Nano and quantum optics; an introduction to basic principles and theory

Graduate Texts in Physics

Ulrich Hohenester

Nano and
Quantum Optics
An Introduction to
Basic Principles and Theory


Graduate Texts in Physics
Series Editors
Kurt H. Becker, NYU Polytechnic School of Engineering, Brooklyn, NY, USA
Jean-Marc Di Meglio, Matière et Systèmes Complexes, Bâtiment Condorcet,
Université Paris Diderot, Paris, France
Sadri Hassani, Department of Physics, Illinois State University, Normal, IL, USA
Morten Hjorth-Jensen, Department of Physics, Blindern, University of Oslo, Oslo,
Norway
Bill Munro, NTT Basic Research Laboratories, Atsugi, Japan
Richard Needs, Cavendish Laboratory, University of Cambridge, Cambridge, UK
William T. Rhodes, Department of Computer and Electrical Engineering and
Computer Science, Florida Atlantic University, Boca Raton, FL, USA
Susan Scott, Australian National University, Acton, Australia
H. Eugene Stanley, Center for Polymer Studies, Physics Department, Boston
University, Boston, MA, USA
Martin Stutzmann, Walter Schottky Institute, Technical University of Munich,
Garching, Germany
Andreas Wipf, Institute of Theoretical Physics, Friedrich-Schiller-University Jena,
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Ulrich Hohenester

Nano and Quantum Optics
An Introduction to Basic Principles
and Theory

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Ulrich Hohenester
Institut fỹr Physik, Theoretische Physik
Karl-Franzens-Universităat Graz
Graz, Austria

ISSN 1868-4513
ISSN 1868-4521 (electronic)
Graduate Texts in Physics
ISBN 978-3-030-30503-1
ISBN 978-3-030-30504-8 (eBook)
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Preface

Nano optics combines the research areas of optics and nanoscience. Through light
we acquire information about the world around us, and the controlled manipulation
of light forms the backbone of numerous optics applications, such as fiber-based

communication or light harvesting. Nanoscience, on the other hand, deals with
the controlled manufacturing and manipulation of matter at the atomic scale, and
has driven the digital revolution that has irrevocably shaped our everyday life, for
instance, in the form of computers or mobile phones. A combination of these areas is
expected to bring together the best of two worlds. Yet, optics and nanoscience don’t
come together easily. The diffraction limit dictates that light cannot be squeezed
into volumes with dimensions smaller than the wavelength, which are on the order
of micrometers rather than nanometers, and conversely in optical microscopy only
objects further apart than approximately the light wavelength can be spatially
distinguished.
Nano optics deals with the manipulation of light at length scales comparable
or smaller than the light wavelength, ideally down to the nanometer scale. In
the last decades, scientists have succeeded in devising schemes to let optics
go nano. For instance, localization microscopy and optical tweezers have been
awarded the Nobel Prizes 2014 and 2018 for light manipulations in the threshold
region of the diffraction limit. To overcome the limit, one can collect light at the
nanoscale, for instance, in scanning nearfield optical microscopy, or bind within the
field of plasmonics light to electron charge oscillations at the surface of metallic
nanostructures, hereby squashing light into extreme subwavelength volumes.
This book provides an introduction to nano optics and plasmonics. It is based on
a lecture series I have taught over several years at the University of Graz and other
places. My main focus is on the basic principles and the theoretical tools needed
in nano optics, whereas applications are discussed only exemplary. In this respect,
the book is expected to differ from most related textbooks. I have kept references
to the current research literature somewhat sparse, but have tried to cite the many
excellent review articles for further information, whenever possible. I have also tried
to keep the discussion self-contained, and have refrained from using the phrase “it
can be shown” or equivalent whenever possible. This has made the presentation
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Preface

considerably longer than I initially thought, but it will hopefully facilitate reading
the book.
The book is separated into two parts. The first one deals with classical nano
optics, where classical can be understood both in terms of classical electrodynamics
and in terms of the classical, canonical presentation of the subject. The second part
brings nano optics to the quantum realm. I have tried hard to make the presentation
as entertaining as possible, but reading through the final version I realize that it has
become somewhat technical and busy—apologies for that. As always, the natural
way of reading a book is from the beginning to the end, and in principle there
is nothing wrong with this traditional approach. However, since I rarely stick to
this order myself, I will not provide any particular advice for using the book: start
reading wherever it looks interesting, and go back to the basics if needed.
Many colleagues and students have helped me to bring the book to its present
form; they are acknowledged separately below. I sincerely hope that this book will
be helpful to both experienced researchers seeking for selected information and to
beginners who are interested in a first glance of the topic. For the field of nano optics
as a whole, I hope that its future will be as bright and shiny as its past has been.
Graz, Austria
June 2019

