Photonic Chern insulator through homogenization of an array of particles

Photonic Chern insulator through homogenization of an array of particles

Meng Xiao and Shanhui Fan* 1Department of Electrical Engineering, and Ginzton Laboratory, Stanford University, Stanford, California 94305, USA *Corresponding E-mail: shanhui@stanford.edu

Abstract: We propose a route towards creating a metamaterial that behaves as a photonic Chern insulator, through homogenization of an array of gyromagnetic cylinders. We show that such an array can exhibit non-trivial topological effects, including topologically non-trivial band gaps and one-way edge states, when it can be homogenized to an effective medium model that has the Berry curvature strongly peaked at the wavevector k=0. The non-trivial band topology depends only on the parameters of the cylinders and the cylinders’ density, and can be realized in a wide variety of different lattices including periodic, quasi-periodic and random lattices. Our system provides a platform to explore the interplay between disorder and topology and also opens a route towards synthesis of topological meta- materials based on the self-assembly approach.

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The classification of band structure in terms of band topology in classical waves has led to the newly emerging field of topological photonics [1,2] and phononics [3]. Numerous realizations of topological nontrivial systems in different classical waves have been proposed and experimentally verified [4-32] after the initial theoretical prediction of one-way edge mode in a photonic crystal with a nontrivial band topology [4]. Recent progress also includes the investigation of classical analogue of three- dimensional (3D) topological insulators [24,25] and semimetals [26,27]. The vast majority of these works use periodic lattices, either explicitly as in topological photonic crystal and phononic crystals, or implicitly in topological meta-materials, where in order to achieve a prescribed effective permittivity and permeability, the underlying physical structure is typically periodic.

Complementing the works on periodic systems, in this Letter, we propose a route towards creating a meta-material that behaves as a photonic Chern insulator through homogenization of an array of gyromagnetic particles as shown in Fig. 1(a). The construction of such a physical metamaterial structure is motivated by its corresponding effective medium model that supports a non-trivial topological band gap. For any physical metamaterial system consisting of a periodic array of sub- wavelength elements, its effective medium model is only valid near the center of the Brillouin zone. Yet topological properties are global properties across the entire Brillouin zone. Therefore, even though non-trivial topological effects have been noted in uniform systems with certain bulk parameters, [8,28- 31], in general it is not obvious, a priori, that the topology of a physical metamaterial system consisting of an array of sub-wavelength elements can be understood in terms of its effective medium model, especially when such an effective medium model is local [28]. In our case, the key idea is to construct an effective medium in which the Berry curvature of its band structure is strongly peaked at 0 k  , where k is the wavevector. Since the effective medium model should describe the physical structure very well near 0 k  , we expect that the non-trivial band topology of the effective medium model should also manifest in the physical metamaterial systems. And indeed, through numerical simulations, we find that our metamaterial systems with a wide variety of lattices, including periodic, quasiperiodic, and random lattices, all of which homogenize to the same effective medium model, all possesses complete non-trivial topological band gap.

From a fundamental physics perspective, the systems that we consider here may provide a platform to 3

explore the interplay between order or disorder and topology. From an experimental perspective, creating a disordered system with non-trivial topology may relax the stringent requirements for fabricating topologically non-trivial photonic and phononic systems. While there have been several very recent works on nontrivial topological photonic structures utilizing aperiodic lattices [33-35], our construction differs in that it is not based on any specific lattice [33,34] or needs to engineer the local connections [35]. The structure reported here may therefore open a route towards synthesis of topological meta-material based on the self-assembly approach.[36-38]

We first introduce an effective medium model that exhibits non-trivial topology in its band structure, and has the Berry curvature of its band structure strongly peaked at 0 k  . We consider electromagnetic waves in two dimensions and focus on the TM polarization, which has the electric field z E along the z-direction and the magnetic field x H , y H in the xy− plane. For such a TM pola

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