Surface second-harmonic generation from coupled spherical plasmonic nanoparticles: Eigenmode analysis and symmetry properties

The surface second-harmonic generation from interacting spherical plasmonic nanoparticles building different clusters (symmetric and asymmetric dimers, trimers) is theoretically investigated. The plasmonic eigenmodes of the nanoparticle clusters are first determined using an ab initio approach based on the Green's functions method. This method provides the properties, such as the resonant wavelengths, of the modes sustained by a given cluster. The fundamental and second-harmonic responses of the corresponding clusters are then calculated using a surface integral method. The symmetry of both the linear and nonlinear responses is investigated, as well as their relationship. It is shown that the second-harmonic generation can be significantly enhanced when the fundamental field is such that its second harmonic matches modes with suitable symmetry. The role played by the nanogaps in second-harmonic generation is also underlined. The results presented in this article demonstrate that the properties of the second-harmonic generation from coupled metallic nanoparticles cannot be fully predicted from their linear response only, while, on the other hand, a detailed knowledge of the underlying modal structure can be used to optimize the generation of the second harmonic.

Figure 1

(a) The symmetric gold nanodimers studied in this work. The nanoparticle diameter is 20 nm, and the spacing is 4 nm. The blue plane corresponds to the plane discussed in the main text ($x$ = 0). This plane is alternatively a symmetry or antisymmetry plane for the electromagnetic field (see the discussion in the main text). The excitation conditions and the mesh used for the SIE computations are also shown. The scattering angle θ is defined relatively to the $z$ axis. (b) Diagram describing the resonant wavelength of the eigenmodes arising from the coupling between the dipolar modes of two 20 nm gold nanoparticles with a nanogap of 4 nm. The arrows indicate the orientation of the dipolar moments for the corresponding eigenmodes. (c)–(f) Normalized real part of the $x$-component of the electric field evaluated for the eigenmode with a resonant wavelength of (c) λ = 711 nm, (d) λ = 678 nm, (e) λ = 645 nm, and (f) λ = 622 nm.

Figure 2

Near-field distribution of (a) the fundamental and (b) the SH intensities close to a nanodimer composed of two 20 nm gold nanoparticles. The spacing between the two nanoparticles is 4 nm. The incident wavelength is λ = 1244 nm. Real part of the $x$-component for (c) the fundamental and (d) the SH electric fields, as well as the real part of the $z$-component for (e) the fundamental and (f) the SH electric fields. (g) Scattered SH intensity as a function of the scattering direction θ (Fig. 1) computed for different incident wavelengths increasing from λ = 1150 nm to 1422 nm.

Figure 3

The symmetric gold nanodimers studied in this work. The nanoparticle diameter is 20 nm, and the spacing is 4 nm. The blue plane corresponds to the plane discussed in the main text ($z$ = 0). Contrary to Fig. 2, an incident plane wave propagating along the $x$ axis and polarized along the $z$ axis is considered. The scattering angle θ is defined relative to the $z$ axis.

Figure 4

Near-field distribution of (a) the fundamental and (b) the SH intensities close to a nanodimer composed of two 20 nm gold nanoparticles. The spacing between the two nanoparticles is 4 nm. An incident plane wave propagating along the $x$ axis and polarized along the $z$ axis is considered. The incident wavelength is λ = 1244 nm. Real part of the $x$-component for (c) the fundamental and (d) the SH electric fields, as well as the real part of the $z$-component for (e) the fundamental and (f) the SH electric fields. (g) Scattered SH intensity as a function of the scattering direction θ (Fig. 3) computed for different incident wavelengths increasing from λ = 1244 nm to 1492 nm.

Figure 5

(a) The two asymmetric nanodimers considered in this work. Diagrams describe the resonant wavelength of the eigenmodes arising from the coupling between the dipolar modes of an asymmetric dimer composed of a 20 nm gold nanoparticle with (b) a 30 nm or (c) a 40 nm gold nanoparticle. The spacing between the two nanoparticles is 4 nm.

Figure 6

Near-field distribution for (a) the fundamental and (b) the SH intensities close to a nanodimer composed of a 20 nm and a 30 nm gold nanoparticle. The spacing between the two nanoparticles is 4 nm. The incident wavelength is 1456 nm. Real part of the $x$-component for (c) the fundamental and (d) the SH electric fields. (e) Scattered SH intensity as a function of the scattering direction computed for fundamental wavelengths increasing from λ = 1150 nm to λ = 1456 nm.

Figure 7

Near-field distribution for (a) the fundamental and (b) the SH intensities close to a nanodimer composed of a 20 nm and a 40 nm gold nanoparticle. The spacing between the two nanoparticles is 4 nm. The incident wavelength is λ = 1492 nm. Real part of the $x$-component for (c) the fundamental and (d) the SH electric fields. (e) Scattered SH intensity as a function of the scattering direction computed for fundamental wavelengths increasing from λ = 1150 nm to λ = 1492 nm.

Figure 8

The gold nanotrimers studied in this work. The nanoparticle diameter is 20 nm, and the spacing is 4 nm. The blue plane corresponds to the plane discussed in the main text ($x$ = 0). This plane is alternatively a symmetry or antisymmetry plane for the electromagnetic field (see the discussion in the main text). The scattering angle θ is defined relative to the $z$ axis.

Figure 9

(a) Diagram describing the resonant wavelength of the eigenmodes arising from the coupling between the dipolar modes of three 20 nm gold nanoparticles with nanogaps of 4 nm. The three nanoparticles are lined up along the $x$ axis. The arrows indicate the orientation of the dipolar moments for the corresponding eigenmodes. (b)–(g) Real part of the $x$-component of the electric field evaluated for the eigenmode with a resonant wavelength of (b) 742.5 nm, (c) 685 nm, (d) 662 nm, (e) 653 nm, (f) 638 nm, and (g) 611 nm.

Figure 10

Near-field distribution of (a) the fundamental and (b) the SH intensities close to a linear nanotrimer composed of three 20 nm gold nanoparticles. The spacing between the nanoparticles is 4 nm. The incident wavelength is λ = 1306 nm. Real part of the $x$-component for (c) the fundamental and (d) the SH electric fields. (e) The scattered SH intensity as a function of the scattering direction computed for incident wavelength increasing from λ = 1222 nm to λ = 1484 nm.