Vermelding onderdeel organisatie 1 Janne Brok & Paul Urbach CASA day, Tuesday November 13, 2007 An analytic approach to electromagnetic scattering problems

2 Currently: Consultant LIME PhD Optics ( ) MA Ethics ( ) Applied Physics ( ) Short CV

3 Solving Maxwell’s equations for specific geometries Analytical solutions exist for: infinitely thin perfectly conducting half plane (Sommerfeld, 1896) sphere (real metal or dielectric, any size) (Mie, 1908) infinitely thin perfectly conducting disc (Bouwkamp, Meixner, 1950) infinitely thin perfectly conducting plane with circular hole (idem) Introduction MethodResults Measurements An analytic approach to electromagnetic scattering problems

4 Infinitely thin perfectly conducting half plane (Sommerfeld, 1896) Introduction MethodResults Measurements Pulse incident on perfectly conducting half plane

5 My thesis subject: finite thickness, perfect conductor, 3D, multiple pits or holes (finite or periodic). Introduction MethodResults Measurements Solving Maxwell’s equations for specific geometries Analytical solutions exist for: Sommerfeld half plane: infinitely thin, perfect conductor, 2D Mie sphere:any diameter, real metal / dielectric, 3D Bouwkamp disc:infinitely thin, perfect conductor, 3D Bouwkamp hole:idem

6 Mode expansion technique Diffraction from layer with 3D rectangular holes Perfectly conducting layer, finite thickness Finite number of rectangular holes Incident field from infinity Brok & Urbach, Optics Express, vol. 14, issue 7, pp – ) Matching at interfaces Typically 400 unknowns per hole per frequency 1) Inside holes: expansion in waveguide modes 2) Above and below layer as: expansion in plane waves Introduction MethodResults Measurements

7 Step 1: Linear superposition of waveguide modes  = (  1,  2,  3,  4 )  1 : pit number  2 : polarization TE / TM  3 : mode m x, m y  4 : up / down The discrete set of propagating and evanescent waveguide modes is complete: description of field inside pits/holes is rigorous Mode expansion technique Diffraction from layer with 3D rectangular holes Introduction MethodResults Measurements Normalization

8 Mode expansion technique Diffraction from layer with 3D rectangular holes Introduction MethodResults Measurements  = (  1,  2 )  1 : polarization S / P  2 : propagation direction (k x,k y ) Step 2: Linear superposition of plane waves The continuous set of propagating and evanescent plane waves is complete: description of field inside pits/holes is rigorous Normalization

9 Step 3: Match tangential fields at interfaces Use Fourier operator… And substitute Mode expansion technique Diffraction from layer with 3D rectangular holes Introduction MethodResults Measurements

10 Mode expansion technique Diffraction from layer with 3D rectangular holes Introduction MethodResults Measurements Valid for all points (x,y)  holes, z = ± D/2 Deriving a system of equations Normalization Valid for all waveguide modes  System of equations for coefficients of waveguide modes only: small system Scattered field is calculated in forward way

11 Mode expansion technique Diffraction from layer with 3D rectangular holes Introduction MethodResults Measurements Interaction integral  I a = h i + F a

12 Mode expansion technique Diffraction from layer with 3D rectangular holes Introduction MethodResults Measurements Small system of equations:  400 per hole

13 Scattering from single, square hole Incident field: short pulse through thick layer Introduction MethodResults Measurements quicktime movie Field amplitude as a function of time (ps); above, inside & below hole input pulse above hole below hole

14 D = L x = L y = /4, linearly polarized light, from above distance between holes is varied two setups: two holes (A) and three holes (B) Normalized energy flux through a hole as a function of distance between the holes A B Scattering from multiple square holes Incident field: linearly polarized plane wave Introduction MethodResults Measurements

15 1 THz  300 μ m Metals  perfect conductors (f.i.  copper = -3.4e e5 i) Comparison with THz measurements Introduction MethodResults Measurements

16 Sample placed on top of electro-optic crystal Scattered THz field changes birefringence of crystal Birefringence changes polarization of optical probe beam THz near field measurement setup Introduction MethodResults Measurements

17 Planken & Van der Valk, Optics Letters, Vol. 29, No. 19, pp – Differential detector Polarization of optical probe beam proportional to THz field Orientation of crystal determines component of THz field: E x, E y or E z Size of optical probe beam determines resolution THz near field measurement setup Introduction MethodResults Measurements

18 Metal layer Thickness 80 μ m Size square holes 200 μ m THz pulse EzEz z y x polarization THz near field measurement setup E z underneath metal layer with rectangular holes Introduction MethodResults Measurements

19 Introduction MethodResults Measurements Near field of holes Calculated with mode expansion technique Size hole: width = 0.2 mm, thickness = 0.08 mm

20 Experiment Calculation single frequency: 1.0 THz (300  m) Comparison theory & experiments Top view: (x,y)-plane, E z underneath metal layer with multiple square holes Introduction MethodResults Measurements

21 Thanks to … An analytic approach to electromagnetic scattering problems Aurèle Adam Paul Planken Minah Seo (Seoul National University) Roland Horsten

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23 Comparison theory & experiments Frequency spectrum at shadow side Introduction MethodResults Measurements

24 Sphere (real metal or dielectric, any size) (Mie, 1908) Ex, dominant polarizationEz Pulse incident on perfectly conducting sphere Introduction MethodResults Measurements

25 dipole orientation Spontaneous emission Incident field: dipole near scattering structure Introduction MethodResults Measurements dipole orientation

26 Near field of holes Calculated with mode expansion technique Introduction MethodResults Measurements ExEx EzEz

27 Scattering from single, square hole Incident field: linearly polarized plane wave Introduction MethodResults Measurements Energy flux through hole, normalized by energy incident on hole area

28 metal: real(  )  -  dielectric Surface plasmon  perfectly conducting metal

29 Dipole source near scattering structure Coefficients for waveguide modes Expression for scattered field