Perturbation theory for plasmonic modulation and sensing

We develop a general perturbation theory to treat small parameter changes in dispersive plasmonic nanostructures and metamaterials. We specifically apply it to dielectric refractive index and metallic plasma frequency modulation in metal-dielectric nanostructures. As a numerical demonstration, we verify the theory’s accuracy against direct calculations for a system of plasmonic rods in air where the metal is defined by a three-pole fit of silver’s dielectric function. We also discuss new optical behavior related to plasma frequency modulation in such systems. Our approach provides new physical insight for the design of plasmonic devices for biochemical sensing and optical modulation and future active metamaterial applications.

Figure 1

(a) Comparing tabulated data for the real part of silver’s permittivity[25] against the Drude model and three-pole fit used in this paper at optical wavelengths. The three-pole fit is accurate to 250 nm. (b) The imaginary and real parts of this three-pole fit over relevant normalized frequencies.

Figure 2

Computed band structures for a square lattice of 2D plasmonic rods ($s=0.45a$) in air for the (a) TM and (b) TE polarization between $(k_x=0,k_y=0)$ and $(k_x=\pi /a,k_y=0)$ points. We note the presence of low-group-velocity modes in both TE and TM band structures and flat surface modes in the TE band structure.

Figure 3

Comparing the perturbation theory prediction (Perturbed) and direct solution (Direct) of ${\omega }_1$ in the region identified in Fig. 2b for dielectric $\Delta \varepsilon (\mathbf{r})=0.02$. The (a) lossless metal (${\Gamma }_n=0$) and (b) lossy metal cases show excellent agreement for both the flat surface mode and nonsurface mode.

Figure 4

Reflectivity spectrum of a finite 2D square lattice of square plasmonic rods ($s=0.45a$) in air, consisting of 50 layers. The results are obtained with a full-field FDFD simulation (shown in the inset schematic) and show a reflectivity dip corresponding to the lowest frequency propagating mode in the system [the mode at ${\omega }_c$ in Fig. 2a]. By altering the air region by $\Delta \varepsilon =1\times {10}^{-3}$ to simulate a perturbation, a shift of $\Delta \omega =-9.89\times {10}^{-5}$ in the dip is observed, matching the theoretical prediction of Eq. (22), ${\omega }_1=-9.55\times {10}^{-5}$, well.

Figure 5

Comparing the perturbation theory prediction (Perturbed) and direct solution (Direct) of ${\omega }_1$ in the same zoomed-in region identified in Fig. 2b, for $\Delta {\omega }_{p,1}=0.08{\omega }_{p,1}$. The (a) lossless (${\Gamma }_n=0$) and (b) lossy cases show excellent agreement for both the flat surface mode and nonsurface mode. Note the substantially greater ${\omega }_1$ for the flat plasmon mode, which has a stronger mechanical component to its eigenmode, and that ${\omega }_1<0$ as predicted by Eq. (29).

Figure 6

Comparing the perturbation theory prediction (Perturbed) and direct solution (Direct) of ${\omega }_1$ in the region above ${\omega }_{0,1}$, identified in Fig. 2b, for $\Delta {\omega }_{p,1}=0.08{\omega }_{p,1}$. The (a) lossless (${\Gamma }_n=0$) and (b) lossy cases show ${\omega }_1>0$ (as opposed to ${\omega }_1<0$ in Fig. 5), verifying the prediction of Eq. (29).