A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals

Otani, Yoshihiro; Nishimura, Naoshi

doi:10.1016/j.jcp.2008.01.029

Cited by 59 publications

(84 citation statements)

References 20 publications

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“…These have a rigorous mathematical foundation [11,34], and have the advantages of discretizing the material interfaces alone, of automatically enforcing radiation conditions, and of high-order convergence, which allows high accuracies to be reached very efficiently. The standard way to adapt integral equations to solve periodic scattering problems is by replacing the free-space kernel (see (2.7)) by the quasi-periodic Green's function (the kernel appearing in (3.4) with P = ∞); see [35,36,43]. Considerable effort has been spent on the tricky task of computing this quasiperiodic Green's function efficiently (see, for example, [3,12,25,28,31]).…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

“…In the related case of periodic surface scattering, Zhang and Chandler-Wilde [47] modified the Green's function to that of a half-space, which cures this divergence (this was implemented in [2]); however, this idea fails to help in our case of disconnected obstacles. A final problem is that the quasi-periodic Green's function is often computed using lattice sums [25,26], which is natural when using fast multipole acceleration in large-scale scattering problems [36]. However, since this representation converges in discs (or spheres in 3D), it becomes cumbersome for high aspect ratio geometries.…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

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A new integral representation for quasi-periodic scattering problems in two dimensions

2010

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Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial boundary conditions on a truncated computational domain. By using the quasi-periodic Green's function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are two drawbacks to this approach: (i) the quasi-periodic Green's function diverges for parameter families known as Wood's anomalies, even though the scattering problem remains well-posed, and (ii) the lattice sum representation of the quasi-periodic Green's function converges in a disc, becoming unwieldy when obstacles have high aspect ratio.In this paper, we bypass both problems by means of a new integral representation that relies on the free-space Green's function alone, adding auxiliary layer potentials on the boundary of the unit cell strip while expanding the linear system to enforce quasi-periodicity. Summing nearby images directly leaves auxiliary densities that are smooth, hence easily represented in the Fourier domain using Sommerfeld integrals. Wood's anomalies are handled analytically by deformation of the Sommerfeld contour. The resulting integral equation is of the second kind and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls are handled easily and automatically. We include an implementation and simple code example with a freely-available MATLAB toolbox.Communicated by Per Lötstedt.

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Section: Introduction and Problem Formulationmentioning

confidence: 99%

Section: Introduction and Problem Formulationmentioning

confidence: 99%

A new integral representation for quasi-periodic scattering problems in two dimensions

2010

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“…Stabilization of MLFMA for low-frequency applications has attracted the interest of many researchers, leading to development of diverse approaches to solve the low-frequency breakdown problem [12][13][14][15][16][17][18][19][20][21][22][23]. In one approach, spectral representation of the Green's function is used so that electromagnetic waves are divided into propagating and evanescent parts [12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

Efficient Solutions of Metamaterial Problems Using a Low-Frequency Multilevel Fast Multipole Algorithm

Erguel¹,

Gürel²

2010

PIER

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Abstract-We present fast and accurate solutions of electromagnetics problems involving realistic metamaterial structures using a lowfrequency multilevel fast multipole algorithm (LF-MLFMA). Accelerating iterative solutions using robust preconditioning techniques may not be sufficient to reduce the overall processing time when the ordinary high-frequency MLFMA is applied to metamaterial problems. The major bottleneck, i.e., the low-frequency breakdown, should be eliminated for efficient solutions. We show that the combination of an LF-MLFMA implementation based on the multipole expansion with the sparse-approximate-inverse preconditioner enables efficient and accurate analysis of realistic metamaterial structures. Using the robust LF-MLFMA implementation, we demonstrate how the transmission properties of metamaterial walls can be enhanced with randomlyoriented unit cells.

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“…In general, when the Maxwell equations are numerically solved in a large-scale system, the fast multipole method, which is the boundary element method to which multipole expansion is applied, is well used, where, in the boundary element method, the differential equations, i.e., the Maxwell equations, are transformed into an integral equation by means of a tensor Green function. The periodic FMM is a numerical method whose Green function is extended for the case that scattering domains exist periodically in a system (Otani and Nishimura, 2008). In the periodic FMM, a system which is finite in the direction stacking layers, i.e., x-direction, but infinite in the y-and z-directions is considered.…”

Section: Numerical Methods and Numerical Modelmentioning

confidence: 99%

Photonic-Crystal Like Approach to Structural Color of the Earthworm Epidermis

Ueta¹,

Otani²,

Nishimura³

2014

Forma

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The epidermis of an earthworm has a log-pile structure of fibers, and exhibits a structural color. The structure is considered as a photonic crystal. Optical properties of the epidermis of an earthworm have been investigated numerically by means of the analogy of the photonic crystal. The numerical method employed here is the periodic fast multipole method. The structural color is reproduced with the RGB color from the reflection spectrum. The reflection spectrum and the reproduced color are compared with those by a multilayer model and it is confirmed that the treatment as a photonic crystal is important.

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A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals

Cited by 59 publications

References 20 publications

A new integral representation for quasi-periodic scattering problems in two dimensions

A new integral representation for quasi-periodic scattering problems in two dimensions

Efficient Solutions of Metamaterial Problems Using a Low-Frequency Multilevel Fast Multipole Algorithm

Photonic-Crystal Like Approach to Structural Color of the Earthworm Epidermis

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