Laguerre and Hermite Soliton Clusters in Nonlocal Nonlinear Media

We introduce novel classes of higher-order spatial optical solitons in analogy with Laguerre-Gaussian and Hermite-Gaussian linear eigenmodes. We reveal that stable higher-order optical solitons can exist in nonlocal nonlinear media in the various forms of soliton necklaces and soliton matrices. Modulational instability can lead to nontrivial transformations between energetically close solitons with different symmetries through the intermediate states resembling generalized Hermite-Laguerre-Gaussian modes.

Figure 1

Variational parameters of the one-dimensional ${\mathrm{HN}}_n$ solitons, Eq. (2). The index $n$ is shown next to the curves.

Figure 2

(a,b) Variational diagrams and (c–f) exact profiles for (a, c, d) Laguerre- and (b, e, f) Hermite-nonlocal solitons. Solitons in (c–f) have the power $P\simeq 200$.

Figure 3

Coexistence of ${\mathrm{HN}}_{22}$ matrix (top row, dashed line) and ${\mathrm{LN}}_{04}$ necklace (bottom row, solid line). Variational (lines) and exact (dots and profiles) solutions are shown for $k=2.12$ (a), 11.5 (b), 30.34 (c), and 54.45 (d).

Figure 4

(a,b) Energy crossings chosen from the overlapping of two ($\mathcal{H}$, $P$) diagrams in Figs. 2a, 2b. Examples of the propagation dynamics of (c) ${\mathrm{LN}}_{10}$ soliton with $P=100$, (d) ${\mathrm{HN}}_{11}$ (quadrupole, same as ${\mathrm{LN}}_{02}$) with $P=75$, (e) ${\mathrm{LN}}_{20}$ with $P=250$, and (f) ${\mathrm{HN}}_{22}$ with $P=210$.