The Mie Theory: Basics and Applications

267 Pages·2012·5.633 MB·English

by  Wolfram Hergert (auth.), Wolfram Hergert, Thomas Wriedt (eds.)

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Wolfram Hergert Thomas Wriedt
•
Editors
The Mie Theory
Basics and Applications
123
Editors
Wolfram Hergert ThomasWriedt
Martin-Luther-Universität StiftungInstitut fürWerkstofftechnik
Halle-Wittenberg Badgasteiner Str. 3
FakultätfürNaturwissenschaften II 28359Bremen
Von-Seckendorff-Platz 1 Germany
06120Halle/Saale
Germany
ISSN 0342-4111 ISSN 1556-1534 (electronic)
ISBN 978-3-642-28737-4 ISBN 978-3-642-28738-1 (eBook)
DOI 10.1007/978-3-642-28738-1
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Preface
The small conference we arranged to celebrate the 100th anniversary of Mie’s
theorywasafascinatingevent.ThetwonephewsofGustavMiewerepresent,one
of them presenting a real live account on his uncle. We got in touch with them
when one of us (TW) started to redesign a web page on a collection of Mie
scatteringprogramsbackin2000.AphotoofGustavMiescannedfromasourceat
thelocaluniversitylibraryandsomeinformationabouthislifewasincludedinthis
webpage.ThisphotocanstillbefoundattheScattPortlightscatteringinformation
portal[1]andsincethenthephotohasbeenwidelydistributedontheInternet.One
of the nephews called by phone introducing himself as the caretaker of the
scientific inheritance of his uncle and asked for the source of the photo. Appar-
ently,hedidnotknowaboutthisspecificphoto.Sincethattimeourinterestonthe
historyofscienceinthetopicoflightscatteringaroseandwecollectedpapersand
other sources of information. Somehow the idea to arrange a conference on the
anniversary in 2008 of Gustav Mie’s original paper arose. After two attempts we
managed to secure some funding to arrange this conference. Scientists from all
over the world presented talks on the current state of light scattering by particles
includingsimulationsandapplications.Thecurrenteditedvolumeisacompilation
of selected extended versions of papers presented at this conference arranged in
Halle(Saale)in2008.Wewouldliketothankalltheauthorsfortheircontributions
and acknowledge the support of the conference by Deutsche Forschungsgeme-
inschaft (DFG).
Halle, Bremen, Germany Wolfram Hergert
Thomas Wriedt
Reference
1. Information on Gustav Mie at ScattPort, http://www.scattport.org/index.
php/gustav-mie-special.
v
Contents
1 Gustav Mie: From Electromagnetic Scattering to an
Electromagnetic View of Matter. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Life and Scientific Career. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Childhood in Rostock. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Study in Rostock and Heidelberg (1886–1890) . . . . . . . 6
1.2.3 Karlsruhe: Beginning of the Scientific
Career (1892–1902). . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.4 Greifswald: The productive Period (1902–1917) . . . . . . 14
1.2.5 Halle: Intermediate Station (1917–1924). . . . . . . . . . . . 17
1.2.6 Freiburg: Ordinarius and Retirement
in Baden (1924–1957) . . . . . . . . . . . . . . . . . . . . . . . . 20
1.3 Scientific Fields of Activity . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.1 Transport of Energy and Thermodynamics . . . . . . . . . . 24
1.3.2 Electrotechnics and Electrodynamics . . . . . . . . . . . . . . 25
1.3.3 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.4 Theory of Matter and Theory of Relativity. . . . . . . . . . 33
1.4 Gustav Mie as University Teacher. . . . . . . . . . . . . . . . . . . . . 37
1.4.1 Lectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.4.2 Disciples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.4.3 Textbooks and Popular Scientific Brochures. . . . . . . . . 39
1.5 Scientific Community and Society. . . . . . . . . . . . . . . . . . . . . 41
1.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Publications of Gustav Mie. . . . . . . . . . . . . . . . . . . . . . . . . . 44
Biographic and Bibliographic Communications
about Gustav Mie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
vii
viii Contents
2 Mie Theory: A Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 Nonspherical Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 History of Mie’s Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Mie Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5 Spheres in an Absorbing Medium . . . . . . . . . . . . . . . . . . . . . 57
2.6 Coated Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.7 Distorted Spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.8 Magnetic Spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.9 Chiral and Anisotropic Spheres . . . . . . . . . . . . . . . . . . . . . . . 58
2.10 Scattering by a Short Pulse. . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.11 Nanosized Spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.12 Gaussian Beam Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.13 Near Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.14 Longitudal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.15 Aggregates of Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.16 Parallelisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.17 Further Topics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.18 Further Reading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.19 Available Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.20 Sample Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.21 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 From Theories by Lorenz and Mie to Ontological
Underdetermination of Theories by Experiments . . . . . . . . . . . . . 73
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2 Quine’s and Ontological Underdeterminations,
with Precursors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3.1 From Classical Mechanics . . . . . . . . . . . . . . . . . . . . . 81
3.3.2 From Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 82
3.4 An Escape to Underdetermination: Ampliative Arguments . . . . 90
