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

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

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54 OPTOINFORMATICS’05 MODELLING RIGOROUS DIFFRACTION FROM 3D SUB-WAVELENGTH STRUCTURES J.M. Brok a & H.P. Urbach a,b a Delft University ong>ofong> Technology, PO Box 5046, 2600 GA Delft, The Netherlands b Philips Research Laboratories, Prong>ofong>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 ong>ofong> diffraction from 3D sub-wavelength metallic structures, the methods that are ong>ofong>ten used are based on meshing the region ong>ofong> 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 ong>ofong> only a single scatterer (such as a pit or hole) or else ong>ofong> a periodic (2D) array ong>ofong> identical scattering objects. However, when we want to understand for example the influence ong>ofong> neighbouring pits in optical recording or the extraordinary transmission ong>ofong> 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 ong>ofong> 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 ong>ofong> 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 ong>ofong> the infinitely long waveguide with the same cross-section. The tangential components ong>ofong> these two expansions are matched at the upper and lower surfaces ong>ofong> 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 ong>ofong> equations and hence, we end up with a fairly small system for only the coefficients ong>ofong> the waveguide modes. Solving this system is a matter ong>ofong> 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

SAINT-PETERSBURG, October 17 – 20, 2005 55 First setup Incident E perpendicular to line that connects centers ong>ofong> holes 2.6 two holes 2.4 three holes Second Figure 2 shows some results for two setups: in the first we have two identical holes and in the second we have three identical holes, both with their centers aligned. The holes are cubic: L x = L y = D = λ/4, with λ the wavelength ong>ofong> the incident light. The distance between the centers ong>ofong> the holes is varied. We show the energy flux through one hole (the center hole for the second case), normalised by the energy flux through an identical, solitary hole. Note that this energy need not reach the far field, it may scatter into evanescent contributions below the layer. In the top figure, the incident field is a perpendicular incident, S-polarised plane wave. In the lower figure, the incident field is P-polarised. Enhancement ong>ofong> the energy flux occurs only for holes that are very near for the S-polarised incident field, whereas for P-polarisation, both enhancement and attenuation occur for larger distances between 0.8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 distance (units ong>ofong> wavelengths) between centers ong>ofong> holes Fig. 2. Normalised energy flux through a hole as a function ong>ofong> distance between the holes. Solid line for the case ong>ofong> two identical holes, dotted line for three identical holes. the holes. This points to the contribution ong>ofong> plasmon effects in the latter case, whereas for S-polarisation enhancement is mainly due to evanescent fields. This research was supported by the Dutch Technology Foundation STW. normalized energy flux normalized energy flux 2.2 1.8 1.6 1.4 1.2 1. T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Extraordinary optical transmission through sub-wavelength hole arrays, Nature 391, 667-669 (1998). 2. A. Roberts, Electromagnetic theory ong>ofong> diffraction by a circular aperture in a thick, perfectly conducting screen, J. Opt. Soc. Am. A, Vol. 4, No. 10, 1970-1983, (1987). 3. J.M. Brok and H.P. Urbach, A mode expansion technique for rigorously calculating the scattering from 3D structures in optical recording, J. Mod. Opt., Vol. 51, No 14, 2059- 2077 (2004). 2 1 1.6 1.4 1.2 1 0.8 0.6 0.4 Incident E parallel to line that connects centers ong>ofong> holes two holes three holes 0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 distance (units ong>ofong> wavelengths) between centers ong>ofong> holes

SAINT-PETERSBURG, October 17 – 20, 2005 55<br />

First setup<br />

Incident E perpendicular to line that c<strong>on</strong>nects centers <str<strong>on</strong>g>of</str<strong>on</strong>g> holes<br />

2.6<br />

two holes<br />

2.4<br />

three holes<br />

Sec<strong>on</strong>d<br />

Figure 2 shows some results for two<br />

setups: in the first we have two<br />

identical holes and in the sec<strong>on</strong>d we<br />

have three identical holes, both with<br />

their centers aligned. The holes are<br />

cubic: L x = L y = D = λ/4, with λ the<br />

wavelength <str<strong>on</strong>g>of</str<strong>on</strong>g> the incident light. The<br />

distance between the centers <str<strong>on</strong>g>of</str<strong>on</strong>g> the<br />

holes is varied. We show the energy<br />

flux through <strong>on</strong>e hole (the center<br />

hole for the sec<strong>on</strong>d case), normalised<br />

by the energy flux through an<br />

identical, solitary hole. Note that this<br />

energy need not reach the far field, it<br />

may scatter into evanescent<br />

c<strong>on</strong>tributi<strong>on</strong>s below the layer. In the<br />

top figure, the incident field is a<br />

perpendicular incident, S-polarised<br />

plane wave. In the lower figure, the<br />

incident field is P-polarised.<br />

Enhancement <str<strong>on</strong>g>of</str<strong>on</strong>g> the energy flux<br />

occurs <strong>on</strong>ly for holes that are very<br />

near for the S-polarised incident<br />

field, whereas for P-polarisati<strong>on</strong>,<br />

both enhancement and attenuati<strong>on</strong><br />

occur for larger distances between<br />

0.8<br />

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />

distance (units <str<strong>on</strong>g>of</str<strong>on</strong>g> wavelengths) between centers <str<strong>on</strong>g>of</str<strong>on</strong>g> holes<br />

Fig. 2. Normalised energy flux through a hole as a functi<strong>on</strong><br />

<str<strong>on</strong>g>of</str<strong>on</strong>g> distance between the holes. Solid line for the case <str<strong>on</strong>g>of</str<strong>on</strong>g><br />

two identical holes, dotted line for three identical holes.<br />

the holes. This points to the c<strong>on</strong>tributi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> plasm<strong>on</strong> effects in the latter case, whereas for<br />

S-polarisati<strong>on</strong> enhancement is mainly due to evanescent fields.<br />

This research was supported by the Dutch Technology Foundati<strong>on</strong> STW.<br />

normalized energy flux<br />

normalized energy flux<br />

2.2<br />

1.8<br />

1.6<br />

1.4<br />

1.2<br />

1. T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio and P.A. Wolff, Extraordinary<br />

optical transmissi<strong>on</strong> through sub-wavelength hole arrays, Nature 391, 667-669<br />

(1998).<br />

2. A. Roberts, Electromagnetic theory <str<strong>on</strong>g>of</str<strong>on</strong>g> diffracti<strong>on</strong> by a circular aperture in a thick,<br />

perfectly c<strong>on</strong>ducting screen, J. Opt. Soc. Am. A, Vol. 4, No. 10, 1970-1983, (1987).<br />

3. J.M. Brok and H.P. Urbach, A mode expansi<strong>on</strong> technique for rigorously calculating the<br />

scattering from 3D structures in optical recording, J. Mod. Opt., Vol. 51, No 14, 2059-<br />

2077 (2004).<br />

2<br />

1<br />

1.6<br />

1.4<br />

1.2<br />

1<br />

0.8<br />

0.6<br />

0.4<br />

Incident E parallel to line that c<strong>on</strong>nects centers <str<strong>on</strong>g>of</str<strong>on</strong>g> holes<br />

two holes<br />

three holes<br />

0.2<br />

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2<br />

distance (units <str<strong>on</strong>g>of</str<strong>on</strong>g> wavelengths) between centers <str<strong>on</strong>g>of</str<strong>on</strong>g> holes

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