Observation of a group of dark rogue waves in a telecommunication optical fiber

Over the past decade, the rogue wave debate has stimulated the comparison of predictions and observations among different branches of wave physics, particularly between hydrodynamics and optics, in situations where analogous dynamical behaviors can be identified, thanks to the use of common universal models. Although the scalar nonlinear Schrödinger equation (NLSE) has constantly played a central role for rogue wave investigations, moving beyond the standard NLSE model is relevant and needful for describing more general classes of physical systems and applications. In this direction, the coupled NLSEs are known to play a pivotal role for the understanding of the complex wave dynamics in hydrodynamics and optics. Benefiting from the advanced technology of high-speed telecommunication-grade components, and relying on a careful design of the nonlinear propagation of orthogonally polarized optical pump waves in a randomly birefringent telecom fiber, this work explores, both theoretically and experimentally, the rogue wave dynamics governed by such coupled NLSEs. We report, for the first time, the evidence of a group of three dark rogue waves, the so-called dark three-sister rogue waves, where experiments, numerics, and analytics show a very good consistency.

Figure 1

Experimental setup: red and blue lines depict the two orthogonally polarized pump waves with distinct frequencies. CP: 50/50 fiber couplers; EDFA: Erbium doped fiber amplifier; EOM: electro-optic (intensity) modulator; PC: polarization controller; BS: 10/90 beam splitter; PBS: polarization beam splitter; PM: power-meter.

Figure 2

Contour plot of the two orthogonal polarization waves (a) $|E_1(t^{\prime },z)|$ and (b) $|E_2(t^{\prime },z)|$, describing the analytical three dark rogue wave solution of the coupled NLSEs ($t^{\prime }=t-z/v_g$). Parameters ${\beta }^{″}=18{\mathrm{ps}}^2/\mathrm{km}$, $\gamma =2.4{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, $P_{1,2}=1.25\mathrm{W}$, $\mathrm{\Delta }f=100\mathrm{GHz}$. Moreover: $a=1$, $v_1=-v_2=\delta =-0.77$, $\eta =-0.986$; ${\varepsilon }_1=3.25i,{\varepsilon }_2=-4.25i,{\varepsilon }_3=7i,{\varepsilon }_4=0$.

Figure 3

Contour plot of the two orthogonal polarization waves (a) $|E_1(t^{\prime },z)|$ and (b) $|E_2(t^{\prime },z)|$, describing the numerical excitation of dark rogue waves. Parameters ${\beta }^{″}=18{\mathrm{ps}}^2/\mathrm{km}$, $\gamma =2.4{\mathrm{W}}^{-1}{\mathrm{km}}^{-1}$, $P_{1,2}=1.25\mathrm{W}$, $\mathrm{\Delta }f=100\mathrm{GHz}$. As to the modulating signal, $\epsilon =0.13$ and $f_m=35\mathrm{GHz}$.

Figure 4

Comparison between analytical (red line) and numerical (black dashed line) temporal profiles of $|E_1\left(t^{\prime }\right)|$ and $|E_2\left(t^{\prime }\right)|$ at the input (a),(b), after $3\mathrm{km}$ (c),(d) and after $5\mathrm{km}$ (e),(f). Theoretical results refer to the case reported in Fig. 2, numerical ones to Fig. 3.

Figure 5

Comparison between the experimental (green line) and numerical (black dashed line) temporal profiles of $|E_1\left(t^{\prime }\right)|$ and $|E_2\left(t^{\prime }\right)|$ at the input (a),(b), after $3\mathrm{km}$ (c),(d) and after $5\mathrm{km}$ (e),(f). Numerical results refer to the case reported in Fig. 3.

Figure 6

Comparison between the experimental (green line) and numerical (black circles) spectral profiles of $P_1$ and $P_2$ at the input (a),(b), after $3\mathrm{km}$ (c),(d) and after $5\mathrm{km}$ (e),(f). Numerical results refer to the case reported in Fig. 3.