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We investigate the ultrafast nonlinear dynamics of light emitted in an activeplasmonic waveguide composed of a thin film of gold sandwiched by two siliconlayers immersed in externally pumped Al$_2$O$_3$:Er$^{3+}$. We model opticalpropagation in such a dissipative system through a generalized cubicGinzburg-Landau equation accounting for the amplification of the active mediumand the effect of absorption and thermo-modulational nonlinearity of gold. Wefind that heating heavily affects the propagation of temporal dissipativesolitons in such a plasmonic waveguide by producing a soliton self-frequencyredshift accompanied by soliton deceleration in the time domain. By adopting asemi-analytical variational approach, we evaluate the dependence of theself-induced redshift by deriving a set of coupled differential equations forthe pulse parameters. These equations provide physical insight into the complexnonlinear dynamics through simple approximate analytic expressions for temporaland frequency shifts. Such analytical predictions are found in excellentagreement with direct numerical simulations of the generalized cubicGinzburg-Landau equation. Our results provide a general understanding ofultrafast nonlinear dynamics in gold-based active plasmonic waveguides, as inparticular the spectral shaping properties of propagating optical pulses.

Heat-induced soliton self-frequency redshift in the ultrafast nonlinear dynamics of active plasmonic waveguides