PHYSICAL REVIEW B 103, 205302 (2021) Itinerant versus localized plasmons in an assembly of metal-dielectric parallel ﬂat slabs in the presence of a perpendicular magnetic ﬁeld: Faraday and magneto-optical Kerr effects Yakov M. Strelniker 1 , * and David J. Bergman 2 , † 1 Department of Physics, Bar-Ilan University, IL-52900 Ramat-Gan, Israel 2 Raymond and Beverly Sackler School of Physics and Astronomy, Faculty of Exact Sciences, Tel Aviv University, IL-69978 Tel Aviv, Israel (Received 13 January 2021; revised 24 March 2021; accepted 24 March 2021; published 3 May 2021) The inﬂuence of the surface running and surface localized plasmons on the optical properties (including Faraday and magneto-optical Kerr effects) is studied and compared for homogeneous as well as perforated metal- dielectric, sandwichlike structures. Closed-form exact expressions are derived for the macroscopic permittivity tensor of an assembly of metal-dielectric parallel ﬂat slabs in the presence of an externally applied static magnetic ﬁeld perpendicular to the slab plane. The structure-dependent resonance associated with running surface plasmon appears only in the perpendicular to the surface zz macroscopic permittivity tensor component and does not depend on the perpendicularly applied magnetic ﬁeld. The other diagonal tensor components vanish at the so- called “hybrid cyclotron-plasma frequency,” which depends on the slab thickness and the applied magnetic ﬁeld. All analytical results are veriﬁed by numerical simulations. DOI: 10.1103/PhysRevB.103.205302 I. INTRODUCTION In the case of conducting ﬁlms with dielectric inclusions (holes), most articles in plasmonics take into account so- called surface plasmons which are running on the ﬂat surfaces of the ﬁlm. However, in the case of dielectric ﬁlms with conducting islands, most authors consider so-called localized surface plasmons (because on the dielectric ﬁlms plasmons cannot exist). In our recent paper [1] we found that both types of plasmons are equally important in optical properties when an in-plane magnetic ﬁeld is applied to a system of metal-dielectric parallel ﬂat slabs. Now we study this subject in the case where the magnetic ﬁeld is perpendicular to the parallel slabs. The recent boom in plasmonics began largely as a result of the pioneering work of Ebbesen at al. [2], where a new phenomenon was reported in perforated metal ﬁlms, called extraordinary optical transmission (EOT). Subsequently it was also reported that a periodic array of holes substantially enhances the Faraday effect [3–7]. Most of those publica- tions follow the ﬁrst Ebbesen article [2] and explain this enhancement by excitation of so-called surface plasmons which propagate along the ﬂat surface of the ﬁlm (i.e., running surface plasmons, RSP) [8]. By contrast, in our attempts to understand EOT [9–12] and related optical properties includ- ing Faraday and Voigt effects [13–15], we considered surface plasmons localized on the surface of circular holes or inclu- sions (LSP) but not on the ﬂat ﬁlm surface. When the holes are circular it is impossible, in practice, to distinguish between these two types of plasmons in an experiment, since both have the same resonance frequency [ω p / √ 2 for RSP and ω p √ n for * strelnik@mail.biu.ac.il † bergman@tauex.tau.ac.il LSP, where n = 1/2 is the transverse depolarization factor of a circular cylinder [9–12] and ω p is the bulk plasma frequency of the metal constituent—see the sentence right after Eq. (2)]. However, if the holes are not circular but elliptical [11], or when a magnetic ﬁeld is applied [9–12], then these resonance frequencies will differ. The effect of an applied static magnetic ﬁeld on EOT was discussed by us in Refs. [9–12]. It was also shown that such a ﬁeld strongly affects the optical properties of those metamaterials, just as it affects the magnetoresistance properties of such materials (both when the nanostructure is periodic [16–20] and when it is disordered [21,22]). As was pointed out in Ref. [1], the application of a magnetic ﬁeld H converts an initially isotropic permittivity tensor into an anisotropic one [see Eq. (2) below]. As a result, the Laplace equation [see Eq. (5) below] for the electric potential also becomes anisotropic. If a coordinate transformation is used to return the Laplace equation back to an isotropic form, then this problem can be solved analytically. However, the shape of the inclusions will be also transformed. Thus, the depo- larization factor n(H) of these deformed inclusions depends on the magnetic ﬁeld H and differs from its original value. As a result of this, the magnetic ﬁeld shifts the surface plasmon resonance ω res = ω p √ n(H) [where n(H) = 1/2] to other frequencies. Therefore it is possible to change the opti- cal transmissivity of the considered sample by changing the magnitude of the magnetic ﬁeld [9–18,23,24]. In a meta- material medium with a periodic nanostructure, an applied magnetic ﬁeld can induce a strong anisotropy and make most of the optical properties depend not only on the magnitude but also on the direction of the applied magnetic ﬁeld [9,10]. This is similar to what we recently predicted for the magnetore- sistance [16–18,20,21] and thermoelectricity [25,26] in such a medium (what was already veriﬁed experimentally [27–30] as well as by other theoretical research [31]). 2469-9950/2021/103(20)/205302(13) 205302-1 ©2021 American Physical Society