Ulrich Hohenester

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Acknowledgements

This book presents my personal account of nano optics and plasmonics, but my
way of seeing the field has been strongly influenced by many colleagues and
predecessors. I thank all of them. My first encounter with the topic has been
through the NANOOPTICS group headed by Joachim Krenn at the University of
Graz who has worked in the field of plasmonics long before it has become
fashionable and has been given this name. I have strongly benefited from their
deep insights as well as their serenity in judging novel developments in the light
of the long history of the field. I am indebted to my long-term collaborator Andi
Trügler who has accompanied me for more than a decade on our joint nano optics
and plasmonics activities. Many thanks also to the numerous experimental and
theoretical collaborators for sharing their results and opinions, as well as for making
science such an exciting and pleasant undertaking.
Teaching the subject has been always important to me, unfortunately, it has never
come easy. In 2011, Jussi Toppari invited me to teach a course at a summer school
in Jyväskylä, Finland, where I enjoyed both the interactions with the students in
the class room and the traditional Finnish sauna. However, I had to realize that I
should probably spend more time with the basic things, which one often erroneously
calls “simple” after having employed them for a sufficiently long time. For several
years, I have used the loose collection of slides compiled for this summer school
as lecture notes for a course I have taught at Graz University. The kind invitation
of Guido Goldoni and Elisa Molinari for teaching a course on “Nano and Quantum
Optics” at the University of Modena and Reggio Emilia in the spring of 2018 finally
triggered my (surprisingly spontaneous) decision to start writing a textbook on the
subject. The conveniences of the superb Modenese food, the beautiful bike tours to
the Apennin, as well as a class of extremely bright students helped me to make the
start as enjoyable as possible. Of course, I have completely underestimated writing a

textbook, and after having worked on it for about one and a half years my emotions
towards the project have remained as mixed and diverse as they have been from the
beginning.
Special thanks go to my wife Olga Flor, among many other things for organizing
during my stay at Modena, a memorable trip to the eroding castle of Canossa,
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Acknowledgements

which has been given up by the Italian state but is kept alive by a few brave
volunteers, as well as a visit together with Elisa to the Osteria Francescana, and
for showing me how to write real books. I am indebted to numerous colleagues
who have read through specific parts of the book and have given most valuable
feedback. In alphabetic order, I wish to thank Javier Aizpurua, Stefano Corni,
Hari Ditlbacher, Hans Gerd Evertz, Antonio Fernández-Domínguez, Christian Hill,
Mathieu Kociak, Joachim Krenn, Olivier Martin, Walter Pötz, Stefan Scheel, and
Gerhard Unger. They have helped me to detect the most obvious errors and mistakes
in the manuscript. I will provide an updated list of errata on my homepage at the
University of Graz, and I invite all readers to inform me about possible errors and
to provide feedback on how the presentation could be made even more clear.

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Contents

1

What Is Nano Optics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Wave Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Evanescent Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
The Realm of Nano Optics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
8
14

2

Maxwell’s Equations in a Nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
The Concept of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
Maxwell’s Equations in Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Time-Harmonic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
Longitudinal and Transverse Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19
19
27
31
37
40

3

Angular Spectrum Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
Fourier Transform of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Far-Field Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Field Imaging and Focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
Paraxial Approximation and Gaussian Beams . . . . . . . . . . . . . . . . . . . . . .
3.5
Fields of a Tightly Focused Laser Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
Details of Imaging and Focusing Transformations . . . . . . . . . . . . . . . . .

45
46
47
51
55
58
60

4

Symmetry and Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Optical Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Poynting’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
Optical Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6
Optical Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71
71
80
81
84
86
90

5

Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
What Are Green’s Functions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2
Green’s Function for the Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . .

