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Pioneering 21st Century Electromagnetics and Photonics

The Finite-Difference Frequency-Domain Method

Raymond C. Rumpf, Ph.D.

“I am always doing that which I cannot do,in order that I may learn how to do it.”

– Pablo Picasso

Short Course Outline

• Background Topics– Numerical methods, electromagnetics, linear algebra,

finite-difference approximations– Break

• The Finite-Difference Frequency-Domain Method– Yee grid, Maxwellmatrix, formulation, PML, TF/SF,

solution, post processing– Break

• Implementation– 3D2D, grids and materials, testing, convergence– Code examples: GMR, photonic crystal, and wire-grid

polarizer– Done!

Short Course on Finite-Difference Frequency-Domain

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Background

Background: Numerical MethodsGolden Rule #1

1. All numbers should equal 1

(1.234567…) + (0.0123456…) = Lost two digits of accuracy!!

Why?

Solution: NORMALIZE EVERYTHING!!!

0

0

E E

0

0

H H

or

0x k x

0y k y

0z k z

00 1 m


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2. Never perform calculations

1. Golden Rule #1.2. Finite floating point precision introduces round-off errors.

Why?

Solution: MINIMIZE NUMBER OF COMPUTATIONS!!!

1. Take problems as far analytically as possible.2. Avoid unnecessary computations.

2 2

2exp

R x y

Rg R

2 2

2

2exp

r x y

rg r



3. Follow rules 1 and 2

1. Improves accuracy by reducing numerical error.2. Enables codes to model larger and more intensive problems.

Why?

Solution

1. Normalize all parameters.2. Minimize computations.


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Background: Numerical MethodsBenefits and drawbacks

Frequency-Domain Time-Domain

Semi-AnalyticalFully Numerical

Fourier-SpaceReal-Space

+ wideband simulations+ scales near linearly+ active & nonlinear devices+ easily locates resonances

- longitudinal periodicity- sharp resonances- memory requirements- oblique incidence

+ resolves sharp resonances+ handles oblique incidence+ longitudinal periodicity+ can be very fast

- scales at best NlogN- can miss sharp resonances- active & nonlinear devices

+ better convergence+ scales better than SA+ complex device geometry

- memory requirements- long uniform sections

+ very fast & efficient+ layered devices+ less memory

- convergence issues- scales poorly- complex device geometry

+ high index contrast+ metals+ resolving fine details+ field visualization

- slow for low index contrast + moderate index contrast+ periodic problems+ very fast and efficient

- field visualization- formulation difficult- resolving fine details

Unstructured GridStructured Grid

+ easy to implement+ rectangular structures+ easy for divergence free

- less efficient- curved surfaces

+ most efficient+ handles larger structures+ conforms to curved surfaces

- difficult to implement- spurious solutions


Background: Numerical MethodsFinite-Difference Frequency-Domain

H j E

E j H

Maxwell’s Equations Matrix Equation

A x b

Numerical Solution

1x A b

Fields fit to a discrete grid


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Background: ElectromagneticsGauss’s law in differential form

Electric fields diverge from positive charges and converge on negative charges.

-+

vD

yx zDD D

Dx y z

If there are no charges, electric fields must form loops.


Background: ElectromagneticsNo magnetic charge

Magnetic fields always form loops.

0B

yx zBB B

Bx y z


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Background: ElectromagneticsConsequence of Zero Divergence

The divergence theorems force the electric and magnetic fields to be perpendicular to the propagation direction of a plane wave.

k D

0

0jk r

D

de

d

no charges

0

0

jk d

k d

k

k

k B

0

0jk r

B

be

b

no charges

0

0

jk b

k b

k

k


Background: ElectromagneticsAmpere’s law in differential form

DH J

t

ˆ ˆ ˆy yx xz zx y z

H HH HH HH a a a

y z z x x y

Circulating magnetic fields induce currents and/or time varying electric fields.Currents and/or time varying electric fields induce circulating magnetic fields.


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Background: ElectromagneticsFaraday’s law in differential form

BE

t

ˆ ˆ ˆy yx xz zx y z

E EE EE EE a a a

y z z x x y

Circulating electric fields induce time varying magnetic fields.Time varying magnetic fields induce circulating electric fields.


Background: ElectromagneticsConsequences of curl equations

The curl equations predict electromagnetic waves.

HE k

The curl equations force the electric and magnetic field components of a plane wave to be perpendicular.

k

Electric Field

Magnetic Field

Electric Field

Magnetic Field


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Background: ElectromagneticsMaxwell’s equations

Divergence Equations

0

v

B

D

DH J

t

BE

t

Curl Equations

Constitutive Relations

D t t E t

B t t H t

What produces fields

How fields interact with materialsmeans convolution


Background: ElectromagneticsSimplifying Maxwell’s equations

0

0

B

D

H D t

E B t

D t t E t

B t t H t

1. Assume no charges or current sources: 0v 0J

0

0

B

D

H j D

E j B

D E

B H

2. Transform Maxwell’s equations to frequency-domain:

0

0

H

E

H j E

E j H

3. Substitute constitutive relations into Maxwell’s equations:

Convolution becomes multiplication

Note: It is helpful to retain μ and ε and not replace with refractive index n.


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Background: ElectromagneticsPhysical boundary conditions

Fields tangential to the interface are continuous across it.

