quadridirectionalmodeexpansionmodeling inintegratedoptics · quadridirectionalmodeexpansionmodeling...

Quadridirectional mode expansion modelingin integrated optics

Manfred Hammer∗

MESA+ Research InstituteUniversity of Twente, The Netherlands

Kleinheubacher Tagung, Miltenberg, Germany, 29.09. – 02.10.2003

∗ Department of Applied Mathematics, University of Twente P.O. Box 217, 7500 AE Enschede, The NetherlandsPhone: +31/53/489-3448 Fax: +31/53/489-4833 E-mail: [email protected]

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1

http://www.math.utwente.nl/~hammerm/

http://www.mesaplus.utwente.nl/

http://www.utwente.nl/

mailto:[email protected]

http://www.utwente.nl/

http://www.mesaplus.utwente.nl/

http://www.math.utwente.nl/mpcm/aamp/

Omnidirectional light propagation

z

gPin

x

0

PD

PRPT

PU

nbng

w

W

x

z

PT

0

w

pr

ngnb

Pin PR

1 Nr2 . . .

nr

∆n 6� 1

x

z

2

Mode expansion modeling of omnidirectional light propagation ?

PD

PU

PT

Pin

PR

nb ng

x

zh

v

0

Directions x and z deservethe same treatment.

QUEP: An eigenmode expansion scheme that implements that constraint.

3

Outline

• Quadridirectional mode propagation:• Problem setting• Eigenmode expansion• Algebraic procedure

• Numerical results:• Gaussian beams in free space• Bragg grating• Waveguide crossing• Square resonator with perpendicular ports• Photonic crystal bend

4

Problem setting

z

x

x0

x1

1

1 2

...

0

2

Nx + 1

xNx

Nx

z0 z1 . . .

Nz + 1Nz

zNz

• 2D problem, Cartesian coordinates x, z,TE / TM polarization.

• Piecewise constant, rectangularrefractive index distribution; linear,lossless materials.

• Fixed frequency simulations,vacuum wavelength λ = 2π/k.

• Rectangular interior computationaldomain, influx & outflux across all fourboundaries, external regions arehomogeneous along x or z.

• Assumption Ey = 0, Hy = 0 on thecorner points and on the external borderlines is reasonable for the problems un-der investigation.

5

Modal basis fields

Basis fields,defined by Dirichlet boundary conditions Ey = 0 (TE) or Hy = 0 (TM):

z

x0

1 20

x

xNx

z0 z1 . . . zNz

Nz + 1Nz

Horizontally traveling eigenmodes:

Mz profiles ψds,m(x)

and propagation constants ±βs,m

of order m, on slice s,for propagation directions d = f, b,

6

Modal basis fields

Basis fields,defined by Dirichlet boundary conditions Ey = 0 (TE) or Hy = 0 (TM):

z

x0

x1

1

...

0

2

z0

x

Nx + 1

xNx

Nx

zNz

. . . and vertically traveling fields:

Mx profiles φdl,m(z)

and propagation constants ±γl,m

of order m, on layer l,for propagation directions d = u, d.

7

Eigenmode expansion

z

x

x0

x1

1

1 2

...

0

2

Nx + 1

xNx

Nx

z0 z1 . . .

Nz + 1Nz

zNz

Ansatz for the optical field,for zs−1 ≤ z ≤ zs, s = 1, . . . , Nz ,and xl−1 ≤ x ≤ xl, l = 1, . . . , Nx:(

E

H

)

(x, z, t) = Re

{

Mz−1∑

m=0

Fs,mψfs,m(x) e−iβs,m(z − zs−1)

+

Mz−1∑

m=0

Bs,mψbs,m(x) e+iβs,m(z − zs)

+

Mx−1∑

m=0

Ul,mφul,m(z) e−iγl,m(x − xl−1)

+

Mx−1∑

m=0

Dl,mφdl,m(z) e+iγl,m(x − xl)

}

eiωt.

8

Eigenmode expansion

z

x

x0

x1

1

1 2

...

0

2

Nx + 1

xNx

Nx

z0 z1 . . .

Nz + 1Nz

zNzMode products ↔ normalization, projection:

(E1,H1;E2,H2)x =1

4

∫

(E∗

1,xH2,y − E∗

1,yH2,x + H∗

1,yE2,x − H∗

1,xE2,y)dx ,

(E1,H1;E2,H2)z =1

4

∫

(E∗

1,yH2,z − E∗

1,zH2,y + H∗

1,zE2,y − H∗

1,yE2,z)dz .

