Effects of non- δ coupling between one-dimensional waveguides and side optical cavities

We study the effects of the non-δ coupling between a one-dimensional waveguide (1DW) and a side optical cavity (SOC), which is a development of previous well-studied δ-coupling situations. We first find the real-space Hamiltonian of the non-δ coupling system of the 1DW and SOC, and then derive out an equation determining the energy of the hybridization state between the 1DW and SOC modes and an analytic formula for the single-photon transmission. Through them, we recognize that the single-photon transmission-dip position can be changed by adjusting the SOC size and the coupling strength between the 1DW and the SOC. The transmission spectrum strongly depends on the coupling types between the 1DW and the SOC and holds an asymmetric line shape. At last, we simulate the asymmetry of the single-photon transmission by a gold-based waveguide coupled with a rectangular SOC.