# Two Polarization Comb Dynamics in VCSELs Subject to Optical Injection
## 3 Theoretical Bifurcation Analysis  
In addition to experimental results, theoretical studies have been performed to provide further insight into the polarization dynamics of comb injected VCSELs. We have used the Spin-Flip Model (SFM) [42], which is widely used to describe polarization dynamics in free-running VCSELs and supplemented it with additional term describing optical comb injection. SFM parameters are chosen such that the dominant polarization mode (normally depressed polarization mode) of the VCSEL in free-running emits along the X-axis ((Y-axis). Model equations are given in Equations (1)-(4). In these equations, \(E_{x}\) and \(E_{y}\) are the two linearly polarized slowly varying components of the complex fields in the X and Y directions, respectively. \(D\) and \(n\) are the sum and the difference between the population inversions for spin-up and spin-down radiation channels. \(\mu\) is the normalized current injection.  
\[\frac{dE_{x}}{dt}=-(\kappa+\gamma_{a})E_{x}-i(\kappa\alpha+\gamma_{p})E_{x}+ \kappa(1+i\alpha)(DE_{x}+inE_{y})+E_{M}, \tag{1}\]  
\[\frac{dE_{y}}{dt}=-(\kappa-\gamma_{a})E_{y}-i(\kappa\alpha-\gamma_{p})E_{y}+ \kappa(1+i\alpha)(DE_{y}-inE_{x}), \tag{2}\]  
\[\frac{dD}{dt}=-\gamma[D(1+\left|E_{x}\right|^{2}+\left|E_{y}\right|^{2})-\mu +in(E_{y}E_{x}^{*}-E_{x}E_{y}^{*})], \tag{3}\]  
\[\frac{dn}{dt}=-\gamma_{s}n-\gamma[n(\left|E_{x}\right|^{2}+\left|E_{y}\right|^ {2})+iD(E_{y}E_{x}^{*}-E_{x}E_{y}^{*})]. \tag{4}\]  
The meaning of the remaining SFM parameters is the following: \(\gamma_{p}\) and \(\gamma_{a}\) correspond to linear birefringence and linear dichroism, respectively; \(\gamma_{s}\) is the spin-flip relaxation rate, \(\gamma\) is the decay rate of \(D\), \(\kappa\) is the field decay rate and \(\alpha\) is the linewidth enhancement factor.  
The complex electrical field of the injected comb is written as follows:  
\[E_{M}=\kappa_{inj}\sum_{j}E_{j}(t)e^{i(2\pi\
u_{j}t+\varphi_{j}(t))} \tag{5}\]  
with angular frequency \(\omega_{j}\) and amplitude \(E_{j}\) corresponding to \(j_{th}\) comb lines and coupling coefficient \(\kappa_{inj}\). We consider an optimal coupling between the VCSEL output and the injected comb, which correspond to \(\kappa_{inj}=\kappa\). We consider a comb with three frequency lines and we simplify the calculations by supposing \(\varphi_{j}=0\), i.e., the phase of the individual injected comb lines is zero. The detuning frequency \(\Delta\
u_{j}\) is the difference between \(\
u_{j}\) and the intermediate frequency between those of X and Y polarizations, \(\frac{2\pi\
u_{x}+2\pi\
u_{y}}{2}\), with \(2\pi\
u_{x}=\alpha\gamma_{a}-\gamma_{p}\) and \(2\pi\
u_{y}=\gamma_{p}-\alpha\gamma_{a}\), the frequency corresponding to the linear polarization mode X and Y, respectively.  
SFM parameters are taken as \(\kappa=33\) ns\({}^{-1}\), \(\gamma_{a}=-0.1\) ns\({}^{-1}\), \(\alpha=2.8\), \(\mu=2.29\), \(\gamma_{p}=9\) GHz, \(\gamma=2.08\) ns\({}^{-1}\) and \(\gamma_{s}=2100\) ns\({}^{-1}\). These numerical values are taken from [38]. In the following, we shall use \(\kappa_{inj}\) for injection strength, with \(\kappa_{inj}=\frac{E_{inj}}{\text{F}_{0}}\), where \(E_{inj}\) and \(E_{0}\) are the total amplitude of the injected field and the total amplitude of the VCSEL in free-running mode, respectively. Rate Equations (1)-(4) are numerically integrated using a fourth-order Runge-Kutta method. Numerical simulations are typically performed with a times series of 200 ns and a time step equal to 1.2 ps. The detuning frequency value is defined from the central injected comb line and is referred to as \(\Delta\
u\). In this section, we model the experiment by considering that the difference between the amplitude of the central injected comb line and the side comb lines is 12 dB.