Unidirectional lasing in nonlinear Taiji micro-ring resonators

Abstract

We develop a general formalism to study laser operation in active micro-ring resonators supporting two counterpropagating modes. Our formalism is based on the coupled-mode equations of motion for the field amplitudes in the two counterpropagating modes and a linearized analysis of the small perturbations around the steady state. We show that the devices including an additional S-shaped waveguide establishing an unidirectional coupling between both modes -- the so-called Taiji resonators (TJR) -- feature a preferred chirality on the laser emission and can ultimately lead to unidirectional lasing even in the presence of sizable backscattering. The efficiency of this mode selection process is further reinforced by the Kerr nonlinearity of the material. This stable unidirectional laser operation can be seen as an effective breaking of $\mathcal{T}$-reversal symmetry dynamically induced by the breaking of the $\mathcal{P}$-symmetry of the underlying device geometry. This mechanism appears as a promising building block to ensure non-reciprocal behaviors in integrated photonic networks and topological lasers without the need for magnetic elements.

Figures (11)

• Figure 1: Schematic diagrams of the ring (a) and “Taiji” (b) microresonators. The field amplitudes of the clockwise (CW) and counterclockwise (CCW) modes are denoted by $a_{+}$ and $a_{-}$, respectively. In the Taiji microresonator (TJR) directional couplers of transmittance (coupling) amplitude $t_{\rm S}$ ($ik_{\rm S}$) are signalized with the dashed green rectangles. The loss rate $\gamma_{\rm A}$ accounts for absorption and radiative couplings, for instance with a bus waveguide.
• Figure 2: (a,b) Steady-state intensity of the CW ($|\tilde{a}^{(0)}_{+}|^2$) and CCW ($|\tilde{a}^{(0)}_{-}|^2$) modes for a backscattering-free ring resonator laser as a function of the pump rate $P_{0}$. Panel (a) refers to a nonlocal thermo-optic nonlinearity ($g=1$), while panel (b) corresponds to a local Kerr nonlinearity ($g=2$). The black dashed line represents the total intensity summed over the two directions. (c,d) Real part of the system eigenvalues $\lambda$ for a ring resonator laser as a function of the pump rate $P_{0}$. Panel (c) refers to a nonlocal thermo-optic nonlinearity ($g=1$), while panel (d) corresponds to a local Kerr nonlinearity ($g=2$). The numbers below the curves indicate their degeneracy.
• Figure 3: (a,b) Steady-state intensity of the CW ($|\tilde{a}^{(0)}_{+}|^2$) and CCW ($|\tilde{a}^{(0)}_{-}|^2$) modes for a backscattering-free TJR laser as a function of the pump rate $P_{0}$. Panel (a) refers to a nonlocal thermo-optic nonlinearity ($g=1$), while panel (b) corresponds to a local Kerr nonlinearity ($g=2$). The black dashed line represents the total intensity summed over the two directions. (c,d) Real part of the system eigenvalues $\lambda$ for a TJR laser as a function of the pump rate $P_{0}$. Panel (c) refers to a nonlocal thermo-optic nonlinearity ($g=1$), while panel (d) corresponds to a local Kerr nonlinearity ($g=2$). The numbers below the curves indicate their degeneracy.
• Figure 4: (a) Pump rate $P_{0}$ vs. square root of the S-coupling losses $\sqrt{\gamma_{\rm S}}$ diagram for a backscattering-free TJR with $g=2$ and $n_{\rm NL}=0$. The region where only the trivial solution is present is depicted in white. The light blue region represents the parameter range where only CCW lasing is possible. In the dark blue region two solutions exist: single-mode lasing in the preferred CCW direction and bidirectional lasing with $|\tilde{a}^{(0)}_{+}|\gg|\tilde{a}^{(0)}_{-}|$. Each region is labeled by the numbers 0, 1, and 2, respectively. (b) Normalized steady-state intensity $|\tilde{a}^{(0)}_{\pm}|^2$ obtained by numerically solving Eq. \ref{['eq:MotionEqsModes']} as a function of $\sqrt{\gamma_{\rm S}}$ for a fixed $P_{0}=5\gamma_{\rm A}$ as indicated by the dashed horizontal line in (a). (c) Real part of the system eigenvalues $\lambda$ as a function of $\sqrt{\gamma_{\rm S}}$ corresponding to the path shown as the dashed horizontal line of panel (a). The points are numerically calculated by diagonalizing the matrix $A$ in Eq. \ref{['eq:DiffEqsMatrix']} for the steady-state solutions displayed in panel (b). Where not visible, $Re\{\lambda_{3}\}$ lies below $Re\{\lambda_{4}\}$. Above $\sqrt{\gamma_{\rm S}/\gamma_{\rm A}}=2$ all real parts are degenerate. The dashed lines correpond to the analytic eigenvalues (\ref{['eq:EigenvaluesBelowThreshold12']}-\ref{['eq:EigenvaluesAboveThreshold4']}).
• Figure 5: Ring resonator with $g=2$ and backscattering parameters $|n|=0.1\gamma_{\rm A}$ and $h=0$. (a) Steady-state intensity $|\tilde{a}^{(0)}_{\pm}|^2$ in each mode as a function of the pump rate $P_{0}$. The dashed line is the backscattering-free intensity. (b) Real part of the fluctuation dynamics eigenvalues $\lambda$ as a function of $P_{0}$. The dashed lines represent the real part of the backscattering-free eigenvalues (\ref{['eq:EigenvaluesBelowThreshold12']}-\ref{['eq:EigenvaluesAboveThreshold4']}). Below threshold, $Re\{\lambda_{1}\}=Re\{\lambda_{2}\}$ and $Re\{\lambda_{3}\}=Re\{\lambda_{4}\}$.