Self-frequency shift and nonlinear interaction of equilibrium and pulsating solutions in the presence of linear and nonlinear gain, spectral filtering, and intrapulse Raman scattering

Abstract

In this work we study numerically the self-frequency shift of the equilibrium and pulsating solutions in the presence of linear and nonlinear gain, spectral filtering, and intrapulse Raman scattering (IRS). It has been established that the stationary value of the central frequency of solutions, which appears as a result of the collective action of these physical effects, is a linearly increasing function of the Raman parameter c for the solution of Tsoy and Akhmediev (Phys Lett A 343:17-422, 2005), Tsoy et al. (Phys Rev E 73:036621, 2006), and the quadratic function of c for the parameters used in Uzunov et al. (Phys Rev E 90:042906, 2014). We have found a complete suppression of the self-frequency shift of the equilibrium and pulsating solutions in the presence of linear and nonlinear gain, spectral filtering, and IRS below and at the point of the Poincare-Andronov-Hopf bifurcation (PAHB) (Uzunov et al. in Phys Rev E 90:042906, 2014). We have numerically observed stable pairs and sequences of equidistant equilibrium solutions and pulsating solutions propagating in the presence of linear and nonlinear gain, spectral filtering, and IRS below and at the point of the PAHB. The pairs and equidistant sequences of pulsating solutions at the point of bifurcation require larger initial separation and exist at smaller distances of propagation than the pairs and equidistant sequences of the equilibrium solutions below the point of bifurcation.

Paths from stationary to chaos in passively mode-locked fiber lasers: research progress of soliton pulsations and soliton explosions

“…Even though numerical simulations have demonstrated the existence of pulsating solutions and their period bifurcations, the search for analytical expressions of pulsating solutions and bifurcation boundaries has been elusive. The moment method [66][67][68], the Lagrange equation [69], or the variational method [67,70,71] can be used to reduce the infinite-dimensional CQGLEs to the five-dimensional model, then the nonlinear part is solved by the RK4IP method and the linear part is solving in Fourier domain [36,66,72,73], which implements the modeling of passively mode-locked fiber lasers. Researchers uncovered that all soliton solutions of CQGLEs had regular evolution patterns of amplitude, frequency, position, width, and chirp [70].…”

“…The mutual transformation between stationary solitons and soliton pulsations may be realized by changing higher-order effects such as IRS, SS, and TOD [67,69], and the former two effects (IRS, SS) have a stronger influence on soliton pulsations than the third one (TOD) [66]. In the presence of linear gain, nonlinear gain, gain saturation effect, spectral filtering effect, and IRS, the self-frequency shift of stationary solitons and soliton pulsations can be completely suppressed [73]. Moreover, the extreme dissipative soliton pulsations were observed with the change of quintic nonlinear gain coefficient [36], demonstrating that the intra-cavity gain saturation effect can affect the character of soliton pulsations [76].…”