finite element method for general three-dimensional time-harmonic electromagnetic problems of optics...
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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:
• TE: E = (0, 0, Ez) • TM: H = (0, 0, Hz)
TE TM
x
y
zEz 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: 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.
TE polarization TM polarization
w = 180 nm w = 370 nmw = 370 nm w = 180 nm
d = 800 nm
A. Sommerfeld 1868-1951
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.
Configurations 2D or 3D
Non-periodic structure (Isolated pit in
multilayer)
Periodic in one direction
(row of pits)
Periodic in two directions (bi-gratings)
Periodic in three directions (3D crystals)
Sources Sources outside the scatterers:
Incident field , e.g.: plane wave, focused spot, etc.
Sources inside scatterers: Imposed current density.
Materials
Linear.
In general anisotropic, (absorbing) dielectrics and/or conductors:
Magnetic anisotropic materials (for completeness):
Materials could be inhomogeneous:
.
zzyzxz
yzyyxy
xzxyxx
r
.
zzyzxz
yzyyxy
xzxyxx
r
).(),( rr rr
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.
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 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*
rar
E
• Higher order elements• Hexahedral meshes and mixed formulation (Cohen’s method)• Iterative Solver
Research subjects: