Identification of local field-enhancement in third-harmonic generation via nonlinear <i>S</i>-matrix

The research presented in this study aims to establish nonlinear <i>S</i>-matrix methods to address the enhancement of third-harmonic generation induced by field localization effects in aperiodic metastructures. The strong resonant modes generated by optical Tamm states within the band gap lead to an increase in localized electromagnetic energy and group delay, enhancing the frequency conversion efficiency at the resonance of the fundamental wave. By comparing two scenarios, introducing a nonlinear media or vacuum layer with a thickness significantly exceeding the coherent wavelength in the central defect layer, the algorithm clearly demonstrates the achievement of third-harmonic quasi-phase matching within the defect layer and adjacent periods. The conversion efficiency is enhanced by three orders of magnitude. Furthermore, there exists a correlation between the resonant frequency and the thickness of the nonlinear layer, enabling the achievement of a conversion frequency range with a certain bandwidth.