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An analytic approach
to electromagnetic
scattering problems
Janne Brok & Paul Urbach
CASA day, Tuesday November 13, 2007
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Vermelding onderdeel organisatie
Short CV
Applied Physics (1996 - 2001)
MA Ethics (2001 - 2002)
PhD Optics (2002 - 2007)
Currently: Consultant LIME
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An analytic approach to electromagnetic
scattering problems
Solving Maxwell’s equations for specific geometries
Analytical 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
3
Infinitely thin perfectly conducting half plane (Sommerfeld, 1896)
Pulse incident on perfectly conducting half plane
Introduction
Method
Results
Measurements
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Solving Maxwell’s equations for specific geometries
Analytical solutions exist for:
• Sommerfeld half plane:
• Mie sphere:
• Bouwkamp disc:
• Bouwkamp hole:
infinitely thin, perfect conductor, 2D
any diameter, real metal / dielectric, 3D
infinitely thin, perfect conductor, 3D
idem
• My thesis subject:
finite thickness, perfect conductor, 3D,
multiple pits or holes (finite or periodic).
Introduction
Method
Results
Measurements
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Mode expansion technique
Diffraction from layer with 3D rectangular holes
• Perfectly conducting layer, finite thickness
• Finite number of rectangular holes
• Incident field from infinity
1) Inside holes: expansion in waveguide modes
2) Above and below layer as: expansion in plane waves
3) Matching at interfaces
Typically 400 unknowns per hole per frequency
Brok & Urbach, Optics Express, vol. 14, issue 7, pp. 2552 – 2572.
Introduction
Method
Results
Measurements
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Mode expansion technique
Diffraction from layer with 3D rectangular holes
Step 1: Linear superposition of waveguide modes
 = (1, 2, 3, 4)
1: pit number
2: polarization TE / TM
3: mode mx, my
4: up / down
Normalization
The discrete set of propagating and evanescent waveguide modes is
complete: description of field inside pits/holes is rigorous
Introduction
Method
Results
Measurements
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Mode expansion technique
Diffraction from layer with 3D rectangular holes
Step 2: Linear superposition of plane waves
 = (1, 2)
1: polarization S / P
2: propagation direction (kx,ky)
Normalization
The continuous set of propagating and evanescent plane waves is
complete: description of field inside pits/holes is rigorous
Introduction
Method
Results
Measurements
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Mode expansion technique
Diffraction from layer with 3D rectangular holes
Step 3: Match tangential fields at interfaces
Use Fourier operator…
And substitute
Introduction
Method
Results
Measurements
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Mode expansion technique
Diffraction from layer with 3D rectangular holes
Deriving a system of equations
Valid for all points (x,y)  holes, z = ± D/2
Normalization
Valid for all waveguide modes 
System of equations for coefficients of waveguide modes only: small system
Scattered field is calculated in forward way
Introduction
Method
Results
Measurements
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Mode expansion technique
Diffraction from layer with 3D rectangular holes
 I a = hi + F a
Interaction integral
Introduction
Method
Results
Measurements
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Mode expansion technique
Diffraction from layer with 3D rectangular holes
Small system of equations:  400 per hole
relative error in energy
0.06
0.05
0.04
0.03
0.02
0.01
0
0
500
1000
1500
2000
number of waveguide modes
Introduction
Method
Results
Measurements
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Scattering from single, square hole
Incident field: short pulse through thick layer
quicktime
movie
input pulse
Field amplitude as a function of time (ps); above, inside & below hole
above hole
below hole
Introduction
Method
Results
Measurements
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Scattering from multiple square holes
A
Incident field: linearly polarized plane wave
• D = Lx = Ly = /4, linearly polarized light, from above
B
• 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
Incident E perpendicular to line that connects centers of holes
Incident E parallel to line that connects centers of holes
2.6
1.6
two holes
three holes
2.4
two holes
three holes
1.4
normalized energy flux
normalized energy flux
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
1.2
0.4
1
0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0.2
0.2
distance (units of wavelengths) between centers of holes
Introduction
Method
Results
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
distance (units of wavelengths) between centers of holes
Measurements
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Comparison with THz measurements
Electronics
THz
visible
microwaves
10
0
10
3
kilo
6
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Photonics
12
15
x-ray
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-ray
21
10
10
10
10
10
10
mega
giga
tera
peta
exa
zetta
1 THz  300 μm
Metals  perfect conductors
(f.i. copper = -3.4e4 - 6.6e5 i)
Introduction
Method
Results
Measurements
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THz near field measurement setup
• Sample placed on top
of electro-optic crystal
• Scattered THz field changes
birefringence of crystal
• Birefringence changes
polarization of optical probe
beam
Introduction
Method
Results
Measurements
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THz near field measurement setup
• 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
Differential
detector
Planken & Van der Valk, Optics Letters, Vol. 29, No. 19, pp. 2306 – 2308.