95
95
97

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5.3
5.4
5.5

Green’s Function for the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Optical Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Details for Representation Formula of Wave Equation . . . . . . . . . . . . . 108

6

Diffraction Limit and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1
Imaging a Single Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2

Diffraction Limit of Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
Scanning Nearfield Optical Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
Localization Microscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115
115
121
126
130

7

Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1
Drude–Lorentz and Drude Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
From Microscopic to Macroscopic Electromagnetism . . . . . . . . . . . . .
7.3
Nonlocality in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4
Reciprocity Theorem in Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139
141
147
150
158

8

Stratified Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1
Surface Plasmons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2
Graphene Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3
Transfer Matrix Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4
Negative Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5
Green’s Function for Stratified Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161
161
171
174
187
192

9

Particle Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1
Quasistatic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2
Spheres and Ellipsoids in the Quasistatic Limit . . . . . . . . . . . . . . . . . . . .
9.3
Boundary Integral Method for Quasistatic Limit . . . . . . . . . . . . . . . . . . .

9.4
Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5
Mie Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6
Boundary Integral Method for Wave Equation . . . . . . . . . . . . . . . . . . . . .
9.7
Details of Quasistatic Eigenmode Decomposition. . . . . . . . . . . . . . . . . .

207
207
209
221
235
244
247
251

10

Photonic Local Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Decay Rate of Quantum Emitter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Quantum Emitter in Photonic Environment. . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Surface-Enhanced Raman Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Förster Resonance Energy Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Electron Energy Loss Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

259
259
266

273
278
281

11

Computational Methods in Nano Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1 Finite Difference Time Domain Simulations . . . . . . . . . . . . . . . . . . . . . . .
11.2 Boundary Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Galerkin Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 Boundary Element Method Approach (Galerkin) . . . . . . . . . . . . . . . . . .
11.5 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6 Details of Potential Boundary Element Method . . . . . . . . . . . . . . . . . . . .

297
297
309
314
320
324
334

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12

Quantum Effects in Nano Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1 Going Quantum in Three Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 The Quantum Optics Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Summary of Book Chaps. 13–18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341
343
347
349

13

Quantum Electrodynamics in a Nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Canonical Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Coulomb Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.4 Canonical Quantization of Maxwell’s Equations . . . . . . . . . . . . . . . . . . .
13.5 Multipolar Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.6 Details of Lagrange Formalism in Electrodynamics. . . . . . . . . . . . . . . .

351
351
357
370
372
391
397

14

Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1 Statistical Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Kubo Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.3 Correlation Functions for Electromagnetic Fields . . . . . . . . . . . . . . . . . .
14.4 Correlation Functions for Coulomb Systems . . . . . . . . . . . . . . . . . . . . . . .
14.5 Quantum Plasmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.6 Electron Energy Loss Spectroscopy Revisited . . . . . . . . . . . . . . . . . . . . . .

407
408
411
418
422
433
456

15

Thermal Effects in Nano Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1 Cross-Spectral Density and What We Can Do with It . . . . . . . . . . . . . .
15.2 Noise Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3 Cross-Spectral Density Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.4 Photonic Local Density of States Revisited . . . . . . . . . . . . . . . . . . . . . . . . .
15.5 Forces at the Nanoscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.6 Heat Transfer at the Nanoscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.7 Details of Derivation of Representation Formula . . . . . . . . . . . . . . . . . . .

467
469

473
479
485
493
501
505

16

Two-Level Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.1 Bloch Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2 Two-Level Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.3 Relaxation and Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.4 Jaynes–Cummings Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

511
511
514
521
527

17

Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1 Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2 Master Equation of Lindblad Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3 Solving the Master Equation of Lindblad Form . . . . . . . . . . . . . . . . . . . .
17.4 Environment Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

533

533
540
543
552

18

Photon Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.1 Photon Detectors and Spectrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 Quantum Regression Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3 Photon Correlations and Fluorescence Spectra . . . . . . . . . . . . . . . . . . . . .
18.4 Molecule Interacting with Metallic Nanospheres . . . . . . . . . . . . . . . . . . .

567
568
575
577
588

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xii

Contents

A

Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593
A.1 Cauchy’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593

A.2 Residue Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595

B

Spectral Green’s Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.1 Spectral Decomposition of Scalar Green’s Function . . . . . . . . . . . . . . .
B.2 Spectral Representation of Dyadic Green’s Function . . . . . . . . . . . . . .
B.3 Sommerfeld Integration Path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

597
597
600
603

C

Spherical Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.1 Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3 Spherical Bessel and Hankel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

609
611
612
614

D

Vector Spherical Harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
D.1 Vector Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621

D.2 Orthogonality Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622

E

Mie Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E.1 Multipole Expansion of Electromagnetic Fields . . . . . . . . . . . . . . . . . . . .
E.2 Mie Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E.3 Plane Wave Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E.4 Dipole Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

F

Dirac’s Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
F.1
Transverse and Longitudinal Delta Function . . . . . . . . . . . . . . . . . . . . . . . 646

627
627
629
632
637

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659

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Chapter 1

What Is Nano Optics?

