Nonlocality Induces Chains of Nested Dissipative Solitons

Dissipative solitons often behave as quasiparticles, and they may form molecules characterized by well-defined bond distances. We show that pointwise nonlocality may lead to a new kind of molecule where bonds are not rigid. The elements of this molecule can shift mutually one with respect to the others while remaining linked together, in a manner similar to interlaced rings in a chain. We report experimental observations of these chains of nested dissipative solitons in a time-delayed laser system.

Figure 1

(a) Sketch of a DS with a pointwise nonlocal term $\mathrm{\Delta }$. (b) Covalent molecule where the rightmost DS is linked to the leftmost one via its echo. Note that the echo of the rightmost DS falls outside the panel. (c) Molecule with nested elements.

Figure 2

(a) Double delay experimental situation featuring DSs with nonlocal coupling. VCSEL output is split into two arms by a beam splitter (BS). Along the first (polarization selective feedback, PSF), $Y$ polarization is selected, and it is fed back after a delay ${\tau }_f$ at a rate $\eta$. Along the second (crossed polarization reinjection, XPR), $Y$ polarization is selected, rotated into the orthogonal polarization direction ($X$) and reinjected after a delay ${\tau }_r$ at a rate $\beta$. (b) Experimental time trace of the $X$ polarization intensity showing a single DS per period.

Figure 3

Numerical integration of Eq. (2). Evolution of the Stokes parameters $(S_1,S_2)$ and intensities $(I_x,I_y)$ in the cases of (a),(c) a covalent and (b),(d) a catenane molecule. Pure emission along the $X$ and $Y$ directions correspond to the red and blue lines with $(S_1,S_2)=(1,0)$ and $(S_1,S_2)=(-1,0)$, respectively. The steady state of Eq. (2), ${\mathrm{\Phi }}_s$, is depicted in pink, while the green line represents $-{\mathrm{\Phi }}_s$.

Figure 4

Numerical analysis of the catenane molecule in Figs. 3 and 3. (a) Temporal profile (black line) and the associated neutral eigenvectors (dotted blue and dashed-dotted red lines). (b) Floquet multipliers ${\mu }_i$ and (c) an enlargement around ${\mu }_i=1$ where one distinguishes two quasidegenerate multipliers. (d) Space-time diagram of $I_x$ in the presence of additive white noise of amplitude $\xi =3\times {10}^{-2}$. The position of each DS is given by the strong intensity peak, followed by the smaller intensity dip echoing at distance $\mathrm{\Delta }$. The intensity of $I_x$ grows from white to black.

Figure 5

Experimental space-time diagrams taking the first DS as a time reference for visually enhancing the relative motion of other DSs. In all of the cases, the bias current is $J=10J_{\mathrm{th}}$ and $T\sim {\tau }_f=10.8\mathrm{ns}$. (a) Two independent DSs, $\mathrm{\Delta }=0.68\mathrm{ns}$ ($0.06T$). (b) Covalent molecule with $\mathrm{\Delta }=3.05\mathrm{ns}$ ($0.28T$) and where the binding occurs via the interaction with the nonlocal echo. (c),(d) Two different cases of catenane molecules with (c) $\mathrm{\Delta }=2.08\mathrm{ns}$ ($0.19T$) and (d) $\mathrm{\Delta }=7.2\mathrm{ns}$ ($0.67T$). The intensity of $I_x$ grows from white to black.