Pseudospectral Methods

Pseudospectral Methods

Sahar SargheiniLaboratory of Electromagnetic Fields and Microwave Electronics (IFH)

ETHZ

1

7th Workshop on Numerical Methods for Optical Nano Structures, ETH Zurich, July 4-6, 2011


2

Numerical methods

for solving PDEs

Approximate the

differential operatore

Approximate the

solution

Finite difference Spectral methods

)()(L xfxu

fuaaaxR N L),...,,,( 21

0),...,,()(U

1 dxaaxRxw Ni

N

nnnaxu

1

)(

3


Weighted residues

Galerkin method Least square

Pseudospectral

or

Collocation

or

method of selected pointsiiw

)( ii xxw

iiw L

Pseudospectral Methods Finite Elements Method:

4

0x 1x 1Nx2Nx2x Nx. . .

ixxdx

du

• Finite Difference Method:

0x 1x 1Nx2Nx2x Nx. . .

ixxdx

du

1 ixxdx

du

1 ixxdx

du

• Pseudospectral Methods

0x 1x 1Nxnxmx Nx. . .

ixxdx

du

. . ... .

Domain 1 Domain 2 Domain 3

N point

method


Pseudospectral methods Created by Kreiss and Oliger in 1972. Were first introduced to the electromagnetic community around

1996 by Liu.

5

Error )( NhO ])/1[( NNONh /1

Infinite order / Exponential convergence

Memory usage and

time consumption

will be reduced

significantly


Basis functions

6

Periodic functions

Trigonometric

Non periodic functions

Chebyshev or Legendre

Semi-Infinite functions

Laguerre

Infinite functions

Hermite

7

Fourier PSFD

Fourier PSFD

8

• Liu extended the pseudospectral methods to the frequency domain (2002).

• All proposed PSFD methods used Chebyshev basis functions.

• However for periodic structures, trigonometric basis functions will be much more suitable. In addition,using trigonometric functions, we can benefit from characteristics of Fourier series, and that is why we call this method Fourier PSFD

• Conventional single-domain PSFD methods suffer from staircasing error.

• This error will not be reduced unless the number of discretization points increases.

• To overcome this difficulty in a multidomain method, curved geometries should

be divided into several subdomains whereas this method is complicated and time

consuming to some extend.• We used a new technique to overcome the staircasing error in a single-domain

PSFD method.

• We formulate the constitutive relations with the help of a convolution in the spatial

frequency domain.

Fourier PSFD

Constitutive relation

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ED r 0

• C-PSFD method

• Conventional PSFD method

),( nmz yxe

),( nmr yx

1 2 m. . . . . .m+1m-10

1

2

.

n

n-1

n+1

.

.

.

.

.

• Conventional PSFD method

ed FCF 10

Bloch-Floquet:

rkjerhrH )()(

rkjererE )()(Periodic functions

• C-PSFD method

)(rr

),( nmz yxe

1 2 m m+1m-1. . . . . .

1

2

n

n-1

n+1

.

.

.

.

.

.

0

Fourier PSFD

Photonic crystals

ar 08.0 ar 4.0

10

ar 4.01

1r

9.82r

TMz mods

C-PSFD: 6×6

Conventional PSFD: 6×6

Rel

ativ

e er

ror

of n

orm

aliz

ed f

requ

ency

(%)

Number of discrete points in one direction (N)

C-PSFDConventional PSFD

5 10 15 20 25 30 350

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Rel

ativ

e er

ror

of n

orm

aliz

ed f

requ

ency

(%)


5 10 15 20 25 30 350

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Rel

ativ

e er

ror

of n

orm

aliz

ed f

requ

ency

(%)


1002r 9.8

2r

TEz mods

r X M r0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Nor

mal

ized

fre

quen

cy

Wave vector

PWEC-PSFD

C-PSFD: 10×10

5 10 15 20 25 30 350

2

4

6

8

10

12

14

16

18

Rel

ativ

e er

ror

of n

orm

aliz

ed f

requ

ency

(%)


Error: Second band at

the M point of the first Brillouin zone

Fourier PSFD

11

• Photonic crystals4/1 ax

4/1 ay 5/1 ar

4/2 ax 6/2 ay

7/2 ar

11r

9.82r

TMz modes

C-PSFDConventional PSFD

Rel

ativ

e er

ror

of n

orm

aliz

ed f

requ

ency

(%)


Error: Second band at

the M point of the first Brillouin zone0

0.5

1

0 0

a/a/

a/ a/ xkyk

Nor

mal

ized

fre

quen

cy

C-PSFD: 8×8

Fourier PSFD

12

• Photonic crystals

a0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Wave vectorXr M r

Nor

mal

ized

Fre

quen

cy

TMz modes

C-PSFD: 12×12

5 10 15 20 25 30 350

5

10

15

20

25


Rel

ativ

e er

ror

of n

orm

aliz

ed f

requ

ency

(%

)

Error: seventeenth band atak x / ak y /2.0

11r

4.112r

Fourier PSFD

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• Left-handed binary grating

iE

d

L

1d

2d

Left handed material

),( 11

),( 22

y

z

1

2

1R

2R

3R

),( 11

),( 22

n

zynr zjkynL

kjCxzyEn

))2

(exp(ˆ),(

n

zynt zjkynL

kjDxzyEn

))2

(exp(ˆ),(

n

n

zjn

zjnnp

nn eBeAyExzyE )(ˆ),(

L

),( 11 ),( 22

z

y0 d )1,1(,

11rr

)2,2(,22

rr 2/Ld

0yk

-1 -0.5 0 0.5 1-25

-20

-15

-10

-5

0

5

10

15

20

25

)/(Real 0k

3.1L

0 d L-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

y

Am

plit

ude

6322.00546.1/ 0 jk

Fourier PSFD

4.1L

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• Left-handed binary grating

y

z

)1,1(,11

rr

)2,2(,22

rr

2/Ld

0yk

Ld 1

0 0 d1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

z

Am

plit

ude

9345.0y

Thank you for your attention

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Pseudospectral Methods

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