A Model Order Reduction Method for Efficient Band Structure Calculations of Photonic Crystals (original) (raw)
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An Efficient Method for Band Structure Calculations in 3D Photonic Crystals
Journal of Computational Physics, 2000
A method for computing band structures for three-dimensional photonic crystals is described. The method combines a mixed finite element discretization on a uniform grid with a fast Fourier transform preconditioner and a preconditioned subspace iteration algorithm. Numerical examples illustrating the behavior of the method are presented.
Computation of the band structure of two-dimensional Photonic Crystals with hp Finite Elements
The band structure of 2D photonic crystals and their eigenmodes can be efficiently computed with the finite element method (FEM). For second order elliptic boundary value problems with piecewise analytic coefficients it is known that the solution converges extremly fast, i.e. exponentially, when using p-FEM for smooth and hp-FEM for polygonal interfaces and boundaries. In this article we discretise the variational eigenvalue problems for the transverse electric (TE) and transverse magnetic (TM) modes in scalar variables with quasi-periodic boundary conditions by means of p-and hp-FEM. Our computations show exponential convergence of the numerical eigenvalues for smooth and polygonal lines of discontinuity of dielectric material properties.
EFFICIENT MODEL ORDER REDUCTION FOR FEM ANALYSIS OF WAVEGUIDE STRUCTURES AND RES- ONATORS
—An efficient model order reduction method for three-dimensional Finite Element Method (FEM) analysis of waveguide structures is proposed. The method is based on the Efficient Modal Order Reduction (ENOR) algorithm for creating macro-elements in cascaded subdomains. The resulting macro-elements are represented by very compact submatrices, leading to significant reduction of the overall number of unknowns. The efficiency of the model order reduction is enhanced by projecting fields at the boundaries of macro-elements onto a subspace spanned by a few low-order waveguide modes. The combination of these two techniques results in considerable saving in overall computational time and memory requirement. An additional advantage of the presented method is that the reduced-order system matrix remains frequency-independent, which allows for very fast frequency sweeping and efficient calculation of resonant frequencies. Several numerical examples for driven and eigenvalue problems demonstrate the performance of the proposed methodology in terms of accuracy, memory usage and simulation time.
Finite Elements in Analysis and Design
The wave finite element (WFE) method is investigated to describe the dynamic behavior of periodic structures like those composed of arbitrary-shaped substructures along a certain straight direction. Emphasis is placed on the analysis of non-academic substructures that are described by means of large-sized finite element (FE) models. A generalized eigenproblem based on the so-called S+S^-1 transformation is proposed for accurately computing the wave modes which travel in right and left directions along those periodic structures. Besides, a model reduction technique is proposed which involves partitioning a whole periodic structure into one central structure surrounded by two extra substructures. In doing so, a few wave modes are only required for modeling the central periodic structure. An error indicator is also proposed to determine in an a priori process the number of those wave modes that need to be considered. Their computation hence follows by considering the Lanczos method, wh...
A WAVE-BASED REDUCTION TECHNIQUE FOR THE DYNAMIC BEHAVIOR OF PERIODIC STRUCTURES
The wave finite element (WFE) method is investigated to describe the dynamic behavior of periodic structures like those composed of arbitrary-shaped substructures along a certain straight direction. A generalized eigenproblem based on the so-called S + S −1 transformation is proposed for accurately computing the wave modes which travel in right and left directions along those periodic structures. Besides, a model reduction technique is proposed which involves partitioning a whole periodic structure into one central structure surrounded by two extra substructures. In doing so, a few wave modes are only required for modeling the central periodic structure. A comprehensive validation of the technique is performed on a 2D periodic structure. Also, its efficiency in terms of CPU time savings is highlighted regarding a 3D periodic structure that exhibits substructures with large-sized FE models.
