Using surface lattice resonances to engineer nonlinear optical processes in metal nanoparticle arrays

Collective responses of localized surface plasmon resonances, known as surface lattice resonances (SLRs) in metal nanoparticle arrays, can lead to high quality factors ($\sim 100$), large local-field enhancements, and strong light-matter interactions. SLRs have found many applications in linear optics, but little work of the influence of SLRs on nonlinear optics has been reported. Here we show how SLRs could be utilized to enhance nonlinear optical interactions. We devote special attention to the sum-frequency, difference-frequency, and third-harmonic generation processes because of their potential for the realization of novel sources of light. We also demonstrate how such arrays could be engineered to enhance higher-order nonlinear optical interactions through cascaded nonlinear processes. In particular, we demonstrate how the efficiency of third-harmonic generation could be engineered via cascaded second-order responses.

Figure 1

Schematic energy-level diagrams of the studied nonlinear processes. (a) In the process of SFG, an output field oscillating at the frequency ${\omega }^{\prime }={\omega }_1+{\omega }_2$ is created. (b) An output field oscillating at the frequency ${\omega }^{\prime }={\omega }_1-{\omega }_2$ is created in the process of DFG. (c) In the process of direct (noncascaded) THG, the three incident field components at the frequency ${\omega }_1$ are converted into the output field oscillating at the tripled frequency $3{\omega }_1$. (d) Cascaded THG can occur by sequential SHG and SFG processes.

Figure 2

Incident plane wave propagating along the $z$ direction illuminates a nanoparticle array. The particles are assumed to be equilateral triangular nanoprisms belonging to the symmetry group $D_{3h}$. The side width of the nanoprisms is $w$ and their height is $h$. The particles are arranged into rectangular arrays in the $xy$ plane with array periods $p_x$ and $p_y$.

Figure 3

Nanoparticle array with periods $p_x=p_y=526$ nm exhibits enhanced DFG and OR response near 800 nm, resulting in THz-frequency generation. The results are presented for a single particle at the center of the array. (a) $x$-polarized incident fields with the wavelengths near 800 nm result in an over 2400-fold enhancement in the dipole moment corresponding to the OR (shown with white arrow). The DFG response of the array for the incident fields with the wavelengths near 800 nm is also noticeably enhanced and exhibits generation of THz radiation (see the area confined by the red dotted lines). (b) The calculated spectral amplitude of the THz signal field exhibits a broadband response between 0.1 and 6.0 THz with an enhancement factor of the order of 30–1000 in comparison with an individual nanoparticle's response.

Figure 4

Nanoparticle array with periods $p_x=759$ and $p_y=792$ nm, giving rise to SLRs near the wavelengths of 1160 and 1205 nm, respectively, exhibits enhanced SFG and SHG responses when the wavelengths of the incident fields coincide with the SLRs. We plot the SFG dipole moment amplitude for the particle at the center of the array for the incident fields, linearly polarized along the (a) $y$, (b) $x$, and (c) $(x+y)$ direction. The dipole moment is scaled with respect to that of a single isolated particle.

Figure 5

Calculated THG dipole moment amplitude for the nanoparticle at the center of the array with periods $p_x=p_y=505$ nm, giving rise to a SLR near the wavelength of 770 nm. When the cascaded contribution is neglected (red dotted line), the THG response of the array does not markedly depend on the incident fundamental wavelength because there are no strong resonances either near the fundamental wavelength (1500–1600 nm) or near the THG signal (500–533 nm). When the cascaded nonlinear effects are taken into account (black solid line), a clear peak near the fundamental wavelength of 1540 nm (highlighted with the vertical blue line) appears, demonstrating a 20-fold enhancement of the overall dipole moment component corresponding to THG.