Giant nonreciprocity near exceptional-point degeneracies

We show that gyrotropic structures with balanced gain and loss that respect antilinear symmetries exhibit a giant nonreciprocity at the so-called exact phase where the eigenfrequencies of the isolated non-Hermitian setup are real. The effect occurs in a parameter domain near an exceptional- point (EP) degeneracy, where mode orthogonality collapses. The theoretical predictions are confirmed numerically in the microwave domain, where a nonreciprocal transport above 90 dB is demonstrated, and are further verified using lumped-circuitry modeling. The analysis allows us to speculate the universal nature of the phenomenon for any wave system where EP and gyrotropy can coexist.

Figure 1

Schematic of the photonic structure: two half-wave microstrip resonators are end-coupled to a bus waveguide. A uniformly distributed gain or loss material property augments the region beneath each of the resonators within the YIG substrate. The substrate is exposed to an external bias field, $H_0$, in the $y$ direction. For an appropriate value of the gain and loss parameter $\gamma$ the transmission in the forward direction takes values of the order of unity (a), while it is essentially zero in the backward direction (b).

Figure 2

Parametric evolution of the real and the imaginary parts of the eigenfrequencies vs $\gamma$ for the isolated setup of Fig. 1. A uniform magnetic field $H_0$ is imposed on the substrate. At $\gamma =0$ we have a nonzero imaginary part due to leakage from the cavities. At $\gamma ={\gamma }_{\tilde{\mathcal{P}}T}\approx 0.26$ an EP degeneracy occurs.

Figure 3

Three representative cases of nonreciprocal transport: (a) $\gamma =0$ where $\text{NR}=17.5\text{dB}$; (b) $\gamma =0.1675$ where $\text{NR}=42.6\text{dB}$; and (c) $\gamma =0.18$ where $\text{NR}=30.4\text{dB}$. The maximum nonreciprocity is observed in the domain around ${\omega }_s$ and it is nonmonotonic with respect to $\gamma$.

Figure 4

(a) We show the dependence of simulated resonant modes $\left[\text{Re}\right(\omega ),•/\text{Im}\left(\omega \right)$, $\blacksquare ]$ on the gain or loss parameter $\gamma$ for the setup with $H_0=0$ only in the domain between the two microcavities. A fitting using Eq. (4) (solid line) gives ${\omega }_0\approx 8.545,{\mathrm{\Omega }}_0\approx 0.2576$, and $\rho \approx 1.445$ (all measured in ${\text{ns}}^{-1}$) corresponding to ${\gamma }_{\tilde{\mathcal{P}}T}^0\approx 0.178$. (b) Nonreciprocity (NR) obtained by calculating the difference between the FWD and BWD transmittance $T$ from the simulations ($•$) and from the theoretical expressions, Eqs. (6) ($\circ$). The green line is obtained using Eq. (7). The inset shows the analogous simulated NR for a 15% reduction in the bias field. The vertical dashed line indicates the position of the EP in this case.

Figure 5

Exploration of the gain and loss $\gamma =\frac{1}{2}(R_1^{-1}+R_2^{-1})\sqrt{L/C}$ and gyration strength $g=G\sqrt{L/C}$ parameter space of the lumped circuit model shown in (a). In (b) we plot the nonreciprocity NR as intensity (high values of NR correspond to bright areas, while low values of NR to dark areas) in the map. Due to the limitation of the resolution, the narrow peaks representing high NR ($>30$ dB) in (c) are not resolved by the color bar. (c) We show some indicative “cuts” from the density map at several gyration strengths [shown in (b)] for $Z_0\sqrt{C/L}=0.82,kd\approx \pi$ at the $LC$ resonant frequency, $C_c/C=0.3,M/L=0.03$, and $\eta =\frac{1}{2}(R_2^{-1}-R_1^{-1})\sqrt{L/C}=0.03$ for the intrinsic loss. (d) shows the corresponding real and imaginary parts of the balanced, isolated ($\eta =C_c=0$) dimer mode frequencies illustrating the relation of the exceptional points to the singularities of the giant nonreciprocity. The solid line through the NR density plot shows the position of the isolated system exceptional point, slightly beyond the singularity.