Green’s Function Integral Equation Methods in Nano-Optics

Chapter 1 Introduction ..........................................................................................1 1.1 Overview of methods and scattering problems...........................1 1.2 Optics versus microwaves...........................................................3 1.3 Examples of nano optics.............................................................4 1.4 Notation, abbreviations and symbols..........................................6 Chapter 2 Theoretical foundation .........................................................................9 2.1 Maxwells equations ....................................................................9 2.1.1 Boundary conditions ....................................................11 2.1.2 Wave equations ............................................................11 2.1.3 Poynting vector ............................................................12 2.2 Planar layered structures...........................................................14 2.2.1 Fresnel reflection and transmission..............................14 2.2.2 Planar waveguides and guided modes..........................18 2.3 Scattering theory.......................................................................22 2.3.1 Scatterer in homogeneous material (2D) .....................23 2.3.2 Scatterer on a layered structure (2D) ...........................26 2.3.3 Scatterer in homogeneous media (3D).........................32 2.3.4 Scatterer on a layered structure (3D) ...........................35 2.4 Exercises ...................................................................................37 Chapter 3 One-dimensional scattering problems................................................39 3.1 Greens function integral equations ...........................................39 3.2 Numerical approach..................................................................41 3.3 Example of a simple barrier......................................................42 3.4 Iterative FFT-based approach for large structures ....................43 3.5 Guide lines for software implementation .................................45 3.6 Exercises ...................................................................................46 Chapter 4 Surface integral equation method for 2D scattering problems...........49 4.1 Scatterer in a homogeneous medium........................................50 4.1.1 Greens function integral equations ..............................50 4.1.2 Finite-element-based discretization approaches ..........54 4.1.3 Pulse expansion and point-matching ...........................60 4.1.4 Linear-field expansion and point-matching .................65 4.1.5 Higher-order polynomial field expansion and point matching.......................................................................67 4.1.6 Fourier expansion methods ..........................................75 4.1.7 Calculating electric and magnetic field distributions...77 4.1.8 Examples of metal nanostrip resonators ......................79 4.1.9 Guidelines for software implementation......................90 4.1.10 Exercises ......................................................................92 4.2 Scatterer on or near planar surfaces..........................................93 4.2.1 Greens function for a layered reference structure with planar surfaces .....................................................94 4.2.2 GFSIEM for a layered reference structure.................105 4.2.3 Calculation of fields using the angular spectrum representation.............................................................107 4.2.4 Example: Gold nanostrip on a dielectric substrate ....116 4.2.5 Example: Silver nanostrip above a silver surface ......123 4.2.6 Example: Single groove in metal ...............................129 4.2.7 Example: Silver nanostrip on a thin-film-siliconon- silver waveguide ...................................................133 4.2.8 Example: Microstructured gradient-index lens for THz photoconductive antennas..................................143 4.2.9 Guidelines for software implementation....................154 4.2.10 Exercises ....................................................................156 4.3 Periodic structures ..................................................................158 4.3.1 Bloch waves ...............................................................159 4.3.2 Greens function for periodic structures......................159 4.3.3 GFSIEM for periodic structures.................................161 4.3.4 Derivatives of periodic Greens function and tabulation ..........................................................................164 4.3.5 Calculating the fields..................................................166 4.3.6 Calculating reflection and transmission .....................167 4.3.7 Multilayer periodic structures ....................................169 4.3.8 Transfer-matrix method for large structures ..............175 4.3.9 Example: Photonic crystal .........................................184 4.3.10 Example: Anti-reflective groove array in a dielectric 189 4.3.11 Example: broadband-absorber ultra-sharp groove array in a metal...........................................................195 4.3.12 Guidelines for software implementation....................202 4.3.13 Exercises ....................................................................203 Chapter 5 Area integral equation method for 2D scattering problems .............205 5.1 Greens function integral equations .........................................206 5.1.1 s polarization..............................................................206 5.1.2 p polarization .............................................................207 5.2 Discretization with square-shaped elements...........................209 5.3 Discretization with triangular elements ..................................211 5.4 Scatterer in a homogeneous medium......................................214 5.4.1 s-polarization..............................................................214 5.4.2 p-polarization.............................................................222 5.5 Scatterer on or near planar surfaces........................................229 5.5.1 s polarization..............................................................229 5.5.2 p polarization .............................................................229 5.6 Periodic surface microstructures.............................................234 5.6.1 s-polarization..............................................................235 5.6.2 p-polarization.............................................................238 