Mode-Division-Multiplexing of Multiple Bessel-Gaussian Beams Carrying Orbital-Angular-Momentum for Obstruction-Tolerant Free-Space Optical and Millimetre-Wave Communication Links

We experimentally investigate the potential of using ‘self-healing’ Bessel-Gaussian beams carrying orbital-angular-momentum to overcome limitations in obstructed free-space optical and 28-GHz millimetre-wave communication links. We multiplex and transmit two beams (l = +1 and +3) over 1.4 metres in both the optical and millimetre-wave domains. Each optical beam carried 50-Gbaud quadrature-phase-shift-keyed data, and each millimetre-wave beam carried 1-Gbaud 16-quadrature-amplitude-modulated data. In both types of links, opaque disks of different sizes are used to obstruct the beams at different transverse positions. We observe self-healing after the obstructions, and assess crosstalk and power penalty when data is transmitted. Moreover, we show that Bessel-Gaussian orbital-angular-momentum beams are more tolerant to obstructions than non-Bessel orbital-angular-momentum beams. For example, when obstructions that are 1 and 0.44 the size of the l = +1 beam, are placed at beam centre, optical and millimetre-wave Bessel-Gaussian beams show ~6 dB and ~8 dB reduction in crosstalk, respectively.


S.1 Numerical Model for Optical Propagation
In this section, we present the numerical model used to calculate the results presented in Figures 3(a), 3(b), 6(a) and 6(b).
The electric field of a Laguerre-Gaussian (LG) beam at its waist is given by 1 : in which is the beam waist, ! ! are the generalized Laguerre polynomials, l and p are the azimuthal and radial mode indices, and ! and ! are the Cartesian coordinates in the waist plane at ! . In order to transform an LG beam into a BG of same mode order, the LG beam is passed through an axicon. The electric field just after the axicon can be given by: where ! is the transmission function of the axicon as given below 2 : and = !! ! is the wave number, is the refractive index, and is the axicon opening angle.
The field given in (2) is numerically propagated to an obstruction plane located at a distance !"# > ! using Huygens-Fresnel diffraction integral. The numerical calculation is carried out by FFT-based implementation of the diffraction integral as described below 3 : where !"# is the propagation distance from the axicon plane to the obstruction plane, FFT[ . ] is the Fast In order to model obstructed beam path, an opaque circular obstruction is introduced in the model such that the field just after the obstruction is given by: The transmission function of the opaque obstruction can be given by 3 : After passing through the obstruction, the field is numerically propagated to the receiver plane using FFT method described in equation (4).
We also calculate the unobstructed field at the receiver plane as given below: Finally, to calculate the received power in the desired mode and power coupled in the neighbouring modes, overlap integral is calculated using the expression given below:

S.2 Design of Metamaterials-based Axicon for 28 GHz Frequencies
This section presents the details of the millimetre-wave generation of BG beams using metamaterials-based axicon. Supplementary Figs. 1(a) and 1(b) show the geometric parameters and schematic structure of the polarized beam with Jones vector !" = 1 0 ! as the input beam, the output beam can be given by: Equation 10 shows that the output beam consists of three polarization states, namely right circular polarization 1 − ! , left circular polarization 1 ! and a linear polarization state 1 0 ! . We also notice that the output components of right and left circular polarization states have phases e j 2α and e − j 2α related to angel α, respectively, suggesting that rectangular slit arrays with spatially varying angle can be used as a desired phase mask 6 . Supplementary Fig. 1(b) shows a schematic structure of the metamaterials-based axicon to generate the BG beams. The axicon can be regarded as a phase mask whose phase shift decreases linearly with the radial distance . The transmission function of the axicon can be written as 7 : where λ is the wavelength of the incident beam. The parameter β determines the dependence of phase shift on the radial distance . In order to use the rectangular slit arrays as an axicon, we set the angle α as a function of the radial distance , namely, α = βr/2λ. Consequently, if !" = 1 0 ! is used as the input beam, the output beam becomes: Equation 12 shows that the designed axicon introduces a phase shift of !!"#/!! to the left circularly polarized components such that left circularly polarized output beam forms a BG beam. In order to separate the generated