Subharmonic Entrainment Breather Solitons in Ultrafast Lasers

We study theoretically and experimentally the subharmonic entrainment (SHE) breather soliton in mode-locked lasers for the first time, in which the ratio of the breather period to the round-trip time is an integer. We build a non-Hermitian degeneracy map of breather soliton, and illustrate that SHE arises between the two exceptional points (EPs). We obtain SHE at the ratio of 20, observe the evolution of breather soliton when tuning the gain and/or cavity loss, and prove that this phenomenon can improve the stability of breather soliton. Our research brings insight into the EP physics of ultrafast lasers and makes the mode-locked laser a powerful test bed for non-Hermitian degeneracy, which may open a new course in ultrafast laser research.

Figure 1

Dynamics of non-Hermitian degeneracy in breather soliton laser. (a) Breather soliton train (the gray line). The blue dots are the breather amplitude at breather period $T_b$, and the period of the red line is $n{T}_r$. (b) Conceptual graph of the modes $f_{nr0}$. The adjacent modes have phase difference $2\pi /n$. The gray line is the soliton train. (c) The blue solid curve (red dashed line) shows the dependence of $f_b$ on $f_{b0}$ when $\kappa \ne 0$ ($\kappa =0$) as per Eq. (2). The upper left (lower right) inset represents the dynamics of breather frequency shift between (out) the two EPs.

Figure 2

Breather soliton laser. (a) The experimental setup. The left part is the mode-locked fiber laser and the right part is the real-time detection system. (b)–(d) Pulse train, frequency spectrum, autocorrelation trace of the generated pulses when the pump power is 404 mW. (d) is the autocorrelation of time-averaged profile of the breather soliton because the profile oscillates at the breather frequency. OC, optical coupler; WDM, wavelength division multiplexer; EFs, extra fibers; PDs, photodetectors, 40 GHz; OC1, $50:50$; OC2, $50:50$; OC3, $35:65$.

Figure 3

Observation of breather solitons when the pump power is 404 mW (a)–(c) and 414 mW (d)–(f). (a),(d) DFT recording single-shot spectra over 100 consecutive round-trips. The black lines are the corresponding pulse intensity. (b),(e) Interferograms of adjacent pulses. (c),(f) Interferograms picked from (b),(e) with interval of 20 round-trips.

Figure 4

Retrieved parameters of breather soliton. (a),(b) Evolution of breather soliton when the pump power is 404 mW (a) and 414 mW (b). The upper curves are the pulse energy evolution, the middle and the lower curves are the time separation and relative phase retrieved from interferograms respectively. (c),(d) Relation of retrieved time separation and relative phase in (a),(b) over 3000 consecutive round-trips. The radius and angle represent $\tau$ and $\phi .\delta \tau =\tau -{\tau }_0$ represents the change of $\tau$ (${\tau }_0$ is a reference).

Figure 5

Evolution of breather solitons. (a) $f_b$ vs the pump power. The blue points are the detected frequency, and the gray curve represents the approximations of $f_{b0}$. The five red points are picked to calculate the breather frequency spectrum. (b) The breather profiles at different pump power in the plateau of (a). Each profile is averaged over 4300 period. (c) The relative change of different round-trips in (b). (d) Frequency spectra near the breather frequency. The five spectra correspond to the five red points in (a), and the pump power increases from bottom to top. The $y$ axis is in arbitrary unit and the offset is 15 dB.

Figure 6

Simulation of SHE breather soliton. (a),(b) Peak power evolution over 3000 round-trips in the free-running (a) and SHE (b) cases. $x(y)$ axis represents the peak power at the $m$ ($m+1$) round-trip. (c) Breather frequency $f_b$ vs ${\theta }_2$. (d) $f_b$ vs $f_{b0}$. The blue dots are obtained from the simulations, and the red line is the fit using Eq. (2). (e) The close-up of (d) at the vicinity of the plateau. The red line is the parabolic fit of the blue dots.