Reflection phase shift of one-dimensional plasmon polaritons in carbon nanotubes

We investigated, both experimentally and theoretically, the reflection phase shift (RPS) of one-dimensional plasmon polaritons. We launched one-dimensional (1D) plasmon polaritons in carbon nanotubes and probed the plasmon interference pattern using a scanning near-field optical microscopy technique, through which a nonzero phase shift was observed. We further developed a theory to understand the nonzero phase shift of 1D polaritons, and found that the RPS can be understood by considering the evanescent field beyond the nanotube end. Interestingly, our theory shows a strong dependence of RPS on polaritons wavelength and nanotube diameter, which is in stark contrast to two-dimensional plasmon polaritons in graphene where the RPS is a constant. In the short-wave region, the RPS of 1D polaritons only depends on a dimensionless variable—the ratio between polaritons wavelength and nanotube diameter. These results provide fundamental insights into the reflection of polaritons in the 1D system, and could facilitate the design of ultrasmall 1D polaritonic devices, such as resonators and interferometers.

Figure 1

One-dimensional plasmon polaritons in carbon nanotubes. (a) Generating 1D plasmon polaritons with s-SNOM. (b) Near-field image of launched-reflected plasmons interference. The incident wavelength is $10.6\mu \text{m}$. Scale bars, 150 nm.

Figure 2

Reflection phase shift (RPS) of edge-reflected 1D plasmon polaritons. (a) AFM topography image (left) and near-field image (right) of a representative nanotube sample. The incident wavelength is $7.8\mu \text{m}$. Scale bars, 150 nm. Inset is the line profile across the nanotube along the yellow line. From the height profiles, the diameter of the nanotube was determined to be 2 nm. (b) Near-field response line profile taken along the nanotube in the near-field image. (c) RPS of 1D plasmons with the wavelength of incident infrared light ranging from $7.7\mu \text{m}\text{to}9.6\mu \text{m}$.

Figure 3

Simulation of 1D plasmon polaritons in a carbon nanotube system. (a) Interference patterns of the $r$ component of electric fields of launched-reflected 1D plasmon polaritons in a carbon nanotube. (b) Schematic of the modeled system. (c) RPS of 1D polaritons derived from (a).

Figure 4

Theoretical results of 1D polaritons reflection. (a) Analytically calculated RPS for various polariton wavelengths and diameters of nanotubes. The wavelength of incident infrared light is 10.6 $\mu \mathrm{m}$. (b) Plot of the corresponding reflection amplitudes $\left|R\right|$. (c) The comparison between analytical and simulation results. (d) The results for short polaritons wavelength. In short-wave approximation, the RPS only depends on the relative size between the polaritons wavelength and the diameter of the nanotube.