1,TE 2,TE

1,TH 2,TH

1 1 and 2 2and

Fields normal to the interface are discontinuous across it.

1 1,NE

1 1,NH

2 2,NE

2 2,NH

Note: normal components of D and B are continuous across an interface.

These are more complicated boundary condition that have numerical consequences.


Background: ElectromagneticsSign Convention for Waves

Forward Propagation Along +x

SIGN CONVENTION #1

ikxeRefractive Index

N n i

00

0 0

oscillatory Decayingterm in exponential

ik n i xik Nxikx

ik nx k x

x

e e e

e e

Forward Propagation Along +x

SIGN CONVENTION #2

ikxeRefractive Index

N n i

00

0 0

oscillatory Decayingterm exponential

ik n i xik Nxikx

ik nx k x

x

e e e

e e

Use

d H

ere


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Background: Linear AlgebraMatrices represent sets of equations

11 12 13 14 1

21 22 23 24 2

31 32 33 34 3

41 42 43 44 4

a w a x a y a z b

a w a x a y a z b

a w a x a y a z b

a w a x a y a z b

11 12 13 14 1

21 22 23 24 2

31 32 33 34 3

41 42 43 44 4

a a a a bw

a a a a bx

a a a a by

a a a a bz

A set of linear algebraic equations can be written in “matrix” form.


Background: Linear AlgebraInterpretation of matrices

11 12 13 1a x a y a z b

21 22 23 2a x a y a z b

31 32 33 3a x a y a z b

11 12 13 1

21 22 23 2

31 32 33 3

a a a x b

a a a y b

a a a z b

11 12 13

21 22 23

31 32 33

a a a

a a a

a a a

Equation for x

Equation for y

Equation for z

EQUATION FOR… RELATION TO…

11 12 13

21 22 23

31 32 33

a a a

a a a

a a a


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Background: Linear AlgebraMatrices have a compact notation

Ax b

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

a a a a

a a a a

a a a a

a a a a

A

w

x

y

z

x

1

2

3

4

b

b

b

b

b

Matrices and vectors can be represented and treated as single variables.

A x b

or

square matrixcolumnvector

columnvector

11 12 13 14 1

21 22 23 24 2

31 32 33 34 3

41 42 43 44 4

a a a a bw

a a a a bx

a a a a by

a a a a bz


Background: Linear AlgebraMatrices require a special algebra

AB BA

A B B A

Commutative Laws

AB C A BC

A B C A B C

Associative Laws

A B A B

AB A B A B

Multiplication with a Scalar

A B C AB AC

A B C AC BC

Distributive Laws

11 1

1

?

n

m mn

a a

a a

A

I A

Addition with a Scalar

AB BA


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Background: Linear AlgebraSpecial matrices

Zero Matrix

0 0

0 0

0

Identity Matrix

1

1

0

0

I

0 A A 0 0

I A A I A

0 A A 0 A

A A 0


Background: Finite-DifferencesWhat is a finite-difference approximation?

1.5 2 1df f f

dx x

1f2f

df

dx

x

second-order accuratefirst-order derivative


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Background: Finite-DifferencesTypes of finite-difference approximations

Backward difference

1.5 2 1df f f

dx x

Central difference

1 2 1df f f

dx x

Forward difference

2 2 1df f f

dx x


Background: Finite-DifferencesGeneralized finite-difference

n

ni

ii

d fa

xf

d


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Background: For more information…

• Electromagnetics– Matthew Sadiku, Elements of Electromagnetics, Saunders, 2000

– Constantine Balanis, Advanced Engineering Electromagnetics, Wiley, 1989

• Linear Algebra– Howard Anton, Elementary Linear Algebra 8th Ed., Wiley, New Jersey, 2000

– Gilbert Strang, Linear Algebra and Its Applications 4th Ed., Thomson, California, 2006

• http://web.mit.edu/18.06/www/Video/video-fall-99.html


Finite-Difference Frequency-Domain

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FDFD Method: Yee GridWhat is a discrete grid?

A grid is constructed by dividing space

into discrete cells

Example physical

(continuous) field profile

Field is known only at discrete

points

Representation of what is actually

stored in memory


FDFD Method: Yee GridWhat is a grid unit cell?

y

x

A field component is assigned to a specific point within the grid unit cell.

Whole Grid

A Single Unit Cell


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FDFD Method: Yee GridUnit cells of Yee grids

• Field components are in physically different locations• Field components may reside in different materials even if they are in the

same unit cell• Field components will be out of phase

xy

z

xEyE

zE

xHyH

zH

3D Yee Grid2D Yee Grids1D Yee Grid

zE

xHyH

xy

xy

zH yExE

Ez Mode

Hz Mode

z

xE

yH

yExH

Ey Mode

Ex Mode

z


FDFD Method: Yee GridAnother interpretation of the Yee Grid


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FDFD Method: Yee GridReasons for using a Yee grid

0E

0H

1. Divergence-free 3. Elegant arrangementto approximate curl equations

2. Physical boundary conditions are naturally satisfied


FDFD Method: Yee GridExtended grids

y

x

ij

222 Grid

44 Grid (Ez Mode)


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FDFD Method: Maxwell MatrixNormalize the magnetic field