9

Algebraic procedure

z

x

x0

x1

1

1 2

...

0

2

Nx + 1

xNx

Nx

z0 z1 . . .

Nz + 1Nz

zNz

• Consistent bidirectional projection at all interfaces−→ linear system of equations in {Fs,m, Bs,m, Ul,m,Dl,m}.

• Influx: F 0, BNx+1, U 0, DNz+1 −→ RHS.

• Outflux: B0, FNx+1, D0, UNz+1.

10

Algebraic procedure

• “Exact” mode profiles −→ interior problems decouple:

Solve for F 2, . . . ,FNzandB1, . . . ,BNz−1

in terms of F 1 andBNz

−→ BEP.

Solve for U 2, . . . ,UNxandD1, . . . ,DNx−1

in terms of U 1 andDNx

−→ BEP.

• Continuity of E and H on outer interfaces:

Interior BEP solutions+ equations at z = z0, zNz

, x = x0, xNx

−→B0, FNx+1, D0, UNz+1.

“QUadridirectional Eigenmode Propagation method” (QUEP).11

Gaussian beams in free space

x

n

0 z

x0

x1

z0 z1

n = 1.0, λ = 1.0 µm,TE polarization,x, z ∈ [0, 15.1] µm,Mx = Mz = 150.

Ey(x, z) :

z [µm]

x [µ

m]

0 5 10 15

0

5

10

15

Top: 4 µm beam width, 10◦ tilt angle, 2 µm offset;

bottom: 45◦ tilt angle, initially centered beams.

12


x

n

0 z

x0

x1

z0 z1


Ey(x, z) :

z [µm]

x [µ

m]

0 5 10 15

0

5

10

15

z [µm]

x [µ

m]

0 5 10 15

0

5

10

15

z [µm]

x [µ

m]

0 5 10 15

0

5

10

15

Top: 4 µm beam width, 10◦ tilt angle, 2 µm offset;

bottom: 45◦ tilt angle, initially centered beams.

12


x

n

0 z

x0

x1

z0 z1


Ey(x, z) :

z [µm]

x [µ

m]

0 5 10 15

0

5

10

15

z [µm]

x [µ

m]

0 5 10 15

0

5

10

15

z [µm]

x [µ

m]

0 5 10 15

0

5

10

15

z [µm]

x [µ

m]

0 5 10 15

0

5

10

15

z [µm]

x [µ

m]

0 5 10 15

0

5

10

15

z [µm]

x [µ

m]

0 5 10 15

0

5

10

15

Top: 4 µm beam width, 10◦ tilt angle, 2 µm offset;bottom: 45◦ tilt angle, initially centered beams.

12

Bragg grating: COST268 benchmark problemx

z

Pin

PR

PT

Λg1 2 . . . . . . Ng

ns

na

0

dedg ng

(∗)dg = 0.500 µm,de = 0.125 µm,ns ≈ 1.45 (SiO2),ng ≈ 1.99 (Si3N4),na = 1.0,Λ = 0.430 µm,g = Λ/2, Ng = 20.

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

0.2

0.4

0.6

0.8

1

λ [µm]

PR

, T

PR

PT

1−PR−P

T

BEP, b.c. E y = 0

x ∈ [−4, 2] µm, 60 modes (QUEP, BEP Ey = 0 b.c.), z ∈ [−1.8, 10.185] µm, 120 modes (QUEP).∗J. Ctyroký et. al., Optical and Quantum Electronics 34, 455–470, 2002; reference: BEP2 (PML b.c.).

13


z

Pin

PR

PT

Λg1 2 . . . . . . Ng

ns

na

0

dedg ng


0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

0.2

0.4

0.6

0.8

1

λ [µm]

PR

, T

PR

PT

1−PR−P

T

QUEP BEP, b.c. E y = 0


13


z

Pin

PR

PT

Λg1 2 . . . . . . Ng

ns

na

0

dedg ng


0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.80

0.2

0.4

0.6

0.8

1

λ [µm]

PR

, T

PR

PT

1−PR−P

T

QUEP BEP, PML b.c.*


13

Waveguide crossings

PD

PU

PT

Pin

PR

nb ng

x

zh

v

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

v [µm]

PR

, T, U

, D

PT

PR

PU

PD

ng = 3.4, nb = 1.45, λ = 1.55 µm, h = 0.2 µm, TE, x, z ∈ [−3, 3] µm, Mx = Mz = 120.Power conservation: error < 10−3.

