Enhanced spin orbit interaction of light in highly confining optical fibers for mode division multiplexing

Light carries both orbital angular momentum (OAM) and spin angular momentum (SAM), related to wavefront rotation and polarization, respectively. These are usually approximately independent quantities, but they become coupled by light’s spin-orbit interaction (SOI) in certain exotic geometries and at the nanoscale. Here we reveal a manifestation of strong SOI in fibers engineered at the micro-scale and supporting the only known example of propagating light modes with non-integer mean OAM. This enables propagation of a record number (24) of states in a single optical fiber with low cross-talk (purity > 93%), even as tens-of-meters long fibers are bent, twisted or otherwise handled, as fibers are practically deployed. In addition to enabling the investigation of novel SOI effects, these light states represent the first ensemble with which mode count can be potentially arbitrarily scaled to satisfy the exponentially growing demands of high-performance data centers and supercomputers, or telecommunications network nodes.


Supplementary Note 1: Modal Notation and Conventional Vector Modes
Although the traditional waveguide definition for HE and EH modes refers to whether the mode is more "TE-like" or more "TM-like" with respect to the longitudinal field components 1 , in the fiber community it has also traditionally implied a choice of sines and cosines as azimuthal basis functions instead of complex exponential functions 2 . The spin orbit interaction (SOI) is theoretically nonzero in all fibers, but is trivially small in many standard fibers used for telecommunications. In fibers where the SOI is non-negligible, such as those in 3 , it causes a lifting of the degeneracy between +1, and −1, modes, but leaves the modal fields unperturbed to a good approximation. Thus, the radial and azimuthal transverse electric field components have nearly identical magnitudes, as shown in Supplementary Figure 2. In this regime, conventionally called the "weakly guiding" regime, one can use the basis set of traditional HE (EH) vector modes, and the basis set of spin-orbit aligned (anti-aligned) OAM modes interchangeably, some apparent advantages of the OAM set, such as ease of free space excitation and a spatially-independent polarization state, notwithstanding.

Supplementary Figure 2 | Weak-SOI Electric Field Profiles
Transverse electric field profiles for a fiber similar to that shown in 3 . Note that the electric field radial and azimuthal profiles are nearly identical in magnitude, even though this mode is separated from its nearest neighbors in neff by ~10 −4 . However, in the regime of strong SOI described in this manuscript, the radial and azimuthal electric field components begin to take on different relative magnitudes, causing the pseudoradial or pseudo-azimuthal polarization states of the mode profiles shown in Fig. 2  components indicates that the modes used in the weak guidance approximation, whether the basis is HE/EH modes or spin orbit aligned/anti-aligned OAM modes, are no longer adequate to describe these strong-SOI modes.

Supplementary Figure 3 | Strong SOI Electric Field Profiles
Transverse field distributions for (a) the J=6, ↑↑ mode from the thin ring fiber described in the manuscript, and (b) the J=4, ↑↓ mode. The obvious difference in radial and azimuthal field components differentiates these modes from those in the "weak guidance" approximation.

Supplementary Note 2: Derivation of Equations (2), (3a), and (3b)
To calculate the effects of the Spin Orbit Interaction according to Perturbation Theory 4 , we begin with the scalar vector wave equations (Equations 32-19a and 32-19b from 2 ): (∇ 2 + 2 2 )Ψ =̃2Ψ (1) Here Ψ is the scalar solution with propagation constant, ̃, Ψ is the exact vector solution with propagation constant, , and although no bolding is used, it is understood that all modes and all Laplacians are vector in nature, although Ψ is easily factorable into a vector and scalar part. We introduce a scale parameter, , assumed small, and expand the vector mode solution and eigenvalue in order of .

Substituting Supplementary Equations
We now make the following assumptions: (a) Radial mode order m=1 assumed for all modes.
(b) We use the scalar vortex beams as a basis for Ψ : l is the mode's vortex order and can be positive or negative, s denotes the sign of the mode's spin and can be either +1 or -1, and j is a dummy index for notational convenience, running from 1-4 for |L|>0. In this notation, s depends on j. Since all OAM modes of a given |l| are degenerate in the scalar picture, the scalar solution can be an arbitrary sum of these modes, with weights cn,j in Supplementary Equation (8). The second index, n, represents the fact that there must still be 4 orthonormal modes. Supplementary equation (8) is introduced only for notational convenience. Note that for a passive waveguide, | | ( ) is a real function.