Modelling in photonic crystal structures

Transcription

1 Modelling in photonic crystal structures Kersten Schmidt MATHEON Nachwuchsgruppe Multiscale Modelling and Scientific Computing with PDEs in collaboration with Dirk Klindworth (MATHEON, TU Berlin) Holger Brandsmeier, Christoph Schwab (ETH Zurich) Sonia Fliss (POEMS, ENSTA Paris) DFG Research Center MATHEON Mathematics for key technologies 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February 2013

2 0.5 Motivation Photonic band gap material Energy states for a certain wavevector In the band gap no propagating modes Photonic crystal wave-guides Propagating Bloch waves in the guide Tailoring the dispersion of guided modes Flats bands lead to slow light. ωa 2πc System of interest: planar Photonic crystal wave-guide index guiding in vertical direction, band gap guiding in horizontal direction Γ K M Γ F. Robin, Communication Photonics Group, ETH Zurich K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

3 Applications of Wave-guides with Tailored Dispersion Flat bands Modes with low group velocity vg = dω dk. reduction of velocity of the light energy light stays longer in the waveguide (by slow down factor). simultaneously: the light is enhanced by the slow down factor (light is compressed spatially) T.F. Krauss, Nature Photonics 2, p , Aug 2008 F. Robin, Communication Photonics Group, ETH Zurich Applications of the Intensity Enhancement of Slow Light Modes. lowers the power threshold of the input light for obtaining non-linear effects. can potentially decrease the length of semiconductor optical amplifiers by the slow down factor (higher integration density) K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22nd February / 18

4 Outline 1 Reflection and transmission on periodic half-space 2 Computation of Bloch waves using Dirichlet cell problems 3 Guided modes in photonic crystal wave-guides 4 Multiscale FEM for finite periodic bands

5 Reflection and transmission on a dielectric half-space Reflection and transmission on a periodic half-space Helmholtz equation u(x 1, x 2 ) + ω 2 n 2 (x 1, x 2 )u(x 1, x 2 ) = 0. n(x 1, x 2 ) = n n(x 1, x 2 ) = n + u inc Decomposition of u inc into plane waves by continuous Fourier transform in x 2 Reflected and transmitted plane waves for each k 2 given by Snell s law (dt. Brechungsgesetz) Apply inverse Fourier transform u inc (x 1, x 2 ) = R u trans (x 1, x 2 ) = û inc (x 1, k 2 )e ik 2x 2 dk 2 = û inc (k 2 )e i((ω2 n 2 k 2) 1/2 x 1 +k 2 x 2 ) dk 2, x 1 < 0 R T (k 2 )ûinc (k 2 )e i((ω2 n+ 2 k 2) 1/2 x 1 +k 2 x 2 ) dk 2, x 1 > 0. R K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

6 Reflection and transmission on a dielectric half-space Reflection and transmission on a periodic half-space Helmholtz equation u(x 1, x 2 ) + ω 2 n 2 (x 1, x 2 )u(x 1, x 2 ) = 0. n(x 1, x 2 ) = n n(x 1, x 2 ) = n + u inc Decomposition of u inc into plane waves by continuous Fourier transform in x 2 Reflected and transmitted plane waves for each k 2 given by Snell s law (dt. Brechungsgesetz) Apply inverse Fourier transform u inc (x 1, x 2 ) = R u trans (x 1, x 2 ) = R û inc (x 1, k 2 )e ik 2x 2 dk 2 = R T (k 2 )ûinc (x 1, k 2 )e ik 2x 2 dk 2 = R û inc (k 2 )e i((ω2 n 2 k 2) 1/2 x 1 +k 2 x 2 ) dk 2, x 1 < 0 T (k 2 )ûinc (k 2 )e i((ω2 n 2 + k 2) 1/2 x 1 +k 2 x 2 ) dk 2, x 1 > 0 K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

