I Photonic Crystal Wave-Guides I - TU...

Modelling of TE and TM Modes in

Photonic Crystal Wave-Guides

K. Schmidt1,3,4 and R. Kappeler2

1 Group POEMS, INRIA, France, 2 Institute for Electronics, ETH Zurich, Switzerland,3 Seminar for Applied Mathematics, ETH Zurich, Switzerland, 4 Hausdorff Center for Mathematics, Bonn University, Germany

WAVES 2009, Pau (France), 15th June 2009

Motivation

ωa

2πc

Γ K M Γ0

0.1

0.2

0.3

0.4

0.5

Photonic band gap materialI Energy states for a certain wavevectorI Band gap in which no waves are allowed

Photonic crystal wave-guidesI Propagating Bloch waves in the guideI Tailoring the dispersion of guided modesI Flats bands lead to slow light.

System of interest: 2D planar Photonic crystal wave-guideI index guiding in vertical direction,I “band gap” guiding in horizontal direction.

F. Robin, Communication Photonics Group, ETH Zurich

K.Schmidt and R. Kappeler WAVES 2009, Pau (France), 15th June 2009 p. 1 / 24

Applications of Wave-guides with Tailored Dispersion

Flat bands ⇒ Modes with low group velocity vg = dωdk

I reduction of velocity of the light energy⇒ light stays longer in the waveguide(by slow down factor)

I simultaneously: the light is enhancedby the slow down factor(light is compressed spatially)

T.F. Krauss, Nature Photonics 2, p. 448–450, Aug 2008 F. Robin, Communication Photonics Group, ETH Zurich

Applications of the Intensity Enhancement of Slow Light ModesI lowers the power threshold of the input light for obtaining non-linear effectsI can potentially decrease the length of semiconductor optical amplifiers

by the slow down factor (higher integration density)


Outline

Modes in photonic crystals

Modes in photonic crystal wave-guides

Supercell approachQuadratic eigenvalue problemDiscretisation by p-FEMNumerical results

Spectral discretisation of infinite stripsBasis functions based on Laguerre functionsNumerical results


Crystal in 2D with hexagonal symmetry

a

a

Reciprocal lattice to order the modes

K = (4π3a , 0)

M = (πa ,√

3π3a )

Γ = (0, 0)

ΓMK

Maxwell’s equations in R2

(TM) − div grad e(x) =

„ωTM

c0

«2

ε(x)e(x)

(TE) − div

„1

ε(x)grad h(x)

«=

„ωTE

c0

«2

h(x)

h, e ... component of magnetic and electric fieldperpendicular to the plane of periodicity

ω ... angular frequency,c0 ... vacuum speed of light,ε(x) ... (relative) dielectric constant

Floquet-Bloch transformation

I Maxwell’s eq. in Ω× B

I with quasi-periodic bc. on ∂Ω

ωa

2πc

Γ K M Γ0

0.1

0.2

0.3

0.4

0.5

I Band structure = eigenvalues ω(k), k ∈ B

I Band gap = ω-range without modesK.Schmidt and R. Kappeler WAVES 2009, Pau (France), 15th June 2009 p. 4 / 24



a

a

Photonic crystal band sructure

ωa

2πc

Γ K M Γ0

0.1

0.2

0.3

0.4

0.5

Family of eigenvalue problems in the unit-cell Ω

1. Quasi-periodic eigenmodes

For each k ∈ B find (ek, ωTM) and (hk, ωTE) with

ek(x + aj ) = ei k·aj ek(x), hk(x + aj ) = e

i k·aj hk(x), x on ∂Ω, j = 1, 2

satisfying

(TM)

ZΩ

grad ek(x) · grad e′k(x) d x =

„ωTM

c0

«2 ZΩ

ε(x) ek(x) e′k(x) d x

(TE)

ZΩ

1

ε(x)grad hk(x) · grad h′k(x) d x =

„ωTE

c0

«2 ZΩ

hk(x) h′k(x) d x

FEM discretisation with quasi-periodic basis functionsK.Schmidt, P.Kauf, CMAME 198, p. 1249–1259, Mar 2009.




