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

Theremoval of the latter waves in conventional optics is responsible for the diffractionlimit of light, which we will explain in terms of the scalar wave equation.. The same analysis app

Nano and

Quantum Optics

Ulrich Hohenester

An Introduction to

Basic Principles and Theory

Graduate Texts in Physics

Graduate Texts in Physics

Series Editors

Kurt H Becker, NYU Polytechnic School of Engineering, Brooklyn, NY, USAJean-Marc Di Meglio, Matière et Systèmes Complexes, Bâtiment Condorcet,Université Paris Diderot, Paris, France

Sadri Hassani, Department of Physics, Illinois State University, Normal, IL, USAMorten Hjorth-Jensen, Department of Physics, Blindern, University of Oslo, Oslo,Norway

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Richard Needs, Cavendish Laboratory, University of Cambridge, Cambridge, UKWilliam T Rhodes, Department of Computer and Electrical Engineering andComputer Science, Florida Atlantic University, Boca Raton, FL, USA

Susan Scott, Australian National University, Acton, Australia

H Eugene Stanley, Center for Polymer Studies, Physics Department, BostonUniversity, Boston, MA, USA

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Andreas Wipf, Institute of Theoretical Physics, Friedrich-Schiller-University Jena,Jena, Germany

and advanced-level undergraduate courses on topics of current and emerging fieldswithin physics, both pure and applied These textbooks serve students at theMS- or PhD-level and their instructors as comprehensive sources of principles,definitions, derivations, experiments and applications (as relevant) for their masteryand teaching, respectively International in scope and relevance, the textbookscorrespond to course syllabi sufficiently to serve as required reading Their didacticstyle, comprehensiveness and coverage of fundamental material also make themsuitable as introductions or references for scientists entering, or requiring timelyknowledge of, a research field.

More information about this series athttp://www.springer.com/series/8431

Ulrich Hohenester

Nano and Quantum Optics

An Introduction to Basic Principles

and Theory

Institut für Physik, Theoretische Physik

Karl-Franzens-Universit¨at 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)

https://doi.org/10.1007/978-3-030-30504-8

© Springer Nature Switzerland AG 2020

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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-basedcommunication or light harvesting Nanoscience, on the other hand, deals withthe controlled manufacturing and manipulation of matter at the atomic scale, andhas driven the digital revolution that has irrevocably shaped our everyday life, forinstance, in the form of computers or mobile phones A combination of these areas isexpected to bring together the best of two worlds Yet, optics and nanoscience don’tcome together easily The diffraction limit dictates that light cannot be squeezedinto volumes with dimensions smaller than the wavelength, which are on the order

of micrometers rather than nanometers, and conversely in optical microscopy onlyobjects further apart than approximately the light wavelength can be spatiallydistinguished

Nano optics deals with the manipulation of light at length scales comparable

or smaller than the light wavelength, ideally down to the nanometer scale Inthe last decades, scientists have succeeded in devising schemes to let optics

go nano For instance, localization microscopy and optical tweezers have beenawarded the Nobel Prizes 2014 and 2018 for light manipulations in the thresholdregion of the diffraction limit To overcome the limit, one can collect light at thenanoscale, for instance, in scanning nearfield optical microscopy, or bind within thefield of plasmonics light to electron charge oscillations at the surface of metallicnanostructures, 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 otherplaces 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 manyexcellent review articles for further information, whenever possible I have also tried

to keep the discussion self-contained, and have refrained from using the phrase “itcan be shown” or equivalent whenever possible This has made the presentation

v

considerably longer than I initially thought, but it will hopefully facilitate readingthe book.

The book is separated into two parts The first one deals with classical nanooptics, where classical can be understood both in terms of classical electrodynamicsand in terms of the classical, canonical presentation of the subject The second partbrings 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 hasbecome somewhat technical and busy—apologies for that As always, the naturalway 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 tothis order myself, I will not provide any particular advice for using the book: startreading 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 presentform; they are acknowledged separately below I sincerely hope that this book will

be helpful to both experienced researchers seeking for selected information and tobeginners 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

