Geometry dependence of surface lattice resonances in plasmonic nanoparticle arrays

Plasmonic nanoarrays which support collective surface lattice resonances (SLRs) have become an exciting frontier in plasmonics. Compared with the localized surface-plasmon resonance in individual particles, these collective modes have appealing advantages such as angle-dependent dispersions and much narrower linewidths. Here, we investigate systematically how the geometry of the lattice affects the SLRs supported by metallic nanoparticles. We present a general theoretical framework from which the various SLR modes of a given geometry can be straightforwardly obtained by a simple comparison of the diffractive order vectors and orientation of the nanoparticle dipole given by the polarization of the incident field. Our experimental measurements show that while square, rectangular, hexagonal, honeycomb, and Lieb lattice arrays have similar spectra near the $\mathrm{\Gamma }$ point $\left(k=0\right)$, they have remarkably different SLR dispersions. Furthermore, their dispersions are highly dependent on the polarization. Numerical simulations are performed to elucidate the field profiles of the different modes. Our findings extend the diversity of SLRs in plasmonic nanoparticle arrays, and the theoretical framework provides a simple model for interpreting the SLRs features, and vice versa, for designing the geometrical patterns.

Figure 1

Polarization dependence of the dipolar radiative couplings between the metallic nanoparticles. The radiative direction is orthogonal to polarization. Therefore, if the metallic nanoparticles are placed parallel with the light polarization, there is no radiative coupling between them (a). If the nanoparticles form an array orthogonal to the polarization direction, they will couple with each other through the dipolar radiative interactions (b). Therefore, in rectangular arrays the SLRs depend only on one dimension (c). But in a more complex lattice—for example, a honeycomb array (d)—the radiative dipolar interactions would depend on both directions due to the nonzero interaction between the nearest particles along a combination of $x$ and $y$ directions, as the red arrow in (d) indicates. Thus the modes of the nanoparticle array are determined not only by the geometry of the lattice but also by its interplay with the polarization direction. We demonstrate the resulting rich mode structure in experiments on square, rectangular, hexagonal, honeycomb, and Lieb nanoparticle geometries, and provide a simple approach to interpret and predict the observed dispersions, such as the one for a hexagonal lattice (e) (here $E$ is the light energy and $k_{\parallel }$ the in-plane wave vector).

Figure 2

(a) Scattering of a plane wave by a 2D particle array. Here $k_0$ and $k$ indicate the incoming and outgoing wave vectors, respectively, and $\mathit{s}=\mathit{k}-{\mathit{k}}_{\mathit{0}}$ denotes the scattering vector. (b) The schematic of a square lattice in reciprocal space and the four lowest DOs (black arrows). The DOs $(0,1)$ and $(0,-1)$ have been slightly shifted horizontally to be better distinguished. The fuchsia arrows indicate the electric polarizations for TE and TM modes (TE and TM are defined with respect to $k_y)$. (c) The calculated DOs $(0,1),(0,-1)$ and the degenerate $(1,0)/(-1,0)$.

Figure 3

(a)–(f) SEMs of square, rectangular, ${45}^{\circ }$ rotated square, hexagonal, honeycomb, and Lieb lattice arrays. The scale bar in (d) is $300\text{nm}$ and the same for all. (g) The measurement setup. The angle-resolved transmission spectra are collected by focusing the image of the back focal plane of the objective to the entrance slit of the spectrometer. The entrance slit is parallel to the $y$ axis of the sample. For transmission measurements, a white LED light source is used.

Figure 4

(a) The schematic of a square lattice in reciprocal space and the four lowest DOs (black arrows). The DOs $(0,1)$ and $(0,-1)$ have been slightly shifted horizontally to be better distinguished. The fuchsia arrows indicate the electric polarizations for TE and TM modes (TE and TM are defined with respect to $k_y)$. The measured extinction spectra for (b) TE- and (c) TM-polarized lights. Dashed and dotted white lines represent the calculated DOs $(0,1)$ and $(0,-1)$, respectively. The dashed red line represents the calculated degenerate DOs $(1,0)$ and $(-1,0)$. The broad feature around $2.4\text{eV}$ is the LSPR, here and in Figs. 5, 6, 7, 8, 9. FDTD simulated extinction spectra for square lattice under normal and ${10}^{\circ }$ incidence with (d) TE- and (e) TM-polarized light.

Figure 5

(a) The schematic of a rectangular lattice in reciprocal space and the four lowest DOs. For clarity of the schematic, the distances along the $x$ and $y$ axes in the reciprocal lattice are not in scale with those in the real sample. Measured extinction and calculated DOs for (b) TE- and (c) TM-polarized light. Dashed and dotted white lines represent the DOs $(0,1)$ and $(0,-1)$, respectively. The dashed red line represents the degenerate DOs $(1,0)/(-1,0)$.

