Supplementary material for the article “Soliton generation in active nonlinear
metamaterials” by A.B. Kozyrev, I.V. Shadrivov and Yu.S. Kivshar
1. Internal structure of the circulating pulse
We additionally attach the animations (see Fig. S1) showing measured time evolution of
the signal voltage as well as envelope of the intensity profile along the transmission line.
Interestingly, one can also see the backward nature of such a pulse, when the phase fronts
move in the direction opposite to that of the envelope.
FIG S1. (animation) Real electric field (top) and intensity envelope dynamics along the
transmission line. Green squares show experimentally obtained data points (enhanced
online).
2. Stationary oscillating structure
The internal structure of the stationary modes is better illustrated by the profile of the
voltage drops across the diodes in the line taken when the circulating pulse is outside the
nonlinear line and when all the inner dynamic of nonlinear line is due to stationary
oscillating structure. Figs. S2(a-c) shows spatial profiles, corresponding to the regimes
shown in Figs. 3(a-c). On “microscopic” level, the stationary mode corresponds to the
nonlinearity-induced high-frequency branch of the split dispersion curve when adjacent
cells oscillate in phase. The profile shown in Fig. S2(a) has a dip on a background
consisting of out-of-phase oscillations which appear right in the center of the
transmission line. The observed dips represent themselves strongly localized modes in
homogeneous nonlinear lattice which may be understood as self-induced dark-profile
envelope solitons. Overall, these structures are stable over long period of time. However,
we observed some ripples in its shape (occurring over time) which can be either due to
the interaction with residue of the circulating pulse or due to small oscillation of the
position of the soliton near its stationary states. The stationary mode shown in Fig. S2(b)
is asymmetric (left and right sides oscillate out of phase). The dark-profile dip is also
formed in the center of the transmission line. However, in contrast to Fig. S2(a), it splits
into two dips that start to oscillate out-of-phase with respect to the center of transmission
line. The profile in Fig. S2(c) is stable over time like the one shown in Fig. S2(a).
However, it has distinct maximum in the center and two dark-profile solitons.
FIG. S2: Snapshots of the voltage distribution in the nonlinear transmission line for (a)
one-, (b) two-, and (c) three-soliton regimes.