# Two Polarization Comb Dynamics in VCSELs Subject to Optical Injection
## 2 Experimental Polarization Dynamics
### Impact of the Injected Comb Spacing  
We first analyze the nonlinear polarization dynamics of a VCSEL under a narrow optical comb injection. To this end, we fix the bias current at I = 6 mA and then scan the plane of injection parameters, namely the detuning frequency and injected power that leads to various polarization mode competition. That bias current corresponds to a relaxation oscillation frequency of \(\omega_{RO}=4.2\) GHz. Figure 2 describes the sequence of bifurcations leading to the excitation of the normally depressed polarization mode (Y-PM) accompanied by unlocked time-periodic dynamics, which extends the injected comb to a much broader optical spectrum. Figure 2 is obtained for a small injected comb spacing, \(\Omega=1\) GHz and fixed detuning frequency \(\Delta\
u=1.6\) GHz when varying the injected power, \(P_{inj}\). The VCSEL output shows stable free-running operation at a very low injected power (see Figure 2a). A small increase in the injected power destabilizes the VCSEL output to modulated signal as a result of nonlinear wave-mixing in the dominant polarization mode (X-PM) (Figure 2b). In agreement with our previous publications [26; 27], this nonlinear wave mixing takes place at the detuning frequency and a new frequency that depends on the injected comb spacing. The number of frequency lines involved in the nonlinear wave-mixing decreases with the injected power, as shown in Figure 2c. The VCSEL output is then an unstable comb with a high noise pedestal in X-PM at the injected comb repetition rate. By fine-tuning the injected power, instead of stabilizing the comb in a single polarization mode, a new bifurcation scenario results in the excitation of the free-running normally depressed polarization mode  
Figure 1: (**a**) Setup for frequency comb injection into a single-mode VCSEL. TL: Tunable Laser, EDFA: amplifier, P.C: Polarization Controller, AWG: Arbitrary Waveform Generator, MZM: Mach–Zehnder Modulator, VOA: Variable Optical Attenuator, OSA: Optical Spectrum Analyser, PD: photodiode, ESA: Electrical spectrum analyzer. (**b–d**) correspond to the optical spectra of the injected comb for comb spacing of \(\Omega=1\) GHz, \(\Omega=2\) GHz and \(\Omega=4\) GHz, respectively. (**e**) shows the optical spectrum of the VCSEL in free running. of the VCSEL. Interestingly, this polarization mode competition induces in the VCSEL two optical frequency combs with orthogonal polarization directions, as seen in Figure 2d. The two polarization combs have comparable power and the same repetition rate as the injected one with nine frequency lines in the X-PM comb and eight in the Y-PM comb at \(-30\) dB from the maximum. The strongest comb line in the optical spectrum of Y-PM appears at the frequency position of the free-running Y-PM, i.e., its frequency position is ruled by the VCSEL linear birefringence. When we keep increasing the injected power, harmonic frequency comb dynamics takes place simultaneously in the two polarization combs, as shown in Figure 2e,f. The harmonic comb dynamics result in a reduction in comb repetition rate. Subharmonic bifurcations are well studied and understood phenomena that occur naturally in periodically forced oscillators [41]. Subharmonic dynamics have been expected to be a form of frequency locking different from the Adler type in diode laser optically injected with optical comb [24; 25]. They appear when the detuning frequency is close to the rational fraction of the injected comb spacing. When we increase the injected power, the harmonic polarization comb destabilizes to complex polarization dynamics in X-PM and Y-PM (Figure 2g). It is worth observing that the X-PM still shows some remarkable comb lines at the repetition rate of the injected comb. A further increase in injected power results in a progressive suppression of all the power in the depressed polarization mode Y-PM accompanied by the stabilization of the comb dynamics in the dominant polarization mode X-PM, as seen in Figure 2h,i.