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Nonreciprocal and Topological Plasmonics

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Nonreciprocal and Topological Plasmonics hv photonics Review Nonreciprocal and Topological Plasmonics Kunal Shastri , Mohamed Ismail Abdelrahman and Francesco Monticone * School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA; [email protected] (K.S.); [email protected] (M.I.A.) * Correspondence: [email protected] Abstract: Metals, semiconductors, metamaterials, and various two-dimensional materials with plasmonic dispersion exhibit numerous exotic physical effects in the presence of an external bias, for example an external static magnetic field or electric current. These physical phenomena range from Faraday rotation of light propagating in the bulk to strong confinement and directionality of guided modes on the surface and are a consequence of the breaking of Lorentz reciprocity in these systems. The recent introduction of relevant concepts of topological physics, translated from condensed-matter systems to photonics, has not only given a new perspective on some of these topics by relating certain bulk properties of plasmonic media to the surface phenomena, but has also led to the discovery of new regimes of truly unidirectional, backscattering-immune, surface-wave propagation. In this article, we briefly review the concepts of nonreciprocity and topology and describe their manifestation in plasmonic materials. Furthermore, we use these concepts to classify and discuss the different classes of guided surface modes existing on the interfaces of various plasmonic systems. Keywords: plasmonics; nanophotonics; nonreciprocity; topological physics; surface waves 1. Introduction Various metals, gas plasmas, highly doped semiconductors, and metamaterials that Citation: Shastri, K.; Abdelrahman, can be modelled as weakly interacting gases of charged particles do not support light M.I.; Monticone, F. Nonreciprocal and propagation within their bulk at frequencies lower than their plasma frequency, a direct Topological Plasmonics. Photonics consequence of their negative permittivity in this regime. However, at the interface be- 2021, 8, 133. https://doi.org/ tween such a plasmonic material and a dielectric, collective charge oscillations coupled to 10.3390/photonics8040133 electromagnetic fields result in surface-plasmon-polariton modes that are strongly confined to the two-dimensional (2D) interface and evanescently decay in the bulk. Analogous to Received: 22 February 2021 the everyday experience of seeing surface gravity waves propagating in all directions on Accepted: 12 April 2021 the surface of a water body that is perturbed by a dropped pebble, these surface plasmon- Published: 20 April 2021 polaritons do not have any preferential direction when propagating on a homogenous isotropic 2D interface. Conversely, if the plasmonic material is magnetized, the resulting Publisher’s Note: MDPI stays neutral Lorentz force affects the motion of the charged particles and breaks this isotropy and sym- with regard to jurisdictional claims in metry, modifying the dispersion of bulk and surface modes. Notably, a magnetic bias opens published maps and institutional affil- iations. a frequency window where backward-propagating surface waves are prohibited [1–6]. These unidirectional surface plasmon-polariton modes on magnetized plasma-dielectric interfaces, known as surface magneto-plasmons, were first discovered several decades ago, and are just one example of the host of unidirectional surface modes existing on com- plex (anisotropic, chiral, nonreciprocal) plasmonic materials. Such plasmonic platforms Copyright: © 2021 by the authors. combine the advantages of subwavelength light confinement and enhancement offered Licensee MDPI, Basel, Switzerland. by plasmonics with the backscattering-immune transport characteristics of unidirectional This article is an open access article waves, and they are being increasingly investigated for novel or enhanced functionalities, distributed under the terms and conditions of the Creative Commons such as compact Faraday rotation, isolation, and circulation [7–10] and extreme light-matter Attribution (CC BY) license (https:// interactions, field enhancement, and giant nonlinear effects [11–14], to name a few. In this creativecommons.org/licenses/by/ article, we review the rich physics of unidirectional surface modes on complex plasmonic 4.0/). Photonics 2021, 8, 133. https://doi.org/10.3390/photonics8040133 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 133 2 of 22 Photonics 2021, 8, 133 2 of 21 a few. In this article, we review the rich physics of unidirectional surface modes on com- plex plasmonic materials, relating them to the properties of the bulk modes, and distin- materials,guishing relatingbetween them various to the classes properties