Figure 6

(a) The schematic of a ${45}^{\circ }$ rotated square lattice in reciprocal space and the four lowest DOs. Measured extinction and calculated DOs for (b) TE- and (c) TM-polarized light. Dashed and dotted white lines represent the degenerate DOs $(1,0)/(0,1)$ and $(-1,0)/(0,-1)$, respectively.

Figure 7

(a) The schematic of a hexagonal lattice in reciprocal space and the six lowest DOs. Measured extinction and calculated DOs for (b) TE- and (c) TM-polarized light. Dashed and dotted white lines represent the DOs $(1,1)$ and $(-1,-1)$, respectively. Dashed and dotted red lines represent the degenerate DOs $(1,0)/(0,1)$ and $(0,-1)/(-1,0)$, respectively.

Figure 8

The schematics of (a) a honeycomb lattice, with its unit cells that constitute a hexagonal lattice, and (b) a hexagonal lattice in reciprocal space and the six lowest DOs. Measured extinction and calculated DOs for (c) TE- and (d) TM-polarized light. Dashed and dotted white lines represent the DOs $(1,1)$ and $(-1,-1)$, respectively. Dashed and dotted red lines represent the degenerate DOs $(1,0)/(0,1)$ and $(0,-1)/(-1,0)$, respectively.

Figure 9

The schematics of (a) a Lieb lattice, with its unit cells that constitute a square lattice, and (b) a square lattice in reciprocal space and the four lowest DOs. Measured extinction and calculated DOs for (c) TE- and (d) TM-polarized light. Dashed and dotted white lines represent the DOs $(0,1)$ and $(0,-1)$, respectively. The dashed red line represents the degenerate DOs $(1,0)/(-1,0)$.

Figure 10

FDTD simulated extinction spectra for a square lattice under normal and ${10}^{\circ }$ incidence with (a) TE- and (e) TM-polarized light. The field distributions $\mathrm{\Phi }\left(E_x\right)$ for TE-polarized light at $583\text{nm}$ wavelength under normal incidence (b), under ${10}^{\circ }$ incidence at $553\text{nm}$ (c) and $643\text{nm}$ (d) wavelengths. For TM-polarized light, the field distributions $\mathrm{\Phi }\left(E_y\right)$ at $583\text{nm}$ wavelength under normal incidence (f) and under ${10}^{\circ }$ incidence at $581\text{nm}$ wavelength (g). Here (b) and (f) relate to the SLRs corresponding to the degenerate DOs at the $\mathrm{\Gamma }$ point in Figs. 4 and 4, respectively. (c) corresponds to the $(0,1)$ DO and (d) to the $(0,-1)$ DO modes in Fig. 4. (g) relates to the degenerate modes of DOs $(1,0)/(-1,0)$ of Fig. 4 slightly away from the $\mathrm{\Gamma }$ point.

Figure 11

The extracted dispersion curves from the experimental results and the calculated DOs of square lattice for (a) TE and (b) TM polarizations. The blue and red circles correspond to the peak positions obtained from fitting Gaussian curves to the line cuts of measured extinction spectra while keeping $k_y$ constant, and the dashed curves are the calculated DOs.

Figure 12

The extracted dispersion curves from the experimental results and the calculated DOs of rectangular lattice for (a) TE and (b) TM polarizations. The blue and red circles correspond to the peak positions obtained from fitting Gaussian curves to the line cuts of measured extinction spectra while keeping $k_y$ constant, and the dashed curves are the calculated DOs.

Figure 13

The extracted dispersion curves from the experimental results and the calculated DOs of a ${45}^{\circ }$ square lattice for (a) TE and (b) TM polarizations. The blue and red circles correspond to the peak positions obtained from fitting Gaussian curves to the line cuts of measured extinction spectra while keeping $k_y$ constant, and the dashed curves are the calculated DOs.

Figure 14

The extracted dispersion curves from the experimental results and the calculated DOs of a hexagonal lattice for (a) TE and (b) TM polarizations. The blue, black, green, and red circles correspond to the peak positions obtained from fitting Gaussian curves to the line cuts of measured extinction spectra while keeping $k_y$ constant, and the dashed curves are the calculated DOs.

Figure 15

The extracted dispersion curves from the experimental results and the calculated DOs of a honeycomb lattice for (a) TE and (b) TM polarizations. The blue, black, green, and red circles correspond to the peak positions obtained from fitting Gaussian curves to the line cuts of measured extinction spectra while keeping $k_y$ constant, and the dashed curves are the calculated DOs.

Figure 16

The extracted dispersion curves from the experimental results and the calculated DOs of Lieb lattice for (a) TE and (b) TM polarizations. The blue and red circles correspond to the peak positions obtained from fitting Gaussian curves to the line cuts of measured extinction spectra while keeping $k_y$ constant, and the dashed curves are the calculated DOs.