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People talk about orbital angular momentum (OAM) of photons. Is there some physical example that cannot be explained without assuming that photons have non-zero OAM? Does different photons have different values of OAM? If yes, then what determines the value of the value orbital angular momentum carried by individual photons?

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If you read the wikipedia article on orbital angular momentum of light you will see that in the first place it is a classical electromagnetic concept, where the light has a vorticity, i.e. a helical motion around the axis of the vortex.

When one goes to the quantum detail of photons one can define an OAM against this classical axis for each photon in this specific classical electromagnetic beam. Thus OAM is not an intrinsic characteristic of photons, but only on photons in special beam distributions, as in the figure:

helical em beam

Different columns show the beam helical structures, phase fronts, and corresponding intensity distributions.

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