of surface of the waves bulk modes,based on and the distinguishing concepts of reciprocity between variousand topology. classes of surface waves based on the concepts of reciprocity and topology. BeforeBefore studying studying these these different different types types of surfaceof surface waves waves in detail, in detail, let us let briefly us briefly introduce intro- someduce general some general concepts concepts related related to the principleto the principle of reciprocity of reciprocity and how and ideas how of ideas topological of topo- wave-physics enter the problem. Reciprocity in electromagnetics is the general principle logical wave-physics enter the problem. Reciprocity in electromagnetics is the general that the ratio of the field detected by a detector to the field emitted by a source are the principle that the ratio of the field detected by a detector to the field emitted by a source same if the source and detector are interchanged [15–17]. For reciprocal media, such as a are the same if the source and detector are interchanged [15–17]. For reciprocal media, simple non-biased plasmonic material, the dispersion of any eigenmode (including surface such as a simple non-biased plasmonic material, the dispersion of any eigenmode (includ- modes) does not depend on the sign of the wavevector k, w(k) = w(−k), as shown in ing surface modes) does not depend on the sign of the wavevector k , ωω()kk=− ( ), as Figure1a,b. This strict symmetry condition rules out the possibility of realizing truly shown in Figure 1a,b. This strict symmetry condition rules out the possibility of realizing unidirectional surface waves on reciprocal media, irrespective of any consideration related totruly the topologicalunidirectional nature surface of the wa materialves on reciprocal (more on media, this below). irrespective We stress of any that, consideration although lossyrelated plasmonic to the topological materials breaknature time-reversal of the material symmetry (more on at this the below). macroscopic We stress level, that, they alt- cannothough overcome lossy plasmonic this limitation, materials since break breaking time-reversal time-reversal symmetry symmetry at the macroscopic is not equivalent level, tothey breaking cannot reciprocity overcome (the this two limitation, concepts since are equivalentbreaking time-reversal only for lossless symmetry media), is as not further equiv- elucidatedalent to breaking in [15]. reciprocity Therefore, (the the two only concepts way to realizeare equivalent an asymmetric only for surface-plasmon-lossless media), as polaritonfurther elucidated dispersion in is [15]. to break Therefore, reciprocity, the only as shown way to in realize Figure an1c,d, asymmetric which is achieved surface-plas- by biasingmon-polariton the system dispersion with an externalis to break physical reciprocity, quantity as that shown is odd in upon Figure time 1c,d, reversal, which for is exampleachieved an by external biasing magnetic the system field with or aan drift external current physical [15]. quantity that is odd upon time reversal, for example an external magnetic field or a drift current [15]. Figure 1. Reciprocal/nonreciprocal and topological/nontopological surface plasmon-polariton modes. (a,b) Model disper- Figure 1. Reciprocal/nonreciprocal and topological/nontopological surface plasmon-polariton modes. (a,b) Model dis- sion diagrams of reciprocal surface plasmon-polaritons that can be classified as topologically trivial and non-trivial, re- persion diagrams of reciprocal surface plasmon-polaritons that can be classified as topologically trivial and non-trivial, spectively. The equifrequency contours of the dispersion surfaces for these modes at two different frequency values are respectively.plotted on the The right equifrequency insets. It can contours be seen of that the dispersionthe dispersion surfaces curves for theseare symmetr modesic at in two momentum different frequency space for valuesboth cases, are plottedωω()kk=− on ( the ) right, but contrary insets. It to can (a), be the seen topologically that the dispersion protected curves mode arein ( symmetricb) does not inbackscatter momentum (within space afor certain both angular cases, w(k) = w(−k), but contrary to (a), the topologically protected mode in (b) does not backscatter (within a certain angular range) for defects that preserve the ky component of momentum, as indicated by the dashed blue line. Blue arrows denote range) for defects that preserve the k component of momentum, as indicated by the dashed blue line. Blue arrows denote the the group velocity vector for a certainy value of momentum. (c,d) Examples of non-reciprocal surface plasmon-polariton groupmodes. velocity The equifrequency vector for a certain contours value are of strongly momentum. asymmetric (c,d)
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