E j H

H j E Standard Maxwell’s Curl Equations

Normalized Magnetic Field

377E

nH

0

0

H j H

Normalized Maxwell’s Equations

0 rE k H

0 rH k E

0 0 0k Note:


FDFD Method: Maxwell MatrixExpand Maxwell’s equations

0 rE k H

0 rH k E

0

0

0

yzxx x xy y xz z

x zyx x yy y yz z

y xzx x zy y zz z

HHk E E E

y z

H Hk E E E

z x

H Hk E E E

x y

0

0

0

yzxx x xy y xz z

x zyx x yy y yz z

y xzx x zy y zz z

EEk H H H

y z

E Ek H H H

z xE E

k H H Hx y


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FDFD Method: Maxwell MatrixNormalize the grid

Normalized Maxwell’s Equations

Normalized Grid

0x k x 0y k y 0z k z

0

0

0

0

0

0

yzxx x xy y xz z

x zyx x yy y yz z

y xzx x zy y zz z

yzxx x xy y xz z

x zyx x yy y yz z

y xzx x zy y zz

EEk H H H

y z

E Ek H H H

z xE E

k H H Hx y

HHk E E E

y z

H Hk E E E

z x

H Hk E E E

x y

z

yzxx x xy y xz z

x zyx x yy y yz z

y xzx x zy y zz z

yzxx x xy y xz z

x zyx x yy y yz z

y xzx x zy y zz

EEH H H

y z

E EH H H

z xE E

H H Hx y

HHE E E

y z

H HE E E

z x

H HE E E

x y

z


FDFD Method: Maxwell MatrixAssume diagonal tensor functions

yzxx x

x zyy y

y xzz z

EEH

y z

E EH

z xE E

Hx y

yzxx x

x zyy y

y xzz z

HHE

y z

H HE

z x

H HE

x y


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FDFD Method: Maxwell MatrixFinite-difference equation for Hx

yzxx x

EEH

y z

z

xyxE

, ,i j kyE

, ,i j kzE

, ,i j kxHyH

zH

, 1,i j kzE

, , 1i j kyE

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

i j k i j ki j k i j ky yz i j k

xj k

xz i

x HE E

y

E

z

E


x zyy y

E EH

z x

FDFD Method: Maxwell MatrixFinite-difference equation for Hy

z

xyyE

, ,i j kzE

xH, ,i j kyH

zH, ,i j kxE

1, ,i j kzE

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

i j k i j k i j ki j

i j kx x z z k i k

yyj

yz x

E E E EH

, , 1i j kxE


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y xzz z

E EH

x y

FDFD Method: Maxwell MatrixFinite-difference equation for Hz

z

xy

zE

xHyH

, ,i j kzH

, ,i j kxE

1, ,i j kyE

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

i j k i j k i j ki j

i j ky y x x k i k

zzj

zx y

E E E EH

, 1,i j kxE

, ,i j kyE


yzxx x

HHE

y z

FDFD Method: Maxwell MatrixFinite-difference equation for Ex

, , , ,,

1, 1,, ,

,,

, i j k i j ki j

i j k i j ky yz k

xj

xxz i kH H

EH H

y z

z

xy

, ,i j kxE

yE

zE

xH, ,i j kyH

, ,i j kzH

, , 1i j kyH

, 1,i j kzH


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x zyy y

H HE

z x

FDFD Method: Maxwell MatrixFinite-difference equation for Ey

z

xy

xEyE

zE

yH, ,i j kxH

, ,i j kzH

, , , , 1 , , 1,,

,, ,,

i j k i j k i j k i j kx x z z i j i

yk

yjk

y

H H H H

z xE

, , 1i j kxH

1, ,i j kzH


y xzz z

H HE

x y

FDFD Method: Maxwell MatrixFinite-difference equation for Ez

z

xy

xE

yE

, ,i j kzE

, ,i j kyH

, ,i j kxH

zH

, , 1, , , , , 1,

,, ,,

i j k i j k i j k i j ky y i jx

zi jxzz

kk EH H H H

x y

, 1,i j kxH

1, ,i j k

yH


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FDFD Method: Maxwell MatrixExtended 2D Yee grid (Ez Mode)

12 , , 1,

y y yi j i j i j

x x

H H H

12, , , 1

x x xi j i j i j

H H

y

H

y

12 , 1, ,

z z zi j i j i jE

x x

E E

12, , 1 ,

z z zi j i j i j

y

E

y

E E

y

x

ij


FDFD Method: Maxwell MatrixSummary of FD approximations

x

y

z

yz

x z

y

xx

yy

zz

xx

yy

zz

yz

x z

y

x

x

x

y

z

EE

E E

E E

E

E

E

y z

z x

x y

y z

H

H

H

HH

H H

H H

z x

x y

, , 1, ,

, ,

,

, ,

,

, ,, 1, , ,

, , 1 , , 1, , , ,

1, , , , , 1,

,

,

, ,

, , ,

, ,

1

,

i j kx

i j ky

i j kz

i j k i

i j k i j ki j ki j kxx

i j kyy

i j ky yz z

i j k i j k i j k i j kx x z z

i j k i j k i j k i j ky y x x i j k

j

z

kz z

z

y

E E

y z

E E

E E E E

H

H

H

HH

z x

x

E E E

y

H

y

E

, , , , 1

, , , , 1

, ,

, ,

, ,

, , 1, ,

, , 1, , , ,

, ,

, ,

,, 1

,,

i j kx

i j ky

i

i j k i j ky

i j k i j k i j k i j k

i j kxx

i jx x z z

i j k i j k i j k i j k

kyy

i j kyz

kz

xz

jy x

H

H H H H

H H

z

z x

x y

H

E

E

HE


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FDFD Method: Maxwell MatrixFields are put into column vectors

