Solving the Maxwell Eigenproblem
Finite cell  discrete eigenvalues ωn
Want to solve for ωn(k),
& plot vs. “all” k for “all” n,
1
constraint:
€
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Solving the Maxwell Eigenproblem: 2a
2
1
ω
(∇ + ik) × (∇ + ik) × H n = n2 H n
ε
c
where:
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Limit range of k: irreducible Brillouin zone
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Limit range of k: irreducible Brillouin zone
2
Limit degrees of freedom: expand H in finite basis (N)
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Limit degrees of freedom: expand H in finite basis
(∇ + ik) ⋅ H = 0
H(x,y) ei(k⋅x – ωt)
Solving the Maxwell Eigenproblem: 2b
1
— must satisfy constraint: (∇ + ik) ⋅ H = 0
N
H = H(xt ) = ∑ hm bm (x t )
solve:
Aˆ H = ω 2 H
Planewave (FFT) basis
Photonic Band Gap
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TM bands
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finite matrix problem:
0
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Limit range of k: irreducible Brillouin zone
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Limit degrees of freedom: expand H in finite basis
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Efficiently solve eigenproblem: iterative methods
Ah = ω 2 Bh
Solving the Maxwell Eigenproblem: 3a
3
Bml = bm bl
Efficiently solve eigenproblem: iterative methods
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Limit range of k: irreducible Brillouin zone
1
Limit range of k: irreducible Brillouin zone
2
Limit degrees of freedom: expand H in finite basis
2
Limit degrees of freedom: expand H in finite basis
3
Efficiently solve eigenproblem: iterative methods
3
Efficiently solve eigenproblem: iterative methods
Ah = ω 2 Bh
Ah = ω Bh
Slow way: compute A & B, ask LAPACK for eigenvalues
— requires O(N 2) storage, O(N3) time
Faster way:
— start with initial guess eigenvector h0
— iteratively improve
— O(Np) storage, ~ O(Np2) time for p eigenvectors
(p smallest eigenvalues)
The Iteration Scheme is Important
Many iterative methods:
— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …,
Rayleigh-quotient minimization
for Hermitian matrices, smallest eigenvalue ω0 minimizes:
“variational
theorem”
ω 02 = min
h
h' Ah
h' Bh
minimize by preconditioned
conjugate-gradient (or…)
€
nonuniform mesh,
more arbitrary boundaries,
complex code & mesh, O(N)
[ figure: Peyrilloux et al. ,
J. Lightwave Tech.
21, 536 (2003) ]
Efficiently solve eigenproblem: iterative methods
The Iteration Scheme is Important
(minimizing function of 10 4–10 8+ variables!)
ω 02 = min
h
h' Ah
= f (h)
h' Bh
Steepest-descent: minimize (h + α ∇f) over α … repeat
Conjugate-gradient: minimize (h + α d)
€ — d is ∇f + (stuff): conjugate to previous search dirs
Preconditioned steepest descent: minimize (h + α d)
— d = (approximate A -1) ∇f ~ Newton’s method
Preconditioned conjugate-gradient: minimize (h + α d)
— d is (approximate A -1) [∇f + (stuff)]
The ε-averaging is Important
100
E|| is continuous
1000000
H G ⋅ (G + k) = 0
3
The Boundary Conditions are Tricky
(minimizing function of ~40,000 variables)
Nédélec elements
[ Nédélec, Numerische Math.
35, 315 (1980) ]
uniform “grid,” periodic boundaries,
simple code, O(N log N)
Solving the Maxwell Eigenproblem: 3c
2
€
G
constraint:
Aml = bm Aˆ bl
f g = ∫ f* ⋅g
constraint, boundary conditions:
H(x t ) = ∑ HG eiG⋅xt
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Finite-element basis
m=1
0.6
100000
E
1000 J
Ñ
% error
100
Ñ
J
10
1
0.1
0.01
0.001
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no preconditioning
preconditioned
conjugate-gradient JJ
0.0001
E⊥ is discontinuous
(D⊥ = εE⊥ is continuous)
Any single scalar ε fails:
(mean D) ≠ (any ε) (mean E)
ε?
ε




0.000001
1
10
100
B
J
1
H
backwards averaging
H
H
H
B
B
J
H
H
B
H
B
J
J
J
B
B
J
0.1
H
B
J
B
tensor averaging
H
H
1000
# iterations
H
B
B
J
B
J
J
ε
ε−1
0.01
10
correct averaging
changes order
of convergence
from ∆x to ∆x 2
J
J

 E||

−1 
 E⊥
H
no Baveraging
Use a tensor ε:
no conjugate-gradient
J
J
0.00001
H
10
% error
E
10000
resolution (pixels/period)
J
100
(similar effects
in other E&M
numerics & analyses)
€
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