Discrete reduced-symmetry solitons and second-band vortices in two-dimensional nonlinear waveguide arrays

Considering a two-dimensional lattice of weakly coupled waveguides, where each waveguide may carry two orthogonal modes of dipolar character, we present a nonlinear discrete vector model for the study of Kerr optical solitons with profiles having a reduced symmetry relative to the underlying lattice. We describe analytically and numerically existence and stability properties of such states in square and triangular lattices and also reveal directional mobility properties of two-dimensional gap solitons which were recently observed in experiment. The model also describes one-site peaked discrete vortices corresponding to experimentally observed “second-band” vortex lattice solitons, for which oscillatory instabilities are predicted. We also introduce a concept of “rotational Peierls-Nabarro barrier” characterizing the minimum energy needed for rotation of stable dipole modes and compare numerically translational and rotational energy barriers in regimes of good mobility.

Figure 1

(Color online) Schematic illustration of coupling between the dipole modes with amplitude profiles ${\Psi }_a$ and ${\Psi }_b$, characterized by the coupling coefficients $C^{AA}$, $C^{AB}$, $C^{BA}$, and $C^{BB}$.

Figure 2

(Color online) Schematic illustration of (a) a square lattice and coupling between dipole modes. [(b) and (c)] Horizontal and vertical $C_1$, $C_2$. (d) Diagonal between the modes of the same symmetry $C^{AA}$. (e) Diagonal between dipoles of different symmetries $C^{AB}$.

Figure 3

(a) Real and (b) imaginary parts of eigenfrequencies of linear oscillations for a fundamental vortex (type III) stationary solutions in a square lattice described by Eq. (3) with $\gamma =1$ and $\sigma =1/3$ as a function of coupling constant $C_1$ with $C_2=C_1/2$, pure NN interactions $\left(C^{AA}=C^{AB}=0\right)$, $\beta =4$, and system size $11\times 11$ sites. In (a), the linearly spreading eigenfrequencies represent extended eigenmodes (constituting the continuous linear band when the system size becomes infinite), while the eigenfrequency with quadratic behavior corresponds to the localized internal mode, whose frequency for weak coupling can be approximated as $\text{Re}\left(\omega \right)=\beta -3\left(C_1^2+C_2^2\right)/\beta$ [dashed line in (a)]. (b) illustrates windows of instability [nonzero $\mathrm{Im}\left(\omega \right)$] which become dense as the system size becomes infinite.

Figure 4

Stability threshold for dipole modes in square lattice (3) with NNN interactions $C^{AA}=C^{AB}$ and NN anisotropy $C_2/C_1=1/2$, for $\sigma =1/3$, $\gamma =+1$, and $\beta =8$. Type-I (horizontal/vertical) dipole modes are stable below the line, while type-II (diagonal) dipole modes are stable above. For $C^{AA}/C_1=C^{AB}/C_1\gtrsim 0.431$, type-I dipoles are always unstable. In addition, quasicollapse (VK) instabilities appear for $C_1\gtrsim 1.4$ so that type II dipoles are always unstable for $C^{AA}/C_1=C^{AB}/C_1\lesssim 0.33$.

Figure 5

Unstable eigenvalue for type-I dipole modes in square lattice (3) with NNN interactions $C^{AA}=C^{AB}=0.2$ and NN anisotropy $C_2/C_1=0.99$ for ${\sigma }^{\prime }=1/3$, $\gamma =-1$, and $\left|\beta \right|=8$. Note the stable window for $0.75\lesssim C_1\lesssim 0.95$, where the otherwise stable (diagonal) type-II mode instead becomes unstable. In addition, quasicollapse (VK) instabilities appear for both modes for $C_1\gtrsim 2.1$. The latter is considerably stronger, with max. $\mathrm{Im}\left(\omega \right)\simeq 1.5$ for $C_1\simeq 2.3$, and therefore not visible on the scale of this figure.

Figure 6

(Color online) Schematic illustration of (a) a triangular lattice and coupling between dipole modes. (b) Horizontal ($C_1$ and $C_2$). (c) Diagonal between the modes of the same symmetry ($C_{AA}$ and $C_{BB}$). (d) Diagonal between dipoles of different symmetries $C_{AB}$.

Figure 7

(a) Simultaneous plot of $\text{Re}\left(\omega \right)$ for the stable vertical [type (ii)] $(\times )$ and $\mathrm{Im}\left(\omega \right)$ for the unstable horizontal [type (i)] $(+)$ dipole modes in triangular lattice (11) with $C_1=C_2\equiv C$ and other coupling constants given by Eq. (13) for ${\sigma }^{\prime }=1/3$, $\gamma =+1$, and $\beta =8$. Note that ${\omega }^{\left(i\right)}\simeq i{\omega }^{\left(ii\right)}$ for small $C$. In addition, quasicollapse (VK) instabilities appear for both modes for $C\gtrsim 1.7$. The latter is considerably stronger, with max. $\mathrm{Im}\left(\omega \right)\simeq 1.3$ for $C\simeq 1.85$, and therefore not visible on the scale of this figure. (b) Logarithmic plot of the weak-coupling regime of (a) (points) together with the function $6.35\times {10}^{-4}{C}^{13/2}$ (dashed line).

