Abstract
We numerically study the effect of the quantum spill-out (QSO) on the plasmon mode indices of an ultra-thin metallic slab, using the Fourier modal method (FMM). To improve the convergence of the FMM results, a novel nonlinear coordinate transformation is suggested and employed. Furthermore, we present a perturbative approach for incorporating the effects of QSO on the plasmon mode indices, which agrees very well with the full numerical results. The perturbative approach also provides additional physical insight, and is used to derive analytical expressions for the mode indices using a simple model for the dielectric function. The analytical expressions reproduce the results obtained from the numerically-challenging spill-out problem with much less effort and may be used for understanding the effects of QSO on other plasmonic structures.
Highlights
Surface plasmon polaritons (SPPs) are transverse magnetic (TM) surface modes, which propagate at metal-dielectric interfaces [1]
The quantum spill-out (QSO) effect softens the abrupt changes of the dielectric function at the interfaces and leads to a position-dependent dielectric function across the slab
We propose an alternative numerical approach for finding the plasmon mode indices of a thin metallic slab with QSO, based on the Fourier modal method (FMM)
Summary
Surface plasmon polaritons (SPPs) are transverse magnetic (TM) surface modes, which propagate at metal-dielectric interfaces [1]. Numerical tools are required for calculating the plasmon mode indices when the QSO is included This is in contrast to a classical metallic slab with spatially constant dielectric response, where the short- and long-range plasmon modes are obtained analytically. For a nm-thick metal slab, solving the QSO problem is numerically challenging due to the two vastly different length scales: the slab thickness (∼1 nm) and the mode exponential decay in the dielectric regions (∼1000 nm), for the long-range SPPs. Recently, we have examined the effect of QSO on plasmons propagating in ultra-thin gold films [17, 18], using a transfer matrix method. We propose an alternative numerical approach for finding the plasmon mode indices of a thin metallic slab with QSO, based on the Fourier modal method (FMM). A set of appendices provide checks of the validity of the perturbative approach, present the derivation of analytical expressions and explain the details of the coordinate transformation
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