finite element method for general three-dimensional time-harmonic electromagnetic problems of optics...

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: