Thermo-optic nonlinearity of single metal nanoparticles under intense continuous wave illumination

Over the last few decades, extensive previous studies of the nonlinear response of metal nanoparticles (NPs) report a wide variation of nonlinear coefficients, thus, revealing a highly confused picture of the underlying physics. This naturally prevents rational design of these systems for practical devices. Here we provide a systematic study of the nonlinear response of metal spheres under continuous wave (CW) illumination within a purely thermal model, i.e., whereby the illumination only acts to modify the optical and thermal parameters via their dependence on the temperature. We characterize the strong dependence of the temperature rise and overall thermo-optic nonlinear response on the particle size and permittivity, on the optical and thermal host properties, as well as on the thermo-derivatives of these properties. This dependence on the nonintrinsic parameters explains why it is inappropriate to extract an intrinsic nonlinear coefficient from a specific system; equivalently, it explains the large differences in reported values for such systems, as well as for more complicated metal-dielectric systems and even pulsed illumination schemes. Despite the revealed complex multiparameter dependence, we managed to uncover a rather simple behavior of the nonlinear response. In particular, we show that the nonlinearity coefficients exhibit a dependence on the illumination intensity which mimics the dependence of the temperature itself on the illumination intensity, namely, it grows for small NP sizes, reaches a maximum and then decreases monotonically for larger NPs. The improved modeling allows us to demonstrate an overall nonlinear response which is about a 1000 times higher than in other strongly nonlinear systems (e.g., ϵ-near-zero systems); it also provides an excellent match to experimental measurements of the scattering from a single metal NP, thus, confirming the dominance of the thermal nonlinear mechanism. Our work lays the foundations for an overall evaluation of previous studies of the nonlinear response of metal-dielectric system under general conditions.