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Finite Element Method for General Three-Dimensional

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

  2. 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

  3. 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

  4. 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.

  5. 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.

  6. A. Sommerfeld 1868-1951

  7. 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.

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

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

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

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

  12. 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.

  13. 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.

  14. 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*

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

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