Karlsruhe Institute of Technology

KIT – University of the State of Baden-Württemberg and National Large-scale Research Center of the Helmholtz Association www.kit.edu

Photonic Band Gap ComputationsChristian Wieners

Institut für Angewandte und Numerische Mathematik, Karlsruhe

Karlsruhe Institute of TechnologyOutline

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 Optimization of Photonic Band Gap (work in progress)

The numerical results are obtained in cooperation with A. Bulovyatov.

The verfication method is developed in cooperation with V. Hoang and M. Plum.

The optimization method is developed in cooperation with M. Richter and W. Dörfler.

1

Karlsruhe Institute of TechnologyThe Maxwell Eigenvalue Problem

We consider the Maxwell eigenvalue problem for the magnetic field HHHfor optic waves (where we have µ ≡ 1)

∇∇∇× ε−1

∇∇∇×HHH = λHHH (1a)

∇∇∇ ·HHH = 0 (1b)

in the case that ε is a periodic function with ε(xxx) = ε(xxx +zzz) for all zzz ∈ Z3.

Let Ω = (0,1)3 be a fundamental cell, and K = [−π,π]3 be the Brillouin zone.The ansatz HHH(xxx) = eikkk ·xxxHHH(xxx) with xxx ∈Ω, kkk ∈ K and HHH periodic in Ω yields

∇k∇k∇k × ε−1

∇k∇k∇k ×HHH = λHHH (2a)

∇k∇k∇k ·HHH = 0 (2b)

in Ω, where ∇k∇k∇k = ∇∇∇ + ikkk .

The reduced problem for kkk ∈ K has a discrete spectrum with eigenvalues

0≤ λkkk ,1 ≤ λkkk ,1 ≤ ·· · ≤ λkkk ,N ≤ ·· ·

Theorem Problem (1) has the spectrum σ =⋃

n∈N[ inf

kkk∈Kλkkk ,n,sup

kkk∈Kλkkk ,n].

(application of the Floquet-Bloch theory)

If (supkkk∈K λkkk ,n, inf kkk∈K λkkk ,n+1) is non-empty for some n ∈ N, this is a band gap.2

Karlsruhe Institute of TechnologyApproximation of the Maxwell Eigenvalue Problem

Let XXX = HHHper(curl; Ω)⊂HHH(curl; Ω) be the subspace of periodic vector fields.For uuu,vvv ∈XXX we define the Hermitian bilinear forms

akkk (uuu,vvv) =∫Ω

ε−1

∇k∇k∇k ×uuu ·∇k∇k∇k ×vvv dx , m(uuu,vvv) =∫Ω

uuu · vvv dx

Let Q = H1per(Ω)⊂ H1(Ω) be the subspace of periodic scalar functions. Let

bkkk (vvv ,q) =∫Ω

vvv ·∇k∇k∇k q dx , ckkk (p,q) =∫Ω

∇k∇k∇k p ·∇k∇k∇k q dx , vvv ∈XXX , p,q ∈Q ,

and VVVkkk = vvv ∈XXX : bkkk (vvv ,q) = 0 for all q ∈Q.

Weak FormFind (uuu,λ ) ∈VVVkkk ×R such that

akkk (uuu,vvv) = λ m(uuu,vvv) for all vvv ∈VVVkkk .

Discrete ApproximationLet XXXh,kkk ⊂XXX and Qh,kkk ⊂Q be discrete subspaces.Set VVV h,kkk = vvvh,kkk ∈XXXh,kkk : bkkk (vvvh,kkk ,qh,kkk ) = 0 for all qh,kkk ∈Qh,kkk.Find (uuuh,kkk ,λh,kkk ) ∈VVV h,kkk ×R such that

akkk (uuuh,kkk ,vvvh,kkk ) = λh,kkk m(uuuh,kkk ,vvvh,kkk ) for all vvvh,kkk ∈VVV h,kkk .3

Karlsruhe Institute of TechnologyLowest order conforming finite elements

Let Ω =⋃

c∈Ch

Ωc be a decomposition into hexahedral cells c ∈ Ch, and let

ϕc : Ω = [0,1]3 −→Ωc be the transformation to the reference cell. We define

XXXh,000 = uuu ∈XXX : Dϕ−1c uuu ϕc ∈P0,1,1eeex +P1,0,1eeey +P1,1,0eeez for all c ∈ Ch,

Qh,000 = q ∈Q : q ϕc ∈P1,1,1 for all c ∈ Ch .

