Spontaneous Symmetry Breaking, Instability, and Chaos in Ring Resonators

Michael Woodley
National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK; Heriot-Watt University, Edinburgh Campus, Edinburgh, EH14 4AS, UK.

When a ring resonator is pumped with laser light of sufficient intensity, then the refractive index -- and so the resonant frequency -- of the resonator can be modulated by the intensity of the light within it -- a phenomenon known as the Kerr nonlinearity. If the resonator is pumped with two laser beams, then this effect can give rise to spontaneous symmetry breaking in the two optical modes within the resonator. We present analytical, numerical, and experimental evidence for a rich range of exotic behaviours exhibited by this symmetry-broken light, including oscillations (implying periodic energy exchange between the modes), period-doubling, and chaos. These optical modes are described by the following coupled system of ordinary differential equations:

$$\dot{e}_{1,2}=\tilde{e}_{1,2} -[1+i(A|e_{1,2}|^{2}+B|e_{2,1}|^{2}-\Delta_{1,2})]e_{1,2},$$

where $\tilde{e}_{1,2}$ and $e_{1,2}$ are the input and coupled electric field amplitudes for each beam, respectively, and $\Delta_{1,2}$ are the frequency detunings of the laser beams, with respect to the non-Kerr-shifted cavity resonance frequency. The coefficients $A$ and $B$ denote the strengths of self- and cross-phase modulation, respectively -- i.e., the extent to which the modes interact with themselves and with each other. The physics of this dynamical system is not only of fundamental interest, but is also important for the construction of integrated all-optical circuitry and devices, such as isolators, circulators, logic gates, advanced sensors, oscillators, and scramblers.