7 Reflection and transmission on a periodic half-space Reflection and transmission on a periodic half-space Helmholtz equation u(x 1, x 2 ) + ω 2 n 2 (x 1, x 2 )u(x 1, x 2 ) = 0. n(x 1, x 2 ) = n n(x 1, x 2 ) = n per(x 1, x 2 ) u inc (a 1, a 2 ) periodicity n per(x 1 + na 1, x 2 + ma 2 ) = n per(x 1, x 2 ) for all (n, m) Z 2 Decomposition of u inc into a 2 quasi-periodic waves by Floquet-Bloch transform in x 2 Reflected and transmitted (propagating and evanescent) waves for each k 2 defined on infinite strip Apply inverse Floquet-Bloch transform FB(u)(x 1, x 2, k 2 ) = m Z u(x 1, x 2 ma 2 )e ik 2m a 2, where FB(u)(x 1, x 2 + ma 2, k 2 ) = FB(u)(x 1, x 2, k 2 )e ik 2ma 2 K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

8 Reflection and transmission on a periodic half-space u inc Reflection and transmission on a periodic half-space Helmholtz equation u(x 1, x 2 ) + ω 2 n 2 (x 1, x 2 )u(x 1, x 2 ) = 0. n(x 1, x 2 ) = n n(x 1, x 2 ) = n per(x 1, x 2 ) (a 1, a 2 ) periodicity n per(x 1 + na 1, x 2 + ma 2 ) = n per(x 1, x 2 ) for all (n, m) Z 2 Decomposition of u inc into a 2 quasi-periodic waves by Floquet-Bloch transform in x 2 Reflected and transmitted (propagating and evanescent) waves for each k 2 defined on infinite strip Apply inverse Floquet-Bloch transform FB(u)(x 1, x 2, k 2 ) = m Z u(x 1, x 2 ma 2 )e ik 2m a 2, (x 2, k 2 ) [ 0, a 2 ] [ π a 2, π a 2 ] where FB(u)(x 1, x 2 + ma 2, k 2 ) = FB(u)(x 1, x 2, k 2 )e ik 2ma 2 FB( u)(x 1, x 2 + ma 2, k 2 ) = FB(u)(x 1, x 2, k 2 ) FB(n peru)(x 1, x 2, k 2 ) = n perfb(u)(x 1, x 2, k 2 ) FB(u inc )(x 1, x 2, k 2 ) = û inc (k 2 + 2π j)e i((ω2 n 2 (k 2+ 2π a2 j) 2 ) 1/2 x 1 +(k 2 + 2π j)x a 2 ) 2 j Z a 2 K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

9 Reflection and transmission on a periodic half-space Problem for Floquet-Bloch transform on infinite strip for u k2 (x 1, x 2 ) := FB(u)(x 1, x 2, k 2 ) u k2 (x 1, x 2 ) + ω 2 n 2 (x 1, x 2 )u k2 (x 1, x 2 ) = 0, u k2 (x 1, a 2 ) = u k2 (x 1, 0)e ik 2a 2, 2 u k2 (x 1, a 2 ) = 2 u k2 (x 1, 0)e ik 2a 2, u k2 FB(u inc ) is outgoing to the left, u k2 is outgoing to the right. n(x 1, x 2 ) = n n(x 1, x 2 ) = n per(x 1, x 2 ) FB(u inc ) K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

10 Reflection and transmission on a periodic half-space Problem for Floquet-Bloch transform on infinite strip for u k2 (x 1, x 2 ) := FB(u)(x 1, x 2, k 2 ) e ik 2x 2 ( ) ( ) ( + ik ) ( + ik2 0 )uk2 (x 1 1, x 2 ) + ω 2 n 2 (x 1, x 2 )u k2 (x 1, x 2 ) = 0, u k2 (x 1, a 2 ) = u k2 (x 1, 0), 2 u k2 (x 1, a 2 ) = 2 u k2 (x 1, 0), u k2 FB(u inc ) e ik 2x 2 is outgoing to the left, u k2 n(x 1, x 2 ) = n n(x 1, x 2 ) = n per(x 1, x 2 ) FB(u inc ) e ik 2x 2 is outgoing to the right. Outgoing in the homogenous half-strip? truncate by a perfectly matched layer truncate with Dirichlet-to-Neumann b.c. Fourier series in x 2 (periodic b.c.) ODE in x 1 with two fundamental solutions take Neumann trace of left-decaying or left-propagating modes (plane waves) u Plane,k2 (x 1, x 2 ) e i(k 1x 1 + 2πj x a 2 ) 2 Outgoing in the periodic half-strip? perfectly matched layer cannot be used truncate with Dirichlet-to-Neumann b.c. compute right-propagating and right-decaying eigenmodes (Bloch waves) u Bloch,k2 (x 1 + a 1, x 2 ) = u Bloch,k2 (x 1, x 2 )e ik 1a 1 K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