Let Vh ⊂ Ω be the set of all vertices zzzv , and let Eh be the set of all edgese = (xxxe,yyye) with midpoint zzze = 0.5(xxxe +yyye) and tangent ttte = yyye−xxxe.

A nodal basis ψe,000 : e ∈ Eh ⊂XXXh,000 exists such that for all uuuh,000 ∈XXXh,000

uuuh,000 = ∑e∈Eh

〈ψ ′e,000,uuuh,000〉ψe,000 , 〈ψ ′e,000,uuuh,000〉=∫ yyye

xxxe

uuuh,000 · ttteds ,

and a nodal basis φv ,000 : v ∈ Vh ⊂Qh,000 exists such that for all qh,000 ∈Qh,000

qh,000 = ∑v∈Vh

〈φ ′e,000,qh,000〉φv ,000 , 〈φ ′e,000,qh,000〉= qh,000(zzzv ) .

(curl conforming elements were introduced by Nédélec)

4

Karlsruhe Institute of TechnologyConforming Elements with Shifted Basis

For kkk ∈ K we define modified elements with a phase shift

XXXh,kkk = spanψe,kkk : e ∈ Eh, ψe,kkk (xxx) = e−ikkk ·(xxx−zzze)ψe,000(xxx)

Qh,kkk = spanφv ,kkk : v ∈ Vh, φv ,kkk (xxx) = e−ikkk ·(xxx−zzzv )φv ,000(xxx)

with

uuuh,kkk = ∑e∈Eh

〈ψ ′e,kkk ,uuuh,kkk 〉ψe,kkk , 〈ψ ′e,kkk ,uuuh,kkk 〉=∫ yyye

xxxe

eikkk ·(xxx−zzze)uuuh,000 · ttteds ,

qh,kkk = ∑v∈Vh

〈φ ′e,kkk ,qh,kkk 〉φv ,kkk , 〈φ ′e,kkk ,qh,000〉= qh,kkk (zzzv ) .

We have∇k∇k∇k φv ,kkk (xxx) = e−ikkk ·(xxx−zzzv )

∇∇∇φv ,000(xxx)

∇k∇k∇k ×ψe,kkk (xxx) = e−ikkk ·(xxx−zzze)∇∇∇×ψe,000(xxx)

∇k∇k∇k ·ψe,kkk (xxx) = e−ikkk ·(xxx−zzze)∇∇∇ ·ψe,000(xxx)

akkk (ψe1,kkk ,ψe2,kkk ) = eikkk ·(zzze1−zzze2 )a000(ψe1,000,ψe2,000)

m(ψe1,kkk ,ψe2,kkk ) = eikkk ·(zzze1−zzze2 )m(ψe1,000,ψe2,000)

bkkk (ψe,kkk ,φv ,kkk ) = eikkk ·(zzze−zzzv )b000(ψe,000,φv ,000)

ckkk (φv1,kkk ,φv2,kkk ) = eikkk ·(zzzv1−zzzv2 )c000(φv1,000,φv2,000) 5

Karlsruhe Institute of TechnologyFinite Element Convergence

The finite element spaces XXXh,kkk , Qh,kkk , and VVV h,kkk satisfy

Ellipticity on VVV h,kkkThere exists C > 0 s.t.

akkk (uuuh,kkk ,uuuh,kkk )≥ C ‖uuuh,kkk‖2L2for all uuuh,kkk ∈VVV h,kkk .

Weak approximability of QThere exists ρ1(h) > 0, tending to zero as h goes to zero such that

supvvvh,kkk∈VVV h,kkk

bkkk (vvvh,kkk ,q)

‖vvvh,kkk‖curl≤ ρ1(h)‖q‖H1 for all q ∈Q .

Strong approximability of VVVFor some r > 0 there exists ρ2(h) > 0, tending to zero as h goes to zero suchthat for any uuu ∈VVV ∩H1+r (Ω,C3) there exists uuuh,kkk ∈VVV h,kkk satisfying

‖uuu−uuuh,kkk‖curl ≤ ρ2(h)‖uuu‖H1+r .

