Circular optical resonators, such as micro-rings or microdisks, support whispering gallery modes (WGMs) carrying high OAM [
39]. WGMs are actually angular momentum eigenstates and have discrete azimuthal propagation constants
vWGM =
bpR =
r resulting from the self-consistent phase requirement for resonance under the periodic boundary condition. The integer
p denotes the azimuthal mode number and physically is the number of optical periods around the resonator;
R is the effective radius of the WGM, and
p is light’s propagation constants at
R. The azimuthal propagation constant, frequently used in rotationally symmetric resonant devices [
40], describes the optical phase shift per unit azimuthal angle. It is also a measure of angular momentum as the amount of angular momentum carried by every WGM photon is
vWGM =
r. Additionally, the SAM of WGM is zero, because WGM has a purely linear polarized state in the cylindrical coordinate. Therefore the angular momentum carried by WGM is purely OAM. To extract the confined WGM into free space emission, we embed angular grating (AG) structures into the WGM resonator (Fig. 1(a)) with a periodic modulation of refractive index in the azimuthal direction.