A computer-assisted Band-Gap Proof for 3D Photonic Crystals

Transcription

1 A computer-assisted Band-Gap Proof for 3D Photonic Crystals Henning Behnke Vu Hoang 2 Michael Plum 2 ) TU Clausthal, Institute for Mathematics, Clausthal, Germany 2) Univ. of Karlsruhe (TH), Faculty of Mathematics, 7628 Karlsruhe, Germany

2 Photonic Crystals We consider periodic optic media (photonic crystals). In such media light is absorbed for all frequencies which are not within a band gap. ( In nanotechnology, photonic crystals are fabricated and band gaps can be observed. 2

3 Photonic crystal: periodic dielectric medium such that electromagnetic waves of certain frequencies cannot propagate in it. Range of the prohibited frequencies: (complete) band gap Physical reason: destructive interference Practical interest: Design periodic materials which have band gaps! Analytically very difficult! Here: Computer-assisted proof of band gap. 3

4 Physical model: Homogeneous Maxwell s equations (c = ) curle = B t, D curlh = t, divb = 0, divd = 0, together with the constitutive relations D = εe, B = µh (E electric field, H magnetic field, D displacement field, B magnetic induction field) ε, µ: material tensors. Isotropic material: ε, µ scalar real-valued functions, not time-dependent ε electric permittivity, µ magnetic permeability. Photonic crystal: non-magnetic, i.e. µ, B H. 4

5 Look for monochromatic waves: E(x, t) = e iωt E(x), H(x, t) = e iωt H(x) Maxwell s equations give curle = iωh, curlh = iωεe, divh = 0, div(εe) = 0. Applying curl to the first two equations gives two decoupled systems: curl curl E = ω 2 εe and curl ε curlh = ω2 H div(εe) = 0 divh = 0 5

6 Operator theoretical formulation L 2 div (R3 ) := {u L 2 (R 3 ) 3 : div u = 0} H := L 2 div (R3 ) H(curl, R 3 ) { L 2 (R 3 ) 3 closed H(div, R 3 ) Maxwell s equation for H-field (curl ε curlh = ω2 H, divh = 0) reads, for u := H, u H \ {0}, ε (curlu) (curlv)dx = ω2 u vdx for each v H R 3 R 3 or, using B[u, v] := R 3 ε (curlu) (curlv)dx + R 3 u vdx (u, v H), λ := ω 2 +, u H \ {0}, B[u, v] = λ R 3 u vdx for all v H ( ) Lax-Milgram yields selfadjoint operator T : L 2 div (R3 ) H L 2 div (R3 ), B[T r, v] = r vdx (r L 2 div (R3 ), v H), R 3 D(A) := range(t ) H, A := T selfadjoint. ( ) u D(A) \ {0}, Au = λu 6

7 Now let ε L (R 3 ) (with ε ε min > 0) be periodic with periodicity cell Ω R 3 (bounded parallelogram). Standard crystal: Ω = (0, ) 3 Floquet-Bloch theory gives: The spectrum σ of ( ) has band-gap structure; more precisely: σ = n N I n, where I n are compact real intervals with min I n as n. I n is called the n-th spectral band. Usually, the bands I n overlap. But there might be gaps between them. These are the band-gaps of prohibited frequencies mentioned earlier. Floquet-Bloch theory tells further: I n = {λ k,n : k K} where K is the Brillouin zone (compact set in R 3, determined by Ω, K = [ π, π] 3 if Ω = (0, ) 3 ), and λ k,n n-th eigenvalue of (written formally) curl ( ε curlu) + u = λu on Ω, divu = 0 on Ω, e ik x u(x) satisfies periodic b.c. on Ω λ,n is called the n-th branch of the dispersion relation. 7

8 Precise formulation of (k-dependent) problem on Ω: (Problem with periodic boundary condition: trace of u H only in H 2( Ω).) G discrete lattice associated with Ω (G = Z 3 if Ω = (0, ) 3 ). Extension operator E : L 2 (Ω) 3 L 2 loc (R3 ) 3, (Ev)(x + g) := v(x) (x Ω, g G). Then boundary condition (e ik x u(x) periodic) together with the required smoothness on Ω reads: Let E(e ik u) H loc (curl, R 3 ) H loc (div, R 3 ) H k := {u L 2 (Ω) 3 : divu = 0, E(e ik u) H loc (curl, R 3 ) H loc (div, R 3 )} Eigenvalue problem generated by Floquet-Bloch theory (curl( ε curlu) + u = λu on Ω, divu = 0 on Ω, e ik x u(x) satisfies periodic b.c. on Ω) now reads: u H k \ {0}, ε (curlu) (curlv)dx + u vdx Ω} {{ Ω } =:B Ω (u,v) = λ Ω u vdx for all v H k (EW P k ) 8

