Evaluation of the nonlinear response of plasmonic metasurfaces: Miller’s rule, nonlinear effective susceptibility method, and full-wave computation

In this article, the second-harmonic generation (SHG) from gold split-ring resonators is investigated using different theoretical methods, namely, Miller's rule, the nonlinear effective susceptibility method, and full-wave computation based on a surface integral equation method. The results confirm that Miller's rule is, in general, not well suited for the description of SHG from plasmonic metasurfaces. On the other hand, the comparison of the nonlinear effective susceptibility method with full-wave computations shows that this method permits us to evaluate second-harmonic (SH) emission patterns from noncentrosymmetric nanoparticles with good accuracy. However, the nonlinear effective susceptibility method fails to reproduce the multipolar nature of the SH emission from centrosymmetric nanoparticles. This shortcoming is attributed to the intrinsic nature of the nonlinear effective susceptibility method, which neglects the exact positions of the nonlinear sources. The numerical implementations of these two methods are also discussed in detail, revealing that the main limitation of the nonlinear effective susceptibility method, aside from the inaccuracy observed in specific cases, is its higher numerical requirements when several emitting directions need to be considered. This limitation stands for most of the numerical methods used for solving Maxwell's equations at the nanoscale. This work provides clear insight into the limitations and advantages of the different methods available for evaluation of SHG from plasmonic metasurfaces.