# Two Polarization Comb Dynamics in VCSELs Subject to Optical Injection
## 3 Theoretical Bifurcation Analysis
### Y-PM Comb Dynamics  
Experimental results have demonstrated the possibility of generating two combs with orthogonal polarization in a VCSEL subject to parallel optical injection. We have also shown that this polarization comb performance is limited by the increase in injected comb spacing. In Figure 9, we analyze the two polarization comb numerically by using bifurcation diagrams. Figure 9 is obtained for fixed injected comb spacing \(\Omega=2\) GHz and detuning \(\Delta v_{x}=-9\) GHz. These bifurcation diagrams are plotted by selecting the minima and the maxima of polarization-resolved intensities, \(I_{x,y}=|E_{x,y}|^{2}\), for each injection strength. In Figure 9, the left and right panels correspond to the bifurcation diagrams of X-PM and Y-PM, respectively. Figure 9(a1,b1) show the bifurcation diagrams for fixed linear dichroism \(\gamma_{a}=-0.1\) ns\({}^{-1}\). When increasing the injection strength, the VCSEL output first shows a nonlinear wave-mixing involving the detuning and the injected comb spacing in X-PM. When we kept increasing the injection strength, wave mixing bifurcates to periodic dynamics corresponding to the optical frequency comb in the two linear orthogonal polarization modes. The two polarization comb region is referred to as \"comb\" in Figure 9. The X-PM shows a larger number of lines than the Y-PM, but the number of lines in Y-PM can be controlled with injection parameters. The two comb dynamics remain stable with a simultaneous increase in the number of lines over a large range of injection strengths. Interestingly, by finetuning the injection strength, the VCSEL output shows harmonic comb dynamics simultaneously in the two orthogonal combs at the boundary of the comb regions. A further increase in injection strength results in complex dynamics in X-PM and Y-PM accompanied by the abrupt suppression of all power in Y-PM. Cascade comb dynamics resulting in a very low comb repetition rate are observed in between complex dynamics depending on injection parameters. The complex polarization dynamics bifurcate to a single polarization comb in the dominant polarization mode (X-PM). We next vary linear dichroism to highlight its impact on the two polarization comb dynamics. Interestingly, when decreasing linear dichroism, the size of the two polarization comb region decreases, as observed in Figure 9(a2,a3,b2,b3). The double horizontal arrows indicate how much the size of the combs' regions decreases with the decrease in linear dichroism. It is worth observing that all bifurcation sequences are similar to the case of \(\gamma_{a}=-0.1\) ns\({}^{-1}\). When linear dichroism reaches \(\gamma_{a}=-0.8\) ns\({}^{-1}\), the two polarization comb disappear and Y-PM is suppressed, as shown in Figure 9(a4,b4). Therefore, with the exception of the disappearance of comb lines in Y-PM, the bifurcation scenarios remain similar for X-PM. The bifurcation scenarios described in Figure 9 are in a very good agreement with the experimental ones in Figures 2 and 4. Figure 10 shows an example comb and complex dynamics in the two polarization modes. Figure 10(a1,b1) show the optical spectra of comb dynamics in X-PM and Y-PM, respectively. We observe clearly in these spectra that the bandwidth of the frequency comb in the injected polarization mode (X-PM) is larger than the comb in the depressed polarization mode, Y-PM. As discussed early in the experimental section, these X-PM and Y-PM combs can combine to form a broad comb in the total output power of the VCSEL. Figure 10(a2,b2) show an example of complex polarization dynamics. Depending on injection parameters and injected comb properties, these complex polarization dynamics can simultaneously bifurcate to a two polarization harmonic comb or single-polarization comb. Figure 9 has shown that the decrease in linear dichroism results in the complete suppression of the comb lines in the normally depressed polarization mode (Y-PM). In Figure 11, we provide a further insight on comb generation, whatever the linear dichroism is.