Wave Optics: Understanding Temporal and Spatial Coherence

It is the coherence of light that makes patterns like the one shown below possible. If you have spent any time around the concepts of wave optics, it is a topic you must have come across. Here is a quick refresher: for two light waves to interfere and produce a maximum, they must be coherent and carry a phase difference equal to an integral multiple of the wavelength of the source. The resulting pattern is ethereal to look at and I have attached an image below:

Figure 1: Interference patterns produced via a double slit experiment.

As you progress in your studies from secondary school to graduate level classes, you will come across the division of coherence into temporal and spatial coherence. This post will discuss these two concepts in detail.

The aim of this article is not to regurgitate the pure theory developed over the years about the coherence of light. Rather, my focus will be more experimental in nature, and I am hoping to provide you with a practical means of understanding how these concepts affect the interference pattern produced by a light source.

Temporal Coherence

We begin by considering a Michelson interferometer as shown below, consisting of a light source S, a beam splitter G, two mirrors M₁ and M₂, the former of which is fixed in place while the latter is moveable forwards and backwards, and finally, the observer O. If you recall, most variations of the Michelson interferometer also keep a glass plate to compensate for the optical path difference introduced while traversing the beam splitter, but that is irrelevant in this scenario. Here is the apparatus we have with us now:

Figure 2: Michelson interferometer assembly.

Normally, the mirrors are kept equidistant from the beam splitter G¹, and what we expect is an excellent interference pattern at the bottom of the setup. But imagine what would happen once I started steadily moving the mirror M₂ away from the light source. Naturally, the contrast of the interference pattern would decrease and in fact, for ordinary light sources like a sodium lamp, as soon as the extra separation becomes a few millimeters, the fringes vanish completely.

So why does this decrease in contrast occur?

The answer is where temporal coherence of light sources enters the picture. One thing you must keep in mind is the fact that no light source in this world can emit continuous, unbroken wave trains. Instead, all practical sources emit electromagnetic radiation that looks something like this:

Figure 3: Wave trains united by a light source; size exaggerated.

Let us suppose that the average duration of the wave train that our source emits is given as τ seconds². It is evident that there will be no predictable phase relationship between each of these wave trains. If the time taken by our light to traverse the distances from G to M₁ and M₂ is significantly smaller than τ, then our resultant pattern will have excellent contrast since it is produced by the interference of two wave fronts emanating out of the same wave train. But once the temporal distance³ from G to M₁ and M₂ becomes larger than τ, the contrast will start to decrease and eventually, our pattern will vanish. This occurs when the interference starts to occur between two entirely separate wave trains with no predictable phase relation.

This phenomenon is exactly what temporal coherence is. No light source can emit an infinitely continuous wave train and thus, even light from a lone source only remains coherent for a particular amount of time.

In the example above, the time τ is referred to as the coherence time of the light source, and the length of each wave train l=cτ is known as the longitudinal coherence length.

Spatial Coherence

To talk about spatial coherence, we turn our attention towards the famous and ubiquitous Young's Double Slit Experiment. In its original form, this remarkable setup was able to demonstrate the wave nature of light and later, it even gave rise to a definitive demonstration of the dual nature of matter. Above all, the probabilistic nature of quantum mechanics can also be described via this experiment, but that is not our target for now.

If you recall, the original experiment utilized a single point-source. We will use two point-sources instead, viz. S and S’. Here is a diagram:

Figure 4: The double slit setup we will use.

Here, S and S’ are two point-sources which have no defined or predictable phase relationship. S₁ and S₂ are the two slits, separated from the plane of the light sources by a distance D, whereas T represents the screen, on which we have selected an arbitrary point O. Note that as far as our interest goes, the areas away from O where the distances S₁O and S₂O differ significantly have no value.

If there were only a single light source, namely, S, then the interference pattern would be easy to predict. However, due to the contribution from S’, the pattern we see will be washed out and fuzzy. Part of the maxima and minima due to S will coincide with that due to S’, leading to a blurred picture. And when we start to move S’ away from S, the contrast will decrease even further. There will finally come a point at which the maxima of one will coincide with the minima of the other and vice versa, leading to an effectively blank screen. Let us fix S’ here, separated from S’ by l. Now, we try to estimate the separations S₁S’ and S₂S’.

Figure 5: Exaggerated view of the sources and slits.

In the figure above, applying Pythagoras’ theorem yields the following relations:

Equations 1, 2: The separations between the second source S’ and the two slits. The approximate expressions may be arrived at by considering D >> d, l.

For the fringes to disappear, the path difference between the light waves that arrive from S’ at the two slits must equal half the wavelength of the source, i.e., λ/2. Thus, we have

S’S₂ — S’S₁ = ld/D = λ/2. — (3)

In other words, if instead of two point-sources, we had a single extended source from S to S’, it would no longer remain coherent beyond a certain distance l, approximately given as λD/2d. This is an effective way to understand what spatial coherence is.

The light waves emitted from various parts of an extended light source will not be coherent and spatial coherence is a measure of the distance up to which, appreciable coherence may be achieved in such a source.

Finally, from equation (3), you can infer that a good interference pattern would be obtained when the following condition is satisfied:

l << λD/d, or d << λD/l — (4)

The angle subtended by the light source at the plane of the slit can be approximated as l/D, and we represent it by θ. Let

lₛ = λ/θ — (5)

The quantity lₛ = λ/θ is known as the lateral coherence width of the given light source. It depends inversely on θ and thus, decreases with the length of the source.

Conclusion

In short, temporal, and spatial coherence may be understood by using a Michelson interferometer and the Young Double Slit Experiment, respectively. They may be defined as follows:

• Temporal coherence is a measure of the time period for which, light emitted from a source remains coherent. It is measured longitudinal to the beam and is related to the interval during which the light source in question emits a continuous wave train.
• Spatial coherence is a transverse measurement. It describes the length up to which an extended light source can go before the waves from its extreme ends become incoherent.

The concept of coherence plays a major role in a variety of optical phenomena and is also one of the major factors that differentiate an ideal source from a practical one. An ideal source would have infinite temporal and spatial coherence. Further, a LASER light has much better temporal and spatial coherence as compared to other sources, so much so that it doesn’t need to be passed through a pinhole to create a coherent source (unlike normal sources).

If you enjoyed reading this article, consider dropping me a tip via the link provided in the end. And for learning more about LASERs, I recommend the excellent LASERS: Fundamentals and Applications by Thyagarajan and Ghatak.

Footnotes

1. As mentioned, there is usually a transparent glass plate placed before the mirror M₁ to compensate for the fact that the light travelling towards it goes through the beam splitter twice. However, in this case, that is not really essential. Instead, when I say that the mirrors are equidistant from the beam splitter, I mean that the optical path traversed by the beams is the same, regardless of the geometrical path.
2. Naturally, the time would be within the nanoseconds range. Don’t be fooled by the SI unit.
3. By temporal distance, I mean time taken to traverse the path. Simply multiply the path by the speed of light and you have temporal separation.