11 Reflection and transmission on a periodic half-space Problem for Floquet-Bloch transform on infinite strip for u k2 (x 1, x 2 ) := FB(u)(x 1, x 2, k 2 ) e ik 2x 2 ( ) ( ) ( + ik ) ( + ik2 0 )uk2 (x 1 1, x 2 ) + ω 2 n 2 (x 1, x 2 )u k2 (x 1, x 2 ) = 0, u k2 (x 1, a 2 ) = u k2 (x 1, 0), 2 u k2 (x 1, a 2 ) = 2 u k2 (x 1, 0), u k2 FB(u inc ) e ik 2x 2 is outgoing to the left, u k2 n(x 1, x 2 ) = n n(x 1, x 2 ) = n per(x 1, x 2 ) FB(u inc ) e ik 2x 2 is outgoing to the right. Outgoing in the homogenous half-strip? truncate by a perfectly matched layer truncate with Dirichlet-to-Neumann b.c. Fourier series in x 2 (periodic b.c.) ODE in x 1 with two fundamental solutions take Neumann trace of left-decaying or left-propagating modes (plane waves) u Plane,k2 (x 1, x 2 ) e i(k 1x 1 + 2πj x a 2 ) 2 Outgoing in the periodic half-strip? perfectly matched layer cannot be used truncate with Dirichlet-to-Neumann b.c. compute right-propagating and right-decaying eigenmodes (Bloch waves) u Bloch,k2 (x 1 + a 1, x 2 ) = u Bloch,k2 (x 1, x 2 )e ik 1a 1 K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

12 Propagating and evanescent Bloch waves Bloch waves u Bloch,k2 (x 1 + a 1, x 2 ) = u Bloch,k2 (x 1, x 2 ) e ik 1a 1 Band diagram ω k2 (k 1 ), here for example for k 2 = 0 computed by high-order FEM within the numerical C++ library Concepts direction of propagative modes determined by group velocity ω k TM ˆω = ˆω = ˆω = TM ˆω = ˆω = ˆω = ˆω = (ωa1)/(2π) ˆω = (ωa1)/(2π) k 1/(π/a 1) (a) propagating modes Im k (b) evanescent modes Computation of the group velocity from the mode ( talk by Dirk Klindworth, 14:20) ω = k 1a 2 1 C u k 2 2 dx 1 dx 2 a 1 Im C u k 2 1 u k2 dx 1 dx 2 k 1 ω C n2 per u k2 2 dx 1 dx 2 K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

13 Computation of Bloch waves using Dirichlet cell problems Dimensionally reduced eigenvalue problem P. Joly, J.-R. Li, S. Fliss, Commun. Comput. Phys. 6: , for fixed (ω, k 2 ) we define the Dirichlet cell problems for u[ϕ L, ϕ R ] with Dirichlet data ϕ L, ϕ R ( + ik 2 ( 0 1 ) ) ( + ik2 ( 0 1 ) )u[ϕl, ϕ R ] ω 2 n 2 per u[ϕ L, ϕ R ] = 0 u[ϕ L, ϕ R ] ΓL = ϕ L u[ϕ L, ϕ R ] ΓR = ϕ R Γ L Γ R unique solution exists except for a countable set of frequencies defines Dirichlet-to-Neumann operators Λ L, Λ R Λ L (ϕ L, ϕ R ) = nu[ϕ L, ϕ R ] ΓL Λ R (ϕ L, ϕ R ) = nu[ϕ L, ϕ R ] ΓR DtN-operators are isomorphisms (Neumann BVP) except for a countable set of frequencies K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

