We developed the theoretical framework based on the coupled-mode theory which describes spectral and scattering properties of the photonic analog of an extended Fano-Anderson model—a waveguide array with two additional side-coupled waveguides. The structure supports a rich spectrum of eigenmodes, including bound state in the continuum (BIC) and other bound and leaky modes, which can be classified according to the relation between the self-coupling coefficients and eigenvalues. We focus on the structures with broken vertical symmetry with their band structures revealing interesting phenomena, such as exceptional points and level repulsion, and offer a lossless platform for $\mathcal{PT}$-symmetry phase transition. We interpreted the resonant features in the scattering spectra through a generalized Weierstrass factorization. The resonance related with quasi-BIC arises from the interference between two leaky modes: one of them representing a continuum spectrum and the other (quasi-BIC) discrete state. The reflectance near the resonance can be rewritten into the form of the Fano formula where the shape parameter $f$ can be expressed in terms of the poles associated with the two modes. Our approach provides a flexible framework which allows to interpret and to engineer the resonant properties of more complex systems.

• Received 31 December 2021
• Accepted 19 May 2022
• Corrected 3 August 2022

DOI:https://doi.org/10.1103/PhysRevA.105.063505