14

Waveguide crossings

z [µm]

x [µ

m]

−2 −1 0 1 2−1.5

−1

−0.5

0

0.5

1

1.5

z [µm]

x [µ

m]

−2 −1 0 1 2−1.5

−1

−0.5

0

0.5

1

1.5

z [µm]x

[µm

]−2 −1 0 1 2

−1.5

−1

−0.5

0

0.5

1

1.5

z [µm]

x [µ

m]

−2 −1 0 1 2−1.5

−1

−0.5

0

0.5

1

1.5

v = 0.218 µm v = 0.374 µm

v = 0.823 µmv = 0.591 µm

PD

PU

PT

Pin

PR

nb ng

x

zh

v

0

15

Square resonator with perpendicular ports

1.53 1.54 1.55 1.56 1.57 1.58 1.590

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ [µm]

PR

, T, U

PT

PU

PR

z

PU

w

Pin

x

0

PRPT

W

ng

gh

ngnb

gv q

w

ng

ng = 3.4, nb = 1.0, W = 1.786 µm, w = 0.1 µm, gh = 0.355 µm, gv = 0.385 µm, q = 0.355 µm,TE, x, z ∈ [−4, 4] µm, Mx = Mz = 100.

16

Square resonator with perpendicular ports

z

PU

w

Pin

x

0

PRPT

W

ng

gh

ngnb

gv q

w

ngng = 3.4, nb = 1.0, W = 1.786 µm, w = 0.1 µm,gh = 0.355 µm, gv = 0.385 µm, q = 0.355 µm,TE, x, z ∈ [−4, 4] µm, Mx = Mz = 100.

λ = 1.55 µm, T = 5.17 fs, Ey(x, z, t) :

z [µm]

x [µ

m]

t = 3T/4

−2 −1 0 1 2−2

−1

0

1

2

z [µm]

x [µ

m]

t = T/2

−2 −1 0 1 2−2

−1

0

1

2

z [µm]

x [µ

m]

t = T/4

−2 −1 0 1 2−2

−1

0

1

2

z [µm]

x [µ

m]

t = 0

−2 −1 0 1 2−2

−1

0

1

2

17

Photonic crystal bend

x

z

PT

0

w

pr

ngnb

Pin PR

1 Nr2 . . .

nr

1.4 1.45 1.5 1.55 1.6 0

0.2

0.4

0.6

0.8

1

λ [µm] P

R, T

PT

PR

nr = 3.4, nb = 1.0, r = 0.15 µm, p = 0.6 µm, ng = 1.8, w = 0.5 µm, Nr = 4,∗

TE, x, z ∈ [−1, 5.95] µm, Mx = Mz = 120.

∗ R. Stoffer et. al., Optical and Quantum Electronics 32, 947–961, 2000J. D. Joannopoulos et. al., Photonic crystals: Molding the Flow of Light, Princeton UP, New Jersey, 1995.

18

Photonic crystal bend

z [µm]

x [µ

m]

λ = 1.461 µm

−1 0 1 2 3 4 5−1

0

1

2

3

4

5

z [µm]

x [µ

m]

λ = 1.551 µm

−1 0 1 2 3 4 5−1

0

1

2

3

4

5

19

Quadridirectional mode expansion

QUEP scheme:• Eigenmode expansion technique,

2D Helmholtz problems with piecewiseconstant, rectangular permittivity.

• Equivalent treatment of the propagationalong the two relevant axes.

• Way to realize transparent boundaries for the interiorregion on a cross-shaped computational domain.

• Basis modes can be restricted to simpleDirichlet boundary conditions.

• Applications: . . .

z

x

x0

x1

1

1 2

...

0

2

Nx + 1

xNx

Nx

z0 z1 . . .

Nz + 1Nz

zNz

z [µm]

x [µ

m]

t = 3T/4

−2 0 2

−1

0

1

z [µm]

x [µ

m]

t = T/2

−2 0 2

−1

0

1

z [µm]

x [µ

m]

t = T/4

−2 0 2

−1

0

1

z [µm]

x [µ

m]

t = 0

−2 0 2

−1

0

1

20

quadridirectionalmodeexpansionmodeling inintegratedoptics · quadridirectionalmodeexpansionmodeling...

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