14 Computation of Bloch waves using Dirichlet cell problems Dimensionally reduced eigenvalue problem P. Joly, J.-R. Li, S. Fliss, Commun. Comput. Phys. 6: , for fixed (ω, k 2 ) we define the Dirichlet cell problems for u[ϕ L, ϕ R ] with Dirichlet data ϕ L, ϕ R ( + ik 2 ( 0 1 ) ) ( + ik2 ( 0 1 ) )u[ϕl, ϕ R ] ω 2 n 2 per u[ϕ L, ϕ R ] = 0 u[ϕ L, ϕ R ] ΓL = ϕ L u[ϕ L, ϕ R ] ΓR = ϕ R Γ L Γ R defines Dirichlet-to-Neumann operators Λ L, Λ R Λ L (ϕ L, ϕ R ) = nu[ϕ L, ϕ R ] ΓL Λ R (ϕ L, ϕ R ) = nu[ϕ L, ϕ R ] ΓR represent any Bloch wave as u Bloch,k2 = u[ϕ L, ϕ R ] with ϕ L = u Bloch,k2 ΓL, ϕ R = u Bloch,k2 ΓR Quasi-periodicity of Dirichlet trace u Bloch,k2 (x 1 + a 1 ) = e ik 1a 1 ϕ R = λ 1 ϕ L }{{} :=ϕ }{{} =:λ 1 u Bloch,k2 (x 1 ) Quasi-periodicity of Neumann trace nu Bloch,k2 (x 1 + a 1 ) = e ik 1a 1 }{{} =:λ 1 nu Bloch,k2 (x 1 ) Λ R (ϕ, λ 1 ϕ) = λ 1 Λ L (ϕ, λ 1 ϕ) Quadratic eigenvalue problem for (λ 1, ϕ) C H 1/2 per(γ L ) λ 2 1 Λ L(0, ϕ) + λ 1 (Λ L (ϕ, 0) + Λ R (0, ϕ)) + Λ R (ϕ, 0) = 0 K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

15 Computation of Bloch waves using Dirichlet cell problems Dimensionally reduced matrix eigenvalue problem for fixed (ω, k 2 ) we define the discrete Dirichlet cell problems for u h [ϕ L, ϕ R ] with discrete Dirichlet data ϕ L,h Sper(Γ p,1 L ), ϕ R,h Sper(Γ p,1 R ), i.e., by mixed variational formulation: Seek (u h, ψ L,h, ψ R,h ) Sper(Ω) p,1 Dper(Γ p,1 L ) Dper(Γ p,1 R ) ( ) ( ) ( + ik )uh ( ik )v ω 2 nper 2 u hv d x ψ L,h v ds ψ R,h v ds = 0 Ω Γ L Γ R u h ψ L ds = ϕ L,h ψ L ds Γ L Γ L u h ψ R ds = ϕ R,h ψ R ds Γ R Γ R high-order FEM-space S p,1 per(ω) on curved meshes of the unit cell Ω K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

16 Computation of Bloch waves using Dirichlet cell problems Dimensionally reduced matrix eigenvalue problem for fixed (ω, k 2 ) we define the discrete Dirichlet cell problems for u h [ϕ L, ϕ R ] with discrete Dirichlet data ϕ L,h Sper(Γ p,1 L ), ϕ R,h Sper(Γ p,1 R ), i.e., by mixed variational formulation: Seek (u h, ψ L,h, ψ R,h ) Sper(Ω) p,1 Dper(Γ p,1 L ) Dper(Γ p,1 R ) ( ) ( ) ( + ik )uh ( ik )v ω 2 nper 2 u hv d x ψ L,h v ds ψ R,h v ds = 0 Ω Γ L Γ R u h ψ L ds = ϕ L,h ψ L ds Γ L Γ L u h ψ R ds = ϕ R,h ψ R ds Γ R Γ R high-order FEM-space Sper(Ω) p,1 on curved meshes of the unit cell Ω high-order FEM-spaces Sper(Γ p,1 L ), Sper(Γ p,1 R ) on its sides, N = dim Sper(Γ p,1 L ) = dim Sper(Γ p,1 R ) high-oder dual FEM-spaces Dper(Γ p,1 L ), Dper(Γ p,1 R ), N = dim Dper(Γ p,1 L ) = dim Dper(Γ p,1 R ) B. Wohlmuth, SIAM J. Numer. Anal. 38: , for basis functions ϕ L,j of Sper(Γ p,1 L ) and ψ L,i of Dper(Γ p,1 L ) Γ L ψ L,i ϕ L,j ds = C i δ ij Dirichlet-to-Neumann maps Λ L (, 0), Λ L (0, ), Λ R (, 0), Λ R (0, ) represented by matrices T 00, T 10, T 01, T 11 K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

