Can a single incoherent source produce interference?

Question

I was wondering if we can setup things in such a manner that a single incoherent source can produce interference with a constant phase difference?

Answer 1

I would say yes, but actually it depends. First, we need some clarification about what a single incoherent source would be. Coherence can be understood as phase correlation that a wave has with itself. We could measure space coherence by considering two different points of the wavefront at the same time and test if they could exhibit interference, or we could measure time coherence by considering a single point and the wavefronts passing through this point in two different times. These concepts are well understood for extreme cases.

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If you consider a point source producing an outgoing spherical wave, it will have infinite spatial coherence. We can measure it through the coherence distance $d_c \rightarrow \infty$ (please, do not confuse with coherence length $l_c$). The easy way to see this is through the dual slit experiment. We can realize dual slit interference with a point source with arbitrarily far slits.

Now if you consider a extensive source, we have finite coherence spatial coherence, so finite coherence distance $d_c<\infty$. We can say that coherence distance (or more precisely, its inverse $d_c^{-1}$) measure how much our source deviates from a point source. This is why with real sources, the slits in the dual slit experiment could not be arbitrary far way from each other, but they should be with distance shorter than $d_c$.

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Similar reasoning is true for time coherence. A source producing perfect monochromatic plane wave has infinite temporal coherence. It can be measure through coherence interval $\tau_c$, so in this case $\tau_c\rightarrow \infty$. The Michelson-Morley interferometer measures this coherence length: for plane waves, doesn't matter the time interval between the two waves in both arms, they will always interfere.

For real sources however, the wave is not perfect monochromatic, but has a frequency width $\Delta \nu$. The coherence interval is defined as $\tau_c = 1/\Delta nu <\infty$. The coherence length is defined as the length travelled by the wave packet in the coherence time $l_c = v\tau_c$. This implies that, in order to show interference in the Michelson-Morley interferometer, the difference in the path should be less than the coherence length.

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To answer your question precisely, it is needed to know how much incoherent this source is. Young realized the dual slit interference experiment using the sunlight, which could be considered incoherent source in many situations. Michelson and his team apparently uses white light, again, something that could be considered incoherent.

Answer 2

Your high school textbook tells you it takes 2 photons to interfere and that they must be out of phase (180) to "interfere". That's fine for highschool but modern physicists now know each photon acts independently.

"Interference" has 2 requirements 1) an apparatus (like a DSE or spectrometer), the apparatus has its own EM field and modes (modes are allowable paths for resonance/photons) and 2) a photon of a certain frequency, phase and direction, the photon being a localized wave in the EM field travelling at c. The EM fields rule all interactions ... the photons are dumb.

In the DSE may photons are reflected by the slits ... the pattern you see are the ones allowed thru.

So back to your question 1) single incoherent sources can't produce constant phase differences and 2) if they could it would not matter as interference would still occur. One can take a spectrometer, tune it to a certain wavelength (say blue) ... all blue photons pass, all other color photons reflect.