# Temporal & Spatial Coherence – LASER

In this post, we will discuss ‘Temporal & Spatial coherence’ in LASER. Let us start it by defining Coherence.

**Coherence :**

A wave that appears to be a pure sine wave for an infinitely large period of time or in an infinitely extended space be said to be a perfectly coherent wave.

In such a wave there is a definite relationship between the phase of wave and given time and at a certain time later for at a point and at a certain distance away.

Coherence describes all properties of the correlation between physical quantities of a single wave or between several waves or wave packets.

No actual light source, however, emits a perfect coherent wave. Light waves which are pure sine waves only for a limited period of time or in a limited space are partially coherent waves.

There are two types of Coherence :

- Temporal
- Spatial

## Temporal and Spatial Coherence

Let us discuss each of them :

**i) Temporal Coherence :**

It’s a measure of the average correlation between the value of a wave and itself delayed by T, at any pair of times. Temporal coherence tells us how monochromatic a source is.

**Example of temporal coherence: **

A wave containing only a single frequency is perfectly correlated with itself at all time delays. On the other hand, a wave whose phase drifts quickly will have a short coherence time.

Similarly, pulses of waves that naturally have a broad range of frequencies also have a short coherence time since the amplitude of the wave changes quickly.

Finally, white light which has a very broad range of frequencies is a wave which varies quickly in both amplitude and phase since it consequently has a very short coherence time it is often called incoherent.

The oscillating Electric field E of a perfectly coherent light wave would have a constant amplitude of vibration at any point, while its phase would vary linearly with time.

As a function of time, the function would appear as shown in fig. below :

However, no light source produces perfect temporal coherence. This is because, “when an excited atom returns to the initial state, it emits a light pulse of short duration such as of the order of 10^-10 s, after which pulse abruptly changes.

Hence the field due to actual light is shown as :

The average time interval for which the field remains sinusoidal is known as coherence time or temporal coherence of light wave. It is denoted by τ.

∴ Velocity = Distance / Time

and, Distance = velocity × Time

The distance for which the field is sinusoidal is given by :

= L= c × τ

= L = τc

where c is the speed of light & L is the distance for which field is sinusoidal or coherence length.

For Sodium light, Coherence time is 10^{-10} s.

**iii) Spatial Coherence :**

The spatial coherence is the phase relationship between the radiation field at different points in space. In some systems such as water waves or optics, wave-like states can extend over one or two dimensions.

Spatial coherence describes the ability for two points in space in the extent of a wave to interfere when averaged over time

The spatial coherence is the cross-correlation between two points in a wave for all times if a wave has only 1 value of amplitude over an infinite length it is perfectly spatially coherent.

Let A and B depend on the distance AB and the temporal coherence.

If AB << L ( Coherence length) there will be a definite phase relationship between A and B i.e. high coherence between A and B. On the other hand if AB >> L, there will be no coherence between A and B.

Let us now consider points A and C which are equidistant from S. If the source S is a true point source, then the waves shall reach the points A & C in exactly the same phase, i.e., the two points will have perfect (spatial) coherence.

If, however, the source S is extended, the points A and C will no longer remain in coherence. This may be demonstrated by Young’s double-slit experiment illustrated in fig.

The light emitting from narrow slit S falls on two slits S1 and S2 placed symmetrically with respect to S.

The beams emerging from S1 and S2, having been derived from the same original beam, maintains a constant phase difference at all points on the screen.

Hence a stationary interference pattern is observed on the screen. If however, the width of the slit S is gradually increased, the pattern becomes poorer and poorer in contrast and finally disappears.

This means that as the size of the source is increased, the situation of spatial coherence on the screen changes into a situation of incoherence.

This happens because when the shit S is wide, the slits S1 and S2 receive waves from different independent parts of S and hence do not remain coherent with respect to each other.

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