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Janne Brok & Paul Urbach

An analytic approach to electromagnetic scattering problems. Janne Brok & Paul Urbach. CASA day, Tuesday November 13, 2007. Short CV. Applied Physics (1996 - 2001). MA Ethics (2001 - 2002). PhD Optics (2002 - 2007). Currently: Consultant LIME. An analytic approach to electromagnetic

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Janne Brok & Paul Urbach

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

  2. Short CV Applied Physics (1996 - 2001) MA Ethics (2001 - 2002) PhD Optics (2002 - 2007) Currently: Consultant LIME

  3. An analytic approach to electromagnetic scattering problems • 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 Method Results Measurements

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

  5. 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 • My thesis subject: finite thickness, perfect conductor, 3D, multiple pits or holes (finite or periodic). Introduction Method Results Measurements

  6. 1) Inside holes: expansion in waveguide modes 2) Above and below layer as: expansion in plane waves 3) Matching at interfaces Typically 400 unknowns per hole per frequency Brok & Urbach, Optics Express, vol. 14, issue 7, pp. 2552 – 2572. Mode expansion techniqueDiffraction from layer with 3D rectangular holes • Perfectly conducting layer, finite thickness • Finite number of rectangular holes • Incident field from infinity IntroductionMethod Results Measurements

  7. Mode expansion techniqueDiffraction from layer with 3D rectangular holes Step 1: Linear superposition of waveguide modes  = (1, 2, 3, 4) 1: pit number 2: polarization TE / TM 3: mode mx, my 4: up / down Normalization The discrete set of propagating and evanescent waveguide modes is complete: description of field inside pits/holes is rigorous IntroductionMethod Results Measurements

  8. Normalization Mode expansion techniqueDiffraction from layer with 3D rectangular holes Step 2: Linear superposition of plane waves  = (1, 2) 1: polarization S / P 2: propagation direction (kx,ky) The continuous set of propagating and evanescent plane waves is complete: description of field inside pits/holes is rigorous IntroductionMethod Results Measurements

  9. Mode expansion techniqueDiffraction from layer with 3D rectangular holes Step 3: Match tangential fields at interfaces Use Fourier operator… And substitute IntroductionMethod Results Measurements

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

  11. Interaction integral Mode expansion techniqueDiffraction from layer with 3D rectangular holes I a = hi + F a IntroductionMethod Results Measurements

  12. Mode expansion techniqueDiffraction from layer with 3D rectangular holes Small system of equations:  400 per hole IntroductionMethod Results Measurements

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

  14. A B Scattering from multiple square holesIncident field: linearly polarized plane wave • D = Lx = Ly = /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 IntroductionMethod Results Measurements

  15. Comparison with THz measurements 1 THz  300 μm Metals  perfect conductors (f.i. copper = -3.4e4 - 6.6e5 i) IntroductionMethod Results Measurements

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

  17. THz near field measurement setup • Polarization of optical probe beam proportional to THz field • Orientation of crystal determines component of • THz field: Ex, Ey or Ez • Size of optical probe beam determines resolution Differential detector Planken & Van der Valk, Optics Letters, Vol. 29, No. 19, pp. 2306 – 2308. IntroductionMethod Results Measurements

  18. polarization z y Ez x THz near field measurement setupEz underneath metal layer with rectangular holes THz pulse Metal layer Thickness 80 μm Size square holes 200 μm IntroductionMethod Results Measurements

  19. Near field of holesCalculated with mode expansion technique Size hole: width = 0.2 mm, thickness = 0.08 mm IntroductionMethod Results Measurements

  20. Comparison theory & experimentsTop view: (x,y)-plane, Ez underneath metal layer with multiple square holes Experiment Calculation single frequency: 1.0 THz (300 m) IntroductionMethod Results Measurements

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

  22. Comparison theory & experimentsFrequency spectrum at shadow side IntroductionMethod Results Measurements

  23. Sphere (real metal or dielectric, any size) (Mie, 1908) Pulse incident on perfectly conducting sphere Ex, dominant polarization Ez Introduction Method Results Measurements

  24. dipole orientation dipole orientation dipole orientation Spontaneous emissionIncident field: dipole near scattering structure IntroductionMethod Results Measurements

  25. Near field of holesCalculated with mode expansion technique Ex Ez IntroductionMethod Results Measurements

  26. Scattering from single, square holeIncident field: linearly polarized plane wave Energy flux through hole, normalized by energy incident on hole area 2 2 2 0 2 0 IntroductionMethod Results Measurements

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

  28. Dipole source near scattering structure • Coefficients for waveguide modes • Expression for scattered field

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