In this chapter we introduce the concepts of propagating and evanescent waves. The
removal of the latter waves in conventional optics is responsible for the diffraction
limit of light, which we will explain in terms of the scalar wave equation. A
discussion within the framework of Maxwell’s equations will be given in later parts
of this book. We start by introducing the one-dimensional scalar wave equation, and
then ponder on the generalization to higher dimensions. Many of these concepts will
be familiar to most readers, but are repeated here for clarity. Once we have set up the
stage, we will focus on the role of evanescent waves and how to live with or without
them in the field of nano optics. We conclude the chapter with a brief summary of
Chaps. 2–11 forming the first part of this book.

1.1 Wave Equation
1.1.1 One-Dimensional Waves
What are waves? I encourage the reader to reflect a while about this question and to
come up with a meaningful answer. After all, waves are abundant in physics, ranging
from water and sound waves to electromagnetic ones, which are the central objects
of this book. However, it seems rather difficult to explain what a wave really is. In
the book “Introduction to Electrodynamics” Griffiths comes up with the following
definition [1]:
A wave is a disturbance of a continuous medium that propagates with a fixed shape at a
constant velocity.

Electronic Supplementary Material The online version of this chapter ( />978-3-030-30504-8_1) contains supplementary material, which is available to authorized users.
© Springer Nature Switzerland AG 2020
U. Hohenester, Nano and Quantum Optics, Graduate Texts in Physics,
/>
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1


2

1 What Is Nano Optics?

( , 0)

Fig. 1.1 Wave in one spatial
dimension. A wave f (x, t)
propagates with a fixed speed
v without changing its shape.
After a time t it has
propagated over a distance vt

( , )

0

This definition leaves a number of open questions (what is the continuous medium
in case of electromagnetic waves? what about dispersive media?), and I will propose
later a modified, albeit more technical definition. To get started, let us take Griffiths’
description and consider waves in one spatial dimension. We denote the wave
disturbance propagating along x with f (x, t), where t is the time. Figure 1.1 shows
a schematic sketch of such a wave propagation. After a time t the initial wave has
been displaced by a distance vt. We can thus write
f (x, 0) = g(x) ,

f (x, t) = g(x − vt) ,

which shows that f is a function of one combined variable u = x − vt rather than of
two independent variables x, t. The same analysis applies to a wave that moves to
the left, and the general solution is a superposition of left- and right-moving waves
f (x, t) = g− (x − vt) + g+ (x + vt) = g− (u− ) + g+ (u+ ) ,

u± = x ± vt .

It is now easy to show that
∂
1 ∂
+
∂x
v ∂t

g− (u− ) =

1 ∂u−
∂u−
+
∂x
v ∂t

dg(u− )
v dg(u− )
= 1−
= 0.
du−
v
du−

Thus, the operator on the left-hand side equates all right-moving waves to zero.
Similarly, we find for the left-moving waves
∂
1 ∂
−
∂x
v ∂t

g+ (u+ ) =

1 ∂u+
∂u−
−
∂x
v ∂t

dg(u+ )
v dg(u+ )
= 1−
= 0.
du+
v
du+

If we apply both operators on the wavefunction f (x, t), we equate the left- and
right-moving waves to zero, and we arrive at the scalar wave equation in one spatial
dimension
Scalar Wave Equation for One Spatial Dimension
∂

1 ∂
+
∂x
v ∂t

∂
1 ∂
−
∂x
v ∂t

f (x, t) =

∂2
1 ∂2
− 2 2
2
∂x
v ∂t

f (x, t) = 0 .
(1.1)

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1.1 Wave Equation

3

In what follows, we consider the most simple wave form, a sinusoidal wave, which
can be written as
f (x, t) = A cos(kx − ωt + δ) .

