Table Of ContentGreen’s Function Integral
Equation Methods in
Nano-Optics
Green’s Function Integral
Equation Methods in
Nano-Optics
Thomas M. Søndergaard
Aalborg University, Aalborg, Denmark
CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Søndergaard, Thomas, author.
Title: Green’s function integral equation methods in nano-optics / Thomas
Søndergaard.
Description: First edition. | Boca Raton, FL : CRC Press/Taylor & Francis
Group, 2019. | Includes bibliographical references and index.
Identifiers: LCCN 2018026592| ISBN 9780815365969 (hardback : acid-free paper)
| ISBN 9781351260206 (ebook)
Subjects: LCSH: Micro-optics--Mathematics. | Green’s functions.
Classification: LCC TA1659.5 .S66 2018 | DDC 621.36--dc23
LC record available at https://lccn.loc.gov/2018026592
Visit the Taylor & Francis Web site at
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Contents
Preface.......................................................................................................................ix
Chapter1 Introduction.....................................................................................1
1.1 Overviewofmethodsandscatteringproblems.......................1
1.2 Opticsversusmicrowaves.......................................................3
1.3 Examplesofnano-optics.........................................................4
1.4 Notation,abbreviationsandsymbols......................................6
Chapter2 Theoreticalfoundation.....................................................................9
2.1 Maxwell’sequations...............................................................9
2.1.1 Boundaryconditions................................................11
2.1.2 Waveequations........................................................11
2.1.3 Poyntingvector........................................................12
2.2 Planarlayeredstructures.......................................................14
2.2.1 Fresnelreflectionandtransmission..........................15
2.2.2 Planarwaveguidesandguidedmodes......................18
2.3 Scatteringtheory...................................................................22
2.3.1 Scattererinhomogeneousmaterial(2D).................23
2.3.2 Scattereronalayeredstructure(2D).......................26
2.3.3 Scattererinhomogeneousmedia(3D).....................32
2.3.4 Scattereronalayeredstructure(3D).......................34
2.4 Exercises...............................................................................37
Chapter3 One-dimensionalscatteringproblems...........................................39
3.1 Green’sfunctionintegralequations......................................39
3.2 Numericalapproach..............................................................41
3.3 Exampleofasimplebarrier..................................................42
3.4 IterativeFFT-basedapproachforlargestructures................43
3.5 Guidelinesforsoftwareimplementation..............................45
3.6 Exercises...............................................................................46
Chapter4 Surfaceintegralequationmethodfor2Dscatteringproblems......49
4.1 Scattererinahomogeneousmedium....................................50
4.1.1 Green’sfunctionintegralequations.........................50
4.1.2 Finite-element-baseddiscretizationapproaches......54
4.1.3 Pulseexpansionandpoint-matching.......................60
4.1.4 Linear-fieldexpansionandpoint-matching.............65
v
vi Contents
4.1.5 Higher-order polynomial field expansion and
pointmatching.........................................................67
4.1.6 Fourierexpansionmethods......................................75
4.1.7 Calculatingelectricandmagneticfield
distributions..............................................................77
4.1.8 Examplesofmetalnanostripresonators..................79
4.1.9 Guidelinesforsoftwareimplementation..................90
4.1.10 Exercises..................................................................92
4.2 Scattereronornearplanarsurfaces......................................93
4.2.1 Green’s function for a layered reference struc-
turewithplanarsurfaces..........................................94
4.2.2 GFSIEMforalayeredreferencestructure.............105
4.2.3 Calculationoffieldsusingtheangularspectrum
representation.........................................................107
4.2.4 Example:Goldnanostriponadielectric
substrate.................................................................116
4.2.5 Example:Silvernanostripaboveasilversurface..123
4.2.6 Example:Singlegrooveinmetal...........................129
4.2.7 Example: Silver nanostrip on a thin-film-
silicon-on-silverwaveguide...................................133
4.2.8 Example: Microstructured gradient-index lens
forTHzphotoconductiveantennas........................143
4.2.9 Guidelinesforsoftwareimplementation................154
4.2.10 Exercises................................................................156
4.3 Periodicstructures..............................................................157
4.3.1 Blochwaves...........................................................158
4.3.2 Green’sfunctionforperiodicstructures................158
4.3.3 GFSIEMforperiodicstructures.............................160
4.3.4 Derivatives of periodic Green’s function and
tabulation................................................................163
4.3.5 Calculatingthefields..............................................165
4.3.6 Calculatingreflectionandtransmission.................167
4.3.7 Multilayerperiodicstructures................................169
4.3.8 Transfer-matrixmethodforlargestructures..........174
4.3.9 Example:Photoniccrystal.....................................183
4.3.10 Example:Anti-reflectivegroovearrayin
adielectric..............................................................188
4.3.11 Example:Broadband-absorberultra-sharpgroove
arrayinametal.......................................................194
4.3.12 Guidelinesforsoftwareimplementation................201
4.3.13 Exercises................................................................202
Chapter5 Areaintegralequationmethodfor2Dscatteringproblems........205
5.1 Green’sfunctionintegralequations....................................206
Contents vii
5.1.1 spolarization..........................................................206
5.1.2 ppolarization.........................................................207
5.2 Discretizationwithsquare-shapedelements.......................209
5.3 Discretizationwithtriangularelements..............................211
5.4 Scattererinahomogeneousmedium..................................214
5.4.1 spolarization..........................................................214
5.4.2 ppolarization.........................................................222
