Fast Analysis of All-Dielectric Structures Using

Rigorous Coupled-Wave Analysis

SEMI-ANALYTICAL FORMULATION OF MAXWELL’S EQUATIONS

Wave Equation for ith Layer

THE UNIVERSITY OF TEXAS AT EL PASO

Pioneering 21st Century Electromagnetics and Photonics

Solution K Matrices

,1,1,1

,1,1,2

, , ,

i

i

i

i P Q R

k

k

k

K

Plane Wave Amplitudes

,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

Rigorous coupled-wave analysis (RCWA) is an extremely efficient algorithm for modeling scattering through all-dielectric structures. It is most efficient for devices with low to moderate dielectric contrast.

THE CONVOLUTION MATRICES

r

ε

22

2

i i

x x

ii i

y y

d

dz

s sΩ 0

s s

2

1 1

, , ,

1 1

, , ,

1 1

, , ,

1 1

, , ,

i i i

x r i y r i x r i x

i

y r i y r i y r i x

x r i y r i x r i x

i

y r i y r i y r i x

Ω PQ

K μ K ε K μ KQ

K μ K ε K μ K

K ε K μ K ε KP

K ε K μ K ε K

x

zy

zx

y

z

z ez

z e

z

λ

λ

s

s W W 0 cψ

u V V 0 c

u

1V QWλ

2

2

eigen-vector matrix

eigen-value matrix

i

i

i

WΩ

λ

THE SEMI-ANALYTICAL SOLUTION

THE SCATTERING MATRIX

11 121 1

2 221 22

i i

i i

S Sc c

c cS S

11 1

11 1 2 2 1 2 2 1 1

11 1

12 1 2 2 1 2 2 2 2

11 1

21 2 1 1 2 1 1 1 1

11 1

22 2 1 1 2 1 1 2 2

i

i i i i i i i i i i i i

i

i i i i i i i i i i i

i

i i i i i i i i i i i

i

i i i i i i i i i i i i

S A X B A X B X B A X A B

S A X B A X B X A B A B

S A X B A X B X A B A B

S A X B A X B X B A X A B

1 1

1 1

ij i j i j

ij i j i j

A W W V V

B W W V V

0i ik L

i e

λ

X

THE SCATTERING MATRIX ALGORITHM

Redheffer Star Product:

a bS S S

1

11 11 12 11 22 11 21

1

12 12 11 22 12

1

21 21 22 11 21

1

22 22 21 22 11 22 12

S a a I b a b a

S a I b a b

S b I a b a

S b b I a b a b

11 1

11

11 1

12

21 12

22 11

i

i i i i i i i i i i i i

i

i i i i i i i i i i i

i i

i i

S A X B A X B X B A X A B

S A X B A X B X A B A B

S S

S S

1 1

0 0

1 1

0 0

i i i

i i i

A W W V V

B W W V V

The formulation for the general asymmetric case is

Our improved formulation offers the following benefits:

1. Symmetric scattering matrices 2. Faster the calculate. 3. More memory efficient. 4. Interchangeable scattering matrices.

, ,p p q q x

p q

a S p q

Assemble convolution matrix from FFT of unit cell.