2-D Systems

1E 5E 9E 13E

2E 6E 10E 14E

3E 7E 11E 15E

4E 8E 12E 16E

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

1

2

3

4

5

E

E

E

E

E

E

1-D Systems

1E2E 3E 4E 5E


FDFD Method: Maxwell MatrixConstruction of field column vectors

1E 5E 9E 13E

2E 6E 10E 14E

3E 7E 11E 15E

4E 8E 12E 16E

1E

2E

3E

4E

5E

6E

7E

8E

9E

10E

11E

12E

13E

14E

15E

16E

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

E

MATLAB ‘reshape’ commandE = E(:);E = reshape(E,Nx,Ny);


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FDFD Method: Maxwell MatrixPoint-by-point multiplication

,r i iE1E

2E 3E 4E 5E

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

r r

r r

r r r

r r

r r

E E

E E

E E

E E

E E

εE

1r 2r 3r 4r 5r



,r i iE1E

2E 3E 4E 5E

1 1r E 1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

r r

r r

r r r

r r

r r

E E

E E

E E

E E

E E

εE

1r 2r 3r 4r 5r


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,r i iE1E

2E 3E 4E 5E

2 2r E

1 1r E 1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

r r

r r

r r r

r r

r r

E E

E E

E E

E E

E E

εE

1r 2r 3r 4r 5r



,r i iE1E

2E 3E 4E 5E

2 2r E

1 1r E

3 3r E

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

r r

r r

r r r

r r

r r

E E

E E

E E

E E

E E

εE

1r 2r 3r 4r 5r


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,r i iE1E

2E 3E 4E 5E

2 2r E

1 1r E

3 3r E

4 4r E

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

r r

r r

r r r

r r

r r

E E

E E

E E

E E

E E

εE

1r 2r 3r 4r 5r



,r i iE1E

2E 3E 4E 5E

2 2r E

1 1r E

3 3r E

4 4r E

5 5r E

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

r r

r r

r r r

r r

r r

E E

E E

E E

E E

E E

εE

1r 2r 3r 4r 5r


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FDFD Method: Maxwell MatrixDerivative operators for Electric fields

1

12

i i

i

E E E

x x

1E2E 3E 4E 5E

1 1.5

2 2.5

3 3.5

4 4.5

5 5.5

1 1 0 0 0

0 1 1 0 01

0 0 1 1 0

0 0 0 1 1

0 0 0 0 1

x

xEx x

x

x

E E

E E

E Ex

E E

E E

DE

x



1

12

i i

i

E E E

x x

1E2E 3E 4E 5E

2 1E E

x

1 1.5

2 2.5

3 3.5

4 4.5

5 5.5

1 1 0 0 0

0 1 1 0 01

0 0 1 1 0

0 0 0 1 1

0 0 0 0 1

x

xEx x

x

x

E E

E E

E Ex

E E

E E

DE

x


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1

12

i i

i

E E E

x x

1E2E 3E 4E 5E

3 2E E

x

2 1E E

x

1 1.5

2 2.5

3 3.5

4 4.5

5 5.5

1 1 0 0 0

0 1 1 0 01

0 0 1 1 0

0 0 0 1 1

0 0 0 0 1

x

xEx x

x

x

E E

E E

E Ex

E E

E E

DE

x



1

12

i i

i

E E E

x x

1E2E 3E 4E 5E

3 2E E

x

2 1E E

x

4 3E E

x

1 1.5

2 2.5

3 3.5

4 4.5

5 5.5

1 1 0 0 0

0 1 1 0 01

0 0 1 1 0

0 0 0 1 1

0 0 0 0 1

x

xEx x

x

x

E E

E E

E Ex

E E

E E

DE

x


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1

12

i i

i

E E E

x x

1E2E 3E 4E 5E

3 2E E

x

2 1E E

x

4 3E E

x

5 4E E

x

1 1.5

2 2.5

3 3.5

4 4.5

5 5.5

1 1 0 0 0

0 1 1 0 01

0 0 1 1 0

0 0 0 1 1

0 0 0 0 1

x

xEx x

x

x

E E

E E

E Ex

E E

E E

DE

x



1

12

i i

i

E E E

x x

1E2E 3E 4E 5E

3 2E E

x

2 1E E

x

4 3E E

x

5 4E E

x

56E E

x

1 1.5

2 2.5

3 3.5

4 4.5

5 5.5

1 1 0 0 0

0 1 1 0 01

0 0 1 1 0

0 0 0 1 1

0 0 0 0 1

x

xEx x

x

x

E E

E E

E Ex

E E

E E

DE

x


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FDFD Method: Maxwell MatrixDerivative operators for Magnetic fields

