Even thought I have been talking so far about "twisted light", there are other common ways to refer to this type of fields. Since there are many papers and books that use only these other terms, learning them will allow you to find, through databases, relevant research in the topic that would otherwise go unnoticed.
I am aware of two other ways to call twisted light or closely related optical fields. The first one is simply by declaring one of its main properties: people write "light carrying orbital angular momentum". Authors that use in some of their papers "twisted light" use also the term "light carrying orbital angular momentum" in other papers. The two terms are quite connected. Articles inspired by Allen et al [1] and Padgett et al [2] use mainly "twisted light" and/or "light carrying orbital angular momentum".
The second term seems less related to the other two, since it is found mainly in papers written by a different group of researchers. The name given is "optical vortex" [3]. It makes reference to the other important property of the fields, that of having a zero of the intensity at the beam axis due to the phase singularity. An electromagnetic field is an optical vortex if
\begin{eqnarray}
\large
\bf{A}(\bf{r},t)\propto e^{i\, \ell\, \varphi} + c.c.
\end{eqnarray}
where the integer ℓ is the topological charge, and A(r,t) is the electric/magnetic field. Clearly, this is the way we mathematically represented the field of twisted light.
It is unfortunate that the "twisted light" and "optical vortex" communities are not well communicated. The use of different names to refer to the same physical process or entity undoubtedly contributes to the situation. Some people in one community fail to trace relevant previous research done in the other community (I know it firsthand). Naturally this leads to the duplication of reported results, and diminishes the quality of the papers, by not providing the reader with interesting and useful connections to previous literature.
So I would like to end this post with a little piece of advice. When looking into databases search for "twisted light", "light carrying orbital angular momentum" AND "optical vortex".
References
[1] Allen L., Beijersbergen M. W., Spreeuw R. J. C. and Woerdman J. P., Phys. Rev. A, 45 (1992) 8185.
[2] "Light´s orbital angular momentum", by Padgett et al, Phys. Today 57(5), 35 (2004).
[3] "Optical vortices" by P. Coullet et al, Optics Communications 73 (5), 403–408 (1989).
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