3.4.1 Undecidability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4.2 Ampliative Arguments. . . . . . . . . . . . . . . . . . . . . . . . 91
3.4.3 The Mechanical Rainbow. . . . . . . . . . . . . . . . . . . . . . 92
3.4.4 On the Nonsingularity Principle . . . . . . . . . . . . . . . . . 93
3.4.5 Newtonian Trajectories of Matter
Points do not Exist . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.6 Deciding Between Undecidables . . . . . . . . . . . . . . . . . 95
Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Contents ix
4 Predicting the Appearance of Materials Using
Lorenz–Mie Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Realistic Image Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.1 Camera. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.2 Geometry of Objects . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2.3 Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.4 Light Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.5 Light Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3 Predicting Appearance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1 Computing Optical Properties . . . . . . . . . . . . . . . . . . . 112
4.3.2 Lorenz–Mie Theory. . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.3.3 Number Density Distributions. . . . . . . . . . . . . . . . . . . 120
4.3.4 Non-Spherical Particles . . . . . . . . . . . . . . . . . . . . . . . 121
4.3.5 Scattering of a Gaussian Beam . . . . . . . . . . . . . . . . . . 122
4.4 Milk as a Case Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.4.1 Particle Composition . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.4.2 Appearance Model. . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.4.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5 Dipole Re-Radiation Effects in Surface Enhanced
Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2.1 Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . 140
5.3 Dipole Re-Radiation Effects in SERS from Isolated
Particles and Dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4 Dipole Re-Radiation Effects in SERS from
Single Sphere Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.5 Dipole Re-Radiation Effects in SERS from Dimer Chains . . . . 148
5.6 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6 Optical Force and Torque on Single and Aggregated Spheres:
The Trapping Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.2 Radiation Force and Radiation Torque . . . . . . . . . . . . . . . . . . 158
6.2.1 Radiation Force by Plane Waves. . . . . . . . . . . . . . . . . 160
6.2.2 Radiation Torque by Plane Waves. . . . . . . . . . . . . . . . 163
6.2.3 Contributions to Radiation Force and Torque . . . . . . . . 167
x Contents
6.3 Radiation Force and Torque on Aggregated Spheres . . . . . . . . 167
6.3.1 Radiation Pressure on Aggregated Spheres. . . . . . . . . . 168
6.3.2 Radiation Torque on Aggregated Spheres. . . . . . . . . . . 169
6.4 Laser Beams as a Superposition of Plane Waves. . . . . . . . . . . 172
6.4.1 The Angular Spectrum Representation. . . . . . . . . . . . . 173
6.4.2 Far Field in the Angular Spectrum Representation. . . . . 174
6.5 Radiation Force from a Highly Focalized Laser Beam . . . . . . . 175
6.6 Trapping and Manipulation of Single and
Aggregated Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.6.1 Trapping of Dielectric Spheres . . . . . . . . . . . . . . . . . . 178
6.6.2 Trapping of Gold Spheres. . . . . . . . . . . . . . . . . . . . . . 181
6.7 Radiation Torque on Elongated Nanostructures . . . . . . . . . . . . 186
6.8 Appendix 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6.9 Appendix 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7 Rainbows, Coronas and Glories . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.2 Rainbows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.3 Coronas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.4 Glories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.5 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8 The Extension of Mie Theory to Multiple Spheres. . . . . . . . . . . . . 223
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
8.2. The Multiple Sphere Solution . . . . . . . . . . . . . . . . . . . . . . . . 225
8.2.1 The Superposition Strategy. . . . . . . . . . . . . . . . . . . . . 225
8.2.2 Translated Fields and the Addition Theorem. . . . . . . . . 227
8.2.3 The Interaction Equations. . . . . . . . . . . . . . . . . . . . . . 229
8.3 The Addition Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
8.3.1 Forms of the Addition Theorem . . . . . . . . . . . . . . . . . 229
8.3.2 Recursive Translations. . . . . . . . . . . . . . . . . . . . . . . . 231
8.3.3 Derivation of the Addition Theorem . . . . . . . . . . . . . . 232
8.3.4 Application to the Plane Wave Expansion . . . . . . . . . . 234
8.3.5 Far-Field Limit of the Translation Matrix. . . . . . . . . . . 236
8.4 Properties of the Multiple Sphere Solution . . . . . . . . . . . . . . . 236
8.4.1 Numerical Issues. . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.4.2 The T Matrix Relationships . . . . . . . . . . . . . . . . . . . . 237
8.4.3 Coordinate Rotation and the Amplitude
and Scattering Matrix. . . . . . . . . . . . . . . . . . . . . . . . . 239
8.4.4 Cross Sections and Energy Conservation . . . . . . . . . . . 241
8.4.5 Random Orientation. . . . . . . . . . . . . . . . . . . . . . . . . . 244
Contents xi
8.4.6 Emission Cross Sections and Inter–Sphere
Energy Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.5 Extensions and Future Challenges . . . . . . . . . . . . . . . . . . . . . 248
8.5.1 Application to Inhomogeneous Media . . . . . . . . . . . . . 249
8.5.2 Future Applications: Direct Simulations and the Bridge
to the RTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
8.6.1 Calculation of the Translation Matrix Elements. . . . . . . 252
8.6.2 The Rotation-Axial Translation Operation . . . . . . . . . . 254
8.6.3 Generalized Spherical Functions . . . . . . . . . . . . . . . . . 255
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257