The capability to fabricate materials with features at the subwavelength scale has opened the door for controlling their macroscopic optical properties. It was shown that non-centrosymmetric shaped nanostructures exhibit a structural nonlinear quadratic susceptibility, e.g. enabling second harmonic generation (SHG). SHG in these nanostructures has been studied extensively, while the more general case of sum frequency generation (SFG) did not receive much attention. The use of this general case gives another degree of freedom to study the nonlinear interaction. In addition, it allows obtaining a much more versatile control of the generated light frequencies, as is already manifested in many applications using conventional nonlinear materials.

In this work, the aim is to study SFG in nanoscale gold split-ring-resonators (SRRs) in order to obtain more information on the nonlinear interaction and to find new ways to enhance and control it. To achieve it we extended the hydrodynamic model that was used before for understanding the mechanism of SHG in SRRs [1] to the generalized case of SFG. The model treats the metal as a free electron gas, whereas the electro-magnetic field is the driving force. The quadratic terms in the hydrodynamic equation at the frequency domain give rise to generation of polarization in any of the possible frequencies, which are the summations of the driving fields frequencies. The nonlinear polarization on the surface of the metal can be treated as the source term for the electromagnetic field in the same frequency. Using finite element simulation of the electromagnetic problem, the nonlinear currents and their respective generated fields can be studied.

Figures 1(a) present the linear transmission for the excitation polarization, which is parallel to the base of the SRR. Similarly, figure 1(b) present the transmission for the light polarized parallel to the arms of the SRR. The dips in the transmission correspond to  localized surface plasmon resonances (LSPR) of the SRR, which can be controlled by modifying  the geometry of the SRR. Figure 1(c), presents the calculated nonlinear emission power for different SFG frequencies combinations (x-y axes). High generation efficiency is achieved for fundamental frequencies that correspond with the LSPR frequency (marked with horizontal and vertical dashed line). This is due to the strong excitation of the fundamental plasmonic mode, which enhances the nonlinear interaction. In addition, high efficiency is achieved also for combinations of frequencies, which give the same specific sum-frequency, i.e. along the cross-diagonal in the figure. This high-efficiency sum frequency corresponds with the LSPR for the emitting mode, i.e., in the polarization along the arms of the SRR. By controlling the geometry of the particle and mode matching the band of efficient nonlinear conversion can be substantially broadened.

Fig. 1 (a) Transmission spectrum for the fundamental frequencies, polarized parallel to the SRR base. (b) Transmission spectrum for the SFG frequencies, polarized parallel to the SRR arms. (c) Log scale of nonlinear conversion efficiency for combination of fundamental frequencies (f₁ and f₂) according to the hydrodynamic model. Dashed lines indicate resonance frequencies for the fundamental resonance and lower frequency emission mode.

[1] C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith "Origin of second-harmonic generation enhancement in optical split-ring resonators," Phys. Rev. B. 85, 201403 (2012).