a

a

Photonic crystal band sructure

ωa

2πc

Γ K M Γ0

0.1

0.2

0.3

0.4

0.5

Family of eigenvalue problems in the unit-cell Ω

2. Periodic eigenmodes : ek → ekei k·x, hk → hke

i k·x

For each k ∈ B find (ek, ωTM) and (hk, ωTE) with

ek(x + aj ) = ek(x), hk(x + aj ) = hk(x), x on ∂Ω, j = 1, 2

satisfying

(TM)

ZΩ

(grad +i k)ek(x) · (grad−i k)e′(x) d x =

„ωTM

c0

«2 ZΩ

ε(x) ek(x) e′(x) d x

(TE)

ZΩ

1

ε(x)(grad +i k)hk(x) · (grad−i k)h′(x) d x =

„ωTE

c0

«2 ZΩ

hk(x) h′(x) d x

I Family of linear eigenvalue problems in ω w.r.t. k ∈ BI Can be regarded as family of quadratic eigenvalue problems in k w.r.t. ω ∈ R+

C. Engstrom, Ch. Hafner, K. Schmidt, J. Comp. Theor. Nanosci. 6, p. 775–783, March 2009C. Engstrom, M. Richter, SIAM J. Appl. Math. 70, p. 231–247, May 2009



Wave-guide with 1D periodicity

a a a

I periodicity cells Ω are infinite strips of diameter a

I Floquet-Bloch transform in 1D, Brillouin zone is [0, 2πa ]

I Maxwell’s eq. in Ω× B with quasi-periodic bc. on ∂Ω andSommerfeld radation condition for |x2| → ∞8>>><>>>:

−∆e(x)− ω2

c20

ε(x)e(x) = 0,

limx2→±∞

|x2|„∂e(x)

∂x2

∓ iω

c0

e(x)

«= 0

(TM)

8>>><>>>:−∇ · 1

ε(x)∇h(x)− ω2

c20

h(x) = 0,

limx2→±∞

|x2|„∂h(x)

∂x2

∓ iω

c0

h(x)

«= 0

(TE)

Crystal modes Guided modes External modes External modes



Wave-guide with 1D periodicity Band structure of guided and crystal modes

a a a

Reformulation with periodic boundary conditions

(TM)

ZΩ

(grad +ik e1)ek (x) · (grad−ik e1)e′(x) d x =

„ωTM

c0

«2 ZΩε(x) ek (x) e′(x) d x

(TE)

ZΩ

1

ε(x)(grad +ik e1)hk (x) · (grad−ik e1)h′(x) d x =

„ωTE

c0

«2 ZΩ

hk (x) h′(x) d x

I where we close the eyes for the infiniteness of Ω and the Sommerfeld radiation conditions atthis moment

I (TM) and (TE) can be interpreted again as family of (linear?) eigenvalue problems for ω(k)or (quadratic?) eigenvalue problems for k(ω)

Goal: Guided modes in the band gap

With eigenvalue problems in k(ω) we can restrict the computation to the band gap



Computation in an infinite strip with radation conditions

a a a

k

ωa

2π

c0

TE

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

I Correct guided modesin the band gap have toseparated from theunwanted surface andexterior modes.

Supercell approach in a finite cell around the channel

a a a

I Approximation of theguided modes without“pollution” of thespectrum by surface orexterior modes.

I Band structure codescan be re-used, just theunit cell is larger.

I No infinite strip andhandling of radiationcondition.

I Eigenvalue problems arequadratic in k.

S.Soussi, SIAM Journal on Numerical Analysis 43, p. 1175—-1201, 2005.


