Matrix representation of the double-curl operator for simulating three dimensional photonic crystals

Tsung Ming Huang, Han En Hsieh, Wen Wei Lin, Weichung Wang

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

Three dimensional photonic crystals can be modeled by the Maxwell equations as a generalized eigenvalue problem (GEVP). Simulations based on the numerical solutions of the GEVP are used to reveal physical properties and boost innovative applications of photonic crystals. However, to solve these GEVP remains a computational challenge in both timing and accuracy. The GEVP corresponding to the photonic crystals with face centered cubic (FCC) lattice is one of the challenging eigenvalue problems. From a viewpoint of matrix computation, we demonstrate how such obstacles can be overcome. Our main contribution is an explicit matrix representation of the double-curl operator associated with the photonic crystal with FCC lattice. This particular matrix represents the degenerate coefficient matrix of the discrete GEVP obtained by Yee's scheme. The explicit matrix leads to an eigendecomposition of the degenerate coefficient matrix and then a fast eigenvalue solver. Promising numerical results in terms of timing and accuracy are reported for solving the discrete GEVP arising in three dimensional photonic crystals with various geometric parameters.

Original languageEnglish
Pages (from-to)379-392
Number of pages14
JournalMathematical and Computer Modelling
Volume58
Issue number1-2
DOIs
Publication statusPublished - 2013 Jul 1

Fingerprint

Generalized Eigenvalue Problem
Curl
Matrix Representation
Photonic crystals
Photonic Crystal
Three-dimensional
Operator
Crystal lattices
Timing
Face
Matrix Computation
Maxwell equations
Coefficient
Physical property
Maxwell's equations
Eigenvalue Problem
Physical properties
Numerical Solution
Eigenvalue
Numerical Results

Keywords

  • Double-curl operator
  • Face centered cubic lattice
  • Matrix representation
  • Photonic crystals
  • The Maxwell equations
  • Yee's scheme

ASJC Scopus subject areas

  • Modelling and Simulation
  • Computer Science Applications

Cite this

Matrix representation of the double-curl operator for simulating three dimensional photonic crystals. / Huang, Tsung Ming; Hsieh, Han En; Lin, Wen Wei; Wang, Weichung.

In: Mathematical and Computer Modelling, Vol. 58, No. 1-2, 01.07.2013, p. 379-392.

Research output: Contribution to journalArticle

@article{518552ff07fc48c6b02dedd5d349d339,
title = "Matrix representation of the double-curl operator for simulating three dimensional photonic crystals",
abstract = "Three dimensional photonic crystals can be modeled by the Maxwell equations as a generalized eigenvalue problem (GEVP). Simulations based on the numerical solutions of the GEVP are used to reveal physical properties and boost innovative applications of photonic crystals. However, to solve these GEVP remains a computational challenge in both timing and accuracy. The GEVP corresponding to the photonic crystals with face centered cubic (FCC) lattice is one of the challenging eigenvalue problems. From a viewpoint of matrix computation, we demonstrate how such obstacles can be overcome. Our main contribution is an explicit matrix representation of the double-curl operator associated with the photonic crystal with FCC lattice. This particular matrix represents the degenerate coefficient matrix of the discrete GEVP obtained by Yee's scheme. The explicit matrix leads to an eigendecomposition of the degenerate coefficient matrix and then a fast eigenvalue solver. Promising numerical results in terms of timing and accuracy are reported for solving the discrete GEVP arising in three dimensional photonic crystals with various geometric parameters.",
keywords = "Double-curl operator, Face centered cubic lattice, Matrix representation, Photonic crystals, The Maxwell equations, Yee's scheme",
author = "Huang, {Tsung Ming} and Hsieh, {Han En} and Lin, {Wen Wei} and Weichung Wang",
year = "2013",
month = "7",
day = "1",
doi = "10.1016/j.mcm.2012.11.008",
language = "English",
volume = "58",
pages = "379--392",
journal = "Mathematical and Computer Modelling",
issn = "0895-7177",
publisher = "Elsevier Limited",
number = "1-2",

}

TY - JOUR

T1 - Matrix representation of the double-curl operator for simulating three dimensional photonic crystals

AU - Huang, Tsung Ming

AU - Hsieh, Han En

AU - Lin, Wen Wei

AU - Wang, Weichung

PY - 2013/7/1

Y1 - 2013/7/1

N2 - Three dimensional photonic crystals can be modeled by the Maxwell equations as a generalized eigenvalue problem (GEVP). Simulations based on the numerical solutions of the GEVP are used to reveal physical properties and boost innovative applications of photonic crystals. However, to solve these GEVP remains a computational challenge in both timing and accuracy. The GEVP corresponding to the photonic crystals with face centered cubic (FCC) lattice is one of the challenging eigenvalue problems. From a viewpoint of matrix computation, we demonstrate how such obstacles can be overcome. Our main contribution is an explicit matrix representation of the double-curl operator associated with the photonic crystal with FCC lattice. This particular matrix represents the degenerate coefficient matrix of the discrete GEVP obtained by Yee's scheme. The explicit matrix leads to an eigendecomposition of the degenerate coefficient matrix and then a fast eigenvalue solver. Promising numerical results in terms of timing and accuracy are reported for solving the discrete GEVP arising in three dimensional photonic crystals with various geometric parameters.

AB - Three dimensional photonic crystals can be modeled by the Maxwell equations as a generalized eigenvalue problem (GEVP). Simulations based on the numerical solutions of the GEVP are used to reveal physical properties and boost innovative applications of photonic crystals. However, to solve these GEVP remains a computational challenge in both timing and accuracy. The GEVP corresponding to the photonic crystals with face centered cubic (FCC) lattice is one of the challenging eigenvalue problems. From a viewpoint of matrix computation, we demonstrate how such obstacles can be overcome. Our main contribution is an explicit matrix representation of the double-curl operator associated with the photonic crystal with FCC lattice. This particular matrix represents the degenerate coefficient matrix of the discrete GEVP obtained by Yee's scheme. The explicit matrix leads to an eigendecomposition of the degenerate coefficient matrix and then a fast eigenvalue solver. Promising numerical results in terms of timing and accuracy are reported for solving the discrete GEVP arising in three dimensional photonic crystals with various geometric parameters.

KW - Double-curl operator

KW - Face centered cubic lattice

KW - Matrix representation

KW - Photonic crystals

KW - The Maxwell equations

KW - Yee's scheme

UR - http://www.scopus.com/inward/record.url?scp=84878613587&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878613587&partnerID=8YFLogxK

U2 - 10.1016/j.mcm.2012.11.008

DO - 10.1016/j.mcm.2012.11.008

M3 - Article

AN - SCOPUS:84878613587

VL - 58

SP - 379

EP - 392

JO - Mathematical and Computer Modelling

JF - Mathematical and Computer Modelling

SN - 0895-7177

IS - 1-2

ER -