Fig. 1. (a) Conceptual illustrations of negative-angle and positive-angle emission via UGRs in grating couplers. Here, S1 and S2 denote the Poynting vectors for the UGR and the emitted light, respectively. (b) Examples of grating coupler applications utilizing positive-angle emission with an optical fiber and a fiber array block.

Fig. 2. (a) Schematic of a conventional 1D grating exhibiting in-plane C2 symmetry. (b) FEM-simulated dispersion curves and radiative Q factors of Bloch modes near the second stop band. Insets depict the spatial Ey field distributions for the two band edge modes. (c) Variation of band edge frequencies as a function of the parameter α=w/Λ. (d) Conceptional illustration of the trajectories traced by LCP and RCP waves, each carrying half-integer topological charges. (e) Transition from quasi-UGR to UGR with increasing in-plane geometric asymmetry.

Fig. 3. Conceptual illustrations of the polarization vector fields around a symmetry-protected BIC, with (a) q=+1 and (b) q=−1.

Fig. 4. (a) A schematic of the type I L-shaped grating. FEM-simulated (b) dispersion curves, (c) radiative Q factors, and (d) radiation ratio η curves of Bloch modes near the second stop band for a symmetric grating with α=0.35>αc. The insets in (b), shown in blue and red, depict the spatial Ey field distributions for the two band edge modes. (e) The evolution of the radiation ratio η curves as wa=ha varies. Dispersion curves are also plotted in the insets. Quasi-UGRs are clearly identifiable in type I L-shaped gratings and gradually evolve into UGR as wa=ha increases. Spatial Ey field distribution for the UGR at kx/K=ku=−0.0831 is plotted in the inset. In the FEM simulations, the structural parameters were set to t=0.6Λ, h=0.48Λ, and w=0.35Λ, with wa=ha.

Fig. 5. (a) Phase-matching condition for determining the emission angle of UGRs. (b) Spatial Ey field distribution illustrating negative-angle emission via the UGR at kx/K=ku=−0.0831, where the upward radiation exhibits a negative emission angle of θ=−9.38°. (c) Spatial Ey field distribution for the non-UGR mode at kx/K=−ku=0.0831, showing light emitted simultaneously in both the upward and downward directions with an emission angle of θ=−9.38°.

Fig. 6. (a) A schematic of the type II L-shaped grating. FEM-simulated (b) dispersion curves, (c) radiative Q factors, and (d) radiation ratio η curves of Bloch modes near the second stop band for a symmetric grating with α=0.75>αc. The insets in (b), shown in blue and red, depict the spatial Ey field distributions for the two band edge modes. (e) The evolution of the radiation ratio η curves as wa varies. Dispersion curves are also plotted in the insets. Quasi-UGRs are clearly identifiable in type II L-shaped gratings and gradually evolve into UGR as wa increases. Spatial Ey distribution for the UGR at kx/K=ku=0.0753 is plotted in the inset. In the FEM simulations, the structural parameters were set to t=0.6Λ, h=0.27Λ, and w=0.75Λ, with wa=2ha.

Fig. 7. (a) Phase-matching condition for determining the emission angle of UGRs. (b) Spatial Ey field distribution illustrating positive-angle emission through the UGR at kx/K=0.0753. The upward radiation exhibits a tilted emission angle of θ=7.47°. (c) Spatial Ey field distribution for the non-UGR mode at kx/K=−ku=−0.0753, showing light emitted simultaneously in both the upward and downward directions with an emission angle of θ=7.47°.

Fig. 8. Formation and radiation properties of UGRs in type II gratings with varying width-to-height ratios. (a) FEM-simulated η curves for six selected width-to-height ratios ranging from 1.6 to 3.6 in discrete increments of 0.4. (b) Spatial Ey field distributions for the six UGRs presented in (a), showing that eigenfrequencies and emission angles increase with the width-to-height ratio. (c)–(e) FDTD-simulated spatial Ey field distributions demonstrating positive-angle emission for width-to-height ratios of 1.6, 2.4, and 3.6.