0.51

1.52

2.53

3.54

4.55

1 0 0 0 0

1 1 0 0 01

0 1 1 0 0

0 0 1 1 0

0 0 0 1 1

x

xHx x

x

x

HH

HH

HHx

HH

HH

D H

1H2H 3H 4H 5H

x 1

12

i i

i

H H H

x x



0.51

1.52

2.53

3.54

4.55

1 0 0 0 0

1 1 0 0 01

0 1 1 0 0

0 0 1 1 0

0 0 0 1 1

x

xHx x

x

x

HH

HH

HHx

HH

HH

D H

1H2H 3H 4H 5H

x 1

12

i i

i

H H H

x x

01H H

x


5/23/2012

32


0.51

1.52

2.53

3.54

4.55

1 0 0 0 0

1 1 0 0 01

0 1 1 0 0

0 0 1 1 0

0 0 0 1 1

x

xHx x

x

x

HH

HH

HHx

HH

HH

D H

1H2H 3H 4H 5H

x 1

12

i i

i

H H H

x x

2 1H H

x

01H H

x



0.51

1.52

2.53

3.54

4.55

1 0 0 0 0

1 1 0 0 01

0 1 1 0 0

0 0 1 1 0

0 0 0 1 1

x

xHx x

x

x

HH

HH

HHx

HH

HH

D H

1H2H 3H 4H 5H

x 1

12

i i

i

H H H

x x

2 1H H

x

01H H

x

3 2H H

x


5/23/2012

33


0.51

1.52

2.53

3.54

4.55

1 0 0 0 0

1 1 0 0 01

0 1 1 0 0

0 0 1 1 0

0 0 0 1 1

x

xHx x

x

x

HH

HH

HHx

HH

HH

D H

1H2H 3H 4H 5H

x 1

12

i i

i

H H H

x x

2 1H H

x

01H H

x

3 2H H

x

4 3H H

x



1

12

i i

i

H H H

x x

2 1H H

x

01H H

x

3 2H H

x

4 3H H

x

5 4H H

x

0.51

1.52

2.53

3.54

4.55

1 0 0 0 0

1 1 0 0 01

0 1 1 0 0

0 0 1 1 0

0 0 0 1 1

x

xHx x

x

x

HH

HH

HHx

HH

HH

D H

1H2H 3H 4H 5H

x


5/23/2012

34

FDFD Method: Numerical BC’sSimplest boundary conditions

1 11

1 12

1 13

1 14

15

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0 0

x x

x x

Ex xx

x x

x

E

E

E

E

E

DE

Dirichlet Boundary Conditions

6Assume 0E

Periodic Boundary Conditions

6 1Assume E E

1E2E 3E 4E 5E

6E

1E2E 3E 4E 5E 1E

1 11

1 12

1 13

1 14

15

1

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

x x

x x

Ex xx

x x

xx

E

E

E

E

E

DE

x

x

51E E

x


FDFD Method: Numerical BC’sPseudo-periodic boundary conditions

6 1Assume x xjkEE e

1E2E 3E 4E 5E 6E

1 11

1 12

1 13

1 14

15

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0jkx x

x x

x x

Ex

ex

xx

x x

x

E

E

E

E

E

DEx


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35

FDFD Method: Maxwell MatrixDerivative operators on a 4x4 grid

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0

0

0

0 0 1 1 0

1Ex x

D

0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 1

0

0 0 0 0

1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 1 0

1Ey y

D

0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0

0 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1 1 0 0

0

0

1Hx x

D

0 0

0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 1 0 0 0 0

1Hy y

D

0

0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1


FDFD Method: Maxwell MatrixFinally in matrix form!!!

