The computation of photonic band gaps Christian Wieners Institut für Angewandte und Numerische Mathematik, Karlsruhe www.kit.edu
Light propagation in periodic structures We consider light propagation in periodic structures. We assume for simplicity that the electro-magnetic fields are described by the linear Maxwell system; the permitivity µ = µ 0 is constant; the permeability ε is periodic; the structure is infinite without boundary conditions; we consider only the time-harmonic case without sources. Optics Express, 16, 14812 (2008) 1
Outline 1. The Mathematics of Photonic Band Gaps (revisited) 2. The Approximation of Photonic Band Gaps (revisited) 3. The Computation of Photonic Band Gaps (state-of-the-art) 4. The Verification of Photonic Band Gap (summarized) 5. The Computation of Photonic Band Gaps (alternative approach) The results are contained in the PhD thesis of A. Bulovyatov and V. Huang, M. Plum, and C. Wieners: A computer-assisted proof for photonic band gaps. Z. angew. Math. Phys. (ZAMP) 60 (2009) 1035-1052 C. Wieners: Calculation of the photonic band structure In: Photonic Crystals. Mathematical Analysis and Numerical Approximation (ed. W. Dörfler) Birkhäuser, New York (2011) 40-62 C. Wieners, J. Xin: Boundary Element Approximation for Maxwell’s Eigenvalue Problem. Mathematical Methods in the Applied Sciences 36 (2013) 2524-2539 2
The Maxwell Eigenvalue Problem We consider the Maxwell eigenvalue problem for the magnetic field H ∇ × ε − 1 ∇ × H = λ H (1a) ∇ · H = 0 (1b) in the case that ε is a periodic function with ε ( x ) = ε ( x + z ) for all z ∈ Z 3 and µ ≡ 1. Let Ω = ( 0 , 1 ) 3 be a fundamental cell, and K = [ − π, π ] 3 be the Brillouin zone. The ansatz H ( x ) = e i k · x � H ( x ) with x ∈ Ω , k ∈ K and � H periodic yields ∇ k × ε − 1 ∇ k × � λ � H = H (2a) ∇ k · � H = 0 (2b) in R 3 / Z 3 , where ∇ k = ∇ + i k . The reduced problem for k ∈ K has a discrete spectrum with eigenvalues 0 ≤ λ k , 1 ≤ λ k , 1 ≤ · · · ≤ λ k , N ≤ · · · � Theorem Problem (1) has the spectrum σ = [ inf λ k , n , sup λ k , n ] . k ∈ K k ∈ K n ∈ N (application of the Floquet-Bloch theory, for photonic crystals see Kuchment and Figotin) If ( sup k ∈ K λ k , n , inf k ∈ K λ k , n + 1 ) is non-empty for some n ∈ N , this is a band gap . 3
Approximation of the Maxwell Eigenvalue Problem Let X = H per ( curl; Ω) ⊂ H ( curl; Ω) be the subspace of periodic vector fields. For u , v ∈ X we define the Hermitian bilinear forms � � ε − 1 ∇ k × u · ∇ k × v d x , a k ( u , v ) = m ( u , v ) = u · v d x Ω Ω Let Q = H 1 per (Ω) ⊂ H 1 (Ω) be the subspace of periodic scalar functions. Let � � b k ( v , q ) = v · ∇ k q d x , c k ( p , q ) = ∇ k ∇ k p · ∇ k ∇ k ∇ k ∇ k q d x , v ∈ X , p , q ∈ Q , Ω Ω and V k = { v ∈ X : b k ( v , q ) = 0 for all q ∈ Q } . Weak Form Find ( u , λ ) ∈ V k × R such that v ∈ V k . a k ( u , v ) = λ m ( u , v ) for all Discrete Approximation Let X h , k ⊂ X and Q h , k ⊂ Q be discrete subspaces. Set V h , k = { v h , k ∈ X h , k : b k ( v h , k , q h , k ) = 0 for all q h , k ∈ Q h , k } . Find ( u h , k , λ h , k ) ∈ V h , k × R such that a k ( u h , k , v h , k ) = λ h , k m ( u h , k , v h , k ) for all v h , k ∈ V V V h , k . 4
Lowest Order Conforming Finite Elements � T = [ 0 , 1 ] 3 − T be a decomposition into cells T ∈ T h , and let ϕ T : ˆ Let Ω = → T T ∈T h be the transformation to the reference cell. In case of hexahedral cells we define X h , 0 = { u ∈ X : D ϕ − 1 T u ◦ ϕ T ∈ P 0 , 1 , 1 e x + P 1 , 0 , 1 e y + P 1 , 1 , 0 e z for all T ∈ T h } , Q h , 0 = { q ∈ Q : q ◦ ϕ T ∈ P 1 , 1 , 1 for all T ∈ T h } . Let V h ⊂ ¯ Ω be the set of all vertices z , and let E h be the set of all edges e = ( x e , y e ) with midpoint z e = 0 . 5 ( x e + y e ) and tangent t e = y e − x e . A nodal basis { ψ e , 0 : e ∈ E h } ⊂ X h , 0 exists such that for all u h , 0 ∈ X h , 0 � y e � � ψ ′ � ψ ′ u h , 0 = e , 0 , u h , 0 � ψ e , 0 , e , 0 , u h , 0 � = u h , 0 · t e ds , x e e ∈E h and a nodal basis { φ z , 0 : z ∈ V h } ⊂ Q h , 0 exists such that for all q h , 0 ∈ Q h , 0 � � φ ′ � φ ′ q h , 0 = z , 0 , q h , 0 � φ z , 0 , z , 0 , q h , 0 � = q h , 0 ( z ) . z ∈V h (curl conforming elements were introduced by Nédélec) 5
