Tailoring of electric dipoles for highly directional propagation in parity-time-symmetric waveguides

Photonic structures offer a flexible platform for studying and demonstrating parity-time ($\mathcal{PT}$) symmetry phenomena. In these platforms, electric dipoles are often used as accurate models for electromagnetic sources, and elliptical dipoles were shown to provide for directional mode excitation. Here we introduce tailored dipole sources for directional excitation in a $\mathcal{PT}$-symmetric structure made of two coupled waveguides. By eliminating one mode in the device, a qualitatively different wave propagation on the two sides of the dipole is achieved. Interestingly, before the exceptional point, a linear dipole suffices to have mode beatings on only one side. Furthermore, beyond the exceptional point, gain can be created on one side only. Finally, at the exceptional point, a near-complete directionality can be achieved due to the mode merging. We explain these effects via a detailed analysis of the modes, and the subsequent mode excitation of the dipole is analytically described. In the end, these various types of contrasting phenomena offer possibilities for integrated photonics applications, routing setups, and lasing behavior.

Figure 1

(a) Schema of the photonic structure used in the simulations. The dielectric gain and loss materials are represented in orange and green, respectively, and the air in white. The waveguides are infinite in the left and right directions ($z\to -\infty$ and $z\to +\infty$). The location of the dipole at (0,0) is marked by a red dot. (b) Real and (c) imaginary propagation constant $\beta$ (in $\mathrm{rad}/\mu \mathrm{m}$) of modes 1 (blue, full curve) and 2 (orange, dot-dashed curve) of the $\mathcal{PT}$-symmetric structure as a function of the gain-loss parameter $\gamma$. The exceptional point is located at ${\gamma }_{EP}=0.1231$ and marked by a gray dashed line. (d) Phase difference and (e) module of the $x$ and $z$ electric field components of modes 1 (blue) and 2 (orange) at the dipole location for $\gamma =0$ to 0.20. The phase difference is calculated as ${\varphi }_{Ez}-{\varphi }_{Ex}$. (f) Electric field profiles of the guided supermodes for $\gamma =0$, 0.05, 0.123, 0.124, and 0.15, for both electric components ($x$ and $z$). The gray dashed line marks the dipole location, $x=0$

Figure 2

(a), (b) Excitation amplitudes of modes 1 (blue) and 2 (orange) by an (a) $x$- or (b) $z$-oriented dipole as a function of the gain-loss parameter $\gamma$. (c) Characteristics of the dipole tailored for the desired contrast at each $\gamma$. The exceptional point is located at ${\gamma }_{EP}=0.1231$ and marked by a gray dashed line.

Figure 3

(a) Excitation amplitude of modes 1 (blue) and 2 (orange) on each side of the dipole tailored for left-right contrast for values of the gain-loss parameter $\gamma =0.01$ to 0.20. The exceptional point is located at ${\gamma }_{EP}=0.1231$ and marked by a gray dashed line. (b)-(f) Magnetic field absolute value in the structure for $\gamma =0.01,0.05,0.10,0.123$, and 0.15, respectively. The dipole is at the center (0,0), the gain waveguide on the top and the loss waveguide on the bottom, as in Fig. 1. In these images, the tailored dipoles are used, with mode 2 canceled on the left side. The insets in (e) and (f) represent (e) the absolute magnetic field or (f) its logarithm in the center of the gain guide for $z=0$ to $80\mu \mathrm{m}$. (g) Magnetic field absolute value in the structure for $\gamma =0.15$, with mode 2 canceled on the right side instead of the left. The dipole tailored for $\gamma =0.15$ is used, with an added phase of $\pi$ compared to image (f).