Fig. 3. Coherent emission of OAM light from high-Q whispering gallery modes.

a (Left) Illustration of the angle θ, which appears in Eq. (1) in describing the OAM ejection rate κ_e, and (right) scanning electron microscope images of devices with θ = 0 (OAM) and θ = π/2 (SMS), respectively. The top image shows ≈5 periods of a PhCR for OAM (N = m = 165), while the bottom image shows ≈10 periods of a PhCR for SMS (N = 2m = 2 × 165). Both devices have A = 32 nm. b In the SMS case, ${\kappa }_{\mathrm{t}}^0$ of the two standing wave modes are not affected by increasing A, as shown in the top panel. The error bars are 95% confidence intervals of the nonlinear fits to the cavity transmission data (see Supplementary Fig. S3 for the details). The mode splitting (β) is nearly linearly dependent on A for the targeted WGM mode, whose experimental data and fit are shown in the bottom panel. The error bars are ≈0.3 GHz and are within the data symbols. c Plot of the OAM cavity linewidth (κ_t) and estimated OAM ejection efficiency (κ_e/κ_t) in the top and bottom graphs, respectively, as a function of l = m−N on the bottom x-axis and θ/(π/2) = ∣l/m∣ on the top x-axis. We investigate three sets of devices having different modulation amplitudes of A = {4, 8, 16} nm, with N varying from 2m to m (i.e., transiting from the SMS regime to the OAM regime). We focus on a specific WGM at ω₀ with mode number m₀ in each set of devices. The error bars represent the 95% confidence intervals from nonlinear least-squares fits to the transmission data. The translucent shaded curves are from theoretical estimates predicted by Eqs. (1)–(3) with q₀ = 2, with all other parameters taken from measurements of OAM and SMS devices. The width of each shaded curve originates from the combined range of six fitted ${\kappa }_{\mathrm{t}}^0$ values from three SMS devices at l = −165. The top inset shows a region of a discrepancy between the experiments and the model, near l = 0. The coupling waveguide in use has a nominal width of 750 nm and a microring-waveguide gap of 500 nm. d Infrared images near the surface of the OAM microrings for different ∣l∣ values, with each displaying 2∣l∣ anti-nodes. The angular intensity distribution is re-scaled and plotted in white to guide viewing. e Predicted patterns from three-dimensional finite-difference time-domain simulations using dipole excitation. While the OAM states are the same as in the experiments, a smaller device size (radius of 12 μm) and symmetric air cladding are used to keep the simulation size tractable.