Nonlinear Surface Lattice Resonance in Plasmonic Nanoparticle Arrays

We study experimentally second-harmonic generation from arrays of split-ring resonators at oblique incidence and find conditions of more than 30-fold enhancement of the emitted second harmonic with respect to normal incidence. We show that these conditions agree well with a nonlinear Rayleigh-Wood anomaly relation and the existence of a surface lattice resonance at the second harmonic. The existence of a nonlinear surface lattice resonance is theoretically confirmed by extending the coupled dipole approximation to the nonlinear case. We further show that the localized surface plasmon modes that collectively contribute to the surface lattice resonance are inherently dark modes that become highly bright due to the collective interaction.

Figure 1

(a) Illustration of the studied SRRs array, incident angle, and polarization. The incident light wave vector lies in the $y\text{−}z$ plane at an angle $\theta$ relative to the normal to the surface. (b) Scanning electron microscope image of the fabricated sample. The inset shows a single SRR. (c) Illustration of the physical dimensions of each SRR.

Figure 2

Second-harmonic emission, NL-RA and NL-SLR. (a) Log scale of the measured zero-order SHG emission vs the FF wavelengths and the incident angle. The black dots indicate the NL-RA of $⟨0,1⟩$ order calculated according to Eq. (11), and the magenta dots indicate the strongest measured response at each wavelength. The arrows diagrams demonstrate the existing diffraction modes in both sides of the NL-RA. (b) SHG enhancement along the position of strongest measured emission relative to normal incidence.

Figure 3

Linear transmission measurements. Angle and wavelength dependent zero-order transmission of the sample at (a) the SH wavelengths range and $y$ polarization, and (b) the FF wavelengths range and $x$ polarization. The black dots at (a) indicate the $⟨0,1⟩$ RA order calculated according to Eq. (10). (c) A cross section of the transmission at (a) for angle of 28.3 degrees [blue dashed line in (a)].

Figure 4

Schematic charge distribution and calculated far-field radiation patterns of (a) quadrupole and (b) dipole modes, in the $x\text{−}y$ plane. These modes exist in our nanostructures in wavelength range of 675–700 and 950–1070 nm, respectively. Angle dependent (c) transmission and (d) SH emission was calculated for quadrupole (green) and dipole (blue) modes. The measured data (red dots) agree with the behavior of a quadrupole mode for both cases. The measured data in (c) and (d) were taken by averaging the response at the wavelengths range of the quadrupole mode.

Figure 5

Simulations of the two different resonance excitations. Surface charge distribution along the SRRs over the full optical cycle is shown in (a) at the quadrupole LSPR and in (b) at the SLR.