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 magnetophotonic crystal with an analytically obtainable solution is considered. We find that a nonreciprocal spectral gap may appear without nonreciprocal transmission and that the imaginary parts of the complex wave vectors $\text{Im}(\mathbf{k})$ may blow up at resonance to give extreme nonreciprocal transmission. We further show that, for a subwavelength 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 $[\ensuremath{\omega}(\mathbf{k})\ensuremath{\ne}\ensuremath{\omega}(\ensuremath{-}\mathbf{k})]$ always implies nonreciprocal transmission.''