# Coherence

Think of a wave propagating through space.

The coherence of that wave is a measure of the correlation that exists between the phases of the wave measured at different points, and that will depend upon the characteristics of the source of the wave.

Let's look at a simple 'thought experiment' example.

Imagine a calm water surface with two tiny rubber ducks floating on the surface a few inches apart.

Now imagine a single stick moved steadily in and out of the water at a constant rate creating a stream of circular waves that emanate from the disturbance. What would the duck do? They too would bob up and down as the wavefronts passed them by.

There exists a perfect correlation between the motion of the two ducks. They may not bob up and down exactly in phase, one may go up while the other one goes down, but the phase difference between the positions of the two ducks is constant in time. This is true because the disturbance in the water causing their motion is harmonic.

We say that the source is perfectly coherent.

A harmonically oscillating point source produces a perfectly coherent wave.

Coherence is attained when there is a constant phase difference in two or more waves over time.

Two waves are said to be in phase if their crests and troughs meet at the same place at the same time, and the waves are out of phase if the crests of one meet the troughs of another.

The waves are incoherent if the crests and troughs meet randomly.

To be coherent waves must be:

of the same frequency - monochromatic - temporally in phase

have a constant phase difference - have a constant spatial phase relationship

be of the same amplitude

Laser light is coherent.

A 'red light' can be monochromatic but still be incoherent because the waves are in random phase... it can also be a range of frequencies in the 'red' region of the spectrum!

White light is incoherent both because the phase of the waves are random and because white light is made of many different frequencies simultaneously.

If a laser is not available for an experiment it is possible to use a sodium lamp (virtually monochromatic) and a fine slit to create the spatial coherence - see Young's Slits.

### Types of coherence

There are two types of coherence - spatial and temporal coherence.These are studied at degree level NOT at 'A' Level.

Spatial coherence is a measure of the correlation between the phases of a light wave at different points at right angles to the direction of propagation. Spatial coherence tells us how uniform the phase of the wave front is. It describes the correlation between signals at different points in space. Spatial coherence means a strong correlation (fixed phase relationship) between the electric fields at different locations. For example, within a cross section of a beam from a laser with diffraction-limited beam quality, the electric fields at different positions oscillate in a totally correlated way, even if the temporal structure is complicated by a superposition of different frequency components. Spatial coherence is the essential prerequisite of the strong directionality of laser beams.

Temporal coherence is a measure of the correlation between the phases of a wave at different points along the direction of propagation or the predictable relationship between signals observed at different moments in time. Temporal coherence tells us how monochromatic a source is. Temporal coherence means a strong correlation between the electric fields at one location but different times. For example, the output of a single-frequency laser can exhibit a very high temporal coherence, as the electric field temporally evolves in a highly predictable fashion.