17 Computation of Bloch waves using Dirichlet cell problems Dimensionally reduced matrix eigenvalue problem for fixed (ω, k 2 ) we define the discrete Dirichlet cell problems for u h [ϕ L, ϕ R ] with discrete Dirichlet data ϕ L,h Sper(Γ p,1 L ), ϕ R,h Sper(Γ p,1 R ) Dirichlet-to-Neumann maps Λ L (, 0), Λ L (0, ), Λ R (, 0), Λ R (0, ) represented by matrices T 00, T 10, T 01, T 11 (pre-computed) Quadratic matrix eigenvalue problem for (λ 1,h, ϕ h ) C C N, N = dim S p,1 per(γ L ) ( λ 2 1 T 10 + λ 1 (T 11 + T 00 ) + T 01 ) ϕh = 0 Linearisation: ϕ h = λ 1 ϕ h ( T00 T 11 ) ( T 01 ϕh } Id {{ 0 } A eigenvalue problem is reduced in dimension (d d 1) ) ( T10 0 = λ ϕ 1 h 0 Id } {{ } M ) ( ) ϕh ϕ h iterative eigenvalue solver to compute the λ 1 with modulus close to 1, implicitely restarted Arnoldi iterations eigenvalues come in pairs, left progapating right propagating, decide by group velocity left decaying-right decaying D. Klindworth, K. Schmidt, S. Fliss, Comput. Math. Appl., accepted, K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

18 Reflection and transmission on a periodic half-space Problem for Floquet-Bloch transform on infinite strip for u k2 (x 1, x 2 ) := FB(u)(x 1, x 2, k 2 ) e ik 2x 2 ( ) ( ) ( + ik ) ( + ik2 0 )uk2 (x 1 1, x 2 ) + ω 2 n 2 (x 1, x 2 )u k2 (x 1, x 2 ) = 0, u k2 (x 1, a 2 ) = u k2 (x 1, 0), 2 u k2 (x 1, a 2 ) = 2 u k2 (x 1, 0), u k2 FB(u inc ) e ik 2x 2 is outgoing to the left, u k2 n(x 1, x 2 ) = n n(x 1, x 2 ) = n per(x 1, x 2 ) FB(u inc ) e ik 2x 2 is outgoing to the right. Outgoing in the homogenous half-strip? truncate by a perfectly matched layer truncate with Dirichlet-to-Neumann b.c. Fourier series in x 2 (periodic b.c.) ODE in x 1 with two fundamental solutions take Neumann trace of left-decaying or left-propagating modes (plane waves) u Plane,k2 (x 1, x 2 ) e i(k 1x 1 + 2πj x a 2 ) 2 Outgoing in the periodic half-strip? perfectly matched layer cannot be used truncate with Dirichlet-to-Neumann b.c. compute right-propagating and right-decaying eigenmodes (Bloch waves) T 00 + T 10 Φ h K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

19 Guided modes in photonic crystal wave-guides Problem for Floquet-Bloch transform on infinite strip for u k2 (x 1, x 2 ) := FB(u)(x 1, x 2, k 2 ) e ik 2x 2 ( ) ( ) ( + ik ) ( + ik2 0 )uk2 (x 1 1, x 2 ) + ω 2 n 2 (x 1, x 2 )u k2 (x 1, x 2 ) = 0, u k2 (x 1, a 2 ) = u k2 (x 1, 0), 2 u k2 (x 1, a 2 ) = 2 u k2 (x 1, 0), u k2 u k2 is outgoing to the left, is outgoing to the right. Outgoing in the periodic half-strip? perfectly matched layer cannot be used approximation by super-cells truncate with Dirichlet-to-Neumann b.c. compute right-propagating and right-decaying eigenmodes (Bloch waves) T 00 (ω, k 2 ) + T 10 (ω, k 2 )Φ h (ω, k 2 ) Non-linear eigenvalue problem in k 2 or ω sample matrices depending non-linear on ω on Chebychev points in some frequency interval interpolation of matrices by polynomial in ω polynomial matrix eigenvalue problem which can be effectively linearised exponential convergence in number of Chebychev points away from band edge D. Klindworth, K. Schmidt, S. Fliss, Comput. Math. Appl., accepted, C. Effenberger, D. Kressner, BIT 52: , K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