Theorem Let (uuu,λ ) a solution of the continuous eigenvalue problem,and let (uuuh,kkk ,λh,kkk ) be the corresponding discrete solution. Then, we have

(uuuh,kkk ,λh,kkk )→ (uuu,λ ) for h→ 0 .(Boffi-Conforti-Gastaldi 2006)

6

Karlsruhe Institute of TechnologyThe Projection

For uuuh,kkk ∈ Xh,kkk we construct ph,kkk ∈Qh,kkk such that uuuh,kkk −∇∇∇kkk ph,kkk ∈VVV h,kkk , i.e.,

0 = bkkk (uuuh,kkk −∇∇∇kkk ph,kkk ,qh,kkk )

= bkkk (uuuh,kkk ,qh,kkk )−ckkk (ph,kkk ,qh,kkk ) for all qh,kkk ∈Qh,kkk

Thus, we have Bh,kkkuuuh,kkk = Ch,kkk ph,kkk and uuuh,kkk = Sh,kkk ph,kkk with operators

"div": Bh,kkk : XXXh,kkk −→Q′h,kkk is defined by

〈Bh,kkkvvvh,kkk ,φv ,kkk 〉= bkkk (vvvh,kkk ,φv ,kkk ), v ∈ Vh

"Laplace": Ch,kkk : Qh,kkk −→Q′h,kkk is defined by

〈Ch,kkk qh,kkk ,φv ,kkk 〉= ckkk (qh,kkk ,φv ,kkk ), v ∈ Vh

"grad": Sh,kkk : Qh,kkk −→XXXh,kkk is given by nodal evaluation

Sh,kkk qh,kkk = ∑e=(xxxe ,yyye)∈Eh

(qh,kkk (yyye)eikkk ·(yyye−zzze)−qh,kkk (xxxe)eikkk ·(xxxe−zzze)

)ψe,kkk .

This defines a projection

Ph,kkk = id−Sh,kkk C−1h,kkk Bh,kkk : XXXh,kkk −→VVV h,kkk .

(special care is required for kkk = 0)7

Karlsruhe Institute of TechnologyModified LOBPCG Method (including projection)

Let Th,kkk : XXX ′h,kkk −→XXXh,kkk be a preconditioner for Aδ

h,kkk = Ah,kkk + δMh,kkk : XXXh,kkk −→XXX ′h,kkk .

S0) Choose randomly uuu1h,kkk , ...,uuuN

h,kkk ∈XXXh,kkk . Compute vvvnh,kkk = Ph,kkkuuun

h,kkk ∈VVV h,kkk .

S1) Ritz-step: Set up Hermitian matrices

A =(

akkk (vvvmh,kkk ,vvvn

h,kkk ))

m,n=1,...,N, M =

(m(vvvm

h,kkk ,vvvnh,kkk ))

m,n=1,...,N∈ CN×N

and solve the matrix eigenvalue problem Azn = λ nMzn.

S2) Compute yyynh,kkk =

N

∑n=1

znmvvvn

h,kkk ∈VVV h,kkk .

S3) Compute rrrnh,kkk = Ah,kkkyyyn

h,kkk −λ nMh,kkkyyynh,kkk ∈XXX ′h,kkk , check for convergence.

S4) Compute uuunh,kkk := Th,kkkrrrn

h,kkk ∈XXXh,kkk and wwwnh,kkk = Ph,kkkuuuh,kkk ∈VVV h,kkk .

S5) Perform Ritz-step for vvv1h,kkk , ...,vvvN

h,kkk ,www1h,kkk , ...,wwwN

h,kkk ⊂VVV h,kkk of size 2N.

S6) Go to step S2).

The full algorithm uses orthogonalization, new random vectors, anda Ritz-step of size 3N. (the LOBPCG method was introduced by Knyazev 2001)

8

Karlsruhe Institute of TechnologyMultigrid Preconditioner for the Maxwell Operator

Let XXX0,kkk ⊂XXX1,kkk ⊂ ·· · ⊂XXX J,kkk be finite element space of mesh size hj = 2−jh0.The multigrid preconditioner Tj ,kkk : XXX ′j ,kkk −→XXX j ,kkk is defined recursively:

For j = 0, define T0,kkk =(

Aδ

0,kkk

)−1.