Introduction
Method
Results
Measurements
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THz near field measurement setup
Ez underneath metal layer with rectangular holes
polarization
z
y
Ez
THz pulse
Introduction
Method
x
Metal layer
Thickness 80 μm
Size square holes 200 μm
Results
Measurements
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Near field of holes
Calculated with mode expansion technique
Size hole: width = 0.2 mm, thickness = 0.08 mm
Introduction
Method
Results
Measurements
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Comparison theory & experiments
Top view: (x,y)-plane, Ez underneath metal layer with
multiple square holes
Experiment
1600
1400
1200
1000
1400
1200
1000
1200
800
1000
800
1000
600
800
600
800
400
600
600
400
400
200
400
200
200
0
-3
0
200
x 10
400
600
800
1000
0
z = 20 m below layerx 10
-3
z = 20 m below
x 10 layer
0
200
400
600
5
800
1000
1200
-3
x 10
12
1
200
5
1400
x0 10
400
600
800
1000
140 200
z = 20 m below layer
0
-3
0
x 10
600
800
z200
= 20x400
10
m5 below
layer1000
5
1200
x 10
14
12
6
1
10
0.5
0.5
1
8
8
4
0.5
8
6
0
5
10
0.5
0
12
1
10
6
3
0
0
4
6
4
2
-0.5
-0.5
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Calculation
2
-1
-0.5
0
0.5
-1
1
2
-0.5
single frequency: 1.0 THz (300 m)
-0.5
0
-3
x 10
0.5
1
-1 -3
x 10-1
-1
-0.5
0
0.5
-11
-0.5
0
0.5
-3
x 10
Introduction
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2
-1
-1
-0.5
Method
Results
Measurements
1
-3
x 10
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An analytic approach
to electromagnetic
scattering problems
Thanks to …
• Aurèle Adam
• Paul Planken
• Minah Seo
(Seoul National University)
• Roland Horsten
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Comparison theory & experiments
Frequency spectrum at shadow side
Introduction
Method
Results
Measurements
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Sphere (real metal or dielectric, any size) (Mie, 1908)
Pulse incident on perfectly conducting sphere
Ex, dominant polarization
Introduction
Method
Results
Ez
Measurements
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Spontaneous emission
Incident field: dipole near scattering structure
dipole orientation
dipole orientation
dipole orientation
Introduction
Method
Results
Measurements
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Near field of holes
Calculated with mode expansion technique
Ex
Introduction
Method
Ez
Results
Measurements
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Scattering from single, square hole
Incident field: linearly polarized plane wave
Energy flux through hole, normalized by energy incident on hole area
Normalized energy flux through solitary hole
Normalized energy flux through solitary hole
1.5
1.5
Lx = Ly = 0.40 
2
Lx = Ly = 0.46 
Lx = Ly = 0.50 
1
0
0.5
2
0.5
Lx = Ly = 0.50 
Lx = Ly = 0.52 
0
0
2
1
2
Lx = Ly = 0.54 
0.5
Introduction
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1.5
thickness of layer D ()
Method
0
2
Results
0.5
1
1.5
thickness of layer D ()
Measurements
2
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Surface plasmon  perfectly conducting metal
dielectric
metal: real()  -
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Dipole source near scattering structure
• Coefficients for
waveguide modes
• Expression for
scattered field
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