Plane Wave Expansion Method and Reduced Bloch Mode Expansion Technique

MAXWELL’S EQUATIONS IN FOURIER SPACE

Real-Space

THE UNIVERSITY OF TEXAS AT EL PASO

Pioneering 21st Century Electromagnetics and Photonics

0

0

0

0

0

0

yzr x

x zr y

y xr z

yzr x

x zr y

y xr z

HHk E

y z

H Hk E

z x

H Hk E

x y

EEk H

y z

E Ek H

z x

E Ek H

x y

, , , , 0 , , ,

, , , , 0 , , ,

, , , , 0 , ,

y pqr z pqr z pqr y pqr p p q q r r x p q r

p q r

z pqr x pqr x pqr z pqr p p q q r r y p q r

p q r

x pqr y pqr y pqr x pqr p p q q r r z

jk U jk U k a S

jk U jk U k a S

jk U jk U k a S

,

, , , , 0 , , ,

, , , , 0 , , ,

, , ,

p q r

p q r

y pqr z pqr z pqr y pqr p p q q r r x p q r

p q r

z pqr x pqr x pqr z pqr p p q q r r y p q r

p q r

x pqr y pqr y p

jk S jk S k b U

jk S jk S k b U

jk S jk

, 0 , , ,qr x pqr p p q q r r z p q r

p q r

S k b U

0

0

0

0

0

0

y z z y r x

z x x z r y

x y y x r z

y z z y r x

z x x z r y

x y y x r z

jk

jk

jk

jk

jk

jk

K u K u ε s

K u K u ε s

K u K u ε s

K s K s μ u

K s K s μ u

K s K s μ u

Fourier Space Matrix Form K Matrices

,1,1,1

,1,1,2

, , ,

i

i

i

i P Q R

k

k

k

K

Plane Wave

,1,1,1 ,1,1,1

,1,1,2 ,1,1,2

, , , , , ,

i i

i i

i i

i P Q R i P Q R

U S

U S

U S

u s

The plane wave expansion method (PWEM) provides a highly efficient numerical solution to Maxwell’s equations for devices with low to moderate dielectric contrast. It expands the field into a set of plane waves and converts Maxwell’s equations to matrix form by assigning each plane wave a complex amplitude. The final matrix equation can be solved using any number of standard eigen-value

THE CONVOLUTION MATRICES

r

ε

3D PWEM FORMULATION 2D PWEM FORMULATION

1 1 2

0x r x y r y z r zk K μ K K μ K s ε s

1 1 2

0x r x y r y z r zk K ε K K ε K u μ u

E Mode

H Mode

1 1

2 2 2 1 1 1 1 21 12

01 1

2 21 2 1 1 2 1 2 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ• • • •

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ• • • •

r r r r

r r r r

P P P Pk

P P P P

K ε P K K ε P K μ P μ Pu u

u uK ε P K K ε P K μ P μ P

ˆ polarization vectors orthogonal to Ki P

BAND DIAGRAMS ISOFREQUENCY CONTOURS

REDUCED BLOCH MODE EXPANSION

Step 1: Calculate the eigen-vector matrices at the key points of symmetry

1 2 3 1 2 3 N N

Γ XΓ Γ Γ Γ X X X XV Vv v v v v v v v

1 2 3 N

M M M M MV v v v v

Step 2: Construct Bloch mode

1 2 3 1 2 3 N N

Γ XΓ Γ Γ Γ X X X XV Vv v v v v v v v

1 2 3 N

M M M M MV v v v v

1 1 1M M M

Γ Γ X X M MU v v v v v v GramSchmidt U U

Step 3: Calculate eigen-value problem using standard PWEM

Ax λBx

Step 4: Calculate and solve a reduced eigen-value problem.

H

H

A U AU

B U BU , Ax λBx V λ

Step 5: If needed, the eigen-vectors can be transformed back to the plane wave basis.

HV UVU

Effect of the number of spatial harmonics.

M. I. Hussein, “Reduced Bloch mode expansion for periodic media band structure calculations,” Proc. Roy. Soc. Lond. Ser. A465, 2825–2848 (2009).