For j > 0, the definition of Tj ,kkk requires:

1) a prolongation operator Ij ,kkk : XXX j−1,kkk −→XXX j ,kkk ;

2) the adjoint operator I′j ,kkk : XXX ′j ,kkk −→XXX ′j−1,kkk ;

3) a smoother Rj ,kkk : XXX ′j ,kkk −→XXX j ,kkk for Aj ,kkk ;

4) a smoother Dj ,kkk : Q′j ,kkk −→Qj ,kkk for Cj ,kkk ;

5) a transfer operator Sj ,kkk : Qj ,kkk −→XXX j ,kkk ;

6) the adjoint transfer operator S′j ,kkk : XXX ′j ,kkk −→Q′j ,kkk .

Now, define Tj ,kkk by

id−Aδ

j ,kkk Tj ,kkk =(

id−Aδ

j ,kkk Ij ,kkk Tj−1,kkk I′j ,kkk)(

id−δ−1Aδ

j ,kkk Sj ,kkk Dj−1,kkk S′j ,kkk)(

id−Aδ

j ,kkk Rj ,kkk).

(this multigrid variant was introduced and analyzed by Hiptmair 1998)

9

Karlsruhe Institute of TechnologyEigenvalue Computation for a Photonic Crystal

We compute Block-Floquet modes for kkk = (3,1,−1) in the unit cube Ω = (0,1)3

with periodic boundary conditions

∇k∇k∇k × ε−1

∇k∇k∇k ×HHH = λHHH , ∇k∇k∇k ·HHH = 000 .

Band structure of a photonic crystal Eigenfunction – (Bloch-Floquet mode)

(configuration from Dobson/Pasciak, Comp. Meth. Appl. Math. 1 (2001) 138–153)

10

Karlsruhe Institute of TechnologyPhotonic Crystals: Parallel Eigenvalue Computation

Calculation of the first 7 eigenvalues (≈ 1 million d.o.f.)

N processor kernels Total time Speed up factor32 5:11 min. 1.7964 2:53 min. 1.85128 1:33 min. 1.42256 1:05 min. 1.39512 0:47 min.

Calculation of the first 7 eigenvalues on 512 processor kernels

Refinement level d.o.f. Total time Scale up factorj = 4 13 872 00:18 min. 1.32j = 5 104 544 00:24 min. 1.91j = 6 811 200 00:47 min. 3.98j = 7 6 390 144 03:08 min. 6.85j = 8 50 725 632 21:31 min.

(every refinement level increases the problem size by a factor of 8)

11

Karlsruhe Institute of TechnologyPhotonic Crystals: Eigenvalue Convergence

Calculation of the first 7 eigenfunctions (up to ≈ 50 million d.o.f.)

j λj1 λ

j2 λ

j3 λ

j4 λ

j5 λ

j6 λ

j7

3 3.4818 3.7189 8.4285 9.0216 13.4387 13.7301 14.24794 3.4449 3.6828 7.9698 8.5039 12.4049 12.8074 13.32875 3.4310 3.6689 7.8410 8.3580 12.0515 12.5959 13.11796 3.4258 3.6637 7.8026 8.3145 11.9466 12.5339 13.06137 3.4239 3.6618 7.7906 8.3008 11.9136 12.5156 13.04518 3.4232 3.6612 7.7868 8.2964 11.9027 12.5100 13.0402

Convergence of the first 7 eigenfunctions meassured by|λ j+1

n −λjn|

|λ jn−λ

j−1n |

|λ 4n −λ 3

n | 0.0369 0.0361 0.4587 0.5177 1.0338 0.9227 0.91922.65 2.60 3.56 3.55 2.93 4.36 4.36

|λ 5n −λ 4

n | 0.0139 0.0139 0.1288 0.1459 0.3534 0.2115 0.21082.67 2.67 3.35 3.35 3.37 3.41 3.72

|λ 6n −λ 5

n | 0.0052 0.0052 0.0384 0.0435 0.1049 0.0620 0.05662.74 2.74 3.20 3.18 3.18 3.39 3.49

|λ 7n −λ 6

n | 0.0019 0.0019 0.0120 0.0137 0.0330 0.0183 0.01622.71 3.17 3.16 3.11 3.03 3.27 3.31

|λ 8n −λ 7

n | 0.0007 0.0006 0.0038 0.0044 0.0109 0.0056 0.0049

12

Karlsruhe Institute of TechnologyPhotonic Crystals: Eigenmodes

Illustration of an isosurface of the field intensity and the field directions.