(1.2)

Here A is the amplitude, k is the wavenumber, ω is the angular frequency, and δ is a
phase factor. The wavenumber and angular frequency are related to the wavelength
λ and the oscillation period T via
k=

2π
,
λ

ω=

2π
.
T

With this, we find that the sinusoidal wave is periodic in λ and T ,
x
+ 1 − ωt + δ = f (x, t)
λ
t
+ 1 + δ = f (x, t) .
f (x, t + T ) = A cos kx − 2π
T
f (x + λ, t) = A cos 2π

It turns out to be convenient to expand our definition of sinusoidal waves to the
complex plane. Starting from Euler’s formula
eiφ = cos φ + i sin φ ,

(1.3)

where φ is some phase argument, we express the sinusoidal wave in the form
f (x, t) = Re A ei(kx−ωt) eiδ = Re A˜ ei(kx−ωt) ,

A˜ = Aeiδ .

˜ which is the
In the last expression we have introduced a complex amplitude A,
iδ
product of A and the phase factor e . It turns out that the use of complex waves
and of the real part operation (in order to get the physically meaningful part) is so
successful that throughout this book we will no longer explicitly indicate the real
part operation. Then, a sinusoidal wave is of the form

Sinusoidal Wave in One Spatial Dimension
f (x, t) = A ei(kx−ωt)
and take real part operation Re . . . at end .

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(1.4)


4

1 What Is Nano Optics?

We have dropped the tilde from the amplitude, which from now on is always
understood as a complex quantity. To make this point clear again: physical waves
are always real, the complex notation is only adopted for simplicity and we assume
implicitly that the “real” wave (that can be compared with experiment) is obtained
by taking the real part of the complex expressions.
So far we have described the sinusoidal wave in terms of the wavenumber k and
angular frequency ω. In the same way as f (x, t) is not a function that depends
independently on x and t, but on a single variable x ∓ vt, ω and k are related to
each other through the so-called dispersion relation. This relation can be obtained
by inserting the sinusoidal wave ansatz into the scalar wave equation, Eq. (1.1),
∂2
1 ∂2
−
∂x 2
v 2 ∂t 2

A ei(kx−ωt) = − k 2 −

ω2
v2

A ei(kx−ωt) = 0 .

To fulfill the wave equation for arbitrary x, t values the term in parentheses must be
zero. We thus find for the dispersion relation of the scalar wave equation

Dispersion Relation for Scalar Wave Equation (1D)

ω(k) = vk .

(1.5)

Here ω(k) is the angular frequency, which is a function of the wavenumber k, and v
is the velocity of the wave propagation. In what follows, we show that any wave can
be decomposed into sinusoidal waves, and that the dispersion relation determines
the propagation properties of waves.
Fourier Transform. An important theorem in mathematics states that any function (which is sufficiently well behaved) can be decomposed into sinusoidal-like
waves through
f (x) =
f˜(k) =

∞

dk
e+ikx f˜(k)
2π
−∞
∞
−∞

e−ikx f (x) dx .

(1.6a)
(1.6b)

Here f˜(k) is called the Fourier transform of f (x). The magic of Eq. (1.6) is that
both f (x) and f˜(k) contain the same information. Thus, if we know f (x) we
can immediately compute f˜(k), and vice versa. Note that the factor of 1/(2π )

in the integration over k could be
√ also shifted to the integration over x, or both
integrations could acquire a 1/ 2π prefactor in a symmetric fashion. In this

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1.1 Wave Equation

5

book we will usually adopt the above definition, but will occasionally deviate
from it.
Wave Propagation. Suppose that we have given a wave f (x, 0) at time zero,
and would like to compute its shape at later time. This can be done easily when
using sinusoidal waves and the Fourier transform. At time zero we decompose
f (x, 0) into sinusoidal waves via Eq. (1.6a). As time goes on, each sinusoidal
wave evolves according to Eq. (1.4), and we get
f (x, t) =

∞
−∞

ei[kx−ω(k)t] f˜(k)

dk
.
2π

(1.7)

With the dispersion relation of Eq. (1.5) for the scalar wave equation we can work
out the integral explicitly, and obtain
f (x, t) =

∞

dk
= f (x − vt, 0) ,
eik(x−vt) f˜(k)
2π
−∞

in agreement to our initial discussion that waves propagate without changing
their shapes.
Dispersion. Let us pause for a moment and consider a more complicated
dispersion relation. We will encounter such modifications in later parts of this
book when discussing dispersive media. In principle we can still use Eq. (1.7) for
the wave propagation, but the evaluation of the integral is now more complicated
because of the modified ω(k) function. As a representative example we consider
for f˜(k) a Gaussian centered around k0 with a width of σ0−1 . For small widths
the function is strongly peaked around k0 , and we can approximate ω(k) through
a Taylor series around k0 ,
ω(k) ≈ ω0 + vg k − k0 +

β
k − k0
2

2

,

vg =

dω
dk

,β=
k0

d 2ω
dk 2

.
k0

Here vg is the group velocity and β a dispersion parameter. As explicitly worked
out in Exercise 1.5, the integral of Eq. (1.7) can be solved analytically for the
Gaussian and the approximated dispersion relation, and we obtain
1

f (x, t) = σ − 2 (t) ei(k0 x−ω0 t) exp −

(x − vg t)2
2σ 2 (t)

,

σ (t) =

σ02 + iβt .

