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We numerically study the effect of the quantum spill-out (QSO) on the plasmonmode indices of an ultra-thin metallic slab, using the Fourier modal method(FMM). To improve the convergence of the FMM results, a novel nonlinearcoordinate transformation is suggested and employed. Furthermore, we present aperturbative approach for incorporating the effects of QSO on the plasmon modeindices, which agrees very well with the full numerical results. Theperturbative approach also provides additional physical insight, and is used toderive analytical expressions for the mode indices using a simple model for thedielectric function. The analytical expressions reproduce the results obtainedfrom the numerically-challenging spill-out problem with much less effort andmay be used for understanding the effects of QSO on other plasmonic structures.

Plasmons in ultra-thin gold slabs with quantum spill-out: Fourier modal method, perturbative approach, and analytical model