Finite Element Method for General Three-Dimensional

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Finite Element Method for General
Three-Dimensional Time-harmonic Electromagnetic
Problems of Optics
Paul Urbach Philips Research
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Simulations
For an incident plane wave with k (kx, ky, 0)
one can distinguish two linear polarizations
TE E (0, 0, Ez) TM H (0, 0, Hz)
y
TM
TE
Hz
Ez
x
z
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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 lt d lt 500 nm.
(Effective) Wavelength 433 nm
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Total near field TM
Hz inside the unit cell for a normally
incident, TM polarized plane wave.
p 740 nm, w 200 nm, 50 lt d lt 500 nm.
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Total near field pit width
TE polarization
TM polarization
w 180 nm
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.
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A. Sommerfeld 1868-1951
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Motivation

• 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
• Optical recording.
• Plasmon at a metallic bi-grating
• Alignment problem for lithography for IC.
• etc.

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• Configurations
• 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
• Sources outside the scatterers
• Incident field , e.g.
• plane wave,
• focused spot,
• etc.
• Sources inside scatterers
• Imposed current density.

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• Materials
• Linear.
• In general anisotropic, (absorbing)
• dielectrics and/or conductors
• Magnetic anisotropic materials
• (for completeness)
• Materials could be inhomogeneous

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• Boundary condition on ??
• Either periodic for periodic structures
• Or surface integral equations on the boundary
• Kernel of the integral equations is the highly
  singular Greens tensor. (Very difficult to
  implement!)
• Full matrix block.

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• 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.
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Nédèlec elements

• Mesh tetrahedron (3D) or triangle (2D)
• 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

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Research subjects

• Higher order elements
• Hexahedral meshes and mixed formulation
  (Cohens method)
• Iterative Solver