, , 1 , ,, 1, , ,

, , 1 , , 1, , , ,

1, , ,

, ,

, ,

,

,

, , 1,

,

, ,

, ,, ,

,

i j k i j ki j k i j ky yz z

i j k i j k i j k

i j

i j

i j kx

i j ky

i j k

kx x z z

i j k i j k

kxx

i j k

i j k i j ky y x

yy

i j kzz z

x

y z

z x

HE EE E

E

x y

H

H

E E E

E E E E

z y

x z

y x

E Ey z xx

E Ez x yy

E Ex y

y

zz

x

z

E E

E E

E E

D D μ

D D μ

D D μ

H

H

H

, , , , 1, , , 1,

, , , , 1 , , 1, ,

,

, ,

, ,

, 1, , , , , 1,

, ,

,

, ,,

,

,

i j k i j ki j k i j ky yz z

i j k i j k i j k i j kx x z z

i j k

i j kxx

i j kyy

i j k i j k i ji j kzz

k

i j kx

i j k

y y x

y

i j kz

x

H HH H

H

y z

z x

H H H

H H

y

HE

H

x

E

E

H Hy z xx

H Hz x yy

H Hx y

z y

x

x

z

y zx zz

y

H H

H H

ED D ε

D D E

ED D εH H

ε

xx

yy

x

y

zy x z

z

z

y

x z

y z

z x

x y

H

H

E E

E E

E E H

xx

yy

zz

z y

x z

y x

x

y

z

y z

z

E

E

E

H H

Hx

yH H

x

H


5/23/2012

36

FDFD Method: Maxwell MatrixSummary

E Ey z z y xx x

E Ez x x z yy y

E Ex y y x zz z

H Hy z z y xx x

H Hz x x z yy y

H Hx y y x zz z

D E D E μ H

D E D E μ H

D E D E μ H

D H D H ε E

D H D H ε E

D H D H ε E

E j H

H j E

0

0

r

r

E k H

H k E

No charges

yzxx x

x zyy y

y xzz z

yzxx x

x zyy y

y xzz z

EEH

y z

E EH

z xE E

Hx y

HHE

y z

H HE

z x

H HE

x y

Normalized H

Normalized Grid Finite-Difference Approximation, , 1 , ,, 1, , ,

, , , ,

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

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

, , , 1,

i j k i j ki j k i j ky y i j k i j kz z

xx x

i j k i j k i j k i j ki j k i j kx x z zyy y

i j k i j k i j k i j ky y i j k i j kx x

zz z

i j k i j kyz z

E EE EH

y z

E E E EH

z x

E E E EH

x y

HH H

y

, , , , 1, , , ,

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

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

i j k i j ky i j k i j k

xx x

i j k i j k i j k i j ki j k i j kx x z zyy y

i j k i j k i j k i j ky y i j k i j kx x

zz z

HE

z

H H H HE

z x

H H H HE

x y

Matrix Form


FDFD Method: Maxwell MatrixHints for other formulations

E Ey z z y xx x

E Ez x x z yy y

E Ex y y x zz z

H Hy z z y xx x

H Hz x x z yy y

H Hx y y x zz z

D E D E μ H

D E D E μ H

D E D E μ H

D H D H ε E

D H D H ε E

D H D H ε E

Fully Numerical Methods BPM, Waveguides, & Semi-Analytical Methods

Ey z y xx x

Ex x z yy y

E Ex y y x zz z

Hy z y xx x

Hx x z yy y

H Hx y y x zz z

d

dzd

dz

d

dzd

dz

D E E μ H

E D E μ H

D E D E μ H

D H H ε E

H D H ε E

D H D H ε E


5/23/2012

37

FDFD Method: Maxwell MatrixBlock Matrix Form

E Ey z z y xx x

E Ez x x z yy y

E Ex y y x zz z

H Hy z z y xx x

H Hz x x z yy y

H Hx y y x zz z

D E D E μ H

D E D E μ H

D E D E μ H

D H D H ε E

D H D H ε E

D H D H ε E

Fully Numerical Methods

E Ez y x xx x

E Ez x y yy yE Ey x z zz z

0 D D E μ 0 0 H

D 0 D E 0 μ 0 H

D D 0 E 0 0 μ H

H Hz y x xx x

H Hz x y yy yH Hy x z zz z

0 D D H ε 0 0 E

D 0 D H 0 ε 0 E

D D 0 H 0 0 ε E

E C E μH

H C H εE

E H

H E


FDFD Method: Formulation3D FDFD method

Block Matrix Form

E

H

C E μH

C H εE

x x

y y

z z

xx xx

yy yy

zz zz

H Hz y

H H H Ez xH Hy x

H E

H H E E

H E

μ 0 0 ε 0 0

μ 0 μ 0 ε 0 ε 0

0 0 μ 0 0 ε

0 D D 0 D

C D 0 D C

D D 0

E Ez y

E Ez xE Ey x

D

D 0 D

D D 0

Matrix Wave Equations

1

1

E H

H E

C ε C μ H 0

C μ C ε E 0

AE 0

For 3D analysis, A is usually too big to solve by simple means.

For information on 3D analysis, see

R. C. Rumpf, A. Tal, S. M. Kuebler, “Rigorous electromagnetic analysis of volumetrically complex media using the slice absorption method,” J. Opt. Soc. Am. A 24, 3123-3134 (2007).


5/23/2012

38

FDFD Method: FormulationReduction to two dimensions

H Hx y y x zz z

Ey z xx x

Ex z yy y

D H D H ε E

D E μ H

D E μ H

For problems uniform along the z-direction,

E Hz z D D 0

Maxwell’s equation decouple into two distinct modes:

Ez Mode Hz Mode

E Ex y y x zz z

Hy z xx x

Hx z yy y

D E D E μ H

D H ε E

D H ε E

z


FDFD Method: Formulation2D matrix wave equations

H Hx y y x zz z

Ey z xx x

Ex z yy y

D H D H ε E

D E μ H

D E μ H

Ez Mode Hz ModeE Ex y y x zz z

Hy z xx x

Hx z yy y

D E D E μ H

D H ε E

D H ε E

1

1

Ex xx y z

Ey yy x z

H μ D E

H μ D E

1

1

Hx xx y z

Hy yy x z

E ε D H

E ε D H

1 1H E H Ex yy x z y xx y z zz z

D μ D E D μ D E ε E 1 1E H E Hx yy x z y xx y z zz z

D ε D H D ε D H μ H

1 1

E z

H E H EE x yy x y xx y zz

A E 0

A D μ D D μ D ε 1 1

H z

E H E HH x yy x y xx y zz

A H 0

A D ε D D ε D μ

AE = DHX/URyy*DEX + DHY/URxx*DEY + ERzz; AH = DEX/ERyy*DHX + DEY/ERxx*DHY + URzz;


5/23/2012

39

FDFD Method: FormulationThese cannot yet be solved

1 Ax b x A b

General solution procedure

Solution of matrix wave equation

1 E z z E A E 0 E A 0 0 trivial solution!!

A source must be incorporated.