(1.8)
Thus, the Gaussian wavepacket propagates with the group velocity vg , but owing
to β it does not conserve its shape but broadens while propagating, as described
by σ (t). In the remainder of this chapter we will not be overly concerned
with dispersive media. However, we have added this brief discussion here to
emphasize that most of our analysis not only applies to wave propagation in free

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6

1 What Is Nano Optics?

space and non-dispersive media, but can be easily extended to more complicated
situations.

1.1.2 Three-Dimensional Waves
So how do things change when we go from one to three spatial dimensions?
Formally not that much. Instead of Eq. (1.1) we get

Scalar Wave Equation for Three Spatial Dimensions
∇2 −

1 ∂2
v 2 ∂t 2

f (r, t) = 0 ,

(1.9)

where f (r, t) is the scalar wavefunction depending on r = (x, y, z), and ∇ 2 is the
usual Laplace operator
∇2 =

∂2
∂2
∂2
+ 2+ 2.
2
∂x
∂y
∂z

Similarly to the decomposition into sinusoidal waves of Eq. (1.4) we introduce plane
waves

Plane Wave in Three Spatial Dimensions
f (x, t) = A ei(k·r−ωt) ,

(1.10)

where A is the amplitude and k = k nˆ is the wavevector that has the length k = 2π/λ
determined by the wavelength λ and points in the direction of the wave propagation,
see Fig. 1.2. With these plane waves we can define in complete analogy to Eq. (1.6)
the three-dimensional Fourier transform
f (r) =

f˜(k) =

∞

d 3k
e+ik·r f˜(k)
(2π )3
−∞
∞
−∞

e−ik·r f (r) d 3 r .

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(1.11a)
(1.11b)


1.1 Wave Equation

,

=

7

−

=

+

,

,

Fig. 1.2 Plane wave in three dimensions. The wavevector k = k nˆ has the length of the
ˆ The lines perpendicular
wavenumber k = 2π/λ, and points in the wave propagation direction n.
to k indicate planes of constant phase

Finally, upon insertion of plane waves into the scalar wave equation of Eq. (1.9) we
get
kx2 + ky2 + kz2 −

ω2
= 0.
v2

In principle, we obtain positive and negative solutions for ω. However, we only keep
the positive ones for reasons to become more clear in Chap. 5 when discussing the
solutions of the wave equation in terms of in- and out-going spherical waves. Only
the latter ones, which correspond to positive frequencies, fulfill the requirement of
causality and must be kept. We thus find for the dispersion relation of waves in three
dimensions

Dispersion Relation for Scalar Wave Equation (3D)
ω(k) = v k = v kx2 + ky2 + kz2 .

(1.12)

We emphasize that this dispersion relation has been directly derived from the scalar
wave equation, and no approximations have been adopted. For this reason, the
dispersion relation must be always fulfilled—it is strict and not negotiable. We will
come back to this in a moment when discussing evanescent waves. A few further
comments might be in place.
Linearity. First, the wave equation is linear which has the consequence that if f1
and f2 are two solutions of the wave equation, then also the sum f1 + f2 is a
solution of the wave equation. The Fourier transform of Eq. (1.11) is a special
case of this, where we have decomposed the wave into particularly simple plane
waves. If we know how a single plane wave evolves in time, we can describe the
time evolution of a more complicated wave by decomposing it into such simple
plane waves. This always works because plane waves form a complete basis, as
stated by Fourier’s theorem.

