TY - JOUR
T1 - A finite element based fast eigensolver for three dimensional anisotropic photonic crystals
AU - Chou, So Hsiang
AU - Huang, Tsung Ming
AU - Li, Tiexiang
AU - Lin, Jia Wei
AU - Lin, Wen Wei
N1 - Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2019/6/1
Y1 - 2019/6/1
N2 - The standard Yee's scheme for the Maxwell eigenvalue problem places the discrete electric field variable at the midpoints of the edges of the grid cells. It performs well when the permittivity is a scalar field. However, when the permittivity is a Hermitian full tensor field it would generate un-physical complex eigenvalues or frequencies. In this paper, we propose a finite element method which can be interpreted as a modified Yee's scheme to overcome this difficulty. This interpretation enables us to create a fast FFT eigensolver that can compute very effectively the band structure of the anisotropic photonic crystal with SC and FCC lattices. Furthermore, we overcome the usual large null space associated with the Maxwell eigenvalue problem by deriving a null-space free discrete eigenvalue problem which involves a crucial Hermitian positive definite linear system to be solved in each of the iteration steps. It is demonstrated that the CG method without preconditioning converges in 37 iterations even when the dimension of a matrix is as large as 5,184,000.
AB - The standard Yee's scheme for the Maxwell eigenvalue problem places the discrete electric field variable at the midpoints of the edges of the grid cells. It performs well when the permittivity is a scalar field. However, when the permittivity is a Hermitian full tensor field it would generate un-physical complex eigenvalues or frequencies. In this paper, we propose a finite element method which can be interpreted as a modified Yee's scheme to overcome this difficulty. This interpretation enables us to create a fast FFT eigensolver that can compute very effectively the band structure of the anisotropic photonic crystal with SC and FCC lattices. Furthermore, we overcome the usual large null space associated with the Maxwell eigenvalue problem by deriving a null-space free discrete eigenvalue problem which involves a crucial Hermitian positive definite linear system to be solved in each of the iteration steps. It is demonstrated that the CG method without preconditioning converges in 37 iterations even when the dimension of a matrix is as large as 5,184,000.
KW - Face-centered cubic lattice
KW - Finite element method
KW - Maxwell's equations
KW - Null-space free eigenvalue problem and fast Fourier transforms
KW - Three-dimensional anisotropic photonic crystals
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U2 - 10.1016/j.jcp.2019.02.029
DO - 10.1016/j.jcp.2019.02.029
M3 - Article
AN - SCOPUS:85062980130
SN - 0021-9991
VL - 386
SP - 611
EP - 631
JO - Journal of Computational Physics
JF - Journal of Computational Physics
ER -