Quadratic eigenvalue problem for the supercell approach

Quadratic eigenvalue problems in k(ω)

(TM)

ZΩ

(grad +ik e1)ek (x) · (grad−ik e1)e′(x) d x =

„ωTM

c0

«2 ZΩε(x) ek (x) e′(x) d x

(TE)

ZΩ

1

ε(x)(grad +ik e1)hk (x) · (grad−ik e1)h′(x) d x =

„ωTE

c0

«2 ZΩ

hk (x) h′(x) d x

can be written as (` = 0 for the TM and ` = −1 for the TE modes)

k2b`(u, v) + kc`(u, v) + a`(u, v)−“

ωc0

”2b`+1(u, v) = 0

with the bilinear forms

a`(u, v) :=

ZΩε`(x)∇u(x) · ∇v(x) d x b`(u, v) :=

ZΩε`(x)u(x) · v(x) d x

c`(u, v) := i

ZΩε`(x)

u(x)

∂v(x)

∂x1−∂u(x)

∂x1v(x)

!d x

Quadratic matrix eigenvalue problem by choosing a basis of finite subspace of H1(Ω)`k2M3 + kM2 + M1(ω)

´~u = 0

with M1(ω) :=à`(φj , φi )−

“ωc0

”2b`+1(φj , φi )

Ńi,j=1

,

M2 :=`c`(φj , φi )

Ńi,j=1

, M3 :=`b`(φj , φi )

Ńi,j=1

.


Quadratic eigenvalue problem for the supercell approach

Quadratic matrix eigenvalue problem by choosing a basis of finite subspace of H1(Ω)`k2M3 + kM2 + M1(ω)

´~u = 0

with M1(ω) :=à`(φj , φi )−

“ωc0

”2b`+1(φj , φi )

Ńi,j=1

,

M2 :=`c`(φj , φi )

Ńi,j=1

, M3 :=`b`(φj , φi )

Ńi,j=1

.

Linearisation of the quadratic matrix eigenvalue problemDefine ~x := (k~u, ~u).

„−M2 −M1(ω)

I 0

«| z

=:A(ω)

~x = k

„M3 00 I

«| z

=:M

~x

F.Tisseur and K.Meerbergen, SIAM Review 43, p. 235–286, 2001.

I Matrix M is symmetric positive definite.

I We can use iterative linear eigenvalue solvers for sparse matrices, like Arpack.


Comparison with transparent boundary conditions

Supercell approach

I Discretisation of the channel and some rows of holes.

I Quadratic eigenvalue problem in k(ω) which can belinearised (or linear eigenvalue problem in ω(k)).

I Iterative linear eigenvalue solvers for sparse matrices,like Arpack, can be used.

Transparent bc. for semi-infinite periodic mediaS.Fliss and P.Joly, Applied Numerical Mathematics 59, p. 2155–2178, Sep 2009.→ Thu 17.00, Room GabardM.Ehrhardt and C.Zheng, J. of Computational Physics 227, p. 6877–6894, Jul 2008.S.Soussi and Th. Hohage,→ Thu 17.30, Room Gabard

I Discretisation of the channel and a PC unit cell only.

I Dirichlet-to-Neumann maps are non-local and wouldlead to almost dense, very small matrices.

I Dirichlet-to-Neumann maps are neither linear norquadratic, and one would have to use a non-lineareigenvalue solver for both eigenvalue problems (ink(ω) or ω(k)), e.g. based on Newtons method

D.Kressner, SAM Report 2009-05, ETH Zurich, Feb 2009.


Discretisation by p-FEMp-FEM in 1D

TE mode at k = 0, for ε(x) = 9 in [0, 1]

−“

1ε(x)

h′(x)”′

= ω2h(x) in [−1, 1]

with first non-zero eigenvalue

ω = arccos(1/4)

♯ dof

|ωh−

ω|/ω

h-FEM p = 1h-FEM p = 2h-FEM p = 3p-FEM

3 10 30 100 30010−8

10−6

10−4

10−2

100

p = 1 p = 2 p = 3

h = 1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

=⇒ p-FEM withexponential convergence

h = 1/2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

h = 1/4

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1

1.5

2

♯ dof

|ωh−

ω|/ω

h-FEM p = 1h-FEM p = 2h-FEM p = 3p-FEM

10 20 30 40 50 60 70 80 90 10010−8

10−6

10−4

10−2

100


Discretisation by p-FEMRegularity in 2D

Dielectric pattern typically piecewise constantLet us assume piecewise analytic pattern in two subdomains. Ω1, Ω2

ε(x)˛Ωi∈ A(Ωi )

Interface Γ := ∂Ω1 ∩ ∂Ω2 p.w. analytic.

