Supplementary Information. Holographic Detection of the Orbital Angular Momentum of Light with Plasmonic Photodiodes

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1 Supplementary Information Holographic Detection of the Orbital Angular Momentum of Light with Plasmonic Photodiodes Patrice Genevet 1, Jiao Lin 1,2, Mikhail A. Kats 1 and Federico Capasso 1,* 1 School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA 2 Singapore Institute of Manufacturing Technology, Singapore , Singapore *corresponding author: capasso@seas.harvard.edu Supplementary Figure S1. Evolution of the transmission and OAM selectivity as a function of beam misalignment. Full-wave simulation of the intensity transmitted through an array of 15 holes for normally incident light onto a plasmonic coupler designed to detect L=-1 vortices, as a function of misalignment. Misalignments (lateral x and y beam shifts) are measured from the edge of the dislocated fringe of the hologram with respect to the position of the peak intensity for the Gaussian beam and with respect to the phase singularity for the vortex beams. The inset displays the ratio between the transmitted intensity for both beams as a function of the displacement in both directions. This figure presents the overall performance of the detector with respect to small misalignment of the incident beams. We observe several interesting features. First, the transmitted intensity ratio between different OAM doesn t change drastically for shifts which are smaller than a wavelength. Note that the simulated beam diameter is 3 microns. Secondly, we observe that small misalignments may actually increase the selectivity (inset). However, we cannot use this effect as a means to improve the detector. The reason is that the ratio may increase for L -1 /L 0 but will decrease for L -1 /L -2 because this effect is actually related to the position of the spurious interference fringe that appears when SPPs are excited with the wrong OAM.

2 Supplementary Figure S2. Numerical calculation of the selectivity of a single charge OAM detector. Results of FDTD simulations showing the intensity collected through the array of holes for a coupler optimized for the detection of an L=-1 vortex beam. The plot shows that the intrinsic cross-talk between adjacent OAM channels is approximately 10%. In our experiment, it could therefore be reduced to ~ 10%, notably by improving the fabrication process.

3 Supplementary Figure S3. Influence of the number of holes on the transmission. Left panel: Full-wave simulation of the ratio of transmitted intensity funneled through the array as a function of the number of holes, for an incident optical vortex beam with OAM L=-1 and L=0. To simplify the discussion L refers to the incident orbital angular momentum noted in the manuscript. Note that the simulated hologram is designed by calculating the interference pattern between the L=-1 vortex beam and the converging SPP. The array of holes is aligned with respect to the axis of the device. The left panel also indicates that there is a critical number of apertures above which the ratio between the transmitted intensity of the light incident with the desired OAM and the intensity for the beam with neighboring topological charge is constant. Right panel: calculated transmitted intensity through the array as a function of the number of holes. This calculation demonstrates that the number of holes participating to the transmission is limited due to the finite propagation length of SPPs. The transmitted intensity varies little as the number of holes is increased beyond about 15. The array of holes has a pitch of 200nm. Adding the focal distance of about 2microns, this corresponds to the propagation length of SPPs at 633nm, which is determined by their losses.

4 Supplementary Figure S4. Estimated performance of a doubly-charged OAM detector. Left panel: FDTD simulations showing the intensity distribution in front of a plasmonic coupler designed for a doubly charged vortex beam (L=-2) and illuminated by different OAM beams. Right panel: Histogram of the transmitted intensity through the array of holes for different incident OAM. The intensity ratio for the two L=-2 and L=-1 beams is about 5 which corresponds to a cross talk of 20%. Note that for this numerical experiment, the size of the holes was kept constant which also contributes to increase the cross-talk for higher OAM. If the OAM of the incident beam does not match the value for which the coupler was designed, the latter tends to excite spurious surface waves which interfere in front of the coupler, giving rise to fringes. If the dimensions of the holes are large compared to the width of the fringes, each hole may capture unwanted signal which decreases the selectivity of the diode. Higher selectivity can be achieved by resizing the holes. The inset shows the design of the double fork-holographic coupler used to launch SPPs using a doubly charged vortex beam. The sidebands of the measured OAM distribution are unwanted signals and would eventually limit the performance of a communication system based on OAM [28,29]. In freespace, the observed broadening of the measured OAM distribution can be attributed to atmospheric turbulence [30,31], misalignment [32, 33], and angular truncation [34, 35]. A number of information coding techniques can be used to overcome this problem [36]. In our case, the sidebands are mostly due to the fact that the intensity along the central line of the propagating surface plasmon wave is not exactly zero when the incident topological charge is not matched to the desired one due to the interference effects. Still, the cross-talk between the different OAM channels in our simulation remains 20% for L=2. In experiments, misalignments and imperfections in fabrication may contribute additional noise.

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