The existence of interface states at the boundary of two semi-infinite photonic crystals (PhCs) with different lattice constants are investigated systematically. Compared to the interface states in the two PhCs with the same period, a band folding effect is observed for the interface states inside the common band gap of the two PhCs with different lattice constants. We demonstrate that these interface states can be predicted by the surface impedance of the two PhCs. The dispersion of interface states can be determined by the condition of impedance matching combined with the band folding effect. Moreover, some part of the folded interface states penetrates the region of projected bulk bands, and they usually leak to the bulk and form resonant states. However, the interface state at the Γ point can be perfectly localized and becomes a bound state in the continuum (BIC) due to the symmetry mismatch. These findings may provide a general scheme for designing BICs in the PhC structures based on the interface states.
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