Origin of shape resonance in second-harmonic generation from metallic nanohole arrays

Second-harmonic generation (SHG) from periodic arrays of subwavelength rectangular air hole with various aspect ratios perforated in gold thin films can get resonantly enhanced for some specific geometric shapes. Here we clarify the physical origin of this shape resonance effect. A nonlinear coupled-mode theory is set up to solve energy conversion from fundamental wave (FW) mode to second-harmonic wave (SHW) mode within the nanoscale air hole. It reveals that several physical mechanisms, including the FW mode excitation amplitude, FW-SHW modal spatial overlap, FW-SHW mode phase mismatch, and SHW mode attenuation, are all geometric shape sensitive and altogether act to induce the SHG shape resonance effect. The theory agrees well with experimental observations and provides an accurate and complete explanation for the long-emphasized but elusive shape effect. The study may stimulate deeper insights to visualize general nonlinear nanophotonic processes and pave the way to engineering high-efficiency nonlinear nanophotonic devices.