Conditions for interference of light

Question

There are the following two conditions for the interference of two light waves:

1. The sources of the waves must be coherent, which means they emit identical waves with a constant phase difference.

2. The waves should be monochromatic - they should be of a single wavelength.

Here is the question, If two sources are coherent it means they are producing waves with the same frequency and since the interference takes place at a common location it means that the waves should have the same speed in there. This implies that they must have the same wavelength which automatically is the second condition. It means if the first condition is met, the second follows immediately. Why do then we have them separately? I mean the first condition was enough for interference.

Answer 1

Monochromaticity makes interference easier to observe, but it is not necessary for interference. "Coherence" between two sources usually means "temporal coherence", which in turn means that there is a constant phase relationship between the two sources. Note that I didn't say "constant phase difference". For example if the phase difference between the two sources is changing at a constant rate the phase relationship is still constant. There will be temporal interference between them. As @flippiefanus said (but in different words), a detector sees the changing phase difference as a beat frequency. If the two are directed at an imaging array, the changing phase difference shows up as moving fringes (each detector in the array sees a beat frequency, but adjacent detectors see the beat frequency out of phase with each other).

If your question is simply, "Does satisfying the first condition imply that the second condition is satisfied?", then the answer is "no". It is possible for waves from two sources to be coherent without being monochromatic. For example, a very short laser pulse is not monochromatic; it is comprised of a superposition of wavelengths. Two pulses, formed by using a beamsplitter to split the pulse, will certainly interfere because they are identical. If two separate sources produce identical pulses, the pulses will interfere.

Note that those two pulses meet the condition I stated above: there is a constant phase relationship between them (because they are identical). They would also have a constant phase relationship (with constantly changing phase difference) if one was bounced off a moving mirror; but if interfered the two pulses would produce a beat frequency proportional to the speed of the moving mirror. Of course the beat frequency would itself be in a pulse, and would only be produced when the two pulses overlap.

Answer 2

Yes, coherence is a necessary condition for interference. It is also true that if a source is monochromatic then it would satisfy the requirement for temporal coherence. However, the general situation is perhaps a bit more complex.

For instance, we know that the light from stars such as our sun is incoherent. Yet, one can use the light from distant stars to obtain interference. How does this work? Well, it turns out one can make the light from incoherent sources coherent.

There are two types of coherent: temporal coherence and spatial coherence. Temporal coherence is related to the wavelength spectrum of the light. The coherence length $d_{\rm coh}$ (relative distance between points along the propagation direction that are still coherent) is given by the inverse of the width of the spectrum $\Delta \nu$: $$ d_{\rm coh} = \frac{c}{\Delta \nu} , $$ where $c$ is the speed of light. One can change the coherence length by changing the width of the spectrum, by using, for instance a wavelength filter.

Spatial coherence is represented by a transverse distance (or a coherence area). This is determined by the transverse size of the source. So, in the far field of a small source one would find that light can have a large coherence area. This is a process described by the Van Cittert-Zernike theorem.

It is worth mentioning that one can also have a form of interference between light sources of different frequencies. If the light from both such sources fall on the same detector one would measure a beat frequency. This is a process called heterodyne detection.