Theory of nanorod antenna resonances including end-reflection phase

We present a fully analytic theory for nanorod resonances including the phase of reflection from the rounded ends using a transmission line approach. It combines the circuit theory response of spherical nanoparticles with standard transmission line theory using the Sommerfeld wave dispersion. The approach agrees well with comprehensive numerical calculations.

Figure 1

(a) A schematic of a nanorod. (b) Its equivalent circuit.

Figure 2

Phase of the reflection Φ of the nanorod for selected values of ${\varepsilon }_d:{\varepsilon }_d=2.8$ (black), ${\varepsilon }_d=5.4$ (orange), and ${\varepsilon }_d=9$ (green). The solid lines show the data calculated by Eq. (2) and the dashed lines show the data from Ref. [20].

Figure 3

Resonant wavelengths of 25 nm diameter nanorods for different lengths. Red dots correspond to resonant wavelengths calculated with FDTD, the green solid line corresponds to the results from analytic theory, and the green dotted line corresponds to the results if we assume $\Phi =0$.

Figure 4

Phases of the reflection Φ as a function of wavelengths for (a) 20, (b) 25, and (c) 30 nm diameter nanorods.

Figure 5

Phase shift between the theory and simulation results as a function of nanorod diameter and $\left(L-2r\right)$. The grayscale shows the value of $\left({\Phi }_{\mathrm{theory}}-{\Phi }_{\mathrm{simulation}}\right)/\pi$.

Figure 6

Bandwidths of the resonance peak as a function of nanorod length. The diameter of the nanorods is 30 nm.

Figure 7

Normalized extinction cross sections as a function of wavelength for various nanorod lengths. The diameter of the nanorods is 30 nm.