambda}{4}$ corresponds to a phase difference of $\dfrac{\pi}{2}$, or $90^\circ$.

Put two identical lamps side by side and you do not see a stable pattern of bright and dark bands. You see uniform illumination.

The reason is that each lamp emits light in countless tiny bursts, and the phase of each burst is random and changes billions of times a second. The interference pattern is real, but it jumps to a new position far too quickly for your eye to follow, so you see the average.

A laser is different: its output is coherent, so the pattern stands still and can be photographed.

The practical trick is to make both waves come from one source — for example, pass a laser beam through two narrow slits. Whatever the source does, both waves do it together, so they stay in step.