# Properties of Phase Solitons in an Optically Driven Semiconductor Ring Laser
## 3 Results  
Starting from the choice of the parameters \(\sigma\) and \(b\) of Figure 2 we investigated how the properties of the phase solitons depend on those two parameters. For fixed \(b=50\) we varied \(\sigma\) from 0.1 to 0.9 by steps of 0.1 and for fixed \(\sigma=0.3\) we varied \(b\) from 10 to 100 by steps of 10. In our study we focused mainly on two properties of the phase solitons: their velocity and their width or duration. As mentioned above, the propagation velocity of the phase soliton is related to the inclination of the straight line that depicts the phase soliton in the space-time diagrams. We calculate the propagation velocity in the following way [13]. Let us call \(\Delta\tau\) the delay accumulated during \(N\) roundtrips. Then the velocity is  
\[v=\frac{NL}{N\tau_{c}+\Delta\tau}=\frac{L/\tau_{c}}{1+\Delta\tau/(N\tau_{c})}= \frac{c}{1+\Delta\tau/(N\tau_{c})} \tag{4}\]  
and  
\[\frac{v}{c}\approx 1-\frac{\Delta\tau}{N\tau_{c}}\,, \tag{5}\]  
since \(\Delta\tau\ll N\tau_{c}\). As shown in panel (b) of Figure 2 in presence of a phase soliton the field intensity is approximately uniform for most of the time and approximately equal to the intensity of the stable fixed point \(A\). This intensity level is the background for the phase soliton and we call it \(l_{\rm{bg}}\). The phase soliton consists of a deep negative pulse where the intensity reaches its minimum value \(I_{\rm{min}}\) followed by a small positive pulse where the intensity reaches its maximum value \(I_{\rm{max}}\). If we introduce the intensities  
\[I_{1}=\frac{I_{\rm{min}}+I_{\rm{bg}}}{2}\,,\qquad I_{2}=\frac{I_{\rm{max}}+I_{ \rm{bg}}}{2}\,, \tag{6}\]  
we can define the duration of the phase soliton or its width, \(\tau_{ps}=\tau_{2}-\tau_{1}\) as the interval between the instant \(\tau_{1}\) when the intensity reaches for the first time the value \(I_{1}\) and the instant \(\tau_{2}\) when the intensity reaches for the second time the value \(I_{2}\). As a first result we found that when \(b\) is fixed and \(\sigma\) is varied the only property of the phase soliton that changes is its velocity. In Figure 3 we show the space-time plots for \(b=50\) fixed and three values of \(\sigma\): \(\sigma=0.1\) in (a), \(\sigma=0.5\) in (b) and \(\sigma=0.9\) in (c). The propagation velocity of the phase soliton decreases with \(\sigma\), while its shape, spectrum and trajectory in the complex plane remain unchanged. Figure 3: Spatio-temporal diagram of the field intensity over 500 roundtrips for \(b=50\) and (a) \(\sigma=0.1\), (b) \(\sigma=0.5\), (c) \(\sigma=0.9\). As \(\sigma\) increases the slope of the vertical line associated with the phase soliton decreases, indicating that the phase soliton moves more slowly. The shape of the pulse, instead, does not change. The situation is different when \(\sigma\) is kept fixed and \(b\) is varied. Figures 4 and 5 show the phase solitons for the same value \(\sigma=0.3\) as in Figure 2 but \(b=10\) in Figure 4 and \(b=100\) in Figure 5. Here both the velocity and the shape of the phase soliton change considerably and consequently also the spectrum. The phase soliton is much narrower and much faster for \(b=100\) and the phase kink is also much steeper. Only the shape of the trajectory in the complex plane is the same in the two cases but it must be considered that in Figure 5 the representative point remains close to the fixed point \(A\) for almost all the roundtrip time and the trajectory around the origin is travelled much faster. The change in shape of the phase solitons is accompanied by a different modal composition. We can compare the panels (e) of Figures 4 and 5. The number of modes with intensity larger than \(-70\) dB is \(39\) for \(b=10\) and \(247\) for \(b=100\). Figure 4: (**a–e**) Same as Figure 2 but with \(b=10\). The phase soliton is wider and slower. Figure 5: (**a–e**) Same as Figure 2 but with \(b=100\). The phase soliton is narrower and faster. An explanation for the above results can be found looking at the structure of the dynamical Equations (1) and (2).