Fig. 1. Linking orbital angular momentum (OAM) emission with selective mode splitting (SMS) in a photonic crystal microring (PhCR).

a–c Schematic of two PhCRs with N = 10 (a) and N = 12 (c), where N is the number of modulation periods in a round trip. Their transmission spectra are illustrated in (b). Here, we focus on a whispering gallery mode (WGM) with an azimuthal mode number m = 6. Similar to a conventional (unmodulated) microring, this WGM has an intrinsic loss rate of κ₀ (from incoherent scattering and absorption) and is evanescently coupled by a waveguide with a coupling rate κ_c. The laser injected from the right side of the waveguide excites the clockwise WGM. In a, the PhCR with N = m + 4 = 10 leads to OAM light emission, at an ejection rate κ_e, and carrying angular momentum of l = −4. In b, the PhCR with N = 12 leads to a coupling of clockwise and counter-clockwise WGMs with m = ±6, at a coupling rate β. d In the OAM device, the grating with N = 10 ejects the clockwise mode with m = 6 into a free-space OAM mode carrying a momentum of l = −4 at a rate of κ_e. The OAM emission leads to a broadening of the cavity linewidth (${\kappa }_{\mathrm{t}}={\kappa }_{\mathrm{t}}^0+{\kappa }_{\mathrm{e}}$). When the WGM is a standing wave with m = ±6, or there is a reflection in the microring or chip facet (e.g., reflecting m = 6 to m = −6), the observed OAM mode has l = ±4, which is manifested in its intensity profile exhibiting 4 pairs of anti-nodes. e Schematic band diagram for OAM and SMS. The purple double-arrow indicates the coupling between the clockwise and counter-clockwise traveling wave WGMs. The red single-arrow indicates the one-way ejection of light from the clockwise mode to the free space OAM mode. The OAM emission is symmetric to clockwise and counter-clockwise modes because the grating is static. f In the SMS device, the grating with N = 12 couples the clockwise and counter-clockwise modes with m = ±6 to each other without introducing excess loss (that is, ${\kappa }_{\mathrm{t}}^0={\kappa }_{\mathrm{0}}+{\kappa }_{\mathrm{c}}$), but introduces a mode splitting of 2β between the two renormalized standing-wave modes.