# Properties of Phase Solitons in an Optically Driven Semiconductor Ring Laser
## 2 Methods  
We consider a semiconductor ring laser made by an antireflection coated semiconductor medium (few mm long) placed in a long ring resonator of length \(L\) (about 1 m). The semiconductor medium is pumped electrically and the laser is driven by a coherent field injected into the cavity. The long cavity combined with the large gainwidth allow highly multimode operation. In Reference [11] the system was studied using a set of effective Maxwell-Bloch equation where the coherent dynamics of the active medium was described by an effective macroscopic polarisation variable. However, very soon it was realised that as long as phase solitons are concerned a rate equation model suffices, where the polarisation variable is adiabatically eliminated [12]. The equations are  
\[\frac{\partial E}{\partial z}+\frac{\partial E}{\partial t} = \sigma\left[E_{I}-\left(1+i\theta\right)E+\left(1-i\alpha\right) DE\right]\,, \tag{1}\] \[\frac{\partial D}{\partial t} = b\left[\mu-\left(1+\left|E\right|^{2}\right)D\right]\,. \tag{2}\]  
The dynamical variables are the dimensionless slowly varying envelope of the electric field \(E\), which obeys a periodic boundary condition and the gain \(D=\zeta(N/N_{0}-1)\), where \(\zeta\) is a dimensionless parameter of order 1 [17], \(N\) is the carrier density and \(N_{0}\) its transparency value, hence \(D\) is proportional to the excess of carriers with respect to transparency. We neglect the dependence of \(E\) and \(D\) on the transverse coordinates. The parameters are the mismatch \(\theta=(\omega_{c}-\omega_{I})\tau_{p}\) between the cavity frequency \(\omega_{c}\) and the frequency of the injected field \(\omega_{I}\) multiplied by the photon lifetime, the dimensionless amplitude \(E_{I}\) of the injected field, the linewidth enhancement \(\alpha\) and the pump parameter \(\mu=\zeta(I/I_{0}-1)\), where \(I\) is the injected current and \(I_{0}\) its transparency value. At the lasing threshold we have \(\mu_{\mathrm{thr}}=1\), which means that the threshold current is \(I_{\mathrm{thr}}=I_{0}(1+\zeta^{-1})\). The longitudinal coordinate \(z\) is scaled to the length of the active medium and time is scaled to the cavity roundtrip time \(\tau_{R}=L/c\), being \(L\) the cavity length and \(c\) the light speed in vacuum. Consequently the decay rates of the electric field and of the population variable are defined as  
\[\sigma=\frac{\tau_{R}}{\tau_{p}}\,,\qquad b=\frac{\tau_{R}}{\tau_{c}}\,, \tag{3}\]  
where \(\tau_{p}\) and \(\tau_{c}\) are, respectively, the photon and carrier lifetimes. Since the photon lifetime is by definition equal to the roundtrip time \(\tau_{R}\) divided by cavity losses, the parameter \(\sigma\) coincides with the cavity losses. The carrier lifetime \(\tau_{c}\) for a semiconductor laser is of the order of 0.1-1 ns and the roundtrip time is on the order of some ns for a cavity length on the order of 1 m, hence the parameter \(b\) is on the order of 10-100. For suitable values of \(\mu\), \(\alpha\) and \(\theta\) the stationary homogeneous solution is bistable. In this paper we fix the values \(\mu=1.1\), which means that the laser is 10% above threshold, \(\alpha=3\), which is a typical value for semiconductor lasers and \(\theta=-2.7\), so that \(\theta+\alpha=0.3\). Since \(\theta+\alpha=(\omega_{L}-\omega_{I})\tau_{p}\), where \(\omega_{L}\) is the laser frequency [18], the latter choice means that the frequency of the driving field is some GHz larger than that of the nearest empty cavity resonance but smaller than the lasing frequency. In Figure 1 we show the S-shaped homogeneous solution together with the branch of stable phase solitons obtained with that choice of the parameters and setting also \(\sigma=0.3\), \(b=50\). The phase solitons are obtained letting the system evolve from an initial state whose amplitude is that of the state \(A\) but to which a phase profile in the form of a positive phase kink from \(0\) to \(2\pi\) is superimposed.