Just solve this: macroscopic Maxwell s equations

Transcription

1 ust solve this: macroscopic Maxwell s equations Faraday: E = B t constitutive equations (here, linear media): Ampere: H = D t + (nonzero frequency) Gauss: D = ρ B = 0 D = ε E B = µ H electric permittivity ε r = ε / ε 0 = relative permittivity or dielectric constant = n 2 (square of refractive index if µ = µ 0 ) ε, µ depend on frequency (dispersion), i.e. * = convolution negligible for transparent media in narrow bandwidth magnetic permeability usually µ 0 at infrared/visible (λ ~ µm) c 2 = 1 / ε 0 µ 0 theorists: often ε 0 = µ 0 = 1 and/or ε r = ε

2 Building Blocks: Eigenfunctions Want to know what solutions exist in different regions and how they can interact: look for time-harmonic modes ~ e iωt E 1 = µ t H = ε t H iω H E + 0 iωε E First task: get rid of this mess 1 ε H = ω 2 H + constraint H = 0 eigen-operator (Hermitian for lossless/real e!) eigen-value eigen-field

3 Periodic Hermitian Eigenproblems [ G. Floquet, Sur les équations différentielles linéaries à coefficients périodiques, Ann. École Norm. Sup. 12, (1883). ] [ F. Bloch, Über die quantenmechanik der electronen in kristallgittern, Z. Physik 52, (1928). ] if eigen-operator is periodic, then Bloch-Floquet solutions: can choose: H ( x,t) = e i ( k x ωt ) H k ( x) planewave periodic envelope Corollary 1: k is conserved, i.e. no scattering of Bloch wave Corollary 2: H k given by finite unit cell, so w are discrete ω n (k)

4 Solving the Maxwell Eigenproblem Finite cell è discrete eigenvalues ω n Want to solve for ω n (k), & plot vs. all k for all n, ( + ik) 1 ( ε + ik ) H n = ω 2 n constraint: where field = H n (x) ei(k x ωt) c 2 ( + ik) H n = 0 H n Photonic Band Gap TM bands 0 1 Limit range of k: irreducible Brillouin zone 2 Limit degrees of freedom: expand H in finite basis

5 Solving the Maxwell Eigenproblem: 1 1 Limit range of k: irreducible Brillouin zone Bloch s theorem: solutions are periodic in k M first Brillouin zone = minimum k primitive cell Γ X 2π a k y k x irreducible Brillouin zone: reduced by symmetry 2 Limit degrees of freedom: expand H in finite basis

6 Solving the Maxwell Eigenproblem: 2a 1 Limit range of k: irreducible Brillouin zone 2 Limit degrees of freedom: expand H in finite basis (N) N m=1 H = H(x t ) = h m b m (x t ) finite matrix problem: solve: ˆ A H = ω 2 H Ah = ω 2 Bh inner product: f g = f * g A ml = b m Galerkin method: ˆ A b l B ml = b m b l

7 Solving the Maxwell Eigenproblem: 2b 1 Limit range of k: irreducible Brillouin zone 2 Limit degrees of freedom: expand H in finite basis must satisfy constraint: ( + ik) H = 0 Planewave (FFT) basis H(x t ) = constraint: G H G e ig x t H G ( G + k) = 0 uniform grid, periodic boundaries, simple code, O(N log N) Finite-element basis [ figure: Peyrilloux et al.,. Lightwave Tech. 21, 536 (2003) ] constraint, boundary conditions: Nédélec elements [ Nédélec, Numerische Math. 35, 315 (1980) ] nonuniform mesh, more arbitrary boundaries, complex code & mesh, O(N)

8 Solving the Maxwell Eigenproblem: 3a 1 Limit range of k: irreducible Brillouin zone 2 Limit degrees of freedom: expand H in finite basis Ah = ω 2 Bh Slow way: compute A & B, ask LAPACK for eigenvalues requires O(N 2 ) storage, O(N 3 ) time Faster way: start with initial guess eigenvector h 0 iteratively improve O(Np) storage, ~ O(Np 2 ) time for p eigenvectors (p smallest eigenvalues)

9 Solving the Maxwell Eigenproblem: 3b 1 Limit range of k: irreducible Brillouin zone 2 Limit degrees of freedom: expand H in finite basis Ah = ω 2 Bh Many iterative methods: Arnoldi, Lanczos, Davidson, acobi-davidson,, Rayleigh-quotient minimization

10 Solving the Maxwell Eigenproblem: 3c 1 Limit range of k: irreducible Brillouin zone 2 Limit degrees of freedom: expand H in finite basis Ah = ω 2 Bh Many iterative methods: Arnoldi, Lanczos, Davidson, acobi-davidson,, Rayleigh-quotient minimization for Hermitian matrices, smallest eigenvalue ω 0 minimizes: variational / min max theorem ω 0 2 = min h h * Ah h * Bh minimize by preconditioned conjugate-gradient (or )

11 a Band Diagram of 2d Model System (radius 0.2a rods, ε=12) 1 frequency ω (2πc/a) = a / λ Photonic Band Gap TM bands irreducible Brillouin zone M k Γ X 0 Γ X M Γ TM E H gap for n > ~1.75:1

12 The Iteration Scheme is Important (minimizing function of variables!) ω 0 2 = min h h * Ah h * Bh = f (h) 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)]

13 % error The Iteration Scheme is Important (minimizing function of ~40,000 variables) E E E E E E EEEEEEEEEEEEE 1000 Ñ 100 Ñ Ñ 10 Ñ Ñ Ñ Ñ Ñ ÑÑÑÑÑ 1 ÑÑÑÑÑÑ E preconditioned Ñ conjugate-gradient no preconditioning no conjugate-gradient # iterations