20 Guided modes in photonic crystal wave-guides Guided modes in hexagonal photonic crystal wave-guide air-holes of radius 0.31a in dielectric substrate with n 2 = 11.4 comparison with the super-cell approach exponential convergence of Chebychev interpolation inside the band-gap (reference values are computed by Newton s method) frequency [ω / 2π] mean error wave vector [k / 2π] number of Chebyshev nodes D. Klindworth, K. Schmidt, S. Fliss, Comput. Math. Appl., accepted, K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

21 Guided modes in hexagonal photonic crystal wave-guide air-holes of radius 0.31a in dielectric substrate with n 2 = 11.4 Guided modes in photonic crystal wave-guides (a) Well confined, odd mode at ω = π. (b) Odd mode close to band edge at ω = π. (c) Even mode at ω = π. K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

22 Reflection and transmission on a (thick) finite periodic band Problem for Floquet-Bloch transform on infinite strip for u k2 (x 1, x 2 ) := FB(u)(x 1, x 2, k 2 ) e ik 2x 2 ( ) ( ) ( + ik ) ( + ik2 0 )uk2 (x 1 1, x 2 ) + ω 2 n 2 (x 1, x 2 )u k2 (x 1, x 2 ) = 0, u k2 (x 1, a 2 ) = u k2 (x 1, 0), 2 u k2 (x 1, a 2 ) = 2 u k2 (x 1, 0), u k2 FB(u inc ) e ik 2x 2 is outgoing to the left, u k2 is outgoing to the right. n(x 1, x 2 ) = n n(x 1, x 2 ) = n per(x 1, x 2 ) n(x 1, x 2 ) = n + FB(u inc ) e ik 2x 2 Outgoing in the homogenous half-strip? truncate by a perfectly matched layer truncate with Dirichlet-to-Neumann b.c. Large number of periods the larger the PhC band the larger the computational effort of standard discretisation schemes Idea: take Bloch waves in a generalised FEM (gfem) basis K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

23 Reflection and transmission on a (thick) finite periodic band Multiscale basis H. Brandsmeier, K. Schmidt, C. Schwab, J. Comput. Phys., 230: , microscopic: Bloch waves oscillates on the scale of one unit cell macroscopic: envelope function oscillating on the scale of the whole crystal How to decrease the error of the multiscale FEM? increase number of Bloch waves evanescent Bloch waves with large Im k 1 leads to ill-conditioned systems refine the macroscopic basis (adaptive mesh refinement, polynomial degree enhancement) piecewise macroscopic polynomials localize Bloch waves Study L 2 best-approximation of fully resolved solution onto multiscale FEM orthogonal incidence (k 2 = 0), ˆω = 0.625, 7 periods best-approximation on whole crystal Ω cr, only on inner cells Ω inner, and on first unit-cell Ω first Ω first Ω inner Ω cr K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

24 Reflection and transmission on a (thick) finite periodic band Study L 2 best-approximation of fully resolved solution onto multiscale FEM orthogonal incidence (k 2 = 0), ˆω = 0.625, 7 periods 0.7 TM ˆω = ˆω = ˆω = TM ˆω = ˆω = ˆω = ˆω = (ωa1)/(2π) ˆω = (ωa1)/(2π) k 1/(π/a 1) (a) propagating modes Im k (b) evanescent modes ˆω = 0.625: n bloch = 4 ˆω = 0.215: n bloch = 2 ˆω = 0.300: n bloch = 2 (growing & decaying Bloch modes) K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

25 Reflection and transmission on a (thick) finite periodic band Study L 2 best-approximation of fully resolved solution onto multiscale FEM orthogonal incidence (k 2 = 0), ˆω = 0.625, 7 periods best-approximation on whole crystal Ω cr, only inner cells Ω inner, and on first unit-cell Ω first Ω cr relative L 2 (Ω cr )-error 10 5 np = 3, ˆω = np = 5, ˆω = np = 9, ˆω = np = 100, ˆω = np = 200, ˆω = np = 100, ˆω = np = 100, ˆω = macro polynomial degree pmac ˆω = 0.625: n bloch = 4 ˆω = 0.215: n bloch = 2 ˆω = 0.300: n bloch = 2 (growing & decaying Bloch modes) no convergence w.r.t. macro polynomial degree K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

