Elliptic solitons in optical fiber media

We examine the evolution of a time-varying perturbation signal pumped into a monomode fiber in the anomalous dispersion regime. We establish analytically that the perturbation evolves into a conservative pattern of periodic pulses whose structures and profiles share a close similarity with the so-called soliton-crystal states recently observed in fiber media [see, e.g., A. Haboucha et al., Phys. Rev. A 78, 043806 (2008); D. Y. Tang et al., Phys. Rev. Lett. 101, 153904 (2008); F. Amrani et al., Opt. Express 19, 13134 (2011)]. We derive mathematically and generate numerically a crystal of solitons using time-division multiplexing of identical pulses. We suggest that at very fast pumping rates, the pulse signals overlap and create an unstable signal that is modulated by the fiber nonlinearity to become a periodic lattice of pulse solitons that can be described by elliptic functions. We carry out a linear stability analysis of the soliton-crystal structure and establish that the correlation of centers of mass of interacting pulses broadens their internal-mode spectrum, some modes of which are mutually degenerate. While it has long been known that high-intensity periodic pulse trains in optical fibers are generated from the phenomenon of modulational instability of continuous waves, the present study provides evidence that they can also be generated via temporal multiplexing of an infinitely large number of equal-intensity single pulses to give rise to stable elliptic solitons.

Figure 1

Elliptic-soliton solution to the nonlinear amplitude equation (2) for two different values of the parameter of pulse dispersion $k$. (a) Periodic network of sharp pulse signals for a large dispersion parameter ($k=0.98$). (b) Elliptic-wave decay into a quasicontinuous signal for a small dispersion parameter ($k=0.6$).

Figure 2

Autocorrelation trace of the periodic train of bound solitons, obtained from analytical reconstruction by time-division multiplexing of identical pulses [Eq. (7)].

Figure 3

Numerically reconstructed signal at the pumping rate ${\tau }_A=23.248{\text{ns}}^{-1}$. (a) Double-time-scale comblike pulse. (b) Apex of the large pulse for a soliton crystal of finite length.

Figure 4

Autocorrelation trace of the reconstructed signal. (a) Strong modulation illustrated by the amplitude gap between the soliton crystal and the large pulse. (b) Minimal modulation at the threshold frequency.

Figure 5

Wave profiles of the five bound states $a_{1j}$ ($j=1,2,3,4,5$) given by Eqs. (13, 14, 15, 16) for $k=0.98$ (left column) and $k=1$ (right column).