The modes are in fashion
Usually, the response of plasmonic nanostructures is studied from a scattering perspective. However, the underlying modes provide much insights into that response. To address this issue, we have developed a full wave analysis of the resonant modes of 3D plasmonic nanostructures. This method, combined with the evaluation of second-harmonic generation, is then used to obtain a better understanding of their nonlinear response. The second-harmonic generation associated with the fundamental dipolar modes of three distinct nanostructures (gold nanosphere, nanorod, and coupled nanoparticles) is computed in the same formalism and compared with the other computed modes, revealing the physical nature of the second-harmonic modes. The proposed approach provides a direct relationship between the fundamental and second-harmonic modes in complex plasmonic systems and paves the way for an optimal design of double resonant nanostructures with efficient nonlinear conversion. In particular, we show that the efficiency of second-harmonic generation can be dramatically increased when the modes with the appropriate symmetry are matched with the second-harmonic frequency.