Subwavelength leaky-wave optical nanoantennas: Directive radiation from linear arrays of plasmonic nanoparticles

We analyze here the leaky-wave properties of linear arrays of plasmonic nanoparticles. It is shown that such periodic arrays may support two orthogonal leaky-wave propagation regimes, respectively, with longitudinal and transverse polarization. Using closed-form dispersion relations derived in the complex domain, we analyze their properties in the leaky-wave regime and we derive general conditions under which a nanoparticle array with subwavelength lateral cross section may support a radiating leaky mode with directive properties, conical radiation, frequency scanning and sufficiently long propagation distance, paving the way to potential applications as a leaky-wave optical nanoantenna with subdiffractive properties. Realistic designs and configurations are presented, considering the material dispersion and absorption of optical materials, for which we determine propagation distance, near-field distribution and far-field leaky-wave radiation pattern.

Figure 1

(Color online) Geometry under consideration: a linear array of polarizable nanoparticles supporting (a) a longitudinal or (b) a transverse eigenmode.

Figure 2

(Color online) Variation of complex $\overline{\beta }$ in (a) longitudinal and (b) transverse polarization versus the normalized inverse polarizability of the nanoparticles composing the array. Here we consider a normalized center-to-center distance $\overline{d}=0.2$.

Figure 3

(Color online) Guided- and leaky-wave regions for longitudinal polarization. The solid blue and dashed red curves are, respectively, the loci of real solutions $\overline{\beta }=\pi /\overline{d}$ and $\overline{\beta }=1$, which delimit the guided-wave regime. The dotted green line defines the upper limit of the leaky-wave regime. The black dots denote the locus $\text{Re}\left[\overline{\beta }\right]={\overline{\beta }}_{\text{min}}$, which may be considered the cut off of the leaky-wave regime.

Figure 4

(Color online) Variation of the ratio $\xi =\text{Re}\left[\overline{\beta }\right]/\text{Im}\left[\overline{\beta }\right]$ for the supported leaky-wave modes of the nanoparticle chain of Fig. 1 for longitudinal polarization, varying the center-to-center distance.

Figure 5

(Color online) Analogous to Fig. 3, guided- and leaky-wave regions for transversely polarized modes.

Figure 6

(Color online) Analogous to Fig. 4, variation in $\xi$ vs $\text{Re}\left[\overline{\beta }\right]$ for transversely polarized leaky modes, varying the center-to-center distance.

Figure 7

(Color online) (a) Guided-wave and leaky-wave regions for longitudinal polarization, as a function of nanosphere permittivity and interparticle distance. The guided-wave regime is supported between the bold lines while the leaky-wave region is bounded by thinner lines. (b) Loci of constant $\xi =\text{Re}\left[\overline{\beta }\right]/\text{Im}\left[\overline{\beta }\right]$ in the leaky-wave region for $\eta =2.2$. Blue solid lines delimit the guided-wave and leaky-wave regions.

Figure 8

(Color online) Analogous to Fig. 7, but for transverse polarization.

Figure 9

(Color online) Variation of $\text{Re}\left[\overline{\beta }\right]$ and $\text{Im}\left[\overline{\beta }\right]$ for $\overline{d}=0.1$ and $\eta =d/a=2.1$ in the longitudinal polarization regime, varying the imaginary part of permittivity. The inset plot shows a zoom in the transition region for the case ${\epsilon }_i=0.1$.

Figure 10

(Color online) Variation of $\xi$ for different levels of material loss ${\epsilon }_i$ in (a) longitudinal and (b) transverse polarization. In both plots, we have considered $\eta =d/a=2.1$ and $\overline{d}=0.1$.

Figure 11

(Color online) $\overline{\beta }d$ vs $k_{0}d$ diagrams and $\xi$ for longitudinal modes supported by silver arrays with $\eta =2.1$.

Figure 12

(Color online) $\overline{\beta }d$ vs $k_{0}d$ diagrams for longitudinal modes supported by dielectric arrays with $\eta =2.1$.

Figure 13

(Color online) Magnetic field and power-flow distribution for a nanoparticle chain operating in the leaky-wave regime [(a) and (c), at 690 nm wavelength] and in the guided propagation regime [(b) and (d), at 600 nm].

Figure 14

(Color online) (a) Far-field radiation patterns vs wavelength of operation. At 722 nm (solid blue line), 714 nm (dashed red line), and 690 nm (dotted green line), directional far-field radiation patterns are obtained, pointing at $20°$, $18°$, and $13°$, respectively. At 600 nm, the guided-wave mode does not significantly contribute to the far-field radiation. (b) Calculated three-dimensional leaky-wave radiation pattern at the wavelength of 690 nm. (c) Scanning of the main lobe radiation pattern (magnitude and main direction) versus wavelength. The highlighted region corresponds to leaky-wave operation.