Strong coupling in an Au plasmonic antenna–${\mathrm{SiO}}_2$ layer system: A hybrid-mode analysis

A detailed analysis of the optical response of a system accommodating several coupled modes is needed for the complete understanding of the strong coupling effect. In this paper, we report on the analysis of scattering cross-section spectra of Au antennas on a ${\mathrm{SiO}}_2$ layer on a Si substrate in the IR region. A classical model of coupled oscillators is used for determining the resonant energies, damping rates, and coupling strengths of four phonon polariton modes in the ${\mathrm{SiO}}_2$ layer coupled to a localized surface plasmon mode in a Au antenna. The calculated Hopfield mixing coefficients then show the contribution of the individual uncoupled modes to the hybrid modes of the coupled system.

Figure 1

(a) Schematic of the investigated system. (b) Schematic of the coupled-oscillator model. (c) Relative dielectric function and functions proportional to the energy loss of the system (maxima positions correspond to bulk and surface phonon modes). (d) Dispersion relation of phonon polaritons for different thicknesses of the ${\mathrm{SiO}}_2$ layer on the Si substrate.

Figure 2

Comparison of reflectance spectra obtained from FTIR measurements (solid lines) with those provided by FDTD simulations (dashed lines) for different antenna lengths $L$ on the ${\mathrm{SiO}}_2$ layer with the thickness $t=50$ nm on the Si substrate. The true $R$ values correspond only to the dark blue (bottom) lines ($L=0.8\mu \mathrm{m}$), while the other lines are shifted for better clarity.

Figure 3

Calculated (solid lines) and fitted (dashed lines) scattering cross-section spectra of gold antennas for different lengths $L$ on the ${\mathrm{SiO}}_2$ layer with the thickness $t=50$ nm on the Si substrate. Splitting is visible around the resonant frequencies of ${\mathrm{SiO}}_2$ phonon polariton modes (vertical lines).

Figure 4

Dependence of the damping rates and coupling strengths (a) on the antenna length $L$ for the constant ${\mathrm{SiO}}_2$ layer thickness $t=50$ nm and (b) on the layer thickness $t$ for the constant antenna length $L=2\mu \mathrm{m}$.

Figure 5

(a) Dependence of the simulated scattering cross section spectra (background), uncoupled mode energies from fitting (magenta lines), and hybrid mode energies from subsequent analysis (red marks) on the inverse values of the antenna length $L$ for the ${\mathrm{SiO}}_2$ layer thickness $t=50$ nm. (b) Hopfield mixing coefficients showing relative contributions of uncoupled modes to the new hybrid modes. The bottom graph corresponds to the bottom branch in the image on the left (similarly for the other graphs and branches), the values for the Hopfield mixing coefficients are between 0 and 1.