Collection and Concentration of Light by Touching Spheres: A Transformation Optics Approach

A general three-dimensional transformation optics approach is presented that yields analytical expressions for the relevant electromagnetic magnitudes in plasmonic phenomena at singular geometries. This powerful theoretical tool reveals the broadband response and superfocusing properties of touching metal nanospheres and provides an elegant physical description of the prominent field enhancement that takes place at the point of contact between a spherical nanoparticle and a flat metallic surface.

Figure 1

Panel (a) renders the problem under study: the interaction of light with two metal nanospheres (radii $R_1$ and $R_2$) that touch at a single point. Panel (b) shows the transformed geometry, illumination, and permittivities obtained from the inversion of the original system.

Figure 2

(a) Numerical absorption cross section ${\sigma }_{\mathrm{abs}}$ versus incident frequency $f=\omega /2\pi$ for twin Ag touching spheres of different $R_1$ (color or shaded dots), and a single 5 nm radius sphere (black dots). Black solid line plots the analytical result obtained from Eq. (5). All the spectra are normalized to $R_1^3$. (b) electric field amplitude enhancement versus angle $\theta$ (see inset) at 500 THz ($R_1=20\mathrm{nm}$). Black (red or gray) line corresponds to analytical (numerical) calculations.

Figure 3

(a) Absorption cross section efficiency, $Q_{\mathrm{abs}}={\sigma }_{\mathrm{abs}}/\pi (R_1^2+R_2^2)$, as a function of the incident frequency and the ratio $\tau =R_2/R_1$ for $R_1=20\mathrm{nm}$. Lower panels: Analytical field enhancement at 500 THz within the symmetry plane of the structure for $\tau =1$ (b) and $\tau =0.5$ (c). $\frac{E_z}{E_0}$ ranges from $-10$ to 10 in these two panels.

Figure 4

Upper panel: Analytical absolute value of the electric field enhancement at the touching point between a spherical nanoparticle and a Ag flat surface as a function of the incident frequency and the angle $\theta$ (see inset). Lower panel: $|E_z/E_0|$ for different particle radii at 500 THz. Black line plots the analytical result (indicated by dashed white line in the upper panel) and small color or shaded dots correspond to simulations.