1 E z z E A E f E A f


FDFD Method: Boundary Cond’sPerfectly matched layer (1 of 2)

PML

PML

PM

L

PM

L ProblemSpace

1ys

1xs 1x y zs s s

1ys

1xs

Perfectly Matched Layer (PML)


5/23/2012

40

FDFD Method: Boundary Cond’sPerfectly matched layer (2 of 2)

0

0

r

r

E k s H

H k s E

0 0

0 0

0 0

y z

x

x z

y

x y

z

s s

s

s ss

s

s s

s

0

0

0

0

0

0

1

1

1

x x x

y y y

z z z

s x a x xjk

s y a y yjk

s z a z zjk

max

max

max

1

1

1

p

x x

p

y y

p

z z

a x a x L

a y a y L

a z a z L

2max

2max

2max

sin2

sin2

sin2

xx

yy

zz

xx

L

yy

L

zz

L

Maxwell’s eqs. with PML Computing PML Parameters

max

max

0 5

3 5

1

a

p


FDFD Method: TF/SF SourceTotal-field / scattered-field framework

Problem rows

tota

l-fie

ldsc

atte

red-

field


5/23/2012

41

FDFD Method: TF/SF SourceCompute source field

1,1

src

,

exp

x y

x y

N N

f

j k k

f

f x y

Compute source as it would exist in a completely homogeneous grid.

incsrc

jk rE r e

This is NOT the “f” term in AE=f

Unit amplitude plane wave


FDFD Method: TF/SF SourceCompute SF masking matrix, Q

tota

l-fie

ldsc

atte

red-

field 1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1

1

0

0

0

Q


5/23/2012

42

FDFD Method: TF/SF SourceCompute TF/SF source vector

Source isolated to the scattered-field:

Source isolated to the total-field:

Quantity that must be subtracted from TF terms:

…but only from SF equations:

Quantity that must be added to SF terms:

…but only from TF equations:

“Corrected” matrix problem:

…or:

scat srcf Qf

tot src f I Q f

totAf

totQAf

scatAf

scatI Q Af

tot scat AE QAf I Q Af 0

src AE QA AQ f

src AE = f f QA AQ f

Corrects SF Equations

Corrects TF Equations


FDFD Method: TF/SF SourceExample simulation (1 of 2)

Materials Field Q

Materials Field Q

PML

PML

PML

PML

PML

PML

PML

PML

PML

PML

PML

PML

Scattered-FieldScattered-Field


Total-FieldTotal-Field


1 1.0n

1 1.0n

2 3.0n


5/23/2012

43

FDFD Method: TF/SF SourceExample simulation (2 of 2)

Materials Field Q

Materials Field Q

PML

PML

PML

PML

PML

PML

PML

PML

PML

PML

PML

PML





1 1.0n

1 1.0n

2 3.0n


FDFD Method: Formulation Summary

1 1H E H EE x yy x y xx y zz

A D μ D D μ D ε

1 1E H E HH x yy x y xx y zz

A D ε D D ε D μ

Matrix Wave Equations

src f QA AQ f

Source Vector

FDFD Matrix Problem

E EA E = f

H HA H = f


5/23/2012

44

FDFD Method: SolutionMatrix division in MATLAB

1E = A f

Matrix Division

WARNING: Do not compute inverse of A

MATLAB Implementation

E = A\f; WARNING: Do not perform

E = inv(A)*f;


FDFD Method: SolutionIterative algorithms

• Good for very large and sparse matrices

• Many algorithms exist

• Arguably more accurate for large matrices

• Usually do not have to explicitly compute matrix A

• Popular iterative algorithms for FDFD– Generalized minimum residue (GMRES)

– Biconjugate gradient (BCG)


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FDFD Method: SolutionSlice absorption method

1 0 1 1 1 2 1 1 0 1 1 1 3 1

2 1 2 2 2 3 2

3 2 3 3 3 4 3 3 1 3 3 3 4 3

a E b E c E f a E b E c E f

a E b E c E f

a E b E c E f a E b E c E f

R. C. Rumpf, A. Tal, S. M. Kuebler, “Rigorous electromagnetic analysis of volumetrically complex media using the slice absorption method,” J. Opt. Soc. Am. A 24, 3123-3134 (2007).


FDFD Method: Post ProcessingDiffraction by periodic structures

Reflected Power

Transmitted Power

Diffraction Efficiency

inc

DE mPmP


5/23/2012

46

FDFD Method: Post ProcessingCompute spatial harmonics

refE

trnE

Step 1: Extract Eref and Etrn Step 2: Remove phase tilt

ref refxjk xE x E x e

trn trnxjk xE x E x e

Step 3: Compute FFT

ref refFFTS m E x

trn trnFFTS m E x

Note: Some FFT algorithms require that you divide by the number of points and shift after calculation.

Eref = fftshift(fft(Fref))/Nx;Etrn = fftshift(fft(Ftrn))/Nx;

SF

TF


FDFD Method: Post ProcessingCompute diffraction efficiencies

Step 2: Compute DE

ref

2 ,ref inc

Re z m

z

kR m S m

k

Step 1: Compute wave vectorcomponents

, ,inc

2x m x

mk k

*

2ref 2, 0 ref ,z m x mk k n k

*

2trn 2, 0 trn ,z m x mk k n k

trn

2 , reftrn inc

trn

Re z mE

z

kT m S m

k

trn

2 , reftrn inc

trn

Re z mH

z

kT m S m

k

Note: These equations assume the source has unit amplitude

,..., 2, 1,0,1,2,...,2 2x xN N

m


5/23/2012

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FDFD Method: Post ProcessingCompute total power

totm

R R mStep 1: Compute total reflected power

totm

T T mStep 2: Compute total transmitted power

Step 3: Verify conservation of energy

tot tot 1R T

Note: This conservation condition is only obeyed when no materials have loss or gain.


