Metal nanospheres under intense continuous-wave illumination: A unique case of nonperturbative nonlinear nanophotonics

We show that the standard perturbative (i.e., cubic) description of the thermal nonlinear response of a single metal nanosphere to intense continuous-wave (CW) illumination is sufficient only for a temperature rise of up to 100 degrees above room temperature. Beyond this regime, the slowing down of the temperature rise requires a nonperturbative description of the nonlinear response, even though the permittivity is linearly dependent on the temperature and despite the deep subwavelength effective propagation distances involved. Using experimental data, we show that, generically, the increase of the imaginary part of the metal permittivity dominates the increase of the host permittivity as well as the resonance shift due to the joint changes to the real parts of the metal and host. Thus, the main nonlinear effect is a decrease of the quality factor of the resonance. We further analyze the relative importance of the various contributions to the temperature rise and thermal nonlinearity, compare the nonlinearity of Au and Ag, demonstrate the potential effect of the nanoparticle morphology, and show that although the thermo-optical nonlinearity of the host typically plays a minor role, its thermal conductivity and its temperature dependence is important. Finally, we discuss the differences between CW and ultrafast thermal nonlinearities.

Figure 1

(a) The temperature rise (4) inside a Au nanosphere ($a=10$ nm) in an oil host (${\epsilon }_h=2.65$) illuminated at $\lambda =550$ nm as a function of the incident intensity (black solid line); see values of additional parameters in the main text. The solutions accounting for up to third, second, and first-order terms are shown by magenta dashed, red dash-dotted, and blue dotted lines, respectively; however, the black and magenta lines are essentially indistinguishable. (b) Real and (c) imaginary parts of the Au permittivity (3) as a function of the incoming intensity, based on the various solutions for $\mathrm{\Delta }T$ shown in (a). (d) Same as in (b) and (c) for the normalized electric field (2) within the Au nanosphere.

Figure 2

Same as Fig. 1 for a ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ host with $\lambda =800$ nm and ${\epsilon }_{h,0}^{\prime }=13$.

Figure 3

Same as Fig. 1 for a Ag nanosphere (permittivity data taken from Ref. [24]) and ${\epsilon }_{h,0}^{\prime }=6.15$.

Figure 4

Same as Fig. 3 but with the permittivity data taken from Ref. [50], hence, with ${\epsilon }_{h,0}^{\prime }\sim 4$.