# What is Temporal Coherence

Coherence is the term used to describe the phase correlation phenomenon between two waves with respect to time or space.

## Temporal Coherence

Temporal coherence is the correlation between the electric field at a point in the space at time t₁ [i.e. E (x, y, z, 1₁)]and the electric field at the same point at time t₂ [i.e. E (x, y, z, 12).

A wave is said to have a temporal coherence if the phase difference between the electric fields at a point at times t1 and t2 is constant during the time interval Δτ = t2 – t1.

No source of light emits a monochromatic wave. In fact, a wave emitted by a source consists of number of wave trains or pulses each of certain duration. The duration of each wave train is about 10-10 s.

This means that the phase difference between the electric fields at a point in space remains constant during the time interval 10-10s. This time interval is known as coherence time (τc). The path length of the wave train corresponding to the coherence time (τc) is known as coherence length of the wave train.

Coherence length of the wave train is given by

Where, c = 3*108 m s-1 is the velocity of light in air.

If Δv be the spread in the frequency v of the wave train, then

Hence eqn. (1) becomes

Temporal coherence can be studied with the help of Michelson’s interferometer.

Let S be the source of light (say sodium vapour lamp) which is nearly a monochromatic source of light. When the light waves reflected from mirrors M₁ and M₂ are seen, interference pattern is observed.

When mirror M1 is displaced through a distance d to new position M₁’, then the interference disappears. This is because the wave reflected from M₁ travel additional path = 2d to reach the eye.

So the wave reflected from the mirror M₂ interferes with the wave reflected from M₁ which had been emitted from the source

seconds earlier.

If,

then the two wave trains (i.e. groups of waves) are incoherent and hence no interference pattern is observed.

On the other hand, if

then the two reflected wave trains have a definite phase relationship and hence interference pattern is formed due to the superposition of these wave trains. So these wave trains are temporally coherent.

Thus, we can conclude that the wave trains superimposing on each other after traveling equal paths give rise to an interference pattern of good contrast. Hence, they have maximum temporal coherence.