Coherent energy exchange between vector soliton components in fiber lasers

We report on the experimental evidence of four wave mixing (FWM) between the two polarization components of a vector soliton formed in a passively mode-locked fiber laser. Extra spectral sidebands with out-of-phase intensity variation between the polarization resolved soliton spectra was firstly observed, which was identified to be caused by the energy exchange between the two soliton polarization components. Other features of the FWM spectral sidebands and the soliton internal FWM were also experimentally investigated and numerically confirmed.

Passive mode-locking of erbium-doped fiber lasers with a semiconductor saturable absorber mirror (SESAM) has been extensively investigated [1,2]. In contrast to the nonlinear polarization rotation (NPR) mode-locking, mode-locking incorporating a SESAM does not require any polarization element inside the laser cavity, thereby under suitable condition of the cavity birefringence, vector solitons could be formed in the lasers [3]. Recently, it was reported that even the polarization-locked vector soliton (PLVS) could be formed in the mode-locked fiber lasers [4,5]. Formation of a PLVS requires not only that the group velocities of the two orthogonal polarization components of a vector soliton are locked but also that their phase velocities are also locked. It is well known that through self-phase modulation (SPM) and cross-phase modulation (XPM), nonlinear interaction between the two polarization-modes of a fiber could result in group velocity locked vector solitons [6]. Although it was also pointed out that the four-wavemixing (also called coherent energy exchange) coupling between the polarization components of a vector soliton could have contributed to the formation of the phase locked vector solitons [4,5], so far no experimental evidence of soliton internal FWM has been presented. In this Letter we report on the experimental observation of FWM between the two orthogonal polarization components of a vector soliton formed in a fiber laser passively mode locked with a SESAM. Energy exchange between the two orthogonal polarization components of vector solitons was observed at specific frequencies on the soliton spectrum. However, our experimental results showed that the existence of FWM didn't guarantee formation of PLVS.
The fiber laser is illustrated in Fig.1. It has a ring cavity consisting of a piece of 4.6 m Erbium-doped fiber (EDF) with group velocity dispersion parameter 10 ps/km/nm and a total length of 5.4 m standard single mode fiber (SMF) with group velocity dispersion parameter 18 ps/km/nm. The cavity has a length of 4.6 EDF +5.4 SMF =10m. Note that within one cavity round-trip the signal propagates twice in the SMF between the circulator and the SESAM. A circulator is used to force the unidirectional operation of the ring and simultaneously to incorporate the SESAM in the cavity. An intra cavity polarization controller is used to change the cavity's linear birefringence.  To verify our experimental observations and determine the extra sideband formation mechanism, we also numerically simulated the FWM in the laser. We used the following coupled Ginzburg-Landau equations to describe the pulse propagation in the weakly birefringent fibers in the cavity: Where, u and v are the normalized envelopes of the optical pulses along the two orthogonal polarized modes of the optical fiber. 2β = 2πΔn/λ is the wave-number difference between the two modes. 2δ = 2βλ/2πc is the inverse group velocity difference. k′ is the second order dispersion coefficient, k ″ is the third order dispersion coefficient and represents the nonlinearity of the fiber. g is the saturable gain coefficient of the fiber and Ω g is the bandwidth of the laser gain. For undoped fibers g=0; for erbium doped fiber, we considered its gain saturation as where G is the small signal gain coefficient and P sat is the normalized saturation energy.
The saturable absorption of the SESAM is described by the rate equation [8]: Where T rec is the absorption recovery time, l 0 is the initial absorption of the absorber, and E sat is the absorber saturation energy. To make the simulation possibly close to the experimental situation, we used the following parameters: γ=3 W -1 km -1 , Ω g =24nm, P sat =100 pJ, k″ SMF =-23 ps 2 /km, k″ EDF =-13 ps 2 /km, k′″=-0.13 ps 3 /km, E sat =1 pJ, l 0 =0.15, and T rec = 6 ps, Cavity length L= 10 m. The numerical simulations well reproduced the extra spectral sidebands and confirmed that their appearance is indeed caused by the FWM between the orthogonal soliton components. The result could also be easily understood. Due to small linear cavity birefringence, coherent coupling between the two polarization components of a vector soliton can no longer be neglected. Its existence causes coherent energy exchange between the two orthogonal soliton polarization components. Nevertheless, as far as the linear cavity birefringence is not zero, energy exchange does not occur at whole soliton spectrum, but only at certain wavelengths where the phase matching condition is fulfilled, which then leads to the formation of the discrete extra spectral sidebands.
In conclusion, we have experimentally observed extra spectral sideband generation on the soliton spectra of the phase locked vector solitons in a passively mode-locked fiber ring laser. Polarization resolved study on the soliton spectrum reveal that they are caused by the coherence energy exchange between the two orthogonal polarization components of the vector solitons. Numerical simulations have confirmed our experimental observation.
Especially, numerical simulations show that FWM always exists under weak cavity birefringence. As far as the net cavity birefringence is not zero, phase matching condition can only be fulfilled at certain wavelengths. Our studies suggest that appearance of the sidebands is not a characteristic of the vector soliton polarization evolution, but the FWM between the components of a vector soliton.