Few-cycle spatiotemporal optical solitons in waveguide arrays

We consider the propagation of Gaussian spatiotemporal wave packets in arrays of parallel optical waveguides, assuming linear and nondispersive coupling between the adjacent guides. The numerical analysis is based on a discrete version of the modified Korteweg–de Vries equation that adequately describes the propagation of ultrashort (few-cycle) spatiotemporal solitons in waveguide arrays. Two kinds of such discrete-continuous localized wave forms, which are discrete solitons in the transverse direction, and few-cycle solitons in the longitudinal one, are put forward, namely breathing solitons and single-humped ones. The conditions of formation of these localized spatiotemporal structures, their time duration and spatial width, as well as their energies, are also investigated.

Figure 1

Formation of solitons from a Gaussian pulse. (a) Input. (b) For a high amplitude, a soliton forms (amplitude $A_0=2.06$, pulse duration $fwhm=3.5$, and propagation distance $z=288$). (c) For a relatively low amplitude, the pulse is spread out by diffraction and dispersion (the input is the same as in (a) above, with same duration but with amplitude $A_0=0.2$. Propagation distance $z=0.72$).

Figure 2

Formation of solitons from a Gaussian input in the amplitude-pulse duration plane, for $w_0=1$. Blue crosses: soliton forms; red circles: dispersion-diffraction occurs.

Figure 3

Formation of solitons from a Gaussian input in the amplitude-energy plane, for $w_0=1$. Blue crosses: soliton forms; red circles: dispersion-diffraction occurs.

Figure 4

Formation of solitons from a Gaussian input in the transverse width-pulse duration plane, for $A_0=2.22$. Blue crosses: soliton forms; red circles: dispersion-diffraction occurs.

Figure 5

Formation of solitons from a Gaussian input in the transverse width-energy plane, for $A_0=2.22$. Blue crosses: soliton forms; red circles: dispersion-diffraction occurs.

Figure 6

Fundamental soliton with amplitude ${max}_{t,n}\left(\right|u_n\left|\right)=2.5667$. (a) The optical field in the $(n,t)$ plane. (b) The temporal profile of the optical field.

Figure 7

Breathing soliton with amplitude ${max}_{t,n}\left(\right|u_n\left|\right)=3.1801$. (a) The optical field in the $(n,t)$ plane. (b) The temporal profile of the optical breather.

Figure 8

The soliton energy $E$ as a function of the soliton's maximum amplitude ${max}_{t,n}\left(\right|u_n\left|\right)$. Blue crosses: breathers; green stars: fundamental solitons.

Figure 9

The soliton duration ${\tau }_0$ as a function of the soliton's maximum amplitude ${max}_{t,n}\left(\right|u_n\left|\right)$. Blue crosses: breathers; green stars: fundamental solitons.

Figure 10

The soliton spatial width $W_0$ as a function of the soliton's maximum amplitude ${max}_{t,n}\left(\right|u_n\left|\right)$. Blue crosses: breathers; green stars: fundamental solitons.