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8

1 What Is Nano Optics?

Time Harmonic Fields. In many cases we are interested in waves oscillating
with a single frequency ω. We may write the wave solution in the form1
f (r, t) = e−iωt f (r) ,

(1.13)

where for notational simplicity we keep the same symbol for f (r, t) and

f (r). The above form is less restrictive than it seems because we can always
decompose a wave into harmonic components (using a Fourier transformation),
and it thus suffices to investigate time-harmonic fields of the form given by
Eq. (1.13) solely. Any more complicated wave form can then be constructed from
the superposition of these simple waves.
Wave Equation. If we insert Eq. (1.13) into the wave equation of Eq. (1.9) we get
∇ 2 + k 2 f (r) = 0 ,
where we have cancelled the common term e−iωt . At the beginning of this
chapter I have promised to come up with a more general definition for the wave
equation. Indeed, we can define a wave as a solution of the generalized wave
equation
∇ 2 + n2 (ω)k 2 f (r) = 0 ,

(1.14a)

where n(ω) is some frequency-dependent refractive index. The above form then
also applies to dispersive media. We could go even a step further and define
waves as solutions of the inhomogeneous wave equation
∇ 2 + n2 (r, ω)k 2 f (r) = 0 ,

(1.14b)

where the refractive index depends on the spatial coordinate and on frequency.
These definitions are not as instructive as the one given by Griffiths, but we shall
find it convenient to refer to “waves” or “wave-like” solutions when dealing with
solutions of Eq. (1.14).

1.2 Evanescent Waves
We are now ready to get real. The situation we have in mind is depicted in Fig. 1.3.
Suppose that we know a scalar field f (x, y, 0) at position z = 0, here the “nano”

that in physics one usually introduces the time-harmonic form e−iωt . In engineering one
usually writes ej ωt , where j is the imaginary unit and the sign in the exponential is reversed in
comparison to the physics convention.

1 Note

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1.2 Evanescent Waves

9

( , , 0)

Fig. 1.3 Suppose that we
know the field f (x, y, 0) at
position z = 0, here the
“nano” letters. How does the
field evolve when propagating
over a distance z?

( , , )

word, and suppose that the field is oscillating with a single frequency ω. In the
following we ask the questions:
– How does the field evolve when propagating over a distance z?
– And how can we compute the field f (x, y, z) at position z?
In principle, with the tools developed in the previous section we can analyze the

situation quite easily.
Plane Wave Decomposition. We start by decomposing the initial field f (x, y, 0)
in a plane wave basis using
f (x, y, 0) = (2π )−2
Wave Propagation.
ansatz

∞
−∞

ei(kx x+ky y) f˜(kx , ky ) dkx dky .

When moving away from z = 0, we can make the general

f (x, y, z) = (2π )−3

∞
−∞

ei(kx x+ky y+kz z) f˜(kx , ky , kz ) dkx dky dkz .

However, we must additionally fulfill the constraint of the wave equation and
therefore cannot chose ω and kx , ky , kz independently. We can thus express one
variable, for instance, kz , in terms of the others and are led to
f (x, y, z) = (2π )−2

∞
−∞

exp i kx x + ky y + kz (kx , ky )z

f˜(kx , ky ) dkx dky ,
(1.15)

where we have explicitly indicated the dependence of kz on kx , ky . From this
expression one observes that for z > 0 each plane wave acquires an additional
phase
f˜(kx , ky ) −→ eikz z f˜(kx , ky ) .
z>0

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10

1 What Is Nano Optics?

Here comes the problematic point. Using the dispersion relation of Eq. (1.3), the kz
component has to be computed from
kz = ± k 2 − kx2 − ky2 ,

k=

ω
.
v

(1.16)

The positive or negative sign has to be chosen for waves propagating in the positive

or negative z direction. We can now distinguish two cases. For kx2 + ky2 ≤ k 2 the
z-component of the wavevector
(Propagating wave)

kz = ± k 2 − kx2 − ky2

(1.17)

is a real number, corresponding to a normal wave propagation. However, for kx2 +
ky2 ≥ k 2 we get
kz = ± k 2 − kx2 − ky2 = ±i kx2 + ky2 − k 2 ≡ ±iκ ,
(1.18)
which corresponds to an imaginary wavenumber! Readers familiar with the concept
of evanescent waves will not be overly surprised by this finding. However, those
unfamiliar with this concept should take their time to check carefully whether
we have done everything properly up to this point, or whether we have missed
something important. However, the only two ingredients in the derivation of
evanescent waves are the plane wave decomposition, which is based on Fourier’s
theorem (and which we better should not question), and the dispersion relation,
which is deeply rooted in the wave equation itself (which forms the basis of our
whole analysis). So obviously there is nothing wrong with evanescent waves, and
they are here to stay.
In order to understand how these evanescent waves propagate, we insert the
imaginary wavenumber into the plane wave ansatz to get
(Evanescent wave)

exp i kx + ky y ± iκz

= exp i kx + ky y ∓ κz .

