Nonlinear plasmonic amplification via dissipative soliton-plasmon resonances

In this contribution we introduce a strategy for the compensation of plasmonic losses based on a recently proposed nonlinear mechanism: the resonant interaction between surface plasmon polaritons and spatial solitons propagating in parallel along a metal/dielectric/Kerr structure. This mechanism naturally leads to the generation of a quasiparticle excitation, the so-called soliplasmon resonance. We analyze the role played by the effective nonlinear coupling inherent to this system and how this can be used to provide a mechanism of quasiresonant nonlinear excitation of surface plasmon polaritons. We will pay particular attention to the introduction of asymmetric linear gain in the Kerr medium. The unique combination of nonlinear propagation, nonlinear coupling, and gain give rise to a scenario for the excitation of long-range surface plasmon polaritons with distinguishing characteristics. The connection between plasmonic losses and soliplasmon resonances in the presence of gain will be discussed.

Figure 1

(a) Scheme of the flux of EM energy in a soliplasmon structure with just metal losses. The SPP component is localized transversely ($x$ axis) with a maximum at the MD interface ($x=0)$, whereas the spatial soliton component is centered at a certain distance $x=a$ from the MD interface within the Kerr medium. Both propagate in parallel to the MD and DK interfaces. (b) Characteristic MDK structure supporting dissipative soliplasmons in which gain is located in the Kerr medium in order to compensate for metal losses.

Figure 2

Nondissipative soliplasmons: (a) 0 and $\pi$ soliplasmon families in the Earth-like Poincaré sphere (red arrows correspond to representative 0-soliplasmons solutions, blue arrows to $\pi$-soliplasmons ones); (b) intensity and phase of the three 0-soliplasmons solutions (red arrows) in the Poincaré sphere; (c) the same for the three $\pi$-soliplasmons solutions (blue arrows).

Figure 3

Representation of the “golden constraint” for increasing values of the gain coefficient $g$ for a given loss coefficient of $l=1.05\times {10}^{-2}$ ($l=758{\mathrm{cm}}^{-1}$, in physical units).

Figure 4

Pair of dissipative soliplasmons in the “Earth-like” Poincaré sphere corresponding to the configuration with $g=1\times {10}^{-3}$ ($g=72{\mathrm{cm}}^{-1}$, in physical units) in Fig. 3: ${\mathit{S}}_a$ (yellow arrow) solution with $C_s=0.295$; ${\mathit{S}}_b$ (orange arrow) solution with $C_s=0.848$. Inset: intensity and phase profiles of the corresponding two dissipative soliplasmon solutions represented in the Earth-like Poincaré sphere. (a) ${\mathit{S}}_a$ yellow arrow solution; (b) ${\mathit{S}}_b$ orange arrow solution.

Figure 5

Nonlinear amplification of a surface plasmon in the Earth-like Poincaré sphere. A purely solitonic input state (green dot) with no plasmonic component at the South pole evolves into a dissipative soliplasmon output state (yellow arrow) with a high plasmon component located near the north pole well above the Earth surface. Trajectory “takes off” from the soliton pole of the Earth surface into the “atmospheric” shell swiftly reaching a point of maximum height and amplification (white spot), then descending smoothly to reach the final output dissipative soliplasmon state.

Figure 6

Normalized intensity plots for the nonlinear SPP amplification process represented in the Earth-like Poincaré sphere of Fig. 5 for two different values of the maximum propagation length $L$. We use normalized values given by $I/I_0={\left|E_x\right|}^2/E_0^2$, where $E_0$ is the normalization value referred to in the text. Physical units are used for distances. Left column graphics show the short distance behavior of the intensity for $L=20\mu \mathrm{m}$: (a) intensity profile of the total soliplasmon solution and (b) intensity profile of the linear SPP mode excited by the soliton. Right column shows the same evolution but for very long distances when $L=1$ cm: (c) full soliplasmon solution and (d) linear SPP corresponding excited mode.

Figure 7

(a) Effective gain as a function of distance showing perfect asymptotic loss compensation—we restore physical units (${\mathrm{cm}}^{-1}$) to gain and loss coefficients and take the same configuration and parameters as in Secs. 6 and 7. (b) Graphical representation of the typical energy flux configuration for a stationary dissipative soliplasmon with gain in the Kerr medium compensating for SPP losses.