$\mathcal{PT}$-symmetry-induced evolution of sharp asymmetric line shapes and high-sensitivity refractive index sensors in a three-cavity array

It is important to control and tune the Fano-resonance spectra to achieve a large slope with, in addition, a relatively high extinction ratio for low-power optical switching and high-sensitivity sensing. Here, we explore the evolution of sharp asymmetric Fano-like line shapes in a three-cavity array with local parity-time ($\mathcal{PT}$) symmetry. In this three-cavity configuration, a single cavity is coupled to a $\mathcal{PT}$-symmetric combination of two cavities via a common waveguide. The influences of local $\mathcal{PT}$ symmetry on the asymmetric Fano-like line shapes are investigated by monitoring the output transmission spectra at various system parameters. It is found that both the slope and the extinction ratio within the sharp asymmetric line shapes can be significantly enhanced by introducing the $\mathcal{PT}$-symmetric unit, compared with the configuration of two indirectly coupled cavities. Subsequently we discuss the application of such a $\mathcal{PT}$-assisted configuration as a family of high-sensitivity refractive index sensors by numerical analysis. For practical parameters based on microring resonators, the best sensitivity of refractive index sensors is more than five orders of magnitude larger than two indirectly coupled lossy cavities. The proposed scheme can be implemented in current state-of-the-art experiments. This investigation can help us to understand the interplay between the Fano resonance and $\mathcal{PT}$ symmetry.

Figure 1

(a) Schematic of a single-mode loss cavity (denoted 1) side coupled to a channel waveguide. $s_{\mathrm{in}}$ and $s_{\mathrm{out}}$ denote the amplitudes of the electromagnetic waves incident from the left side and leaving from the right side of the waveguide, respectively. (b) Schematic of two single-mode loss cavities (denoted 1 and 2) side coupled to a common waveguide. The two cavities are separated by a distance $L$ much longer than the wavelength so that there is no direct coupling between the two cavities. (c) Schematic of the $\mathcal{PT}$-assisted three-cavity array (denoted 1, 2, and 3) derived by introducing an auxiliary single-mode gain cavity in (b). The gain cavity, 3, balances the loss of cavity 2. Such coupled dimer structures with balanced gain and loss are referred to as having $\mathcal{PT}$ symmetry [47, 48, 49, 50, 51].

Figure 2

(a) Power transmission spectrum of a single cavity as shown in Fig. 1. System parameters are chosen as ${\kappa }_{1e}=20{\kappa }_{1i}$. (b) Power transmission spectrum of the two indirectly coupled cavities in Fig. 1. Here the two cavities are identical. System parameters are chosen as ${\kappa }_{2e}={\kappa }_{1e}=20{\kappa }_{1i}, {\kappa }_{2i}={\kappa }_{1i}$, and $\theta =\pi /8$, respectively. (c) Power transmission spectrum of the $\mathcal{PT}$-assisted three-cavity array in Fig. 1. System parameters are chosen as ${\kappa }_{2e}={\kappa }_{1e}=20{\kappa }_{1i}, {\kappa }_{2i}={\kappa }_{1i}, \theta =\pi /8, J=8{\kappa }_{1i}$, and ${\kappa }_3=-21{\kappa }_{1i}$, respectively.

Figure 3

(a) Representative transmission spectra of the two indirectly coupled cavities in Fig. 1 as a function of the detuning $\mathrm{\Delta }/{\kappa }_{1i}$ for nine propagation phase factors $\theta /\pi$. Other system parameters are chosen as ${\kappa }_{2e}={\kappa }_{1e}=20{\kappa }_{1i}$ and ${\kappa }_{2i}={\kappa }_{1i}$, respectively. (b) Representative transmission spectra of the $\mathcal{PT}$-assisted three-cavity array in Fig. 1 as a function of the detuning $\mathrm{\Delta }/{\kappa }_{1i}$ for nine propagation phase factors $\theta /\pi$. Other system parameters are chosen as ${\kappa }_{2e}={\kappa }_{1e}=20{\kappa }_{1i}, {\kappa }_{2i}={\kappa }_{1i}, J=8{\kappa }_{1i}$, and ${\kappa }_3=-21{\kappa }_{1i}$, respectively.

Figure 4

Representative transmission spectra of the $\mathcal{PT}$-assisted three-cavity array in Fig. 1 as a function of the detuning $\mathrm{\Delta }/{\kappa }_{1i}$ for 11 intercavity coupling strengths $J/{\kappa }_{1i}$. Other system parameters are chosen as ${\kappa }_{2e}={\kappa }_{1e}=20{\kappa }_{1i}, {\kappa }_{2i}={\kappa }_{1i}, \theta =\pi /8$, and ${\kappa }_3=-21{\kappa }_{1i}$, respectively.

Figure 5

Sensitivity $S$ (in units of ${\kappa }_{1i}$) as a function of the detuning $\mathrm{\Delta }/{\kappa }_{1i}$ for three configurations: (a) a single cavity, (b) two indirectly coupled cavities, and (c) a $\mathcal{PT}$-assisted three-cavity array. System parameters for the simulation are the same as in Fig. 2.

Figure 6

Maximum sensitivity $S_{\max }$ versus propagating phase factor $\theta /\pi$ with the assisted $\mathcal{PT}$-symmetric structure (solid blue line; $J/{\kappa }_{1i}=8$) and without the $\mathcal{PT}$-symmetric structure, i.e., in two indirectly coupled cavities (dashed red line; $J/{\kappa }_{1i}=0$). Other system parameters for the simulation are chosen as ${\kappa }_{2e}={\kappa }_{1e}=20{\kappa }_{1i}, {\kappa }_{2i}={\kappa }_{1i}$, and ${\kappa }_3=-21{\kappa }_{1i}$, respectively.

Figure 7

Maximum sensitivity $S_{\max }$ versus intercavity coupling strength $J/{\kappa }_{1i}$ for the $\mathcal{PT}$-assisted three-cavity array. Other system parameters for the simulation are chosen as ${\kappa }_{2e}={\kappa }_{1e}=20{\kappa }_{1i}, {\kappa }_{2i}={\kappa }_{1i}, {\kappa }_3=-21{\kappa }_{1i}$, and $\theta =\pi /8$, respectively.