Thus, evanescent waves grow or decay exponentially when moving away from
z. To be physically meaningful, we only keep the decaying waves, this is e−κz
for z > 0 and eκz for z < 0. Evanescent waves are better known in quantum
mechanics. Figure 1.4a shows a quantum mechanical particle that impinges on a
potential barrier. If the kinetic energy of the particle is smaller than the height of the
barrier, it is reflected. However, part of the wave penetrates into the barrier where
its amplitude decays exponentially. This is the analog to evanescent waves for the
scalar wave equation. If the width of the potential barrier is reduced, see panel (c),
the particle can tunnel through the barrier, where it becomes converted again into a
propagating wave.

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1.2 Evanescent Waves

11

a

b

c

d

Fig. 1.4 (a) Transmission of a quantum mechanical particle at a potential barrier. The particle
tunnels into the barrier and becomes reflected. (c) When the width of the barrier is reduced,
the wavefunction penetrates through the barrier and the quantum mechanical particle can tunnel
through a classically forbidden region. (b) Classical wave analog to quantum tunneling. A

light wave propagates through a prism and becomes reflected under conditions of total internal
reflection. The wave “tunnels” into the classically forbidden region at the prism-air side, and the
wave acquires a small phase shift usually known as the Goos–Hänchen shift. (d) When a second
prism is brought close to the first one, with a distance comparable to the light wavelength, light
can “tunnel” through the air gap, and becomes converted on the other side of the prism into a
propagating wave

Tunneling is a general wave phenomenon, and can not only be observed in
quantum mechanics but also in electrodynamics. Panel (c) shows a prism where
an incoming light beam becomes reflected under the condition of total internal
reflection. Similarly to the situation shown in panel (a), the reflection is not abrupt
but part of the light field penetrates to the air side of the prism where it decays
exponentially (evanescent wave). This penetration can be observed as the so-called
Goos–Hänchen phase shift a reflected wave suffers in comparison to an abruptly
reflected wave [2]. When a second prism is brought close to the first one, panel
(d), the exponentially decaying field amplitude of the first prism can “tunnel” to the
second prism, where it becomes converted again into a propagating light field.
While the above examples are of somewhat limited use, evanescent waves play a
more important role in the understanding of the resolution limit of light. We return
to our previous plane wave decomposition of Eq. (1.15), and express the fields at
larger z values in the form

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12

1 What Is Nano Optics?

Scalar Wave Propagation (z > 0)

f (x, y, z) = (2π )−2
+ (2π )−2

k 2 >kx2 +ky2

k 2
e

e

i(kx x+ky y+ k 2 −kx2 −ky2 z)

f˜(kx , ky ) dkx dky

i(kx x+ky y)− kx2 +ky2 −k 2 z

f˜(kx , ky ) dkx dky .
(1.19)

Here the terms in the first and second line correspond to the propagating and
evanescent waves, respectively. The evanescent waves decay exponentially and
contribute only in close vicinity to z = 0. The further we move away from this plane,
the more the evanescent waves become exponentially damped. In later parts of the
book we will show that evanescent waves are always bound to matter. Figure 1.5
demonstrates the impact of the decay of evanescent waves. Panel (a) shows the
Fourier transform of the image formed by the “nano” letters shown in the inset.
When part of the k-space is removed, for instance, through the decay of evanescent
waves, the inverse Fourier transform gives a modified function. Panel (b) shows
the function reconstructed from the Fourier components located inside the circle of

panel (a), and panels (c,d) report images with a further reduced k-space content.
From these images it is clear that the large wavenumber components of f˜(kx , ky )
carry the high-resolution information. The more these waves become removed from
the propagated wave f (x, y, z), the more the function blurs and all fine details are
washed out.

a

=5

b

=1

=2

c

d

Fig. 1.5 Dependence of resolution on cutoff parameter k0 . (a) Fourier transform of “nano” letters
(inset). The red circle reports wavenumbers with k0 = 5. (b–d) Inverse Fourier transform for
different cutoff parameters, with kx2 + ky2 ≤ k02 in arbitrary units

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