Nonlinearity-Induced Reciprocity Breaking in a Single Nonmagnetic Taiji Resonator

We report on the demonstration of an effective, nonlinearity-induced nonreciprocal behavior in a single nonmagnetic multi-mode Taiji resonator. Nonreciprocity is achieved by a combination of an intensity-dependent refractive index and of a broken spatial reflection symmetry. Continuous wave power-dependent transmission experiments show nonreciprocity and a direction-dependent optical bistability loop. These can be explained in terms of the unidirectional mode coupling that causes an asymmetric power enhancement in the resonator. The observations are quantitatively reproduced by a numerical finite-element theory and physically explained by an analytical coupled-mode theory. This nonlinear Taiji resonator has the potential of being the building block of large arrays for the study of topological and/or non-Hermitian physics. This represents an important step towards the miniaturization of nonreciprocal elements for photonic integrated networks.

Figure 1

(a) Optical image of the Taiji resonator (TJR) coupled to a bus waveguide. In the forward configuration (light blue arrows) light enters the sample from the left; in the reverse configuration (dark blue arrows) light is injected from the right. The dashed arrow represents the reflected light in the reverse case. The orange arrows indicate the propagation direction of clockwise ($+$) and counterclockwise ($-$) modes. (b),(c) Simulated intensity $I$ (in units of the input intensity $I_{\mathrm{in}}$) inside a TJR operating in the forward (b) and reverse (c) configurations for the same excitation frequency and power. The intensity maps are produced using the finite-element approach of Eq. (3).

Figure 2

(a) Simulated internal power $P_{\mathrm{ring}}$ for a downwards frequency sweep (black arrow) at two input powers $P$. Light (dark) blue curves refer to the forward (reverse) configuration. (b) Simulated transmitted power $P_T$ for a fixed frequency $\omega -{\omega }_0\simeq -4{\gamma }_T$ in the forward and reverse configurations. Black arrows indicate the sweep direction. Black dashed lines label the threshold powers. The inset shows the ratio $P_T^{\mathrm{(for)}}/P_T^{\mathrm{(rev)}}$ of the transmitted powers in the forward and reverse configurations along a decreasing $P$ ramp. Parameters for both panels: $L\simeq 850\mu \mathrm{m}$, $L_S=L/2$, $k_w=k_S=0.14$, ${\gamma }_A=5.7\times {10}^9{\mathrm{s}}^{-1}$, $n_{\mathrm{L}}=1.83$, $n_T=8.8\times {10}^{-13}{\mathrm{cm}}^2/\mathrm{W}$, $A=0.66\mu {\mathrm{m}}^2$, and $g=1$.

Figure 3

(a) Normalized transmittance $\tilde{T}=T/T_{\max }$ as a function of the detuning $\mathrm{\Delta }\lambda =\lambda -{\lambda }_0$ from the linear resonance wavelength for an upwards wavelength sweep (black arrow). The red dotted line represents the experimental transmittance in the linear regime for an input power $P\simeq 0.01$ W. The light (dark) blue solid curve corresponds to the experimental transmittance in forward (reverse) configuration at a fixed input power $P=0.21$ W. The black dashed lines are theoretical fits of each experimental curve produced using Eq. (3). The inset gives a magnified view of the region of strongest nonreciprocal behavior. (b) Shift $\mathrm{\Delta }{\lambda }_0$ of the resonance wavelength as a function of the input power $P$ for the resonance displayed in panel (a). Light (dark) blue circles correspond to experimental data in forward (reverse) configuration. The black dashed (dotted) line is a linear fit to the theoretically calculated $\mathrm{\Delta }{\lambda }_0$ in forward (reverse) configuration.

Figure 4

Optical bistability in the normalized transmittance $\tilde{T}=T/T_{\max }$ along a round-trip sweep of the laser detuning $\mathrm{\Delta }\lambda$ across the resonance at a fixed input power $P=0.21$ W: (a) forward configuration (light blue); (b) reverse configuration (dark blue). The ramp direction is indicated by the black arrows. Experimental data are shown as solid lines whereas the black dashed lines are the fits obtained with the finite-element theory of Eq. (3).

Figure 5

Normalized transmittance $\tilde{T}$ as a function of the relative wavelength shift with respect to the resonance wavelength ($\mathrm{\Delta }\lambda$) in an upwards ramp (indicated by the black arrow) for a TJR in the forward (light blue) and reverse (dark blue) configurations. The dashed curves are calculated by using the coupled-mode theory (c.m.), whereas the solid lines are obtained with the finite-element model (f.e.). The incident power is taken to be $P=0.21\mathrm{W}$. The parameters of the simulations are those of Fig. 2.

Figure 6

Scheme of the modeled Taiji resonator (TJR) showing the electric field amplitudes $E$ inside each segment $z_j$ (delimited by the short transverse lines) and the transmission and coupling amplitudes at each directional coupler (represented by the long transverse lines) $t_{w,S,m}$ and $i{k}_{w,S,m}$.

Figure 7

Shift of the resonance wavelength $\mathrm{\Delta }{\lambda }_0$ with respect to its linear value $\lambda =1545.76$ nm as a function of the input power $P$. All curves are linear fits of the numerically calculated data using the finite-element model. Dashed lines are the curves displayed in Fig. 3 for the experimental case of a thermo-optic nonlinearity with $n_T=8.8\times {10}^{-13}{\mathrm{cm}}^2/\mathrm{W}$. Solid lines are obtained by employing a purely Kerr nonlinearity ($n_T=0$) whose strength has been artificially increased to $n_K=8.8\times {10}^{-13}{\mathrm{cm}}^2/\mathrm{W}$ to match the strength of the thermo-optic nonlinearity of the experiment. Light (dark) blue lines correspond to the forward (reverse) configuration.