Analytic instability thresholds in folded Kerr resonators of arbitrary finesse

We present analytic threshold formulas applicable to both dispersive (time-domain) and diffractive (pattern-forming) instabilities in Fabry-Perot Kerr cavities of arbitrary finesse. We do so by extending the gain-circle technique, recently developed for counterpropagating fields in single-mirror-feedback systems, to allow for an input mirror. In time-domain counterpropagating systems, walk-off effects are known to suppress cross-phase modulation contributions to dispersive instabilities. Applying the gain-circle approach with appropriately adjusted cross-phase couplings extends previous results to arbitrary finesse, beyond mean-field approximations, and describes Ikeda instabilities.

Figure 1

A FP cavity of length $L$, filled with a nonlinear Kerr medium. A uniform pump field $F_{\text{in}}$ enters at $z=-L$ through a mirror of reflectivity $r$ and drives forward ($F$) and backward ($B$) fields in the cavity, establishing zero-order cavity fields $F_0$ and $B_0$, whose stability depends on relative perturbations $f$ and $b$. A transmitted field $F_{\mathrm{tr}}$ exits the cavity at $z=0$.

Figure 2

Unit circle (thick) centered on the origin of the complex plane. A gain circle (dashed) intersects the unit circle at points ($g_1,g_2$) given by (7). The center $C_g$ and radius $R_g$ of the gain circle are given by (8) for a FP Kerr cavity. Intersection of the gain circle and any loss circle of radius $r, 1>r>r_{\text{crit}}$ enables threshold condition (9) to be satisfied for two different values of $\varphi$ at that reflectivity $r$. At $r=r_{\text{crit}}$, the gain and loss circles touch, marking the limit of instability. $(\theta ,IL)=(0.4,0.1)$.