13

Karlsruhe Institute of TechnologyBand Gap Verification

Analytically, it can be shown that photonic crystals with band gaps exist(by asymptotic investigations).

Numerically, one can check a given photonic crystalsfor band gap candidates.

Our aim is—combining analysis and numerics—to provide a rigorous proof for theexistence of a band gap for a specific test configuration.

This requires the following techniques:

Close finite element approximations of eigenmodes.

Upper spectral bounds (Rayleigh-Ritz).

Lower spectral bounds (Weinstein, Lehmann/Goerisch).

Spectral exclosure for perturbed operators (Hoang, Plum).

Spectral homotopy (Goerisch, Plum).

14

Karlsruhe Institute of TechnologyPolarized Waves in 2D

Now, we consider the Maxwell eigenvalue problem for the electic field EEE

∇∇∇×∇∇∇×EEE = λεEEE , ∇∇∇ · εEEE = 0

in special case that ε is periodic in x1 and x2, constant in x3 and that the waveEEE = (0,0,u) is polarized, which leads to the Helmholtz problem

−∆u = λεu .

We compute in the periodicity cell Ω for finitely many kkk in the Brillouin zone

(ukkk ,n,λkkk ,n) ∈ H1per(Ω)×R :

∫Ω

∇k∇k∇k ukkk ,n ·∇k∇k∇k v dx = λkkk ,n

∫Ω

εukkk ,nv dx , v ∈ H1per(Ω)

(n = 1, ...,N) with guaranteed accuracy.

15

Karlsruhe Institute of TechnologyA Candidate

We set ε(x) = 1 for x ∈ [1/16,15/16]2 and ε(x) = 5 else. By symmetry we havethe same spectrum for kkk = (k1,k2), (π−k1,k2), (k1,π−k2), ...

(−π,−π) (π,−π)

(π,π)(−π,π)

Brillouin zone K periodic ε distribution

16

Karlsruhe Institute of TechnologyA Candidate

(0,0)(π/2,0)(π,0)(π,π/2)(π,π)(π/2,π/2)(0,0)

λkkk ,n

35

30

25

20band gap

15

10

5

eigenvalues λkkk ,1, ...,λkkk ,6 along the test path kkk ∈ P ⊂ K

17

Karlsruhe Institute of TechnologyA Candidate

-4-3

-2-1

0 1

2 3

4

-4-3-2-1 0 1 2 3 4

5

10

15

20

25

30

eigenvalue λkkk ,3 for all kkk ∈ K

18

Karlsruhe Institute of TechnologyA Candidate

-4-3

-2-1

0 1

2 3

4

-4-3-2-1 0 1 2 3 4

5

10

15

20

25

30

eigenvalue λkkk ,4 for all kkk ∈ K

19

Karlsruhe Institute of TechnologyA Candidate

5

10

15

20

25

30

-4-3-2-1 0 1 2 3 4-4-3-2-1 0 1 2 3 4

eigenvalues λkkk ,1, ...,λkkk ,5 for all kkk ∈ K

20

Karlsruhe Institute of TechnologyA Candidate

Bloch modes ukkk ,1, ...,ukkk ,6 for kkk = (π,π)

21

Karlsruhe Institute of TechnologyUpper Spectral Bounds (Ritz-Galerkin)

Theorem

For eigenfunction approximations uk ,1, ..., uk ,N ∈ H1(Ω/Λ,C) define

AAA =(

ak (uk ,m, uk ,n))

m,n=1,...,N,

BBB =(〈uk ,m, uk ,n〉ρ

)m,n=1,...,N

∈ CN,N

and let

ψk ,1 ≤ ψk ,2 ≤ ·· · ≤ ψk ,N

be the eigenvalues of the problem AAAxxx = ψ BBBxxx . Then we have

λk ,n ≤ ψk ,n , n = 1, ...,N .

