Circular array of anisotropic fibers: A discrete analog of a $q$ plate

In this paper, we have studied the propagation of light in a $p$ array, that is, a circular array of strongly anisotropic fibers, orientation of whose anisotropy axes linearly depends on the angular position of the fiber in the array and makes an integer number of full rotations $p$ while tracing along its contour. We have obtained the spectrum and the structure of supermodes for such a system and have shown that they consist of two discrete optical vortices nestled in the opposite circular polarizations. We have found the expressions for topological charges of such vortices. We have also studied the angular momentum carried by these supermodes. We have obtained the expression for the evolution of an arbitrary excitation created at the array's input upon its discrete diffraction in the array. As an example, we have examined the propagation of the set of circularly polarized fundamental modes excited at the input end with equal weights and phases. We have demonstrated that, in certain cross sections, the $p$ array generates a discrete circularly polarized optical vortex, whose topological charge is determined by the array's index $p$. In this way, we have shown that the $p$ arrays enable polarization control over phase singularities being a discrete analog of the $q$ plates.

Figure 1

Geometry of a double-ring fiber array and the scheme of fibers' numeration. Red arrows indicate the anisotropy director's orientation. In the given example, the director vector makes one full rotation, which corresponds to the array's index $p$ $=$ 1.

Figure 2

(a) Specific orbital angular momentum and (b) specific spin angular momentum of certain supermodes vs the array's index $p$. Array's parameters: $N=21$, $r_0/\rho =14$. The number $m$ of the supermode is indicated near the corresponding set of points.