June 2019

This book presents my personal account of nano optics and plasmonics, but myway of seeing the field has been strongly influenced by many colleagues andpredecessors I thank all of them My first encounter with the topic has beenthrough the NANOOPTICS group headed by Joachim Krenn at the University ofGraz who has worked in the field of plasmonics long before it has becomefashionable and has been given this name I have strongly benefited from theirdeep 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 AndiTrügler who has accompanied me for more than a decade on our joint nano opticsand plasmonics activities Many thanks also to the numerous experimental andtheoretical collaborators for sharing their results and opinions, as well as for makingscience such an exciting and pleasant undertaking

Teaching the subject has been always important to me, unfortunately, it has nevercome 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 inthe class room and the traditional Finnish sauna However, I had to realize that Ishould probably spend more time with the basic things, which one often erroneouslycalls “simple” after having employed them for a sufficiently long time For severalyears, 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 QuantumOptics” at the University of Modena and Reggio Emilia in the spring of 2018 finallytriggered my (surprisingly spontaneous) decision to start writing a textbook on thesubject The conveniences of the superb Modenese food, the beautiful bike tours tothe Apennin, as well as a class of extremely bright students helped me to make thestart as enjoyable as possible Of course, I have completely underestimated writing atextbook, and after having worked on it for about one and a half years my emotionstowards the project have remained as mixed and diverse as they have been from thebeginning

Special thanks go to my wife Olga Flor, among many other things for organizingduring my stay at Modena, a memorable trip to the eroding castle of Canossa,

vii

which has been given up by the Italian state but is kept alive by a few bravevolunteers, as well as a visit together with Elisa to the Osteria Francescana, andfor showing me how to write real books I am indebted to numerous colleagueswho have read through specific parts of the book and have given most valuablefeedback 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, andGerhard 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 theUniversity 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

