Spatiotemporal collective excitations in photonic multiwire arrays

We study theoretically the dynamics of collective excitations in one- and two-dimensional arrays of evanescent-wave coupled nonlinear photonic nanowires. This system shows a strong spatiotemporal coupling, which can be observed in its frequency-dependent reciprocal space. We have found that the first Brillouin zone of the system can be separated into two regions characterized by elliptic and hyperbolic nonlinear solutions. Different forms of soliton bullets, vortices, and $X$ waves are estimated, and the effect of the spatiotemporal coupling is identified as a spatiotemporal tilt of the moving solutions. A numerical analysis is performed to support the theoretical calculations.

Figure 1

First Brillouin zone for three different frequency detuning $\delta$. Black and gray regions correspond to the elliptic and hyperbolic cases, respectively. Detuning values are (a) $\delta =0$ ($\Delta \lambda =0$ nm), (b) $\delta =-3$ ($\Delta \lambda \simeq 76$ nm), and (c) $\delta =3$ ($\Delta \lambda \simeq -76$ nm). Other values estimated in Ref. [4] are $c_0^{\left(s\right)}=0.336$, $c_1^{\left(s\right)}=-0.106$, $c_2^{\left(s\right)}=0.0187$, $c_3^{\left(s\right)}=-0.00213$, $d_2=-0.5$, and $d_3=-0.00326$ with $s=x,y$.

Figure 2

Solutions of Eq. (20) [Eq. (29)] for ${\rho }_0=1$ with (a) ${\sigma }_1=1$ (${\sigma }_T=1$) and (b) ${\sigma }_1=-1$ (${\sigma }_T=-1$). The dot-dot-dashed [in panel (a)] and continuous lines (black lines in both panels) correspond to the bounded and unbounded solutions for $\ell =0$, respectively. The dashed lines (red [gray] lines in both panels) correspond to $\ell =1$. The dotted and dot-dashed lines (blue lines in both panels) correspond to $\ell =2$.

Figure 3

Intensity isosurface of initial bullet shapes for the cases (a) $c^{\left(0\right)}=1$, $c_1^{\left(s\right)}=c_2^{\left(s\right)}=c_3^{\left(s\right)}=0$; and (b) $c_{\nu }^{\left(s\right)}$ values are the same as in Fig. 1. $d_{\nu }$ values also are the same as in Fig. 1. $k_n=k_m=0.95\pi /2$ and $\delta =1.5$ ($\Delta \lambda \simeq -38$ nm).

Figure 4

Superposition of intensity isosurfaces of a bullet at different propagation distances: the parameters are the same as those of Fig. 3. The black arrow indicates the propagation direction in the $\{m,n,t^{\prime }\}$ space. The generalized-momentum elements are $k_n=k_m=0.95\pi /2$ and frequency detuning $\delta =1.5$ ($\Delta \lambda \simeq -38$ nm).

Figure 5

Intensity isosurfaces of $X$ waves. For panels (a) and (b) a value $p=3$ was chosen for the super-Gaussian truncation; on the other hand, a value $p=4$ was chosen for panel (c).

Figure 6

First Brillouin zone for three different frequency detunings $\delta$. Black and gray regions correspond to the elliptic and hyperbolic cases, respectively. Other values follow from Fig. 1.

Figure 7

Snapshots of the spatiotemporal intensity distribution of 2D optical excitations during propagation. Left panels [(a), (c), (e), and (g)] correspond to the initial condition, $\xi =0$. Right panels [(b), (d), (f), and (h)] correspond to the shape at some finite propagation distance $\xi >0$. Panels (a) and (b) correspond to a tilted bullet, (c) and (d) correspond to a nontilted bullet, (e) and (f) correspond to a tilted $X$ wave, and (g) and (h) correspond to a nontilted $X$ wave. Constant values for these simulations are the same as those in Figs. 1 and/or 3.