Oscillations and multistability in two semiconductor ring lasers coupled by a single waveguide

We theoretically study the dynamical behavior of semiconductor ring lasers coupled by a single bus waveguide, both when weakly coupled and when strongly coupled. We provide a detailed analysis of the multistable landscape in the coupled system, analyze the stability of all solutions, and relate the internal dynamics in the individual lasers to the field effectively measured at the output of the waveguide. We show that coupling phases close to $\pi /2$ generally promote instabilities. Finally, our analysis enables us to discuss the advantages and disadvantages for optical memory operation of coupled semiconductor ring lasers versus solitary ones.

Figure 1

The counterpropagating fields in SRL A (B) are referred to as $E_{1\mathrm{A}}$ and $E_{2\mathrm{A}}$ ($E_{1\mathrm{B}}$ and $E_{2\mathrm{B}}$). The total field at the left (right) output is referred to as $E_{\alpha }$ ($E_{\beta }$). The coupling amplitude is referred to as $k_c$, and the coupling phase as ${\varphi }_c$.

Figure 2

Possible combinations in the single-waveguide-coupling configuration when the SRLs operate in the unidirectional regime. (a), (b) Symmetric states, equal output powers. $S_{\mathrm{out}}$ has the high-power modes pointing outward, while $S_{\mathrm{in}}$ has them pointing inward. (c), (d) Asymmetric states, unequal output powers. The $A_{\mathrm{CCW}}$ ($A_{\mathrm{CW}}$) asymmetry introduced by the coupling becomes apparent in cases (c) and (d), when both SRLs are lasing in the same direction.

Figure 3

Bifurcation diagrams of fixed points of the total power at the $\alpha$ port versus the dimensionless bias current $\mu$. Panel (a) shows the bifurcation lines related to the asymmetric states ($A_{\mathrm{CCW}}$, $A_{\mathrm{CW}}$) and the symmetric $S_{\mathrm{out}}$ solution, while panel (b) only depicts the bifurcation lines related to the symmetric $S_{\mathrm{in}}$ solution. Solid, dotted, and dash-dotted lines indicate stable fixed points, while dashed lines indicate unstable fixed points. Blue-colored dotted branches belong to the $A_{\mathrm{CCW}}$ solution and red-colored dash-dotted branches belong to the $A_{\mathrm{CW}}$ solution (see also Fig. 4). Pitchfork (saddle-node) bifurcations are denoted by PF (SN). The bifurcation diagram for the total power at the $\beta$ port is identical. Parameter values: $k_c=0.3k$, ${\varphi }_c=0$.

Figure 4

Bifurcation diagrams of intra-SRL mode amplitudes ($|E_1\mathrm{A}|$, $|E_2\mathrm{A}|$, $|E_1\mathrm{B}|$, $|E_2\mathrm{B}|$, see Fig. 1) and inter-SRL optical phase differences (${\chi }_1$, ${\chi }_2$). Solid, dotted, and dash-dotted lines indicate stable fixed points, while dashed lines indicate unstable fixed points. Blue-colored dotted branches belong to the $A_{\mathrm{CCW}}$ solution and red-colored dash-dotted branches belong to the $A_{\mathrm{CW}}$ solution (see also Fig. 3). Pitchfork bifurcations are denoted by PF. Parameter values: $k_c=0.3k$, ${\varphi }_c=0$.

Figure 5

Bifurcation diagrams with conventions as in Fig. 3. In the regions delimited by the Hopf bifurcations $H$${}_1$ to $H$${}_5$, there exists at least one stable limit cycle. Parameter values: $k_c=0.3k$, ${\varphi }_c=\pi /2$.

