Optical Amplification with Large Goos-H$\ddot{a}$nchen Shift Driven by Non-Hermitian Bilayer Meta-Grating

Optical Goos-H$\ddot{a}$nchen shifts can be enhanced by resonant mode with high quality factor, such as quasi-bound states in the continuum in meta-grating. Coexistence of gain and loss in bilayer meta-grating with parity-time symmetry could transfer bound states in the continuum into lasing threshold modes with real resonant frequencies and non-zero far-field radiation. When the incident frequency approaches the resonant frequency of a lasing threshold mode, the reflected and transmitted beams are strongly amplified and undergo large Goos-H$\ddot{a}$nchen shifts. The amplitude of the Goos-H$\ddot{a}$nchen shifts, including the magnitude and sign, are proportional to the reciprocal of the imaginary part of the resonant frequencies. As the incident frequency scan across the resonant frequency of a lasing threshold mode, the imaginary part flip sign, so that the Goos-H$\ddot{a}$nchen shifts diverge as well as flip sign. Simulations of optical responses under incident of Gaussian beams with finite beam width exhibit the sign flipping of the Goos-H$\ddot{a}$nchen shift with large magnitude by fine tuning the incident frequency across the resonant frequency of a lasing threshold mode.