Electromagnetic waves propagating in a periodic array of parallel metallic carbon nanotubes

This is a theoretical study of electromagnetic modes, guided by infinitely long single-wall metallic carbon nanotubes, forming two-dimensional periodic array. It demonstrates that electromagnetic coupling dramatically raises the slow-wave factor of modes propagating in arrays compared to that observed in single carbon nanotubes. It is found a strong dependence of the slow-wave factor on a wave vector of the Floquet-Bloch wave ${\mathbf{k}}_{\perp }$ (the transversal wave vector) propagating in the plane of periodicity. The slow-wave factor varies from unity as $\left|{\mathbf{k}}_{\perp }\right|\to 0$ to more than two hundreds for high values ${\mathbf{k}}_{\perp }$. Considerably low attenuation at terahertz frequencies is demonstrated. Comparison of electromagnetic properties of arrays of metallic carbon nanotubes and conventional wire media is discussed. Transmission of a plane wave through a finite-thickness slab, which can be used as a terahertz polarizer, is analyzed.

Figure 1

Two-dimensional periodic array of infinitely long carbon nanotubes.

Figure 2

(Color online) The real part of the slow-wave factor $\text{Re}\left(\beta /k\right)$, calculated for different periods $d$.

Figure 3

(Color online) The real part of the slow-wave factor $\text{Re}\left(\beta /k\right)$ in plane of ${\psi }_x/\pi =k_{x}d/\pi$, ${\psi }_y/\pi =k_{y}d/\pi$, calculated for period $d=2.5\text{nm}$.

Figure 4

(Color online) Real part of the slow-wave factor $\text{Re}\left(\beta /k\right)$ versus frequency, calculated for the hexagonal lattice at $k_x=0$, $k_y=\pi /\left(3\sqrt{3}d\right)$, and different periods $d$. Dashed curve illustrates the group delay factor $c\frac{d\beta }{d\omega }$, calculated at $d=200\text{nm}$.

Figure 5

Frequency dependence of $\text{Im}\left(\beta \right)/\text{Re}\left(\beta \right)$, calculated at $k_x=0$, $k_y=\pi /\left(3\sqrt{3}d\right)$, and $d=100\text{nm}$.

Figure 6

(Color online) Normalized attenuation constant, calculated at different frequencies: dotted curve (black in color) $-0.5\text{THz}$, dashed (blue in color) $-1\text{THz}$, and solid (red in color) $-2\text{THz}$. Period of the lattice $d=20\text{nm}$ and $m=21$.

Figure 7

(Color online) Transmission ${\left|T\right|}^2$ (black), reflection ${\left|R\right|}^2$ (red), and absorption ${\left|A\right|}^2=1-{\left|T\right|}^2-{\left|R\right|}^2$ (blue). Relaxation time is $\tau =3\times {10}^{-12}\text{s}$. Dashed curves correspond to period of lattice 200 nm, solid curves $-100\text{nm}$.

Figure 8

(Color online) ${\left|T\right|}^2$ (black), ${\left|R\right|}^2$ (red), and ${\left|A\right|}^2$ (blue). Relaxation time is $\tau =3\times {10}^{-13}\text{s}$. The dashed and solid curves correspond to periods of lattice 200 nm and 100 nm, respectively.