Coherent Goos-H$\ddot{a}$nchen shifts of meta-grating with radiation asymmetry

Abstract: The coherent Goos-H$\ddot{a}$nchen shifts of meta-grating are proposed, which is the Goos-H$\ddot{a}$nchen shifts of the two outgoing beams under the simultaneous incidence of two coherent optical beams from opposite sides of the grating with the same lateral wave number. As both of the frequency and lateral wave number are resonant with a topological state of the meta-grating, such as unidirectionally guided resonance or circular polarized states, the energy flux and Goos-H$\ddot{a}$nchen shifts of the two outgoing beams are coherently controlled by the relative phase difference between the two incident beams. By applying stationary-phase method, it is found that the enhancement of coherent Goos-H$\ddot{a}$nchen shifts by the unidirectionally guided resonance and circular polarized states is accompanied by constant and peak transmittance, respectively. Analysis with temporal coupled mode theory shows that the different features are due to difference mechanism of interference between direction scattering and resonant radiation. The coherent Goos-H$\ddot{a}$nchen shifts with incident Gaussian beams are sensitive to the relative phase between the two beams, which can be applied in refractive index sensor.