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Finite Element Method for General Three-Dimensional

This research focuses on developing a general program for solving electromagnetic scattering problems in optics using the Finite Element Method. The study includes applications such as optical recording, plasmonics, lithography alignment, etc., considering various configurations and sources.

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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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