Dark and bright modes, and their coherent control in dipolar metasurface bilayers

Several plasmonic nanoparticles supporting dipolar resonances can couple to form normal modes. Here, we develop an analytical model to explain the formation of nonradiative “dark” and radiative “bright” modes through radiative coupling in bilayers consisting of dipolar nanoantenna arrays that are separated by a subwavelength distance. We also include near-field contributions in our model and show that the absorption and reflectance spectra obtained from our model agree reasonably well with the respective finite-difference time-domain simulation results for both perfectly aligned and misaligned bilayers. The ability to vary the reflection and absorption spectra of these bilayers by changing the material and geometrical parameters has potential applications in the design of efficient spectral filters. We also show that we can selectively excite these modes by adjusting the phase between two counterpropagating normally incident fields, which has applications in all-optical modulators and switches based on purely linear interferometric effects.

Figure 1

Diagram of the bilayer structure of two nanoantenna arrays spaced by a separation $d$ and excited by a normally incident plane wave ${\mathit{E}}_{\mathrm{inc}}$.

Figure 2

Spectra of the (a) real and (b) imaginary parts of polarizabilities normalized to the volume of a cuboidal gold nanoantenna (185 nm long, 105 nm wide, and 20 nm thick and embedded in BK7 glass) in the electrostatic limit (blue solid lines), including only the retardation of the nanoantenna and the array (red dashed lines), and including both retardation and the radiation reaction of the nanoantenna and the array (black dot-dashed lines). The nanoantenna polarizability model is taken to be the same as in [29], and the array lattice constant is assumed to be 250 nm.

Figure 3

Absorption spectra of a bilayer structure with no misalignments between the layers and (a) $d/a=0.4$ and (d) $d/a=0.6$ obtained from Eqs. (17) and (19) without (blue dot-dashed lines) and with (red solid lines) the near-field terms $T_{xx}(\mathit{0},d)$. The black dashed lines are the respective absorption spectra obtained from FDTD simulations. The reflection spectra for the $d/a=0.4$ and $d/a=0.6$ structures are shown in (b) and (e), respectively. The polarization spectra of $P_A$ (solid lines) and $P_M$ (dot-dashed lines) for the two aforementioned bilayers without and with absorption losses in the nanoantennas are shown in (c) and (f), respectively, along with $P_{\mathrm{eff}}$ for a single nanoantenna plane (black dashed lines) without and with the absorption losses.

Figure 4

Normalized real (left panels) and imaginary (middle panels) parts of the field polarization $P_x(x,z)={\epsilon }_0{\epsilon }_r(x,z)E_x(x,z)/a^2$ in a single unit cell of a bilayer structure with $d/a=0.4$ in the $xz$ plane. The field distribution $E_x$ is obtained from FDTD simulations, and ${\epsilon }_r$ is the relative permittivity of the medium at a point $(x,z)$. The top (bottom) panels show the polarization distributions at the resonance of the $A$ ($M$) mode. The panels on the right show linecuts of the respective modes along the $z$ axis (black dotted lines in the left and middle panels), $P_x(0,z)/\left|P_x(0,z)\right|$. The yellow shaded regions represent the $z$ locations of the nanoantennas.

Figure 5

Reflectance spectra of (a) perfectly aligned and (b) misaligned bilayers with $\{{\mathrm{\Delta }}_x,{\mathrm{\Delta }}_y\}=\{0.2a,0.2a\}$ with various separations $d/a$ shown in the legend. The solid reflectance curves were calculated using the full analytical model with the near-field terms $T_{xx}(0.2a,0.2a,d)$, and the dashed curves were obtained from FDTD simulations.

Figure 6

(a) Spectra of $\left|\det \right({\mathrm{S}}_{\mathrm{scat}}\left)\right|$ for various separations $d$ of perfectly aligned bilayers. Absorption spectra of a bilayer with (b) $d/a=0.34$ and (c) $d/a=1.08$ for a single normally incident field (black dashed lines) and two counterpropagating normally incident fields with equal amplitude and the same phase (blue lines) and opposite phases (red lines). (d) The joint absorption $\mathcal{A}$ as the phase difference $\varphi$ between the two fields $E_{1,2}$ is varied (solid lines) for bilayers with $d/a=0.34$ (blue lines) and $d/a=1.08$ (red lines). The respective maximum single-incident-field absorption values are shown by dashed lines.

Figure 7

(a) Scattering, (b) absorption, and (c) extinction cross-section spectra of an individual nanoantenna obtained from FDTD simulations (blue dot-dashed lines) and from the dipolar analytical model (red solid lines).