Figure 8

(Color online) Left figure: Simultaneous plot of the spectrum of $\text{Re}\left(\omega \right)$ for small $\omega$ and large coupling for the stable vertical [type (ii)] dipole mode in triangular lattice (11) with $C_2=C_1/3$ and other coupling constants given by Eq. (13) for ${\sigma }^{\prime }=1/3$, $\gamma =+1$, and $\beta =8$ for four different system sizes: $17\times 17(+)$, $21\times 21(\times )$, $25\times 25(\ast )$, and $29\times 29(▣)$. The isolated eigenvalue asymptotically approaching zero at the left part is the translational mode. Note that the bifurcation point at the right part of the figure approaches the value $C_1=\beta /2$ as the system size increases, as expected for a one-dimensional band-edge mode. No VK instabilities appear for these parameter values. Right figure: Total power ${\left|A_{m,j}\right|}^2+{\left|B_{m,j}\right|}^2$ for the solution with $C_1=3.95$ and system size $29\times 29$.

Figure 9

(a) $\text{Re}\left(\omega \right)$ for small $\omega$ and large coupling and (b) $\mathrm{Im}\left(\omega \right)$ for the stable vertical [type (ii)] dipole mode in triangular lattice (11) with $C_2=0.3C_1$ and other coupling constants given by Eq. (13) for ${\sigma }^{\prime }=1/3$, $\gamma =+1$, and $\beta =8$ and system size $21\times 21$. The isolated eigenvalue asymptotically approaching zero at the lower left part of (a) is the translational mode and the eigenvalue colliding at 0 for $C_1\simeq 3.73$ is the VK mode, yielding the instability shown in (b).

Figure 10

(Color online) Left figure shows translational (solid red line) and rotational (dashed green line) PN barriers as a function of power for a square lattice with pure NN interactions ${\sigma }^{\prime }=1/3$, $C_2=0.15$, and $C_1=1$. Only VK-stable modes are included. Right figures show envelopes of the modes used to calculate the PN barriers at power $P\approx 3.28$ above. (b) Stable one-site vertical dipole mode $\left(A\equiv 0\right)$. (c) Unstable two-site vertical dipole mode $\left(A\equiv 0\right)$. [(d) and (e)] Unstable one-site mixed (diagonal) dipole mode.

Figure 11

(Color online) Translational (left figure) and rotational (right figure) PN barriers as a function of power and anisotropy ratio $C_2/C_1$ for a square lattice with pure NN interactions, ${\sigma }^{\prime }=1/3$ and $C_1=1$. Only VK-stable modes are included.

Figure 12

Evolution along $z$ for an initially perturbed stable one-site vertical dipole mode in a square lattice, with the same parameter values as in Fig. 10 but with $P=9$. Solid (dashed) lines show the $A$ $\left(B\right)$ parts of the central-site intensity. The perturbation is chosen from Eq. (16) with $ϵ=0.0112$ (a) just below rotation threshold and $ϵ=0.0113$ (b) just above rotation threshold. System size $17\times 17$, periodic boundary conditions.

Figure 13

(Color online) Left figure shows translational (solid red line) and rotational (dashed green line) PN barriers as a function of power for a triagonal lattice with $\gamma =+1$, ${\sigma }^{\prime }=1/3$, $C_2=0.35$, $C_1=1$, and other coupling constants given by Eq. (13). Only VK-stable modes are included. Right figures show A and B components of the modes used to calculate the PN barriers at power $P\approx 1.65$. [(b) and (c)] Stable one-site vertical dipole mode. [(d) and (e)] Unstable two-site vertical dipole mode. [(f) and (g)] Unstable one-site horizontal dipole mode.

Figure 14

(Color online) Translational (left figure) and rotational (right figure) PN barriers as a function of power and anisotropy ratio $C_2/C_1$ for a triangular lattice with $\gamma =+1$, ${\sigma }^{\prime }=1/3$, $C_1=1$, and other coupling constants given by Eq. (13). Only VK-stable modes are included.

Figure 15

Evolution along $z$ for an initially perturbed stable one-site vertical dipole mode in a triangular lattice, with the same parameter values as in Fig. 13 but with $P=3$. Solid (dashed) lines show the $A$ $\left(B\right)$ parts of the central-site intensity. The perturbation is chosen from Eq. (16) with $ϵ=0.0306$ (a) just below rotation threshold and $ϵ=0.0307$ (b) just above rotation threshold. System size $17\times 17$, periodic boundary conditions.

Figure 16

Evolution along $z$ for an initially perturbed stable one-site vertical dipole mode in a triangular lattice, with the same parameter values as in Fig. 13 but with $P\approx 1.149545$. Solid lines show the total intensity of the central site (11,11), dashed lines of its horizontal neighbor (11,10), and dotted line of its horizontal next-nearest neighbor (11,9). The perturbation is chosen from Eq. (17) with $ϵ=0.012$ (a) below translation threshold and $ϵ=0.014$ (b) above translation threshold. System size $21\times 21$, periodic boundary conditions.

Figure 17

(Color online) Left figure shows rotational PN barrier as a function of power and anisotropy ratio $C_2/C_1$ for a triangular lattice with $\gamma =-1$, ${\sigma }^{\prime }=1/3$, $C_1=1$, and other coupling constants given by Eq. (13). Only VK-stable modes are included. Right figures show A and B components of the modes used to calculate the PN barrier at $C_2=0.35$ and $P=5$. [(b) and (c)] Stable one-site vertical dipole mode. [(d) and (e)] Unstable one-site horizontal dipole mode.