9 Strategy for proving gap: ) Choose finitely many grid points in K 2) Compute verified eigenvalue enclosures for λ k,,..., λ k,n (N chosen fixed) for k in the grid 3) Use perturbation type argument to deduce from 2) also enclosures for λ k,,..., λ k,n for k between grid-points Together enclosure for λ k,,..., λ k,n for all k K enclosures for the bands I,..., I N If a gap in these enclosures occurs: proof of gap 9

10 Perturbation argument: Let H 0 k H k be given by omitting the condition divu = 0 in H k, i.e. H 0 k := {u L2 (Ω) 3 : E(e ik u) H loc (curl, R 3 ) H loc (div, R 3 )} and consider, besides (EW P k ), the problem with H 0 k instead of H k: u Hk 0 \ {0}, ε (curlu) (curlv)dx+ Ω Ω u vdx = λ Ω u vdx for all v H 0 k (EWP0 k ) λ = is an eigenvalue of infinite multiplicity of (EW Pk 0 ). (For each ϕ H 2 (Ω) s.t. e ik x ϕ(x) satisfies periodic b.c., ϕ is an eigenfunction.) This is the only difference between the spectra of (EW P k ) and (EW P 0 k )! 0

11 Defining w(x) := e ik x u(x), we obtain the equivalent problem w H 0 \ {0}, Ω ε [curlw + ik w] [curlv + ik v]dx + Ω w vdx = λ Ω w vdx for all v H 0 ( EW P 0 k ) where H 0 := {w L 2 (Ω) 3 : Ew H loc (curl, R 3 ) H loc (div, R 3 )} (independent of k!)

12 Let k be one of the gridpoints (to be) chosen in the Brillouin zone K; consider perturbation k + h of k. Theorem: Let [a, b] R be an interval such that, for some n N, ( <)λ k,n < a < b < λ k,n+ (whence [a, b] resolvent set of unperturbed problem (EW Pk 0 )), and let h < δ k, where δ k > 0 is such that { } { } λk,n λ k,n+ δ k max, + δ k max,. ε min a λ k,n λ k,n+ b Then, [a, b] is contained in the resolvent set of the perturbed problem (EW P 0 k+h ). Corollary: Let the assumptions of the Theorem hold for all gridpoints k in K, and suppose that gridpoints k K Ball(k, δ k ) K. Then, [a, b] is contained in a spectral band-gap. 2

13 Remaining task: Compute enclosures for eigenvalues λ k,,..., λ k,n of (EW P k ) for all gridpoints k; N N chosen fixed. Let k K denote a fixed gridpoint now. First step: Compute approximate eigenpairs to u H k \ {0}, ε (curlu) (curlv)dx + u vdx = λ u vdx Ω Ω} {{ Ω } =:B Ω (u,v) (EW P k ) for all v H k by Ritz method with appropriate basis functions in H k Second step: Upper eigenvalue bounds by Rayleigh-Ritz method (with approximate eigenfunctions as basis functions) Third step: Lower eigenvalue bounds by Lehmann-Goerisch method 3

14 Rayleigh-Ritz-Method (upper bounds) Fix k in the grid. Theorem. Let ũ k,,..., ũ k,n H k be linearly independent (approximate eigenfunctions), ( ) A = B Ω (ũ k,n, ũ k,m ) B = n,m=,...,n ( ) ũ k,n, ũ k,m L 2 n,m=,...,n and let Λ k, Λ k,n be the eigenvalues of Ax = ΛBx. Then λ k,n Λ k,n (n =,..., N). 4

15 Lehmann-Goerisch-Method for lower eigenvalue bounds (k in the grid still fixed): Choose a fixed shift parameter γ >. Compute additional approximations σ k,n satisfying, for n =,..., N, ε σ k,n H(curl, Ω), E σ k,n λ k,n + γ curlũ k,n ( e ik ) ε σ k,n H loc (curl, R 3 ), Moreover, suppose that β R is known such that Λ k,n < β γ λ k,n+ 5

16 Theorem (Goerisch). Define ( ) A = B Ω (ũ k,m, ũ k,n ) B = S = ( ) ũ k,m, ũ k,n L 2 ( ε σ k,m, σ k,n L 2 T = ( ũ k,m curl γ + m,n=,...,n m,n=,...,n ) m,n=,...,n ) ( ε σ k,m C N,N, C N,N, C N,N,, ũ k,n curl ( ) ) ε σ k,n L 2 m,n=,...,n If the matrix N = A + (γ 2β)B + β 2 (S + T) is positive definite, and if the eigenvalues θ θ 2 θ N of the eigenvalue problem ( ) A + (γ β)b x = θnx C N,N. are negative, we have β γ β θ n λ n,k for n =, N. 6

17 Spectral Homotopy For determining β such that Λ k,n < β γ λ k,n+, let ε s (x) := ( s)ε max + sε(x) x Ω, 0 s, and consider the family of eigenvalue problems u H k \ {0}, ε s (x) (curlu) (curlv)dx + u vdx = λ (s) Ω Ω Ω for all v H k, 0 s, k still fixed in the grid. Eigenvalues (λ (s) n ) n N. For s = 0: eigenvalues λ (0) n are known For s = : λ () n = λ k,n (n N). Lemma. For each fixed n N, λ (s) n λ (t) n for 0 s t. (Proof by Poincaré s min-max principle.) u vdx 7

