Nonreciprocal Local-Resonance Induced Complex Band Hybridization

We study the complex band hybridization induced by nonreciprocal local resonances in photonic crystals. Composed of trimer unit cells, a two-dimensional (2D) magnetophotonic crystal with an analytically obtainable solution is considered. We find that nonreciprocal spectral gap may appear without nonreciprocal transmission and that the imaginary parts of the complex wavevectors $\text{Im}(\mathbf{k})$ may blow up at resonance to give extreme nonreciprocal transmission. We further show that, for a subwavelegnth lattice, the isolation ratio for the nonreciprocal transmission is determined solely by $\text{Im}(\mathbf{k})$ instead of the extensively studied real part $\text{Re}(\mathbf{k})$. Our finding contradicts the common belief that "spectral nonreciprocity [$\omega(\mathbf{k})
eq\omega(-\mathbf{k})$] always implies nonreciprocal transmission".