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Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis

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Abstract

We describe a fully-vectorial, three-dimensional algorithm to compute the definite-frequency eigenstates of Maxwell’s equations in arbitrary periodic dielectric structures, including systems with anisotropy (birefringence) or magnetic materials, using preconditioned block-iterative eigensolvers in a planewave basis. Favorable scaling with the system size and the number of computed bands is exhibited. We propose a new effective dielectric tensor for anisotropic structures, and demonstrate that Ox 2) convergence can be achieved even in systems with sharp material discontinuities. We show how it is possible to solve for interior eigenvalues, such as localized defect modes, without computing the many underlying eigenstates. Preconditioned conjugate-gradient Rayleigh-quotient minimization is compared with the Davidson method for eigensolution, and a number of iteration variants and preconditioners are characterized. Our implementation is freely available on the Web.

©2001 Optical Society of America

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Figures (6)

Fig. 1.
Fig. 1. Eigenvalue convergence as a function of grid resolution (grid points per lattice constant a) for three different methods of determining an effective dielectric tensor at each point: no averaging, simply taking the dielectric constant at each grid point; averaging, the smoothed effective dielectric tensor of Eq. (12); and backwards averaging, the same smoothed dielectric but with the averaging methods of the two polarizations reversed.
Fig. 2.
Fig. 2. Eigensolver convergence for two variants of conjugate gradient, Fletcher-Reeves and Polak-Ribiere, along with preconditioned steepest-descent for comparison.
Fig. 3.
Fig. 3. The effect of two preconditioning schemes from section 2.4, diagonal and transverse-projection (non-diagonal), on the conjugate-gradient method.
Fig. 4.
Fig. 4. Comparison of the Davidson method with the block conjugate-gradient algorithm of section 3.1. We reset the Davidson subspace to the best current eigenvectors every 2, 3, 4, or 5 iterations, with a corresponding in increase in memory usage and computational costs.
Fig. 5.
Fig. 5. Conjugate-gradient convergence of the lowest TM eigenvalue for the “interior” eigensolver of Eq. (27), solving for the monopole defect state formed by one vacancy in a 2D square lattice of dielectric rods in air, using three different supercell sizes (3×3, 5×5, and 7×7).
Fig. 6.
Fig. 6. Scaling of the number of conjugate-gradient iterations required for convergence (to a fractional tolerance of 10-7) as a function of the spatial resolution (in grid points per lattice constant, with a corresponding planewave cutoff), or the number p of bands computed.

Equations (27)

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× 1 ε × H = 1 c 2 2 t 2 H ,
· H = 0 .
H = e i ( k · x ω t ) H k ,
A ̂ k H k = ( ω c ) 2 H k ,
A ̂ k = ( + i k ) × 1 ε ( + i k ) × .
H k ( n ) | H k ( m ) = δ n , m ,
H k m = 1 N h m b m .
A h = ( ω c ) 2 B h ,
H k ( k n k R k N k ) = { m j } h { m j } e i j , k m j G j · n k R k N k = { m j } h { m j } e 2 π i j m j n j N j .
A m = ( k + G ) × IFFT ε 1 ˜ FFT ( k + G m ) × .
ε 1 ˜ = ε 1 ˜ P + ε - 1 ( 1 P )
ε 1 ˜ = 1 2 ( { ε 1 ¯ , P } + { ε 1 ¯ , ( 1 P ) } ) ,
A ˜ m = k + G m 2 δ , m ,
A ̂ ˜ = × P ̂ T 1 ε P ̂ T × ,
0 = min y 0 A y 0 y 0 B y 0 ,
Y = Z ( Z B Z ) 1 2
t r [ Z A Z U ] ,
G = P A Z U ,
D = K ̂ G + γ D 0 ,
γ = t r [ G K ̂ G ] t r [ G 0 K ̂ G 0 ]
γ = t r [ ( G G 0 ) K ̂ G ] t r [ G 0 K ̂ G 0 ]
Z = Z * + δ Z ,
G P ( A δ Z B δ Z U Z A Z ) U ,
G P A δ Z U .
δ Z K ̂ G = A 1 ˜ G U 1 ,
R = P ( A D B D L ) ,
A ̂ ' k = ( A ̂ k ω m 2 c 2 ) 2 .
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