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) y TE TM Ez Hz x z

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 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.

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:

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*

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