Download - Janne Brok & Paul Urbach
Vermelding onderdeel organisatie
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Janne Brok & Paul Urbach
CASA day, Tuesday November 13, 2007
An analytic approach to electromagnetic scattering problems
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Currently: Consultant LIME
PhD Optics (2002 - 2007)
MA Ethics (2001 - 2002)
Applied Physics (1996 - 2001)
Short CV
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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
An analytic approach to electromagnetic scattering problems
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Infinitely thin perfectly conducting half plane (Sommerfeld, 1896)
Introduction Method Results Measurements
Pulse incident on perfectly conducting half plane
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• My thesis subject: finite thickness, perfect conductor, 3D, multiple pits or holes (finite or periodic).
Introduction Method Results Measurements
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
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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
Introduction Method Results Measurements
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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
Mode expansion techniqueDiffraction from layer with 3D rectangular holes
Introduction Method Results Measurements
Normalization
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z
x
y
LxLy
D
Mode expansion techniqueDiffraction from layer with 3D rectangular holes
Introduction Method Results Measurements
= (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
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Step 3: Match tangential fields at interfaces
Use Fourier operator…
And substitute
z
x
y
LxLy
D
Mode expansion techniqueDiffraction from layer with 3D rectangular holes
Introduction Method Results Measurements
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Mode expansion techniqueDiffraction from layer with 3D rectangular holes
Introduction Method Results Measurements
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
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Mode expansion techniqueDiffraction from layer with 3D rectangular holes
Introduction Method Results Measurements
Interaction integral
I a = hi + F a
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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
Mode expansion techniqueDiffraction from layer with 3D rectangular holes
Introduction Method Results Measurements
Small system of equations: 400 per hole
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Scattering from single, square holeIncident field: short pulse through thick layer
Introduction Method Results Measurements
quicktime movie
Field amplitude as a function of time (ps); above, inside & below hole
input pulse
above hole
below hole
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• 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
Introduction Method Results Measurements
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1 THz 300 μmMetals perfect conductors (f.i. copper = -3.4e4 - 6.6e5 i)
Comparison with THz measurements
Introduction Method Results Measurements
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• 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
Introduction Method Results Measurements
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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
Introduction Method Results Measurements
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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
Introduction Method Results Measurements
19Introduction Method Results Measurements
Near field of holesCalculated with mode expansion technique
Size hole: width = 0.2 mm, thickness = 0.08 mm
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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
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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
Introduction Method Results Measurements
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Thanks to …
An analytic approach to electromagnetic scattering problems
• Aurèle Adam• Paul Planken• Minah Seo (Seoul National University)
• Roland Horsten
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Comparison theory & experimentsFrequency spectrum at shadow side
Introduction Method Results Measurements
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Sphere (real metal or dielectric, any size) (Mie, 1908)
Ex, dominant polarization Ez
Pulse incident on perfectly conducting sphere
Introduction Method Results Measurements
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dipole orientation
dipole orientation
Spontaneous emissionIncident field: dipole near scattering structure
Introduction Method Results Measurements
dipole orientation
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Near field of holesCalculated with mode expansion technique
Introduction Method Results Measurements
Ex Ez
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Scattering from single, square holeIncident field: linearly polarized plane wave
Introduction Method Results Measurements
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
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metal: real() -
dielectricSurface plasmon perfectly conducting metal
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Dipole source near scattering structure• Coefficients for
waveguide modes
• Expression for scattered field