FDFD Method: Algorithm

1. Construct FDFD Problema. Define your problemb. Choose a gridc. Assign materials to the grid

2. Implement PMLa. Compute sx, sy, and sz

b. Incorporate into r and r

3. Construct Matrix Problema. Compute derivative matricesb. Construct diagonal materials

matricesc. Compute Ad. Compute source

i. compute source fieldii. compute Qiii. compute source vector f

4. Solve Matrix Problem: E = A-1f;

5. Post Process Dataa. Extract Eref and Etrn

b. Remove phase tiltc. FFT the fieldse. Compute wave vector termsd. Compute diffraction efficienciesf. Verify conservation of energy


5/23/2012

48

FDFD Method: For more information…

• Literature– W. Sun, K. Liu, C. A. Balanis, “Analysis of Singly and Doubly Periodic

Absorbers by Frequency-Domain Finite-Difference Method,” IEEE Trans. Ant. and Prop. 44, 798-805 (1996)

– S. Wu, E. N. Glytsis, “Finite-number-of-periods holographic gratings with finite-width incident beams: analysis using the finite-difference frequency-domain method,” J. Opt. Soc. Am. A 19, 2018-2029 (2002)

• Raymond Rumpf’s Ph.D. Dissertation– R. C. Rumpf, “Design and optimization of nano-optical elements by

coupling fabrication to optical behavior,” Ph.D. dissertation, University of Central Florida, 2006.

– See chapter 3, pp. 60—81

– http://purl.fcla.edu/fcla/etd/CFE0001159


Implementation

5/23/2012

49

Implementation: 3D2DUniform devices

perio

dic

boun

dary

periodic boundary

Absorbing boundary

Absorbing boundary


Implementation: 3D2DEffective index method

1,effn

2,effn

1,effn

2,effn

Effective indices are best computed by modeling the vertical cross section as a slab waveguide.

A simple average index can also produce good results.


5/23/2012

50

Implementation: Grid & Materials(1) Choose initial grid resolution

Must resolve the minimum wavelength

0min max , 10

n x yN

N

x

Must resolve the minimum structural dimension

min 1d d

d

dN

N

Initial grid resolution is the smallest number computed above

min ,x y d

y


Implementation: Grid & Materials(2) “Snap” grid to critical dimensions

Decide what dimensions along each axis are critical

Compute how many grid cells comprise dc, and round UP

ceil

ceil

x x x

y y y

M d

M d

Adjust grid resolution to fit this dimension in grid EXACTLY

Typically this is a lattice constant or grating period along x Typically this is a film thickness along y

and x yd d

x x x

y y y

d M

d M

initi

al g

rid

critical dimension

adju

sted

grid

critical dimension


5/23/2012

51

Implementation: Grid & Materials(3) Compute total grid size

x

y

yN

xN

• Don’t forget to add cells for PML!

• Must often add “space” between PML and device.

PML space2 2

xx

x

yy

y

N

N N N

Problem Space

Buffer Space

Buffer Space

PML

PML

Note: This is particularly important when modeling devices with large evanescent fields.


Implementation: Grid & Materials(4) Compute 2X grid

1X Grid 2X Grid


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52

Implementation: Grid & Materials(5) Assign materials

Direct Averaged


Implementation: Grid & Materials(6) Extract materials onto 1X grid

2X Grid

x

yzE

zH

,x yH E

,y xH E

zz

zz

,xx yy

,yy xx

Field and materials assignments

zE

xHyH

xy

Ez Mode

xy

zH yExE

Hz Mode

I II

III IV

1X GridsI II

III IV


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Implementation: Grid & Materials(7) Oh yeah, metals!

Perfect Electric Conductors

10000r

1 1

00 0 1 0 0 m

M M

E f

E

E f

or

Include Tangential Fields at Boundary (TM modes!)

0mE

xy

zH yExE

Hz Mode

Bad placement of metals Good placement of metals


Implementation: TestingGuided-mode resonance filters

Both a diffraction grating and a waveguide

Incident

Reflected

Transmitted

1

2

134 nm

314 nm

1.0

1.52

2.0

2.1

0.5

L

L

T

n

n

n

n

f

2n

Ln Hn

1n

Guided-mode resonance filters are excellent devices for benchmarking and testing numerical codes because they are so sensitive to their geometry, refractive indices, and dispersion.

Hz Ez

S. Tibuleac R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14, 1617-1626 (1997)


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Implementation: TestingConvergence

Grid Resolution

Ans

wer

Probably “good enough”


Simulation Examples

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Simulation Examples:Guided-Mode Resonance Filter


Simulation Examples:Photonic Crystal


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Simulation Examples:Broadband Polarizer


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Documents

numerical methodsgolden

mshort course

numerical methodsbenefits

numerical error

fast scales

expr x yrg r short course

x bnumerical solution

spurious solutionsshort