How to understand the topological charge that is not an integer, how would the signal OAM crosstalk if one were to model its transport in turbulence, for example, if the beam carries a topological charge of 1.1, what is the topological charge that it would crosstalk to, and would it be like jumping to an energy level with a difference of 1 like atomic physics?
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$\begingroup$ There isn't really any such thing as a topological charge of 1.1. Fractional OAMs have indeed been discussed, but the details depend on the context. Where did you see this being discussed? $\endgroup$– Emilio PisantyAug 22 at 15:04
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$\begingroup$ Thanks for your reply, I am currently working on the Lommel beam, which is thought to have a continuously variable topological charge, which seems to be realizable as a 1.1 topological charge $\endgroup$– MatthiasAug 22 at 15:13
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As a starting point, topological charges associated with light have integer values as they represent the "holes" in the topology describing the phase space of the light. In this regard they never have fractional values as they are countable discrete entities. One can understand this by comparing integer and fractional OAM states of light. Also, as a comment, we need to consider the light field that represents the photon, since the photon is really just an excitation of this field. (For a good introduction to topology, see the paper by Mermin "The topological theory of defects in ordered media" https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.51.591)
In many cases light that exhibits integer orbital angular momentum (OAM) has a phase vortex with a phase singularity at its centre where the phase is undefined (see e.g. https://en.wikipedia.org/wiki/Orbital_angular_momentum_of_light). Around this point the phase wraps by $2 \pi q$ where $q$ is the integer that determines the orbital angular momentum, often written as $L=\hbar q$. This singularity has a value that does not exist in the parameter space of the phase of the light - in other words, the parameter space has a "hole" in it and this "hole" determines the topology of this space. The "hole" is associated with the $2 \pi q$ phase wrapping and is often referred to as the topological charge with value $q$, since its presence affects the entire phase pattern of the vortex. The sign of the charge relates to the direction in which the phase increases (i.e. clockwise or anticlockwise) about the singularity.
Now light with fractional orbital angular momentum is really a linear combination of waves with integer orbital angular momentum (usually an infinite number) and, as such, is associated with a large number of phase singularities that can be distributed throughout the light field. The sum of all the integer values of the OAM weighted by the probability of them being excited gives a non-integer, or fractional, OAM. For such fields, the topology is fractured and there are usually an infinite number of topological charges, of both positive and negative sign. There is no "fractional" topological charge and there is no relationship between the charges, their sum and the value of the (fractional) orbital angular momentum.
In fact, there is no general relationship between integer orbital angular momentum and topological charge in light fields and the fact that sometimes they are the same is merely fortuitous. This was demonstrated theoretically in 2022 by Michael Berry and Wei Liu in, "No general relation between phase vortices and orbital angular momentum" Journal Of Physics A: Mathematical And Theoretical 2022, 55, 374001. (https://iopscience.iop.org/article/10.1088/1751-8121/ac80de)
