Elliptic Operators with Honeycomb Symmetry: Dirac Points, Edge States and Applications to Photonic Graphene View Full Text


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Article Info

DATE

2019-04

AUTHORS

J. P. Lee-Thorp, M. I. Weinstein, Y. Zhu

ABSTRACT

Consider electromagnetic waves in two-dimensional honeycomb structured media, whose constitutive laws have the symmetries of a hexagonal tiling of the plane. The properties of transverse electric polarized waves are determined by the spectral properties of the elliptic operator LA=-∇x·A(x)∇x, where A(x) is Λh-periodic (Λh denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(x) is PC-invariant (A(x)=A(-x)¯) and 120∘ rotationally invariant (A(R∗x)=R∗A(x)R, where R is a 120∘ rotation in the plane). A summary of our results is as follows: (a) For generic honeycomb structured media, the band structure of LA has Dirac points, i.e. conical intersections between two adjacent Floquet–Bloch dispersion surfaces; (b) Initial data of wave-packet type, which are spectrally concentrated about a Dirac point, give rise to solutions of the time-dependent Maxwell equations whose wave-envelope, on long time scales, is governed by an effective two-dimensional time-dependent system of massless Dirac equations; (c) Dirac points are unstable to arbitrary small perturbations which break either C (complex-conjugation) symmetry or P (inversion) symmetry; (d) The introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell’s equations which are propagating parallel to the line-defect and spatially localized transverse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term; (e) These results imply the existence of unidirectional propagating edge states for two classes of time-reversal invariant media in which C symmetry is broken: magneto-optic media and bi-anisotropic media. More... »

PAGES

1-63

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    http://scigraph.springernature.com/pub.10.1007/s00205-018-1315-4

    DOI

    http://dx.doi.org/10.1007/s00205-018-1315-4

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