Vector-Resonance-Multimode Instability

The modulation and multimode instabilities are the main mechanisms which drive spontaneous spatial and temporal pattern formation in a vast number of nonlinear systems ranging from biology to laser physics. Using an Er-doped fiber laser as a test bed, here for the first time we demonstrate both experimentally and theoretically a new type of a low-threshold vector-resonance-multimode instability which inherits features of multimode and modulation instabilities. The same as for the multimode instability, a large number of longitudinal modes can be excited without mode synchronization. To enable modulation instability, we modulate the state of polarization of the lasing signal with the period of the beat length by an adjustment of the in-cavity birefringence and the state of polarization of the pump wave. As a result, we show the regime’s tunability from complex oscillatory to periodic with longitudinal mode synchronization in the case of resonance matching between the beat and cavity lengths. Apart from the interest in laser physics for unlocking the tunability and stability of dynamic regimes, the proposed mechanism of the vector-resonance-multimode instability can be of fundamental interest for the nonlinear dynamics of various distributed systems.

Figure 1

(a) Erbium-doped fiber laser. EDF, erbium-doped fiber; LD, l480 nm laser diode for pump; POC1 and POC2, polarization controllers; OISO, optical isolator; WDM, wavelength division multiplexer; output C, $80:20$ output coupler. (b) The map of the states of polarization on the Poincaré sphere (b) output power ($S_0$) (c left) and the corresponding phase difference between linearly polarized modes and DOP (c right) for different settings of POC2: ${\theta }_2=-80°$, $-78°$, $-74°$, and $-69°$. IN1, the state of polarization with the ellipticity $\delta$ is equivalent to the linearly polarized pump oriented at an angle $\psi$ [$\tan (\psi )=\delta$] to the axis ${\mathbf{e}}_x$.

Figure 2

The rf spectrum (a1)–(d1) and corresponding oscillograms (a2)–(d2) for different settings of POC2: (a1),(a2) ${\theta }_2=-80°$; (b1),(b2) ${\theta }_2=-78°$; (c1),(c2) ${\theta }_2=-74°$; (d1),(d2) ${\theta }_2=-69°$.

Figure 3

The results of the linear stability analysis of Eqs. (2). (a) Vector-multimode instability in terms of positive real parts of eigenvalues, i.e., $A_0$ (dots), $A_5$ (solid line) ($A_6$ is close to $A_5$ and so is not shown here), and the output signal $S_0$ (dashed line) vs pump power $I_p$ for $\xi =0$ (blue lines) and $\xi =0.1$ (black line). (b) The frequencies of the scalar branch: ${\mathrm{\Omega }}_1$ (empty circles), ${\mathrm{\Omega }}_2-1$ (empty triangles), and ${\mathrm{\Omega }}_3$ (empty squares), and vector branch ${\mathrm{\Omega }}_4$ (empty diamonds) along with the real parts of the scalar branch: $A_1$ (solid circles), $A_2$ (solid triangles), and $A_3$ (solid squares), and the vector branch: $A_4$ (solid diamonds) vs pump power $I_p$ for $\xi =0.1$. (c) Frequencies for branch I, i.e., ${\mathrm{\Omega }}_0=\pm q$, $q=0$,1,2,3 (solid lines), and the vector branch (III), i.e., ${\mathrm{\Omega }}_{5,6}=q\pm \mathrm{\Delta }\mathrm{\Omega }$ (squares, triangles, circles), vs the birefringence strength $2\beta -2\mathrm{\Delta }{\alpha }_1{f}_2/(1+{\mathrm{\Delta }}^2)$ for $\xi =0.1$. The equality ${\mathrm{\Omega }}_0={\mathrm{\Omega }}_{5,6}$ for $2\beta -2\mathrm{\Delta }{\alpha }_1{f}_2/(1+{\mathrm{\Delta }}^2)=1,2,\dots$ means the resonance mode locking shown in Fig. 2. Parameters: $L=615m$, $2{\alpha }_1=\ln (10)6.4$, $2{\alpha }_2=\ln (10)0.5$, ${\chi }_s=3/2$, ${\chi }_p=1/0.7$, $\mathrm{\Delta }=0.1$, $I_p=10$ (c), $\gamma =2\times {10}^{-6}$, $ϵ={10}^{-3}$, $\beta =1$ (a),(b), $q=1$ (b), and $\xi =0.1$ (b),(c).