Goos-H$\ddot{a}$nchen shifts of bilayer meta-grating with unidirectional guide resonance

Bilayer meta-gratings with asymmetric structural parameters could host unidirectional guide resonances. The distribution of unidirectional guide resonances in the space of structural parameters and synthetic parameters is identified. As the incident optical beam being resonant with the unidirectional guide resonance, the Goos-H$\ddot{a}$nchen shifts of the scattered beams exhibit two anomalous behaviors: the resonant peak of the Goos-H$\ddot{a}$nchen shift is accompanied by constant transmittance and reflectance; the magnitude of the Goos-H$\ddot{a}$nchen shift is not always proportional to the quality factor of the unidirectional guide resonance. The temporal coupled mode theory analysis reveals that the first anomalous behavior is due to interference between direct scattering and radiation from the unidirectional guide resonance; the Goos-H$\ddot{a}$nchen shifts are proportional to the group velocity as well as the quality factor of the unidirectional guide resonance. Numerical simulations of incidence of Gaussian beam with finite beam width provide intuitive visualization of the Goos-H$\ddot{a}$nchen shift.