Radiation loss of grating-assisted directional couplers using the Floquet-Bloch theory

The Floquet-Bloch theory (FBT) is used to determine the radiation loss of a grating-assisted directional coupler. Improper waves obtained by FBT represent leakage and produce radiation loss along the transverse direction. By summing the fields of all leaky harmonics, the power flow radiated into the superstrate and substrate can be calculated from Poynting´s theorem. However, the locations at which these summations occur impact the accuracy. An analytic formula for radiation loss is also derived. Results from both the FBT and the analytic formula are compared with results obtained using the coupled-mode theory.