Bloch theory-based gradient recovery method for computing topological edge modes in photonic graphene
Introduction
Graphene has been one of the popular research topics in different theoretical and applied fields in the past two decades [14]. Its success inspires a lot of analogs (referred to as “artificial graphene”) which are two-dimensional systems with similar properties to graphene [21], [26], [32], [33], [36], [37]. Among those analogs, photonic graphene, a photonic crystal with honeycomb structures, has attracted great interest recently [2], [31], [32]. Similar to graphene which admits Dirac fermions and topological edge states, photonic graphene supports novel and subtle propagating localized modes of electromagnetic waves. These modes are the main research objects in topological photonics and have large applications in many optical systems [24], [25], and thus it is crucial to understand such interesting propagating modes. This brings opportunities and challenges to both applied and computational mathematics.
The propagation of electromagnetic waves in media is governed by the Maxwell equations in three spatial dimensions. Thanks to the symmetries of photonic crystals, the in-plane propagating electromagnetic modes can be described by the following eigenvalue problem in [22], Physically, represents the propagating mode of electromagnetic waves, the eigenvalue E is related to the frequency of the wave, and the positive definite Hermitian matrix function corresponds to the material weight of the media; see [20], [22] for details.
If the medium is a perfect photonic crystal, the material weight is periodic. To obtain novel propagation modes, a bulk photonic crystal is often modulated by different types of defects which break the periodicity of the medium. For instance, in this work, we will consider a photonic graphene modulated by a domain wall defect. In this setup, there exist the so-called topological edge states. In some proper asymptotic regimes, the existence and dynamics can be explicitly analyzed with a rigorous asymptotic analysis, see for instance in [1], [3], [22]. However, in a generic parameter regime, one needs to resort to numerical computation to investigate the existence and study the properties of electromagnetic modes.
The numerical challenge of the eigenvalue problem (1.1) in photonic lattice lies in the lattice structure. For bulk geometry, is periodic and the eigenfunction Ψ is quasi-periodic (periodic up to a phase) in each lattice, the spectral method is usually used after applying the Bloch theory [8] when the material weight is smooth. However, when one introduces the domain-wall modulated defect to break the symmetry of geometry which leads to the appearance of edge modes, the spectral method is no longer a good option due to the loss of symmetry and quasi-periodic boundary conditions in the lattice. Since has a divergence form, finite element method comes to be a natural choice. A standard finite element method lead to that the numerical eigenfuntions and their gradients have different accuracy. In applications, the eigenfunction of (1.1) usually represents the longitudinal electric/magnetic components and the transverse components are the gradients of the eigenfunctions. It is very important to accurately compute the mode and its gradient in order to construct the full electromagnetic fields under propagation [22], and therefore a finite element method with high order accuracy in gradient is desired for the computation of (1.1).
Gradient recovery methods are one of the major postprocessing techniques based on finite element methods, which are able to provide superconvergent gradient and asymptotically exact a posteriori error estimators [4], [7], [9], [27], [39], [40], [41], anisotropic mesh adaption [12], [13], [19], and enhancement of eigenvalue approximation [16], [30], [35]. Recently, recovery techniques are used to construct new finite element methods for higher order partial differential equations [10], [17], [18]. A famous example of gradient recovery methods is the Superconvergent Patch Recovery (SPR) proposed by Zienkiewicz and Zhu [40], also known as ZZ estimator, which has become a standard tool in many commercial Finite Element software such as ANSYS, Abaqus, and LS-DYNA. An important alternative is the polynomial preserving recovery (PPR) proposed by Zhang and Naga [38], which improved the performance of SPR on chevron pattern uniform mesh. It has also been implemented by commercial Finite Element software COMSOL Multiphysics as a superconvergence tool. However, direct application of gradient recovery methods to (1.1) leads to huge computational cost due to the existence of lattice structure.
In this paper, we consider the honeycomb lattice structure and develop a gradient recovery method based on Bloch theory. We apply the Bloch theory in the direction that has no domain-wall modulated defect, and then use the gradient recovery method to solve the eigenvalue problem for each wave number. Compared to standard finite element methods, this method provides higher order accuracy with the help of gradient recovery technique. We analyze the accuracy and prove the superconvergence of this method. We also compute the edge modes for the -symmetry and -symmetry breaking cases in honeycomb structures to show the efficiency of the method. Our results are consistent with the analytical results given in [22].
The rest of the paper is organized as follows. In Section 2, we introduce the problem background on photonic graphene, Dirac points and edge modes and the Bloch–Floquet theory; In Section 3, we propose the gradient recovery method based on Bloch theory, analyze the accuracy and prove the superconvergence of the method; numerical examples of computing -symmetry and -symmetry breaking cases in honeycomb structures are presented in Section 4 to show the efficiency, and we give conclusive remarks in Section 5.
Section snippets
Preliminary
In this section, we summarize basic properties of the photonic graphene, Dirac points and edge states as a description of problem background, and refer interested readers to [22] and references therein for more details.
Gradient recovery method
In this section, we introduce the Bloch-theory based gradient recovery method to solve (2.7)–(2.9).
Numerical examples
In this section, we present several numerical examples to show the efficiency of the proposed Bloch theory-based gradient recovery method. Our method and analysis apply for any honeycomb structured media with a domain wall modulation given in Section 2. The material weight is always of the form In the numerical examples, is given in (2.2), is given in (2.4) or (2.5) and . These simple choices of material weights are sufficient enough to demonstrate
Conclusion
Photonic graphene is an “artificial graphene” which admits subtle propagating modes of electromagnetic waves. It is also an important topological material which supports topological edge states. These states propagates along the edge without any back scattering when passing through a defect. So they have wide applications in many optical systems. Unfortunately, only few analytical results which work in a very narrow parameter regime can be obtained, see for example [22]. How to numerically
Acknowledgements
This work was supported by the National Natural Science Foundation of China under grant 11871299, NSF grants DMS-1418936 and DMS-1818592, Andrew Sisson Fund of the University of Melbourne, and Tsinghua University Initiative Scientific Research Program (Grant 20151080424).
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