5.5 Scattereronornearplanarsurfaces....................................228
5.5.1 spolarization..........................................................228
5.5.2 ppolarization.........................................................229
5.6 Periodicsurfacemicrostructures.........................................234
5.6.1 spolarization..........................................................235
5.6.2 ppolarization.........................................................238
5.7 FastiterativeFFT-basedapproachforlargestructures.......241
5.8 Example:Purcellfactorofemitterinaphotoniccrystal....244
5.9 Example: Excitation of surface plasmon polaritons by
secondharmonicgenerationinaorganicsingle
nanofiber.............................................................................252
5.10 Guidelinesforsoftwareimplementation............................261
5.11 Exercises.............................................................................261
Chapter6 Volumeintegralequationmethodfor3Dscatteringproblems....265
6.1 Green’sfunctionintegralequation......................................265
6.2 Scattererinahomogeneousmedium..................................266
6.2.1 Discretizationwithcubicvolumeelements...........268
6.2.2 Discretedipoleapproximation(DDA)...................273
6.3 Scattereronornearplanarsurfaces....................................275
6.3.1 Green’stensorforlayeredreferencestructures.....275
6.3.2 Far-fieldGreen’stensor..........................................282
6.3.3 Green’stensorinCartesianvectorform................287
6.3.4 Opticalcrosssections.............................................288
6.3.5 Example: Scattering by a nanostrip on a thin
metalfilm...............................................................289
6.4 Periodicsurfacemicrostructures.........................................293
6.4.1 Green’stensorforperiodicstructures....................294
6.4.2 Calculatingreflectionandtransmission.................298
6.4.3 Example:2Dperiodicantireflectivesurface
microstructure........................................................299
6.5 Guidelinesforsoftwareimplementation............................303
6.6 Exercises.............................................................................303
viii Contents
Chapter7 Volume integral equation method for cylindrically symmetric
structures......................................................................................305
7.1 Expansion of homogeneous-medium dyadic Green’s
tensorincylindricalharmonics...........................................306
7.1.1 Eigenfunctions.......................................................306
7.1.2 Orthogonalityrelationsandnormalization............308
7.1.3 ConstructingthedirectGreen’stensor...................309
7.2 Green’stensorforalayeredstructure.................................313
7.2.1 IndirectGreen’stensor:Cylindricalharmonics.....314
7.2.2 TransmittedGreen’stensor:Cylindrical
harmonics...............................................................315
7.3 Out-of-planefar-fieldapproximationsofthecylindrical
Green’stensorelements......................................................316
7.3.1 Far-fielddirectGreen’stensor................................319
7.3.2 Far-fieldindirectGreen’stensor............................320
7.3.3 Far-fieldtransmittedGreen’stensor.......................321
7.4 Guided-modefar-fieldapproximationsofthecylindrical
Green’stensorelements......................................................321
7.5 Opticalcrosssections.........................................................324
7.6 Numerical approach: ring elements with rectangular
crosssection........................................................................325
7.7 Example:Nanocylinderonalayeredstructure...................327
7.7.1 Cylindricalscattereronadielectricsubstrate........328
7.7.2 Cylindrical scatterer on a thin-film silicon-on-
silverwaveguide.....................................................330
7.8 Example:Microstructuredgradient-indexlens..................332
7.8.1 Dipolereferencefield.............................................333
7.8.2 Calculationofemittedpower.................................335
7.8.3 Emissionpatternsandemittedpowers...................336
7.9 Guidelinesforsoftwareimplementation............................339
7.10 Exercises.............................................................................340
Chapter8 Surfaceintegralequationmethodforthequasi-staticlimit.........341
8.1 Green’sfunctionintegralequations....................................341
8.2 Numericalapproach:Pulseexpansion................................344
8.3 Finite-element-approach:Linearexpansion.......................347
8.4 Finite-element-approach:Quadraticexpansion..................353
8.5 Examples of absorption cross sections of 3D silver
nanoparticles.......................................................................355
8.6 Guidelinesforsoftwareimplementation............................356
8.7 Exercises.............................................................................357
Contents ix
Chapter9 Surfaceintegralequationmethodfor3Dscatteringproblems....359
9.1 Surfaceintegralequations...................................................359
9.2 Calculatingopticalcrosssections.......................................365
9.3 Numericalapproach:Generalstructure..............................366
9.4 Numericalapproach:Cylindricallysymmetricstructure....369
9.5 Example:Metalnano-discresonators.................................375
9.6 Guidelinesforsoftwareimplementation............................378
9.7 Exercises.............................................................................379
AppendixA Residuetheorem..........................................................................381
AppendixB Conjugategradientalgorithm......................................................383
AppendixC Besselfunctions...........................................................................385
AppendixD Analyticscatteringfromacircularcylinder................................387
AppendixE Analyticscatteringfromasphericalparticle...............................391
AppendixF Calculatingguidedmodesofplanarwaveguides........................395
F.1 Exercises.............................................................................401
AppendixG Plane-wave-expansiontheory......................................................403
G.1 Exercises.............................................................................406
References.............................................................................................................407
Index......................................................................................................................415
