Exact dispersion relation for nonlinear plasmonic waveguides

We derive an exact dispersion relation for the surface plasmon polaritons of a nonlinear plasmonic waveguide using exact field decomposition of TM waves. Our approach generalizes the known linear dispersion relations to the case of a medium nonlinearity of the form ${\ensuremath{\varepsilon}}_{\mathit{NL}}={\ensuremath{\varepsilon}}_{L}+\ensuremath{\alpha}{|\mathbf{E}|}^{2n}$. We apply the unique dispersion relation to a plasmonic waveguide with a Kerr-type nonlinearity $(n=1)$ and show that it enables backward-propagating modes. It also introduces critical points in the energy spectrum of surface plasmon polaritons that result in enhanced interaction of nonlinear modes with each other and external electromagnetic fields.

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