Transcript Document

Finite Element Method for General Three-Dimensional
Time-harmonic Electromagnetic Problems of Optics
Paul Urbach
Philips Research
Simulations
For an incident plane wave with k = (kx, ky, 0) one can distinguish
two linear polarizations:
y
TM
TE
Ez
Hz
x
z
• TE: E = (0, 0, Ez)
• TM: H = (0, 0, Hz)
Aluminum grooves: n = 0.28 + 4.1 i
|Ez| inside the unit cell for a normally incident, TE polarized plane
wave.
p = 740 nm, w = 200 nm, 50 < d < 500 nm.
(Effective) Wavelength = 433 nm
Total near field – TM
|Hz| inside the unit cell for a normally incident, TM polarized plane
wave.
p = 740 nm, w = 200 nm, 50 < d < 500 nm.
Total near field – pit width
TE polarization
w = 180 nm
TM polarization
w = 180 nm
w = 370 nm
w = 370 nm
d=
800
nm
TE: standing wave pattern inside pit is depends strongly on w.
TM: hardly any influence of pit width.
Waveguide theory in which the finite conductivity of aluminum is taken
into account explains this difference well.
A. Sommerfeld 1868-1951
Motivation
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In modern optics, there are often very small
structures of the size of the order of the
wavelength.
We intend to make a general program for
electromagnetic scattering problems in optics.
Examples
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Optical recording.
Plasmon at a metallic bi-grating
Alignment problem for lithography for IC.
etc.
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Configurations
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2D or 3D
Non-periodic structure
(Isolated pit in
multilayer)
Periodic in one direction
(row of pits)
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Periodic in two directions
(bi-gratings)
Periodic in three directions
(3D crystals)
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Sources
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Sources outside the scatterers:
Incident field , e.g.:
 plane wave,
 focused spot,
 etc.
Sources inside scatterers:
 Imposed current density.
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Materials
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Linear.
In general anisotropic,
(absorbing)
dielectrics and/or conductors:
Magnetic anisotropic materials
(for completeness):
  xx  xy

 r    xy  yy

 xz  yz
 xz 

 yz .
 zz 
  xx

r    xy

 xz
 xz 

 yz .
 zz 
 xy
 yy
 yz
 r ( r ),  r ( r ).
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Boundary condition on :
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Either periodic for periodic structures
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Or: surface integral equations on the boundary
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Kernel of the integral equations is the highly
singular Green’s tensor. (Very difficult to
implement!)
Full matrix block.
Example (non-periodic structure in 3D):
Total field is
computed in 
Scattered field is
computed in PML
Note: PML is an approximation, but it seems to be a
very good approximation in practice.
Nédèlec elements
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Mesh: tetrahedron (3D) or triangle
(2D)
Er    a  r 
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For each edge , there is a linear
vector function (r).
Unknown a is tangential field
component along edge  of the mesh
Tangential components are always
continuous
Nédèlec elements can be
generalised without problem to the
modified vector Helmholtz
equation*
Research subjects:
• Higher order elements
• Hexahedral meshes and mixed formulation (Cohen’s method)
• Iterative Solver