Conforming Elements with Shifted Basis For k ∈ K we define modified elements with a phase shift ψ e , k ( x ) = e − i k · ( x − z e ) ψ e , 0 ( x ) X h , k = span { ψ e , k : e ∈ E h } , φ z , k ( x ) = e − i k · ( x − z ) φ z , 0 ( x ) Q h , k = span { φ z , k : z ∈ V h } , with � y e � � ψ ′ � ψ ′ e i k · ( x − z e ) u h , 0 · t e ds , u h , k = e , k , u h , k � ψ e , k , e , k , u h , k � = x e e ∈E h � � φ ′ � φ ′ q h , k = z , k , q h , k � φ z , k , z , k , q h , 0 � = q h , k ( z ) . z ∈V h We have e − i k · ( x − z ) ∇ φ z , 0 ( x ) ∇ k φ z , k ( x ) = e − i k · ( x − z e ) ∇ × ψ e , 0 ( x ) ∇ k × ψ e , k ( x ) = e − i k · ( x − z e ) ∇ · ψ e , 0 ( x ) ∇ k · ψ e , k ( x ) = e i k · ( z e 1 − z e 2 ) a 0 ( ψ e 1 , 0 , ψ e 2 , 0 ) a k ( ψ e 1 , k , ψ e 2 , k ) = e i k · ( z e 1 − z e 2 ) m ( ψ e 1 , 0 , ψ e 2 , 0 ) m ( ψ e 1 , k , ψ e 2 , k ) = e i k · ( z e − z ) b 0 ( ψ e , 0 , φ z , 0 ) b k ( ψ e , k , φ z , k ) = e i k · ( z 1 − z 2 ) c 0 ( φ z 1 , 0 , φ z 2 , 0 ) . c k ( φ z 1 , k , φ z 2 , k ) = 6
Finite Element Convergence The finite element spaces X h , k , Q h , k , and V h , k satisfy Ellipticity on V h , k There exists C > 0 s.t. a k ( u h , k , u h , k ) ≥ C � u h , k � 2 for all u h , k ∈ V h , k . L 2 Weak approximability of Q There exists ρ 1 ( h ) > 0, tending to zero as h goes to zero such that b k ( v h , k , q ) sup ≤ ρ 1 ( h ) � q � H 1 for all q ∈ Q . � v h , k � curl v h , k ∈ V h , k Strong approximability of V For some r > 0 there exists ρ 2 ( h ) > 0, tending to zero as h goes to zero such that for any u ∈ V ∩ H 1 + r (Ω , C 3 ) there exists u h , k ∈ V h , k satisfying � u − u h , k � curl ≤ ρ 2 ( h ) � u � H 1 + r . (for coercivity and regularity see Dauge-Norton-Scheichl 2015) Theorem Let ( u , λ ) a solution of the continuous eigenvalue problem, and let ( u h , k , λ h , k ) be the corresponding discrete solution. Then, we have ( u h , k , λ h , k ) → ( u , λ ) h → 0 . for (Boffi-Conforti-Gastaldi 2006) 7
The Projection For u h , k ∈ X h , k we construct p h , k ∈ Q h , k such that u h , k − ∇ k p h , k ∈ V h , k , i.e., b k ( u h , k − ∇ k p h , k , q h , k ) 0 = = b k ( u h , k , q h , k ) − c k ( p h , k , q h , k ) for all q h , k ∈ Q h , k Thus, we have B h , k u h , k = C h , k p h , k and u h , k = S h , k p h , k with operators → Q ′ "div": B h , k : X h , k − h , k is defined by � B h , k v h , k , φ z , k � = b k ( v h , k , φ z , k ) , z ∈ V h → Q ′ "Laplace": C h , k : Q h , k − h , k is defined by � C h , k q h , k , φ z , k � = c k ( q h , k , φ z , k ) , z ∈ V h "grad": S h , k : Q h , k − → X h , k is given by nodal evaluation � q h , k ( y e ) e i k · ( y e − z e ) − q h , k ( x e ) e i k · ( x e − z e ) � � S h , k q h , k = ψ e , k . e =( x e , y e ) ∈E h This defines a projection P h , k = id − S h , k ◦ C − 1 h , k ◦ B h , k : X h , k − → V h , k . (special care is required for k = 0) 8
Modified LOBPCG Method (including projection) Let T h , k : X ′ → X h , k be a preconditioner for A δ → X ′ h , k − h , k = A h , k + δ M h , k : X X X h , k − h , k . S0) Choose randomly u 1 h , k , ..., u N h , k ∈ X h , k . Compute v n h , k = P h , k u n h , k ∈ V h , k . S1) Ritz-step: Set up Hermitian matrices � � � � ˆ ˆ a k ( v m h , k , v n m ( v m h , k , v n m , n = 1 ,..., N ∈ C N × N A = h , k ) m , n = 1 ,..., N , M = h , k ) z n = λ n ˆ and solve the matrix eigenvalue problem ˆ z n . A ˆ M ˆ N � S2) Compute y n z n m v n ˆ h , k = h , k ∈ V h , k . n = 1 S3) Compute r n h , k = A h , k y n h , k − λ n M h , k y n h , k ∈ X ′ h , k , check for convergence. S4) Compute u n h , k := T h , k r n h , k ∈ X h , k and w n h , k = P h , k u h , k ∈ V h , k . S5) Perform Ritz-step for { v 1 h , k , ..., v N h , k , w 1 h , k , ..., w N h , k } ⊂ V h , k of size 2 N . S6) Go to step S2). The full algorithm uses orthogonalization, new random vectors, and a Ritz-step of size 3 N . (the LOBPCG method was introduced by Knyazev 2001) 9
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