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Article Dans Une Revue Journal of the Optical Society of America B Année : 2019

The energy point of view in plasmonics

Armel Pitelet
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Rémi Pollès
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Emmanuel Centeno
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Antoine Moreau

Résumé

The group velocity of a plasmonic guided mode can be written as the ratio of the flux of the Poynting to the integral of the energy density along the profile of the mode. This theorem, linking the way energy propagates in metals to the properties of guided modes and Bloch modes in a multilayer, provides a unique physical insight in plasmonics. It allows to better understand the link between the negative permittivity of metals and the wide diversity of exotic phenomenon that occur in plasmonics-like the slowing down of guided modes, the high wavevector and the negative refraction. Deeply subwavelength metallic structures give us an unprecedented control of visible light, allowing to focus, concentrate, absorb, scatter light very efficiently, or to even enhance the emission of light by fluorophores[1] at totally new levels. Metals actually present a very peculiar optical response that dielectrics are totally incapable of-which can be linked to the presence of a free electron gas, a plasma, inside even the tiniest metallic nanoparticles[2]. Plasmonic resonators are the smallest optical resonators possible and their resonances can always be linked to the excitation of some kind of plasmonic guided mode. There is thus a large diversity of these modes, from the well known surface plasmon[3] to long and short-range surface plasmons[4], gap-plasmons[5] or modes supported by hyperbolic metallo-dielectric multilayers[6]. Most of them present very high wavevectors which explains the reduced size of the plasmonic resonators[7, 8]: they are essentially cavities for guided modes with very small effective wavelength. One must finally underline that exotic phenomena like negative refraction may also occur in metallo-dielectric multilayers[9-12]. All these features lack a unified view that would enable to give a physical insight into the reasons why large wavevector guided modes and negative refraction are common in plasmon-ics and very exotic in dielectric structures-requiring the careful tailoring of photonic crystals, for instance[13, 14]. We think that considering the way that energy flows when such modes propagate provides this kind of insight. The average flux of the Poynting vector has actually been used in the context of metamaterial and negative index materials as a useful tool to predict in which direction a mode really propagates (i.e. the sign of its group velocity). Such a approach relies on a theorem showing that the energy velocity is equal to the group velocity for modes propagating in non-dispersive, dielectric media[15, 16]. This theorem has been mostly ignored because, except in a few cases like when a mode approaches a cutoff condition, the group velocity does not present any exotic behaviour. Assuming this link holds even in the case of plasmonic or metamaterial waveguides, it can prove very useful to determine the sign of the group velocity by simply using the profile of the guided modes at a given frequency without having to actually compute the dispersion relation[17-19]. However, the optical response of metals is linked to the presence of free electrons, that transport a part of the guided mode energy and whose kinetic energy can not be neglected. For a plane wave propagating in a plasma, provided the energy of the electrons is taken into account both in the energy flux and in the energy density, it has been shown by Bers[20] that the energy and flux velocity are the same. This underlines that Yariv and Yeh's theorem[15] can not be applied to plasmonic waveguides. Since metals are highly dispersive and can be considered as boxes containing a real plasma, it is even surprising that computing the average flux of the Poynting vector could be successful in predicting the sign of the group velocity. Here we show that it is actually possible to generalize Yariv and Yeh's theorem in the context of dispersive media, including metals. This means that there is no need to modify the expression of the energy flux and only to adapt the energy density expression, to make the theorem valid-despite what has been established for plasma[20]. Said otherwise, the group velocity of a guided mode in a plasmonic multilayer is equal to the energy velocity of the electromagnetic field alone, and the energy transported by the free electrons, while not negligible[20], can be completely ignored. We then show using several examples how considering the energy velocity can provide a unifying vision of the optical response of plasmonic multilayers. We consider a multilayered structure invariant in the x and y directions, and a guided mode, solution of Maxwell's equations presenting a e i(k x x−ω t) dependency in x and t. We will assume the mode is p-polarized, because nothing exotic occurs for the s polarization in metallo-dielectric structures. Maxwell's
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Dates et versions

hal-02088217 , version 1 (02-04-2019)

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Rabih Ajib, Armel Pitelet, Rémi Pollès, Emmanuel Centeno, Ziad Ajaltouni, et al.. The energy point of view in plasmonics. Journal of the Optical Society of America B, 2019, ⟨10.1364/JOSAB.36.001150⟩. ⟨hal-02088217⟩
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