New analytic and computational formalism for the band structure of N-layer photonic crystals
Physics Letters A, 2005
This Letter presents a new analytic and computational formalism for the eigenfrequency spectra of arbitrary, one-dimensional, N-layer photonic band gap (PBG) materials. The secular equation is formulated in terms of tangents only, a form that has the following beneficial attributes: (a) a compact, algorithmically simple, N × N Hermitian eigenvalue-eigenvector problem (real symmetric at symmetry points) that can be diagonalized once to find both the eigenfrequencies and associated wave amplitudes, and (b) a transparent analytical structure that can be exploited to gain additional insights such as physically appealing, geometric representations of the eigenfrequency condition and analytic forms not otherwise available. The formalism is demonstrated on the example of an eighth-wave/quarter-wave/half-wave PBG stack.
Computational modelling of photonic crystals
2003
Photonic crystals (PC) are a novel class of complex materials which have properties that make them the optical analogues of semiconductors. Accordingly, PCs will likely be key building blocks for future micro-optical and communication technology, specifically because of their ability to tailor the propagation of light on the scale of optical wavelengths with minimal diffraction losses. This paper, which describes the computational modelling of PC based devices being undertaken within the ARC Centre for Ultrahigh-bandwidth Devices and Optical Systems (CUDOS), commences with an outline of multipole and Bloch mode theory, outlines the implementation of the computational algorithms using MPI and Open MP on parallel computer systems, and presents briefly the results of the modelling of some PC based devices.
Band structure computations of metallic photonic crystals with the multiple multipole method
Physical Review B, 2002
A method for the computation of the band structure of two-dimensional photonic crystals is presented. It is well suited for crystals including materials with arbitrary frequency-dependent dielectric constants. The technique can be applied to study photonic crystals with irregularly shaped ͑noncircular͒ elements. This method is based on the multiple multipole method. In order to find the solutions of the nonlinear eigenvalue problem, a multipolar source is introduced which acts as an excitation. By varying the frequency of the source, the various eigenmodes are excited and can be localized as resonances in an appropriately chosen function. The approach is demonstrated for two systems with different geometries: a square lattice of circular cross-section cylinders, and a triangular lattice of triangular cross-section cylinders. The case of metallic systems in H polarization, where surface plasmons may be excited, is chosen. The localized nature of the surface modes poses problems to other methods whereas the eigenvalues and eigenmodes are accurately computed with the proposed technique.
A Novel Method for Band Structure Analysis of Photonic Crystal Slabs
IEEE Photonics Journal, 2011
We propose a new method to extract the modes of photonic crystal slabs and, thus, obtain their band structures. These slabs, which are 2-D periodic structures with finite thickness, can completely confine light and have the important advantage of simple construction for applications in integrated optic devices. In this paper, reflection pole method (RPM) is utilized to analyze photonic crystal slabs. Modes are poles of reflection and transmission coefficients of multilayered structures. According to this principle, modes can be detected by only pursuing phase variations of transmission coefficients that are equal to rad. Therefore, extraction of modes becomes fast and simple through obtaining these coefficients. Photonic crystal slabs are a kind of crossed gratings, and therefore, we use Fourier modal method (FMM) to obtain their reflection and transmission coefficients. FMM is a popular fast convergence method for the analysis of gratings, which offers ease of implementation. In this paper, slab band structures are calculated for dielectric rods in air and air holes in dielectric. The achieved results are compared with 3-D finite-difference time-domain (FDTD) and plane wave expansion (PWE) methods, demonstrating very good agreement.
IEEE Transactions on Magnetics, 2012
This paper illustrates how to determine the bandgap structure of photonic crystals through MLPG. This method is akin to the Finite Element Method (FEM), as it also deals with the discretization of weak forms and produces sparse global matrices. The major difference is the complete absence of any kind of mesh. We concentrate in a particular version of it, the MLPG4, also known as Local Boundary Integral Equation Method (LBIE). Since the boundary conditions governing the electromagnetic field are periodic in a unit cell, we develop a special scheme to embed this feature on the shape functions used in the discretization process. As a result, boundary conditions do not need to be imposed on the unit cell.