Janne Brok & Paul Urbach

Vermelding onderdeel organisatie

1

Janne Brok & Paul Urbach

CASA day, Tuesday November 13, 2007

An analytic approach to electromagnetic scattering problems

2

Currently: Consultant LIME

PhD Optics (2002 - 2007)

MA Ethics (2001 - 2002)

Applied Physics (1996 - 2001)

Short CV

3

Solving Maxwell’s equations for specific geometriesAnalytical solutions exist for:

• infinitely thin perfectly conducting half plane (Sommerfeld, 1896)• sphere (real metal or dielectric, any size) (Mie, 1908)• infinitely thin perfectly conducting disc (Bouwkamp, Meixner, 1950)• infinitely thin perfectly conducting plane with circular hole (idem)

Introduction Method Results Measurements


4

Infinitely thin perfectly conducting half plane (Sommerfeld, 1896)


Pulse incident on perfectly conducting half plane

5

• My thesis subject: finite thickness, perfect conductor, 3D, multiple pits or holes (finite or periodic).


Solving Maxwell’s equations for specific geometriesAnalytical solutions exist for:

• Sommerfeld half plane: infinitely thin, perfect conductor, 2D• Mie sphere: any diameter, real metal / dielectric, 3D • Bouwkamp disc: infinitely thin, perfect conductor, 3D• Bouwkamp hole: idem

6

Mode expansion techniqueDiffraction from layer with 3D rectangular holes

• Perfectly conducting layer, finite thickness• Finite number of rectangular holes• Incident field from infinity

Brok & Urbach, Optics Express, vol. 14, issue 7, pp. 2552 – 2572.

3) Matching at interfacesTypically 400 unknowns per hole per frequency

1) Inside holes: expansion in waveguide modes2) Above and below layer as: expansion in plane waves


7

Step 1: Linear superposition of waveguide modes = (1, 2, 3, 4)1: pit number2: polarization TE / TM3: mode mx, my

4: up / down

The discrete set of propagating and evanescent waveguide modes is complete: description of field inside pits/holes is rigorous

z

x

y

LxLy

D



Normalization

8

z

x

y

LxLy

D



= (1, 2)1: polarization S / P2: propagation direction (kx,ky)

Step 2: Linear superposition of plane waves

The continuous set of propagating and evanescent plane waves is complete: description of field inside pits/holes is rigorous

Normalization

9

Step 3: Match tangential fields at interfaces

Use Fourier operator…

And substitute

z

x

y

LxLy

D



10



Valid for all points (x,y) holes, z = ± D/2

Deriving a system of equations

Normalization

Valid for all waveguide modes

System of equations for coefficients of waveguide modes only: small system

Scattered field is calculated in forward way

11



Interaction integral

I a = hi + F a

12

0

0.01

0.02

0.03

0.04

0.05

0.06

0 500 1000 1500 2000

number of waveguide modes

rela

tive

err

or

in e

ner

gy



Small system of equations: 400 per hole

13

Scattering from single, square holeIncident field: short pulse through thick layer


quicktime movie

Field amplitude as a function of time (ps); above, inside & below hole

input pulse

above hole

below hole

14

• D = Lx = Ly = /4, linearly polarized light, from above• distance between holes is varied• two setups: two holes (A) and three holes (B)Normalized energy flux through a hole as a function of distance between the holes

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6Incident E perpendicular to line that connects centers of holes

distance (units of wavelengths) between centers of holes

no

rma

lize

d e

ne

rgy

flu

x

two holesthree holes

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.2

0.4

0.6

0.8

1

1.2

1.4

1.6Incident E parallel to line that connects centers of holes

distance (units of wavelengths) between centers of holes

no

rma

lize

d e

ne

rgy

flu

x

two holesthree holes

A

B

Scattering from multiple square holesIncident field: linearly polarized plane wave


15

1 THz 300 μmMetals perfect conductors (f.i. copper = -3.4e4 - 6.6e5 i)

Comparison with THz measurements


16

• Sample placed on top of electro-optic crystal

• Scattered THz field changes birefringence of crystal

• Birefringence changes polarization of optical probe beam

THz near field measurement setup


17

Planken & Van der Valk, Optics Letters, Vol. 29, No. 19, pp. 2306 – 2308.

Differential detector

• Polarization of optical probe beam proportional to THz field

• Orientation of crystal determines component of THz field: Ex, Ey or Ez

• Size of optical probe beam determines resolution

THz near field measurement setup


18

Metal layerThickness 80 μmSize square holes 200 μm

THz pulse

Ez

zy

x

polarization

THz near field measurement setupEz underneath metal layer with rectangular holes


19Introduction Method Results Measurements

Near field of holesCalculated with mode expansion technique

Size hole: width = 0.2 mm, thickness = 0.08 mm

20

0 200 400 600 800 1000 1200 14000

200

400

600

800

1000

1200

0 200 400 600 800 10000

200

400

600

800

1000

0 200 400 600 800 10000

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 12000

200

400

600

800

1000

1200

1400

1600

Experiment

z = 20 m below layer

-1 -0.5 0 0.5 1

x 10-3

-1

-0.5

0

0.5

1

x 10-3

2

4

6

8

10

12

x 105z = 20 m below layer

-1 -0.5 0 0.5 1

x 10-3

-1

-0.5

0

0.5

1

x 10-3

2

4

6

8

10

12

14x 10

5

z = 20 m below layer

-1 -0.5 0 0.5 1

x 10-3

-1

-0.5

0

0.5

1

x 10-3

2

4

6

8

10

12

14

x 105z = 20 m below layer

-1 -0.5 0 0.5 1

x 10-3

-1

-0.5

0

0.5

1

x 10-3

1

2

3

4

5

6

x 105

Calculation single frequency: 1.0 THz (300 m)

Comparison theory & experimentsTop view: (x,y)-plane, Ez underneath metal layer with multiple square holes


21

Thanks to …


• Aurèle Adam• Paul Planken• Minah Seo (Seoul National University)

• Roland Horsten

22

23

Comparison theory & experimentsFrequency spectrum at shadow side


24

Sphere (real metal or dielectric, any size) (Mie, 1908)

Ex, dominant polarization Ez

Pulse incident on perfectly conducting sphere


25

dipole orientation

dipole orientation

Spontaneous emissionIncident field: dipole near scattering structure


dipole orientation

26

Near field of holesCalculated with mode expansion technique


Ex Ez

27

Scattering from single, square holeIncident field: linearly polarized plane wave


Energy flux through hole, normalized by energy incident on hole area

0.5 1 1.5 20

0.5

1

1.5

thickness of layer D ()

Normalized energy flux through solitary hole

Lx = L

y = 0.40

Lx = L

y = 0.46

Lx = L

y = 0.50

0

0 2

0.5 1 1.5 20

0.5

1

1.5

thickness of layer D ()

Normalized energy flux through solitary hole

Lx = L

y = 0.50

Lx = L

y = 0.52

Lx = L

y = 0.54

2

2

2

28

metal: real() -

dielectricSurface plasmon perfectly conducting metal

29

Dipole source near scattering structure• Coefficients for

waveguide modes

• Expression for scattered field

Documents

Janne Brok & Paul Urbach