5.7 Fast iterative FFT-based approach for large structures ...........241 5.8 Example: Purcell factor of emitter in a photonic crystal ........244 5.9 Example: Excitation of surface plasmon polaritons by second harmonic generation in a single organic nanofiber..........252 5.10 Guidelines for software implementation ................................261 5.11 Exercises .................................................................................262 Chapter 6 Volume integral equation method for 3D scattering problems.........265 6.1 Greens function integral equation...........................................265 6.2 Scatterer in a homogeneous medium......................................266 6.2.1 Discretization with cubic volume elements ...............268 6.2.2 Discrete dipole approximation (DDA).......................273 6.3 Scatterer on or near planar surfaces........................................275 6.3.1 Greens tensor for layered reference structures...........275 6.3.2 Far-field Greens tensor...............................................282 6.3.3 Greens tensor in cartesian vector form ......................287 6.3.4 Optical cross sections.................................................288 6.3.5 Example: Scattering by a nanostrip on a thin metal film .............................................................................289 6.4 Periodic surface microstructures.............................................293 6.4.1 Greens tensor for periodic structures .........................294 6.4.2 Calculating reflection and transmission .....................298 6.4.3 Example: 2D periodic antireflective surface microstructure ................................................................299 6.5 Guidelines for software implementation ................................303 vi Contents 6.6 Exercises .................................................................................303 Chapter 7 Volume integral equation method for cylindrically symmetric structures ..................................................................................................305 7.1 Expansion of homogeneous-medium dyadic Greens tensor in cylindrical harmonics .........................................................306 7.1.1 eigenfunctions ............................................................306 7.1.2 Orthogonality relations and normalization ................308 7.1.3 Constructing the direct Greens tensor........................309 7.2 Greens tensor for a layered structure ......................................313 7.2.1 Indirect Greens tensor: cylindrical harmonics ...........314 7.2.2 Transmitted Greens tensor: cylindrical harmonics ....315 7.3 Out-of-plane far-field approximations of the cylindrical Greens tensor elements ...........................................................316 7.3.1 Far-field direct Greens tensor.....................................319 7.3.2 Far-field indirect Greens tensor..................................320 7.3.3 Far-field transmitted Greens tensor............................321 7.4 Guided-mode far-field approximations of the cylindrical Greens tensor elements ...........................................................321 7.5 Optical cross sections .............................................................324 7.6 Numerical approach: ring elements with rectangular cross section .....................................................................................325 7.7 Example: Nanocylinder on a layered structure.......................327 7.7.1 Cylindrical scatterer on a dielectric substrate ............328 7.7.2 Cylindrical scatterer on a thin-film silicon-onsilver waveguide.........................................................330 7.8 Example: Microstructured Gradient-Index Lens ....................332 7.8.1 Dipole reference field.................................................333 7.8.2 Calculation of emitted power.....................................335 7.8.3 Emission patterns and emitted powers.......................336 7.9 Guidelines for software implementation ................................339 7.10 Exercices.................................................................................340 Chapter 8 Surface integral equation method for the quasi-static limit .............341 8.1 Greens function integral equations .........................................341 8.2 Numerical approach: pulse expansion ....................................344 8.3 Finite-element-approach: linear expansion.............................347 8.4 Finite-element-approach: quadratic expansion.......................353 8.5 Examples of absorption cross sections of 3D silver nanoparticles ...........................................................................355 8.6 Guidelines for software implementation ................................356 8.7 Exercises .................................................................................357 Chapter 9 Surface integral equation method for 3D scattering problems.........359 9.1 Surface integral equations.......................................................359 9.2 Calculating optical cross sections...........................................365 9.3 Numerical approach: general structure...................................366 9.4 Numerical approach: cylindrically symmetric structure.........369 9.5 Example: metal nano-disc resonators .....................................375 9.6 Guidelines for software implementation ................................378 9.7 Exercises .................................................................................379 Chapter A Residue theorem...............................................................................381 Chapter B Conjugate gradient algorithm...........................................................383 Chapter C Bessel functions................................................................................385 Chapter D Analytic scattering from a circular cylinder.....................................387 Chapter E Analytic scattering from a spherical particle....................................391 Chapter F Calculating guided modes of planar waveguides .............................395 F.1 Exercises .................................................................................401 Chapter G Plane-wave-expansion theory...........................................................403 G.1 Exercises .................................................................................406 References.............................................................................................................407

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