Enhanced response of non-Hermitian photonic systems near exceptional points

This paper theoretically and numerically studies the response characteristics of non-Hermitian resonant photonic systems operating near an exceptional point (EP), where two resonant eigenmodes coalesce. It is shown that a system near an EP can exhibit a non-Lorentzian frequency response, whose line shape and intensity strongly depend on the modal decay rate and coupling parameters for the input waves, unlike a normal Lorentzian response around a single resonance. In particular, it is shown that the peak intensity of the frequency response is inversely proportional to the fourth power of the modal decay rate and can be significantly enhanced with the aid of optical gain. The theoretical results are numerically verified by a full wave simulation of a microring cavity with gain. In addition, the effects of the nonlinear gain saturation and spontaneous emission are discussed. The response enhancement and its parametric dependence may be useful for designing and controlling the excitation of eigenmodes by external fields.

Figure 1

Microring cavity coupled to a straight waveguide. The input waves from the left and right ports in the straight waveguide couple to the CW and CCW modes in the microring, respectively. The two modes are also coupled to each other by backscattering.

Figure 2

(a), (b) Dynamics of the intensity ${\parallel \mathbf{\psi }\parallel }^2={\sum }_j{\left|a_j\left(t\right)\right|}^2$ at the EP ($\epsilon \ne 0$ and $p=0$) in a microring cavity under (a) a delta-function-like pulse with a period $T=4\times {10}^5$ and (b) a Gaussian noise input (red curves). These results were obtained from Eq. (1) with the 2×2 matrix of the microring cavity for ${\gamma }_0/{\gamma }_n=0.4$. For comparison, the intensity in a ring cavity without backscattering, i.e., a cavity at the DP ($\epsilon =p=0$), is also shown as the blue dotted curve for each case. In (b), $\mathbf{f}={(f_1,f_2)}^T$ is treated as complex white Gaussian noise, $\langle f_i^{*}\left(t^{\prime }\right)f_j\left(t\right)\rangle =2D{\delta }_{ij}\delta (t-t^{\prime })$, where $D$ is the noise variance. (c) ${\gamma }_0/{\gamma }_n$ dependence of $I_{\text{EP}}/I_{\text{DP}}$. $I_{\text{EP}}$ indicates the time average of the response intensity dynamics at the EP, whereas $I_{\text{DP}}$ indicates that at the DP ($\epsilon =p=0$). Gaussian noise is applied to the cavity.

Figure 3

(a) Time-averaged response intensities of the CCW and CW waves at the EP ($\epsilon =1.8\beta$ and $p=0)$, ${\overline{I}}_{\text{CCW}}$ and ${\overline{I}}_{\text{CW}}$, under the CW wave input condition (${\kappa }_1=0$, ${\kappa }_2={10}^{-7}\beta$). ${\mathrm{\Delta }}_0$ is the degenerate resonant frequency at the EP. The theoretical curves of the CCW and CW wave intensities obtained from Eq. (6) are shown as the blue and pink solid curves, respectively. (b) Intensity $I_{\text{EP}}={\overline{I}}_{\text{CCW}}+{\overline{I}}_{\text{CW}}$ at the EP under the same CW wave input condition. For comparison, the intensity $I_{\text{DP}}$ at the DP ($\epsilon =p=0$) is also shown. The theoretical curves obtained from Eq. (6) at the EP and DP are shown as the blue and pink solid curves, respectively. In (a) and (b), $W_{\infty }=0.95\times {10}^{-3}$, and ${\gamma }_0/{\gamma }_s=\frac{1}{18}$.

Figure 4

(a) Time-averaged response intensity ${\overline{I}}_{\text{CCW}}$ under the CCW wave input condition (${\kappa }_1={10}^{-7}\beta$ and ${\kappa }_2=0$). (b)–(d) Time-averaged response intensities ${\overline{I}}_{\text{CCW}}$ and ${\overline{I}}_{\text{CW}}$ under a bidirectional wave input condition with (b) ${\kappa }_1=\epsilon /\left(2{\gamma }_0\right){\kappa }_2$, (c) ${\kappa }_1=-\epsilon /\left(2{\gamma }_0\right){\kappa }_2$, and (d) ${\kappa }_1=-i\epsilon /\left(2{\gamma }_0\right){\kappa }_2$, where ${\kappa }_2={10}^{-7}\beta$. In (a)–(d), $\epsilon =1.8\beta$, $p=0$, $W_{\infty }=0.95\times {10}^{-3}$, and ${\gamma }_0/{\gamma }_s=\frac{1}{18}$. The theoretical curves of the CCW and CW wave intensities obtained from Eq. (6) for each input condition are shown as the blue and pink solid curves, respectively.

Figure 5

Enhancement factor $\eta$ versus ${\gamma }_0/{\gamma }_s$. $\eta$ is measured under the CW wave input condition (${\kappa }_1=0$) at the resonance $\mathrm{\Delta }={\mathrm{\Delta }}_0$. The theoretical curve is given by $\eta ={\gamma }_s^2/{\gamma }_0^2+1$.

Figure 6

Enhancement factor $\eta$ plotted in the $\epsilon \text{−}p$ plane. (a) ${\kappa }_2={10}^{-7}\beta$, (b) ${\kappa }_2={10}^{-4}\beta$, and (c) ${\kappa }_3={10}^{-3}\beta$. In (a)–(c), $W_{\infty }=0.95\times {10}^{-3}$ and $arg(\epsilon /p)=0$.

Figure 7

(a) Each emission spectrum at the EP ($\epsilon =1.8\beta$ and $p=0$) and DP ($\epsilon =p=0$). The spectrum at the EP is fitted by a squared Lorentzian curve, whereas the spectrum at the DP is fitted by a Lorentzian curve. The fitting curves are shown as the blue dotted curves. (b) Emission spectra when an input wave with a frequency of ${\mathrm{\Delta }}_0$ is coupled to the cavity in the CW direction. In (a) and (b), $\kappa$ = ${10}^{-3}\beta$, $W_{\infty }=0.9\times {10}^{-3}$, and ${\gamma }_0/{\gamma }_n=\frac{2}{9}$.

Figure 8

Integrals of the noise spectra shown in Fig. 7. The integrals of the spectra at the EP and DP are fitted by $S_1$ and $S_2$, respectively.