1. Finite Element Method for General Three-Dimensional
Time-harmonic Electromagnetic Problems of Optics
Paul Urbach
Philips Research

11. 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 TE TM Ez Hz x z

12. 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

13. 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.

14. Total near field – pit width
TE polarization
TM polarization
w = 180 nm
w = 370 nm
w = 180 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.

15. A. Sommerfeld

16. 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.

17. Configurations 2D or 3D Non-periodic structure
(Isolated pit in multilayer)
Periodic in one direction
(row of pits)

18. Periodic in two directions
(bi-gratings)
Periodic in three directions
(3D crystals)

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

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

24. Boundary condition on :
Either periodic for periodic structures
Or: surface integral equations on the boundary
Kernel of the integral equations is the highly singular Green’s tensor. (Very difficult to implement!)
Full matrix block.

29. 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.

32. 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*

33. Research subjects:
Higher order elements
Hexahedral meshes and mixed formulation (Cohen’s method)
Iterative Solver