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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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There is "external orbital angular momentum" and "internal orbital angular momentum." The latter is indeed intrinsic. – Eric Walker Oct 29 '15 at 4:41
@EricWalker internal orbital angular momentum for elementary particles is called spin. The photon is an elementary particle. – anna v Oct 29 '15 at 4:59
@EricWalker Light is composed of photons, photons are not light. Buildings are composed out of bricks. Bricks are not buildings. Anyway, the link you quote is on the top of my answer from which I took the illustration. I am talking of internal here, and it is intrinsic to the beam of light, not to the photons. – anna v Oct 29 '15 at 5:07
You may be correct. But if you are, this leaves open another question that has been on my mind. How can a gamma photon emitted from an $I=2$ or $I=3$ nuclear transition carry away only one $1ℏ$ unit of angular momentum (i.e., its intrinsic spin)? – Eric Walker Oct 29 '15 at 5:11

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