22

Karlsruhe Institute of TechnologyLower Spectral Bounds (Lehmann-Goerisch)

TheoremLet γ > 0. For eigenfunction approximations uk ,n ∈ V and scaled dualapproximations σk ,n ≈ 1

γ+λk ,n∇k uk ,n ∈W define

SSS =(〈σk ,m, σk ,n〉

)m,n=1,...,N

,

TTT =1γ

(〈uk ,m +

1ρ

∇k · σk ,m, uk ,n +1ρ

∇k · σk ,n〉ρ)

m,n=1,...,N.

If β > γ satisfies β − γ ≤ λk ,N+1, if NNN = AAA + (γ−2β )BBB + β 2(SSS +TTT ) is positivedefinite, and if the eigenvalues

θ1 ≥ θ2 ≥ ·· · ≥ θN

of the eigenvalue problem (AAA + (γ−β )BBB

)xxx = θ NNN xxx

are negative, we have the lower eigenvalue bound

β − γ− β

1−θn≤ λn,k , n = 1, ...,N.

Remark: A suitable β is obtained by a spectral homotopy.23

Karlsruhe Institute of TechnologyEigenvalue Exclosure by Perturbation AnalysisTheoremFor some λgap, n > 1, and kkk ∈ K assume that δ > 0 exits such that

λkkk ,n−1 + δ < λgap < λkkk ,n−δ .

Then we have

λkkk+hhh,n−1 < λgap < λkkk+hhh,n

for all hhh ∈ R2 with

|hhh|<√

minxxx∈Ω

ε(xxx)(√

λkkk ,n + δ −√

λkkk ,n

).

For the application we compute a guaranteed lower bound λkkk ,n,inf ≤ λkkk ,n and upperbounds λkkk ,n−1,sup ≥ λkkk ,n−1, λkkk ,n,sup ≥ λkkk ,n, and

rkkk =√

minxxx∈Ω

ε(xxx)(√

λkkk ,n,sup + δ −√

λkkk ,n,sup

).

This ensures that the interval (λkkk ,n−1,sup + δ ,λkkk ,n,inf−δ ) is contained in theresolvent set of the elliptic operator associated to the eigenvalue problem for allkkk ′ ∈ Ball(kkk , rkkk ).

24

Karlsruhe Institute of TechnologyA Verified Band Gap

This figure illustrates Brillouin vectors

kkk ′ ∈⋃

kkk∈KBall(kkk , rkkk )

where λgap ∈ (18.2,18.25)is not a spectral value.

By symmetry this proves that for all kkk ∈ K the existence of a band gap

(18.2,18.25)⊂ (λmax,3,λmin,4)

for the eigenvalue problem −∆u = λεu in R2.

The proof requires the close approximation of more than 5000 eigenvalues and eigenfunctions

(for 100 vectors k ∈K and for a homotopy to bound the rest of the spectrum) and takes about 90 h computing time.

25

Karlsruhe Institute of TechnologyTowards the Optimal Design of Photonic Crystals

Assume that for a given periodic electric permittivity ε0 a band gap exists, i.e., forsome λgap ∈ R and some n > 1 we have

λn−1,kkk (ε0) < λgap < λn,kkk (ε

0) , kkk ∈ K .

The purpose is to find a material distribution ε0 ≤ ε(xxx)≤ ε1 such that

g(ε) = minkkk∈K

λgap−λn−1,kkk (ε),λn,kkk (ε)−λgap

is maximal.

Numerically, this can be approximated by a subgradient method definingε1,ε2,ε3, ... by

0 ∈ g(εm−1) + 〈∂g(ε

m−1),εm− εm−1〉 , m = 1,2,3, ...

Here, ∂g(ε) can be evaluated explicitly by the computation of eigenmodes.(realized in 2-d by Cox/Dobson 1999)

26

Karlsruhe Institute of TechnologyTowards the Optimal Design of Photonic Crystals

initial band gap: 0.0444

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Γ X M Γ

ω/(

2π)

result after 1150 optimization steps (subgradient method)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Γ X M Γ

ω/(

2π)

optimized band gap: 0.044427