18 Spectral Homotopy λ λ 0 λ 9 λ 8 λ 7 λ 6 λ 5 λ 4 λ 3 λ 2 λ

19 { x ( Concrete case: Ω = (0, ) 3 if 2,, ε(x) := 2, 2) < 2 25 otherwise Basis functions: combination of a) plane waves: A (k) n e i(2πn+k) x, n Z 3, A (k) n C 3, A n (k) (2πn + k) = 0 b) certain functions which are non-zero only on the ball x ( 2, 2, 2) < 2, constructed via polynomials in r and spherical harmonics in ϕ, θ. By symmetry, only the following part of the Brillouin zone K needs to be considered: Show B, T k 2 3 D k3 C B k2 0 A 9

20 4 ritzwerte kugel ei eo 25 R 0.5 m A B C A D B C D 20

21 50 ritzwerte kugel ei eo 25 R 0.5 m

22 50 ritzwerte kugel ei eo 25 R 0.5 m 3 dicht

23 jointly with V. Hoang, C. Wieners: 2D-situation: ε = ε(x, x 2 ), polarized wave E = (0, 0, u) 0 = div(εe) = x (εu) = ε u 3 x, 3 i.e. u x 3 = 0, u = u(x, x 2 ). curl curl E = 0 0 u Maxwell s equation gives, with λ = ω 2, = 2 x x 2 2 ( ) u = λεu equation on whole of R 2 23

24 A Candidate Let Λ = Z 2, Ω = (0, ) 2 and K = [ π, π] 2. We set ε(x) = for x [/6, 5/6] 2 and ε(x) = 5 else. By symmetry we have the same spectrum for k = (k, k 2 ), ( k, k 2 ), (k, k 2 ), (k 2, k ) ( π, π) (π, π) ( π, π) (π, π) Brillouin zone K periodic material distribution ε 24

25 A Candidate eigenvalue λ k,3 for all k K

26 A Candidate eigenvalue λ k,4 for all k K

27 A Candidate eigenvalues λ k,,..., λ k,5 for all k K 27

28 A Candidate eigenfunctions u k,,..., u k,6 for k = (π, π) 28

29 Spectral Homotopy for k = (2.530, ) λ λ 0 λ 9 λ 8 λ 7 λ 6 λ 5 λ 4 λ 3 λ 2 λ λ (s) 0 λ (s) 9 λ (s) for s /32 λ(s) for s 4/32 λ(s) for s 8/32 λ(s) for s 9/ for s 22/ for s 28/32 29

30 Spectral Homotopy for k = (2.530, ) s 0 4/32 8/32 9/32 λ (.295,.296) (.402,.403) (.528,.529) ( 2.07, 2.08) λ 2 ( 2.875, 2.876) ( 3.4, 3.5) ( 3.396, 3.397) ( 4.526, 4.527) λ 3 ( 8.74, 8.75) ( 8.840, 8.84) ( 9.594, 9.595) (2.89,2.90) λ 4 ( 9.754, 9.755) (0.563,0.564) (.523,.524) (5.397,5.398) λ 5 (0.208,0.209) (.048,.049) (2.09,2.020) (5.575,5.577) λ 6 (.788,.789) (2.783,2.784) (4.09,4.020) (9.920,9.92) λ 7 (5.507,5.508) (6.778,6.779) (8.236,8.237) (23.339,23.373) λ 8 (20.246,20.247) (2.907,2.93) (23.786,23.832) λ 9 (22.386,22.387) (24.20,24.23) λ 0 (24.49,24.420) s 9/32 22/32 28/32 λ ( 2.07, 2.08) ( 2.204, 2.205) ( 2.690, 2.69) ( 3.27, 3.28) λ 2 ( 4.526, 4.527) ( 4.979, 4.980) ( 6.220, 6.22) ( 7.433, 7.434) λ 3 (2.89,2.90) (3.046,3.048) (4.98,4.985) (6.445,6.452) λ 4 (5.397,5.398) (6.653,6.655) (9.383,9.389) (2.422,2.450) λ 5 (5.575,5.577) (7.88,7.90) (22.45,22.465) λ 6 (9.920,9.92) (22.809,22.83) λ 7 (23.339,23.373) 30

31 A Verified Band Gap This figure illustrates the covering K k grid Ball(k, r k ) Eigenvalue bounds (in grid) and perturbation arguments give λ k,3 8.2, λ k, for all k K. This proves the existence of a band gap (8.2, 8.25) (λ max,3, λ min,4 ) for the spectral problem u = λεu in R 2. The proof requires the close approximation of more than eigenvalues and eigenfunctions (for 00 vectors k grid with up to 7 homotopy steps each) and takes about 90 h computing time. 3