Analytical Theory of Second Harmonic Generation from a Nanoparticle with a Non-Centrosymmetric Geometry

We analytically investigate the effect of a non-centrosymmetric geometry in the optical second harmonic (SH) generation from a particle made of a centrosymmetric material, in the interior of which quadratic optical processes are suppressed. We consider a cylindrical particle with a cross-section that is slightly deformed away from a circle and with a radius much smaller than the wavelength. We calculate the induced linear and nonlinear fields perturbatively in terms of the deformation parameter and obtain the nonlinear dipolar and quadrupolar hyperpolarizabilities, whose spectra we evaluate for metallic and dielectric materials. We show that for very small deformations the dipolar contribution to the response competes with the quadrupolar term, and may even be dominant. We explore the spectra of the hyperpolarizability and identify the contributions to its structure for metallic and dielectric particles. We also discuss the nature of SH radiation at various frequencies and find that it may be dominated by the dipolar or the quadrupolar term, or that both may compete yielding non-symmetric radiation patterns. Our calculation may be employed to assess, calibrate and test numerical SH calculations.