Smooth case

I eigenfunctions analytic up to the interface

I p-FEM on a fixed mesh has exponential convergence

Polygonal case

I eigenfunctions analytic up to the interface except the corners

I singularity at interface corners, depends on ε(x) in surrounding

I hp-FEM with geometric meshes and linearly from the cornersincreasing polynomial degrees has exponential convergence

K.Schmidt, P.Kauf, CMAME 198, p. 1249–1259, Mar 2009.


Discretisation by p-FEM

C++ class library Concepts

I Started by Christian Lage during his Ph.D. studies (1995).

I Used and improved by Ph.Frauenfelder, A.-M. Matache, G.Schmidlin, K.Schmidt,P.Kauf, H.Brandsmeier and severals students.

I Concept Oriented Design using mathematical principles [1].

I Currently two parts: hp-FEM (nodal and edge elements), BEM (wavelet andmultipole methods).

I hp-FEM-part (in 2D and 3D) is released under GPL,http://www.concepts.math.ethz.ch

Ph.Frauenfelder and Ch.Lage, Math. Model. Numer. Anal. 36, p. 937–951, 2002.

Basis functions on curved cells

vertex function edge function inner or bubble function


http://www.concepts.math.ethz.ch

Numerical resultsComparison with other codes

I MPB – MIT Photonic-Bands (Group of St.G.Johnson, MIT)Expansion in plane waves, mostly used code for band structure computation.St.G.Johnson and J.D.Joannopoulos, Optics Express 8, p. 173–190, 2001.

I Comsol – FE solver with broad class of applications

Linear eigenvalue problem for ω(k)

Concepts mesh forW1 wave-guidewith 5 rows of holes

(r = 0.31a)and ε(x) = 11.4

degrees of freedom

erro

r

time [s]

erro

r

102 104102 104

10−8

10−6

10−4

10−2

10−8

10−6

10−4

10−2

error taking for uniform sampling on the border of reduced Brillouin zone

Sun 4600, Dual Core AMD Opteron 2.53 GHz, Linux


Numerical results

Exponential convergence for the quadratic eigenvalue problem in k(ω)

Concepts mesh forW1 wave-guidewith 5 rows of holes

(r = 0.31a)and ε(x) = 11.4

Polynomgrad

rel.

erro

r

4 5 6 7 8 9 1010−8

10−6

10−4

10−2

error taken for 20 uniformly distributed frequencies in [0.2, 0.28]2πc0

a


Spectral discretisation of infinite strips

Supercell approach

I approximates the guided modes

I without noticing unwanted surface and exterior modes

Can we compute the guided modes to a higher accuracy ?

Perfect Matched Layer (PML)

I finite absorbing layer, easy for implementation,but shifts the eigenvalues k away from the real axis.

Exact non-reflecting boundary conditions

I expansion of the solution outside the computational domain

I Dirichlet-to-Neumann map is non-local and depends non-linearly on ω and k⇒ need of a non-linear eigenvalue solver

Infinite elements

I infinite cell with decaying basis functions

I basis can be chosen so that matrices stay sparse

I eigenvalue problem is still quadratic in k(ω)

Yes ⇒ infinite elements based on Laguerre functions


Spectral discretisation of infinite stripsExact representation of the eigenmodes in the outer air

Quasi-periodic solutions fulfilling the Sommerfeld radiation condition

limx2→±∞

|x2|„∂u(x)

∂x2

∓iω

c0

u(x)

«= 0

can be written as

uk (x) =X

j∈J(k,ω)

Aj,ke−Kj (k,ω)|x2|ei(k+j 2πa

)x1 .

with

Kj (k, ω) =

s„k + j

2π

a

«2

−ω2

c20

and J(k, ω) the subset of Z such that Kj (k, ω) is real and positive.