Figure 6

Oscillatory solutions from Fig. 5. From top to bottom: power of the modes in SRL A ($P_1\mathrm{A}={\left|E_1\mathrm{A}\right|}^2$ and $P_2\mathrm{A}={\left|E_2\mathrm{A}\right|}^2$), power of the modes in SRL B ($P_1\mathrm{B}={\left|E_1\mathrm{B}\right|}^2$ and $P_2\mathrm{B}={\left|E_2\mathrm{B}\right|}^2$), and power at the outputs $\alpha$ and $\beta$ ($P_{\alpha }$ and $P_{\beta }$). (a) $\mu =1.1$ (related to $H$${}_1$). (b) $\mu =1.38$ (limit cycles related to $H$${}_3$ and $H$${}_5$). Parameter values: $k_c=0.3k$, ${\varphi }_c=\pi /2$.

Figure 7

Two-parameter continuation in $(\mu ,{\varphi }_c)$ of the Hopf bifurcations responsible for the oscillatory regimes in Fig. 5. They are defined as $H$${}_1$ to $H$${}_5$ for increasing $\mu$ in Fig. 5. The solid (dashed) line indicates a Hopf bifurcation of a stable (unstable) structure. RO stands for relaxation oscillations and AO stands for alternate oscillations. The red crosses indicate the bifurcations points of Fig. 5, where ${\varphi }_c=\pi /2$. The red circles are branch points. Parameter values: $k_c=0.3k$.

Figure 8

Numerical simulation of Eqs. (1) depicting the intra-SRL (${\psi }_{\mathrm{A},\mathrm{B}}$) and inter-SRL (${\chi }_{1,2}$) phase differences in the asymmetric state $A_{\mathrm{CCW}}$ as a function of the coupling phase ${\varphi }_c$. The intra-SRL phase differences ${\psi }_{\mathrm{A}}$ and ${\psi }_{\mathrm{B}}$ are relatively insensitive to variations of ${\varphi }_c$. The inter-SRL phase difference between the high-power CCW modes ${\chi }_2+{\varphi }_c$ is kept fixed, while the phase difference between the low-power CW modes ${\chi }_1-{\varphi }_c$ spans the whole $[0,2\pi ]$ interval for ${\varphi }_c$ going from 0 to $\pi$. Parameter values: $\mu =2$, $k_c=0.3k$.

Figure 9

Numerical simulations of Eqs. (1). (a) Steady-state power levels at the $\alpha$ port (left axis, black) and $\beta$ port (right axis, red) of the $A_{\mathrm{CCW}}$ state as a function of the coupling phase ${\varphi }_c$. The minimum value of $P_{\beta }$ is $6\times {10}^{-5}$. (b) Power suppression ratio (PSR) of the $A_{\mathrm{CCW}}$ state as a function of the coupling phase ${\varphi }_c$. Parameter values: $k_c=0.3k$, $\mu =3$.

Figure 10

Bifurcation diagrams with conventions as in Fig. 3. Parameter values: $k_c=20k$, ${\varphi }_c=0$.

Figure 11

Orbit diagram depicting local extrema constructed by numerical simulation of Eq. (1). The orbit diagram was initialized in the asymmetric state $A_{\mathrm{CCW}}$ at $\mu =20$ before decreasing the current to $\mu =1$. Parameter values: $k_c=20k$, ${\varphi }_c=0$.

Figure 12

(a) Two-parameter continuation in $(\mu ,{\varphi }_c)$ of $H$${}_{\mathrm{A}}$ in Fig. 10. Parameter values: $k_c=20k$. (b) Two-parameter continuation in $(\mu ,k_c)$ of $H$${}_{\mathrm{A}}$ in Fig. 10. The solid (dashed) line indicates a Hopf bifurcation of a stable (unstable) structure. Note that the current ${\mu }_{\mathrm{H}}$ at which the bifurcation takes place hardly changes with the coupling phase ${\varphi }_c$, but drastically changes with $k_c$. Parameter values: ${\varphi }_c=0$.

Figure 13

(a) Bifurcation diagram with conventions as in Fig. 3. Parameter values: $k_c=100k$, ${\varphi }_c=0$. (b) Two-parameter continuation in $(\mu ,{\varphi }_c)$ of $S$${}_{\mathrm{A}}$ in (a). Parameter values: $k_c=100k$.