Guided propagation along quadrupolar chains of plasmonic nanoparticles

The employment of periodic arrays of plasmonic nanoparticles has been proposed by several groups for enhanced transmission or absorption and for realizing optical nanowaveguides. Generally, due to their small transverse dimensions, such linear arrays have been operated near their dipolar resonance. However, it has been recently shown that nanoscale plasmonic particles may also support higher-order resonances, which provide some advantages in different applications. Here we derive a full-wave analytical closed-form dispersion equation for the guided and leaky modes supported by linear chains of nanoparticles near a quadrupolar resonance. We show that, despite the vanishing bandwidth of the individual quadrupolar resonance in each of the nanoparticles composing the chain, the overall bandwidth of quadrupolar chain guidance is relatively large due to strong coupling, even considering realistic losses and frequency dispersion of optical materials. Applications for low-damping optical nanotransmission lines and leaky-wave nanoantennas are suggested.

Figure 1

(Color online) Analogy between the (a) dipolar or (b) quadrupolar operation of a linear chain of nanoparticles, its low-frequency interpretation in terms of currents traveling along conducting wires, and corresponding circuit models.

Figure 2

(Color online) Geometry of the problem: an array of quadrupolar nanoparticles supporting a (a) longitudinal or (b) transverse mode.

Figure 3

(Color online) Region of guidance for longitudinally polarized modes [Fig. 2a]. Longitudinal modes are supported for any value of polarizability above the red dashed line.

Figure 4

(Color online) Dispersion plots for the guided mode for the longitudinal polarization in terms of the particle’s quadrupolarizability, for two different values of spacing.

Figure 5

(Color online) Region of guidance for transversely polarized modes [Fig. 2b].

Figure 6

(Color online) Dispersion plots for guided mode in transverse polarization in terms of the quadrupolarizability, for three different spacing values [(a) narrow spacing, (b) anomalous dispersion region, and (c) large spacing].

Figure 7

(Color online) Value of $\overline{\beta }$ for achieving the minimum damping loss factor in the modal propagation in the (a) two polarizations and (b) corresponding damping factor.

Figure 8

(Color online) Range of guidance for the permittivity of homogeneous nanospheres vs their spacing for: (a) longitudinal polarization and (b) transverse polarization.

Figure 9

(Color online) Frequency dispersion for a chain of spherical particles with radius $a={\lambda }_0/20$ spacing $d=2.5a$ and permittivity following the Drude model $\epsilon \left(\omega \right)={\epsilon }_0\left(1-2.5{\omega }_0^2/{\omega }^2\right)$.

Figure 10

(Color online) (a) Dispersion with the permittivity for a chain of homogeneous spherical particles with radius $a=15\text{nm}$ and spacing $d=2.25a$ at the wavelength ${\lambda }_0=500\text{nm}$ for longitudinally polarized modes; (b) dispersion for the same size, spacing, and polarization, but for a silicon carbide sphere covered with silver shell in terms of the ratio of radii $\gamma =a_1/a$.

Figure 11

(Color online) Frequency dispersion for the longitudinal modes supported by a chain of spherical silver particles with $a=20\text{nm}$ and $d=45\text{nm}$.