⇒ Exponentially decaying modes


Spectral discretisation of infinite stripsLaguerre polynomials

Ln(x) orthonormal in [0,∞) w.r.t. weighting function e−xZ ∞0

Ln(x)Lm(x)e−x dx = δnm

First Laguerre polynomials

L0(x) = 1, L1(x) = −x + 1,

L2(x) =1

2

`x2 − 4x + 2

´, L3(x) =

1

6(−x3 + 9 x2 − 18 x + 6),

can be computed by the recursion formula

(n + 1) Ln+1(x) = (2 n + 1− x) Ln(x)− n Ln−1(x).

Boundary values at x = 0 Ln(0) = 1

Derivatives x L′n(x) = n (Ln(x)− Ln−1(x))

Recursion formula L′n(x)− L′n−1(x) = −Ln−1(x)


Spectral discretisation of infinite stripsBasis functions based on Laguerre functions

Laguerre functions Ln(x) = Ln(x) e−x2 ⇒

R∞0 Ln(x)Lm(x) dx = δnm

Definition of a basis in [0,+∞) φn(x) = (Ln(x)− Ln−1(x)) e−x2

J.Shen, SIAM Journal on Numerical Analysis 38, p. 1113–1133, 2001.

Boundary values at x = 0

Ln(0) = 1 ⇒ φ0(0) = 1, φn(0) = 0, ∀n > 0,

Derivatives

φ′n(x) =`L′n(x)− L′n−1(x)| z

−Ln−1(x)

−1

2(Ln(x)− Ln−1(x))

´e−

x2 = −

1

2(Ln(x) + Ln−1(x))e−

x2 .

Stiffness and mass matrix are tridiagonal and explicitely given by

Z ∞0

φn(x)φm(x) dx =

8<:2, n = m,

−1, |n − m| = 1,

0, |n − m| > 1,Z ∞0

φ′n(x)φ′m(x) dx =

8<:12 , n = m,14 , |n − m| = 1,

0, |n − m| > 1,

.


Spectral discretisation of infinite stripsInfinite element in 2D based on Laguerre functions

Basis function in the infinite cell [0, a]× [0,∞)

bi,n(x) = ψi (x1)φn(x2)

where ψi are polynomials in [0, a]

Matrix entriesZK

grad bi,n(x) · grad bj,m(x) d x =

Z a

0ψi (x1)ψj (x1) dx1

Z ∞0

φ′n(x2)φ′m(x2) dx2

+

Z a

0ψ′i (x1)ψ′j (x1) dx1

Z ∞0

φn(x2)φm(x2) dx2,ZK

bi,n(x) bj,m(x) d x =

Z a

0ψi (x1)ψj (x1) dx1

Z ∞0

φn(x2)φm(x2) dx2,ZK

bi,n(x)∂bj,m

∂x1(x) d x =

Z a

0ψi (x1)ψ′j (x1) dx1

Z ∞0

φn(x2)φm(x2) dx2

⇒ Matrices are sparse


Spectral discretisation of infinite stripsNumerical results

Concepts mesh forW1 wave-guide

with 5 rows of holes(r = 0.31a),

and ε(x) = 11.4

Band structure of guided and crystal modes

Infinite Elements withLaguerre functions

ω = 0.24776, k = 2.67040

Supercell approach withperiodic boundaries

ω = 0.24776, k = 2.67035


Conclusion

Goal: Computation of guided modes in photonic crystal wave-guides

Band structure calculation of the infinite crystal

I Knowledge about band-gaps and propagating frequencies

Supercell approach with periodic boundary conditions

I only computation of guided and crystal modes

I quadratic eigenvalue problems in k for each ω for TE and TM modes⇒ linearisation ⇒ linear sparse eigenvalue solver, e.g. Arpack

Discretisation by p-FEM

I resolution of the interfaces by curved elements

I increasing polynomial degree ⇒ exponential convergence

I C++ class library Concepts

Spectral discretisation of the infinite strips

I Sommerfeld radiation condition leads to exponentially decaying solutions

I 1D basis of [0,∞) based on Laguerre functions resulting in sparse matrices

Thanks I H. Brandsmeier, SAM, ETH Zurich,

I S. Belfanti, B. Jin, IfE, ETH Zurich.


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