Fresnel coefficients and Fabry-Perot formula for spatially dispersive metallic layers

Abstract

The repulsion between free electrons inside a metal makes its optical response spatially dispersive, so that it is not described by Drude's model but by a hydrodynamic model. We give here fully analytic results for a metallic slab in this framework, thanks to a two-modes cavity formalism leading to a Fabry-Perot formula, and show that a simplification can be made that preserves the accuracy of the results while allowing much simpler analytic expressions. For metallic layers thicker than 2.7 nm modified Fresnel coefficients can actually be used to accurately predict the response of any multilayer with spatially dispersive metals (for reflection, transmission or the guided modes). Finally, this explains why adding a small dielectric layer [Y. Luo et al., Phys. Rev. Lett. 111, 093901 (2013)] allows to reproduce the effects of nonlocality in many cases, and especially for multilayers. Drude's model, where the electromagnetic response of metals is summarized in a local permittivity, has been very successful in describing the optical response of metals even at the scale of a few nanometers. Electrons are however repulsing each other, making the response of metals spatially dispersive -a phenomenon that is completely overlooked in Drude's model. The response is then said to be non-local because the metal can not be described by a simple permittivity any more. This subject has attracted a lot of interest from a theoretical point of view in the seventies and eighties[1, 2], but an experimental evidence that the Drude model could be inaccurate even in the optical domain has been produced only very recently for very narrow gaps between two metals [3,4]. The hydrodynamic model with hard-wall boundaries[5-7] is a sufficiently accurate framework to take these nonlocal effects into account -even if more complex models taking spill-out corrections have very rencently been proposed [8]. It appears now that nonlocal effects have an impact on metallo-dielectric multilayers with deeply subwavelength thicknesses of dielectric or metal[9] for instance when guided modes are supported [6,10] or when trying to design all kinds of plasmonic flat lenses [11][12][13]. The hydrodynamic model is particularly interesting in the framework of multilayers because the fields have analytic expressions in that case [6,10,12,14,15]. Taking nonlocality into account can be complicated for more complex structures, and there is clearly a need for simpler approaches: it has been recently shown [16], spurring debate [17,18], that adding a very thin dielectric layer could yield results that match very well with the prediction of the hydrodynamic model.In the present work, we first obtain simple analytic expressions using a generalized cavity formalism for a single metallic slab. We then show that a simple assumption, which is valid as soon as the metallic layer is thicker than 2.7 nm in the visible range and 5-6 nm in the near UV, can greatly reduce the complexity of the calculus of the nonlocal response. Using our assumption, simpler * antoine.moreau@uca....

Simulation of three-dimensional nanoscale light interaction with spatially dispersive metals using a high order curvilinear DGTD method

“…Since the impact of nonlocality scales with the effective wave vector [13,14], gap structures are of particular interest, since they are highly sensitive to spatial dispersion. It should be noted that, when the gap size gets smaller than 1 nm, electron spill-out may occur, and the model considered in this work would fail.…”

“…Nano cubes are particularly interesting in the present modeling context because of the high mode confinement in the gap between the cube and the metallic ground plate which is known to be highly sensitive to nonlocality in comparison to e.g. a standard surface plasmon [14].…”

Impact of Nonlocality on Group Delay and Reflective Behavior Near Surface Plasmon Resonances in Otto Structure

“…Moreover, it can be reasonably implemented in numerical calculations and useful for finding the closed-form, analytical result [ 15 ]. Therefore, the hydrodynamic model has attracted considerable attention, and it can be used to study the optical response in the different metallic geometries, such as the field enhancement and extinction in the metallic structures, including plasmonic nanowire dimers [ 16 ], silver nanogroove [ 17 ] and plasmonic tips [ 18 ], the mode confinement of plasmonic waveguides [ 19 ], quantum confinement and grain boundary electron scattering in connected gold nanoprisms structures [ 20 ], second-harmonic generation enhancement in optical split-ring resonators [ 21 ], the size-dependent nonlocal effects in plasmonic semiconductor particles [ 22 ], and the dispersion relation in metallo-dielectric multilayer configurations [ 23 , 24 , 25 ]. Simultaneously, it has been applied to transformation-optics approaches to investigate the optical response of non-trivial plasmonic metasurfaces [ 26 , 27 ].…”

“…Simultaneously, it has been applied to transformation-optics approaches to investigate the optical response of non-trivial plasmonic metasurfaces [ 26 , 27 ]. Up to now, research on the spatial dispersion based on the hydrodynamical mode has concentrated on two directions: (i) developing numerical tools based on the hydrodynamical mode to take account for the phenomena in different structures [ 15 , 23 , 28 , 29 ] and (ii) theoretically studying the effect of the spatial dispersion [ 24 , 30 , 31 ].…”

“…The large wavevector leads to a significant discrepancy between the Drude model (i.e., local theory) and the hydrodynamic model (i.e., nonlocal theory). With reference to previous research available in the literature [ 23 , 24 , 25 ], we pay attention to the behaviors of the reflected light pulse. By analyzing the movements of zeros and poles of the reflection coefficient in the frequency complex domain, the corresponding optical properties, including the amplitude and group delay of the reflected pulse, are investigated.…”