26 Reflection and transmission on a (thick) finite periodic band Study L 2 best-approximation of fully resolved solution onto multiscale FEM orthogonal incidence (k 2 = 0), ˆω = 0.625, 7 periods best-approximation on whole crystal Ω cr, only on inner cells Ω inner, and on first unit-cell Ω first Ω inner relative L 2 (Ω inner )-error 10 5 np = 3, ˆω = np = 5, ˆω = np = 9, ˆω = np = 100, ˆω = np = 200, ˆω = np = 100, ˆω = np = 100, ˆω = macro polynomial degree pmac ˆω = 0.625: n bloch = 4 ˆω = 0.215: n bloch = 2 ˆω = 0.300: n bloch = 2 (growing & decaying Bloch modes) for ˆω = 0.625, p mac = 1: n dof = 16! K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

27 Reflection and transmission on a (thick) finite periodic band Study L 2 best-approximation of fully resolved solution onto multiscale FEM orthogonal incidence (k 2 = 0), ˆω = 0.625, 7 periods best-approximation on whole crystal Ω cr, only inner cells Ω inner, and on first unit-cell Ω first Ω first relative L 2 (Ω first )-error 10 5 np = 3, ˆω = np = 5, ˆω = np = 9, ˆω = np = 100, ˆω = np = 200, ˆω = np = 100, ˆω = np = 100, ˆω = macro polynomial degree pmac ˆω = 0.625: n bloch = 4 ˆω = 0.215: n bloch = 2 ˆω = 0.300: n bloch = 2 (growing & decaying Bloch modes) exponential convergence w.r.t. macro polynomial degree in the first cell K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

28 Reflection and transmission on a (thick) finite periodic band Multiscale FEM macroscopic mesh consists of five cells independent of the number of crystal periods K 4 K 1 K 2 K 3 K 5 Macroscopic (piecewise) polynomial are multiplied with microscropic basis (Bloch waves + 1) u(x 1, x 2 ) b mac b x multi 1 K 1 K 2 x 2 K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

29 Multiscale FEM Reflection and transmission on a (thick) finite periodic band macroscopic mesh consists of five cells independent of the number of crystal periods K 4 K 1 K 2 K 3 K 5 Overlap handling, only micro function 1 is multiplied with polynomials identified to Γ over u(x 1, x 2 ) b ext 3 b multi 1 b multi 2 x 1 K 4 Γ over K 1 K 2 x 2 K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

30 Reflection and transmission on a (thick) finite periodic band Multiscale FEM macroscopic mesh consists of five cells independent of the number of crystal periods K 4 K 1 K 2 K 3 K 5 Convergence of the multiscale FEM for increasing macro polynomial degree np = 3, ˆω = np = 100, ˆω = np = 3, ˆω = relative H 1 (Ω cr )-error np = 100, ˆω = np = 3, ˆω = np = 100, ˆω = number of degrees of freedom supported in Ω cr K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

31 Reflection and transmission on a (thick) finite periodic band Multiscale FEM macroscopic mesh consists of five cells independent of the number of crystal periods K 4 K 1 K 2 K 3 K 5 Convergence of the multiscale FEM in comparison to p-fem n dof = 8, pmac = 1 relative H 1 (Ω cr )-error n dof = 78, pmac = 3 n dof = 434, pmac = 7 n dof = 78, p-fem n dof = 434, p-fem n dof = 2 000, p-fem n dof = , p-fem n dof = , p-fem number of periods np K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18

32 Reflection and transmission on periodic half-space Floquet-Bloch transform of incoming and scattered wave Conclusion Dirichlet-to-Neumann boundary conditions using decomposition in propagating and evanescent Bloch waves Computation of Bloch waves using Dirichlet cell problems Bloch waves are represented as solution of Dirichlet cell problems which defines local Dirichlet-to-Neumann maps Λ L, Λ R Quadratic (matrix) eigenvalue problem of small size in λ 1 = e ik 1a 1 Computation of guided modes in photonic crystal wave-guides Dirichlet-to-Neumann boundary conditions using decomposition in propagating and evanescent Bloch waves Boundary conditions are non-linear, Chebychev interpolation, exponential convergence Multiscale FEM for finite periodic bands Bloch waves as microscopic basis multiplied with polynomial basis on macroscopic cells computation effort and accuracy is independent of the number of periods Thank you for your attention. K.Schmidt Modelling of photonic crystal wave-guides modes 6th Annual Meeting Photonic Devices, Berlin (Germany), 22 nd February / 18