Proceedings of Topical Meeting on Optoinformatics (pdf-format, 1.21 ...

54 OPTOINFORMATICS’05
MODELLING RIGOROUS DIFFRACTION FROM 3D
SUB-WAVELENGTH STRUCTURES
J.M. Brok a & H.P. Urbach a,b
a
Delft University <strong>of</strong> Technology, PO Box 5046, 2600 GA Delft, The Netherlands
b
Philips Research Laboratories, Pr<strong>of</strong>essor Holstlaan 4, 5656 AA Eindhoven, The
Netherlands
E-mail: j.m.brok@tnw.tudelft.nl
We present a rigorous method to calculate the electromagnetic field that is
scattered from a perfectly conducting layer with finite thickness, containing
multiple, rectangular, 3D pits and holes. Plasmon effects and polarisation
phenomena are shown.
In the rigorous modelling <strong>of</strong> diffraction from 3D sub-wavelength metallic structures, the
methods that are <strong>of</strong>ten used are based on meshing the region <strong>of</strong> interest and applying the
finite-difference time-domain or the finite-element method. When calculating a (large) 3D
volume, these methods are computationally (very) costly. Therefore, the modelled
structures usually consist either <strong>of</strong> only a single scatterer (such as a pit or hole) or else <strong>of</strong> a
periodic (2D) array <strong>of</strong> identical scattering objects. However, when we want to understand
for example the influence <strong>of</strong> neighbouring pits in optical recording or the extraordinary
transmission <strong>of</strong> light through sub-wavelength holes [1] , it is important to study the mutual
interaction between two or more scatterers at varying distances. When the scatterers are
rectangular holes or pits in a very good conductor, we describe a mode expansion
technique that is a very efficient alternative to the numerical techniques mentioned above.
Consider a perfectly conducting
metallic layer <strong>of</strong> thickness D,
with rectangular pits and holes.
The materials above and below
the layer, as well as inside the
pits and holes, are homogeneous
dielectrics. The incident field can
be a simple plain wave or a
complicated spot. The field
z
above and below the layer is written as an integral over a continuum <strong>of</strong> propagating and
evanescent plane waves. The pits and holes can be considered finite, metallic waveguides.
The field inside such waveguides can be written as a sum over propagating and evanescent
waveguide modes <strong>of</strong> the infinitely long waveguide with the same cross-section. The
tangential components <strong>of</strong> these two expansions are matched at the upper and lower
surfaces <strong>of</strong> the layer. With this mode expansion technique [2,3] , a 3D diffraction problem is
turned into a 2D numerical problem. It turns out that the coefficients for the plane waves
can be eliminated from the system <strong>of</strong> equations and hence, we end up with a fairly small
system for only the coefficients <strong>of</strong> the waveguide modes. Solving this system is a matter <strong>of</strong>
seconds on a modern desktop computer.
y
x
Fig. 1 Problem definition. Multiple rectangular pits (with depth
smaller than D) or holes are modelled.
D
Lx
Ly