Effects of self- and cross-phase modulation on the spontaneous symmetry breaking of light in ring resonators

Spontaneous symmetry breaking can occur in the powers of two optical modes coupled into a ring resonator, described by a pair of coupled Lorentzian equations, and featuring tunable self- and cross-phase modulation terms. Investigated is a wide variety of nonlinear materials by changing the ratio of the self- and cross-phase interaction coefficients. Static and dynamic effects range from the number and stability of stationary states to the onset and nature of oscillations. Minimal conditions to observe symmetry breaking are provided in terms of the ratio of the self- and cross-phase coefficients, detuning, and input power. Different ratios of the nonlinear coefficients also influence the dynamical regime, where they can induce or suppress bifurcations and oscillations. A generalized description on this kind is useful for the development of all-optical components, such as isolators and oscillators, constructed from a wide variety of optical media in ring resonators.

Figure 1

Ring resonator setups, showing (a) copropagation of two light components with left and right circular polarizations and (b) the counterpropagation of two light beams, linearly polarized along the same axis.

Figure 2

Different graphical representations of spontaneous symmetry breaking when scanning pump power, $I$, shown here for $A=1$, $B=2$, and $\theta =2$. (a) Coupled powers are plotted against each other, Eq. (3). The points $a$ and $b$ indicate the opening and closing of the symmetry-broken bubble given by Eq. (12). Point $c$ is the point at which symmetry breaking occurs at the detuning limit; see Eq. (13). The maximum coupled power difference can be found at points $d$ and $e$; see Eq. (15). (b) The coupled powers are plotted against input power. Note the visible presence of the “symmetric bistability” (the highlighted S-shaped curve), but also the loss of some symmetries seen in (a).

Figure 3

Different graphical representations of spontaneous symmetry breaking when scanning detuning, $\theta$, shown here for $A=1$, $B=2$, and $I=3$. (a) Coupled powers are plotted against each other, Eq. (4). The points $f$ and $h$ indicate the symmetry breaking bifurcation pair for a detuning scan—see Eq. (17). At point $g$, the bubble emerges at the intensity limit—see Eq. (16). (b) Coupled envelope powers are plotted against detuning, $\theta$. Note again the loss of symmetry between (a) and (b).

Figure 4

Example optical bistability of Eq. (6), with limits calculated via Eqs. (7) and (8). Blue dashed line indicates an unstable state; see Sec. 4.

Figure 5

(a) Minimum detuning required to observe symmetry breaking when changing the ratio of the coupling constants $B/A$. The yellow regions indicate where symmetry breaking is possible, with the blue lines indicating the limits where symmetry breaking becomes impossible. (b) Minimum input power required to observe symmetry breaking. The negative values for $I$ are included for mathematical symmetry only—they are clearly not physically attainable.

Figure 6

(a), (b) Plots of the input power scans for $B/A=2.2$, $\theta =1.5$ and $B/A=0.5$, $\theta =4$ respectively. Panels (c), (d) show the special case of $B=0$, $A\in \mathbb{R}$, with $A=1$, $\theta =3$. Panel (c) shows the possibility still for symmetry-broken solutions, while (d) shows how their origin is due to the bistable region of the Lorentzian equation.

Figure 7

Panels (a)–(d) illustrate the coupled power $P_{1,2}$ against the detuning parameter $\theta$ for $A=1$ and $B=0.9$. Stable and nonoscillatory unstable solutions are shown in solid and dotted blue, respectively, whilst oscillatory instabilities are shown in dashed red. The input power, $I$, increases with each frame. (a) $I=9$. No symmetry-broken solutions have yet emerged. (b) $I=18$. Spontaneous symmetry breaking occurs in the unstable symmetric branch. (c) $I=40$. Stable symmetry-broken solutions emerge but quickly lose stability to complex eigenvalues with positive real parts. (d) $I=80$. The entire structure is stretched, including the region of unstable oscillatory and stable symmetry-broken solutions. Note the parameter range with four stable solutions. The right-hand plots illustrate the coupled power, $P_{1,2}$, against the input power, $I$. The key phenomena shown on the left are visible here. (e) $\theta =15$. (f) $\theta =25$. (g) $\theta =75$. (h) $\theta =170$. Note that, in the above plots, growing oscillations are always accompanied by a stable solution so these oscillations may not be experimentally observable.

Figure 8

Coupled power $P_{1,2}$ against the detuning parameter $\theta$ for $A=1$, $B=7$, and $I=3.3$. Stable and nonoscillatory unstable solutions are shown in solid and dotted blue, respectively, whilst oscillatory instabilities are shown in dashed red.

Figure 9

(a) Poincaré sections of the maxima of oscillating coupled power $P_1$ versus the cross-phase modulation coefficient $B$ for constant $A=1$. These points correspond to constant values of the detuning $\theta =5$ and input power $I=3.3$. (b) Illustration of periodic oscillations and deterministic chaos in the Poincaré sections of the maxima of $P_1$ when varying the detuning parameter $\theta$ for a large cross- to self-phase modulation ratio. In this case, $A=1$, $B=7$ and the input power is $I=3.3$. Note the dense columns of chaotic windows. The solid (dashed) black line corresponds to a stable (unstable) symmetric stationary state. The red lines indicate the detuning values used in Fig. 10.

Figure 10

Results of numerical simulations of Eq. (18) for $A=1$, $B=7$, and $I=3.3$ showing oscillatory (a), (b), chaotic switching (c), (d), and periodic switching (e), (f) solutions. Panels (a), (c), and (e) all show the temporal evolutions of the coupled powers, while (b), (d), and (e) show the phase space orbits of the real and imaginary components of the late temporal evolutions of $E_{\pm }$. The detuning parameter for (a), (b) is $\theta =3.6$, for (c), (d) is $\theta =4.31$, and for (e), (f) is $\theta =4.755$. Time is dimensionless.