Towards nanoscale multiplexing with parity-time-symmetric plasmonic coaxial waveguides

We theoretically investigate a nanoscale mode-division multiplexing scheme based on parity-time- ($\mathcal{PT}$) symmetric coaxial plasmonic waveguides. Coaxial waveguides support paired degenerate modes corresponding to distinct orbital angular momentum states. $\mathcal{PT}$-symmetric inclusions of gain and loss break the degeneracy of the paired modes and create new hybrid modes without definite orbital angular momentum. This process can be made thresholdless by matching the mode order with the number of gain and loss sections within the coaxial ring. Using both a Hamiltonian formulation and degenerate perturbation theory, we show how the wave vectors and fields evolve with increased loss/gain and derive sufficient conditions for thresholdless transitions. As a multiplexing filter, this $\mathcal{PT}$-symmetric coaxial waveguide could help double density rates in on-chip nanophotonic networks.

Figure 1

(a) Schematic of the $\mathcal{PT}$-symmetric coaxial waveguide where the channel is perturbed with alternating sections of gain (yellow) and loss (red). (b) The Fourier coefficients describing $\mathrm{\Delta }n$ for each distribution $N=0.5,1,1.5,2$. Due to the symmetry only positive-order coefficients are shown.

Figure 2

(a) Modes of the uniform coaxial waveguide numbered according to the azimuthal order $m$. For the rest of the paper, we focus on $E=2\mathrm{eV}$, which is marked on the first three modes as black circles. (b) Distribution of the Poynting vector component $P_z$ for $m=0,1,2$ at $E=2\mathrm{eV}$.

Figure 3

(a) Real and (b) imaginary parts of the propagation constant of the five lowest order modes of a $\mathcal{PT}$-symmetric coaxial waveguide when $N=0.5$ at $E=2\mathrm{eV}$. The inset shows the schematic of the waveguide cross section.

Figure 4

(a) The out-of-plane component of the Poynting vector $P_z$ for the five lowest order modes of a $\mathcal{PT}$-symmetric coaxial waveguide, when $N=0.5$ at $E=2$ eV is plotted for each mode of the perturbed waveguide at $\kappa =0.02$ and (b) $\kappa =0.2$. (c) and (d) show $P_z$ along the propagation direction for the different modes for a slice in the ring 5 nm from the core interface. The rings are rotated by $5^{\circ }$ for clarity. The length of the propagation direction in (c,d) is 200 nm.

Figure 5

(a) Real and (b) imaginary parts of the propagation constant of the five lowest order modes of a $\mathcal{PT}$-symmetric coaxial waveguide when $N=1$ at $E=2$ eV. The inset shows the schematic of the waveguide cross section.

Figure 6

(a) The out-of-plane component of the Poynting vector $P_z$ for the five lowest order modes of a $\mathcal{PT}$-symmetric coaxial waveguide, when $N=1$ at $E=2$ eV is plotted for each mode of the perturbed waveguide at $\kappa =0.02$ and (b) $\kappa =0.2$. (c) and (d) show the $P_z$ along the propagation direction for the different modes for a slice in the ring 5 nm from the core interface. The rings are rotated by $5^{\circ }$ for clarity. The length of the propagation direction in (c,d) is 200 nm.

Figure 7

(a) $E_r$ for $m=1$ when $\kappa =0$. (b) Phase of $E_r$ for $m=1$ showing OAM. (c) and (d) show $E_r$ and its phase for $\kappa =0.02$ for the $N=1$ waveguide. No OAM is present.

Figure 8

Real (a) and imaginary (b) parts of the wave vectors of the five lowest order modes of a $\mathcal{PT}$-symmetric waveguide for $N=1.5$. Real (c) and imaginary (d) parts of the wave vectors of the five lowest order modes of a $\mathcal{PT}$-symmetric waveguide for $N=2$.

Figure 9

Real and imaginary parts of the wave vector of the $\mathcal{PT}$-symmetric coaxial waveguide as a function of $\kappa$ for $N=0.5$ [(a),(b)] and $N=1$ [(c),(d)] at $E=2$ eV. The empirical Johnson and Christy dataset has been used for this set of calculations.

Figure 10

Modes of the coaxial waveguide in the absence of gain as a function of $\kappa$, where $n_{\mathrm{gain}}=0$ and $n_{\mathrm{loss}}=+i\kappa$ for $N=0.5$ [(a),(c)] and $N=1$ [(b),(d)] at $E=2$ eV.

Figure 11

Expansion coefficients of each mode for $N=0.5$ at $E=2$ eV. Coefficients are plotted for the same five modes shown in Fig. 3 at two values of the non-Hermiticity parameter. Since the coefficients are complex in general, we have plotted the real and imaginary parts separately. In each panel, the continuous horizontal line is 0.

Figure 12

Expansion coefficients of each mode for $N=1$ at $E=2$ eV. Coefficients are plotted for the same five modes shown in Fig. 5 at two values of the non-Hermiticity parameter. Since the coefficients are complex in general, we have plotted the real and imaginary parts separately. In each panel, the continuous horizontal line is 0.