1 What Is Nano Optics? 1

1.1 Wave Equation 1

1.2 Evanescent Waves 8

1.3 The Realm of Nano Optics 14

2 Maxwell’s Equations in a Nutshell 19

2.1 The Concept of Fields 19

2.2 Maxwell’s Equations 27

2.3 Maxwell’s Equations in Matter 31

2.4 Time-Harmonic Fields 37

2.5 Longitudinal and Transverse Fields 40

3 Angular Spectrum Representation 45

3.1 Fourier Transform of Fields 46

3.2 Far-Field Representation 47

3.3 Field Imaging and Focusing 51

3.4 Paraxial Approximation and Gaussian Beams 55

3.5 Fields of a Tightly Focused Laser Beam 58

3.6 Details of Imaging and Focusing Transformations 60

4 Symmetry and Forces 71

4.1 Optical Forces 71

4.2 Continuity Equation 80

4.3 Poynting’s Theorem 81

4.4 Optical Cross Sections 84

4.5 Conservation of Momentum 86

4.6 Optical Angular Momentum 90

5 Green’s Functions 95

5.1 What Are Green’s Functions? 95

5.2 Green’s Function for the Helmholtz Equation 97

ix

5.3 Green’s Function for the Wave Equation 103

5.4 Optical Theorem 107

5.5 Details for Representation Formula of Wave Equation 108

6 Diffraction Limit and Beyond 115

6.1 Imaging a Single Dipole 115

6.2 Diffraction Limit of Light 121

6.3 Scanning Nearfield Optical Microscopy 126

6.4 Localization Microscopy 130

7 Material Properties 139

7.1 Drude–Lorentz and Drude Models 141

7.2 From Microscopic to Macroscopic Electromagnetism 147

7.3 Nonlocality in Time 150

7.4 Reciprocity Theorem in Optics 158

8 Stratified Media 161

8.1 Surface Plasmons 161

8.2 Graphene Plasmons 171

8.3 Transfer Matrix Approach 174

8.4 Negative Refraction 187

8.5 Green’s Function for Stratified Media 192

9 Particle Plasmons 207

9.1 Quasistatic Limit 207

9.2 Spheres and Ellipsoids in the Quasistatic Limit 209

9.3 Boundary Integral Method for Quasistatic Limit 221

9.4 Conformal Mapping 235

9.5 Mie Theory 244

9.6 Boundary Integral Method for Wave Equation 247

9.7 Details of Quasistatic Eigenmode Decomposition 251

10 Photonic Local Density of States 259

10.1 Decay Rate of Quantum Emitter 259

10.2 Quantum Emitter in Photonic Environment 266

10.3 Surface-Enhanced Raman Scattering 273

10.4 Förster Resonance Energy Transfer 278

10.5 Electron Energy Loss Spectroscopy 281

11 Computational Methods in Nano Optics 297

11.1 Finite Difference Time Domain Simulations 297

11.2 Boundary Element Method 309

11.3 Galerkin Scheme 314

11.4 Boundary Element Method Approach (Galerkin) 320

11.5 Finite Element Method 324

11.6 Details of Potential Boundary Element Method 334

Contents xi

12 Quantum Effects in Nano Optics 341

12.1 Going Quantum in Three Steps 343

12.2 The Quantum Optics Toolbox 347

12.3 Summary of Book Chaps 13–18 349

13 Quantum Electrodynamics in a Nutshell 351

13.1 Preliminaries 351

13.2 Canonical Quantization 357

13.3 Coulomb Gauge 370

13.4 Canonical Quantization of Maxwell’s Equations 372

13.5 Multipolar Hamiltonian 391

13.6 Details of Lagrange Formalism in Electrodynamics 397

14 Correlation Functions 407

14.1 Statistical Operator 408

14.2 Kubo Formalism 411

14.3 Correlation Functions for Electromagnetic Fields 418

14.4 Correlation Functions for Coulomb Systems 422

14.5 Quantum Plasmonics 433

14.6 Electron Energy Loss Spectroscopy Revisited 456

15 Thermal Effects in Nano Optics 467

15.1 Cross-Spectral Density and What We Can Do with It 469

15.2 Noise Currents 473

15.3 Cross-Spectral Density Revisited 479

15.4 Photonic Local Density of States Revisited 485

15.5 Forces at the Nanoscale 493

15.6 Heat Transfer at the Nanoscale 501

15.7 Details of Derivation of Representation Formula 505

16 Two-Level Systems 511

16.1 Bloch Sphere 511

16.2 Two-Level Dynamics 514

16.3 Relaxation and Dephasing 521

16.4 Jaynes–Cummings Model 527

17 Master Equation 533

17.1 Density Operator 533

17.2 Master Equation of Lindblad Form 540

17.3 Solving the Master Equation of Lindblad Form 543

17.4 Environment Couplings 552

18 Photon Noise 567

18.1 Photon Detectors and Spectrometers 568

18.2 Quantum Regression Theorem 575

18.3 Photon Correlations and Fluorescence Spectra 577

18.4 Molecule Interacting with Metallic Nanospheres 588

A Complex Analysis 593

A.1 Cauchy’s Theorem 593

A.2 Residue Theorem 595

B Spectral Green’s Function 597

B.1 Spectral Decomposition of Scalar Green’s Function 597

B.2 Spectral Representation of Dyadic Green’s Function 600

B.3 Sommerfeld Integration Path 603

C Spherical Wave Equation 609

C.1 Legendre Polynomials 611

C.2 Spherical Harmonics 612

C.3 Spherical Bessel and Hankel Functions 614

D Vector Spherical Harmonics 619

D.1 Vector Spherical Harmonics 621

D.2 Orthogonality Relations 622

E Mie Theory 627

E.1 Multipole Expansion of Electromagnetic Fields 627

E.2 Mie Coefficients 629

E.3 Plane Wave Excitation 632

E.4 Dipole Excitation 637

F Dirac’s Delta Function 645

F.1 Transverse and Longitudinal Delta Function 646

References 651

Index 659

Chapter 1

What Is Nano Optics?

In this chapter we introduce the concepts of propagating and evanescent waves Theremoval of the latter waves in conventional optics is responsible for the diffractionlimit of light, which we will explain in terms of the scalar wave equation Adiscussion 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, andthen 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 thestage, we will focus on the role of evanescent waves and how to live with or withoutthem in the field of nano optics We conclude the chapter with a brief summary ofChaps.2 11forming the first part of this book

What are waves? I encourage the reader to reflect a while about this question and tocome up with a meaningful answer After all, waves are abundant in physics, rangingfrom 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 Inthe book “Introduction to Electrodynamics” Griffiths comes up with the followingdefinition [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 (https://doi.org/10.1007/ 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,

https://doi.org/10.1007/978-3-030-30504-8_1

1

Fig 1.1 Wave in one spatial

dimension A wave f (x, t)

propagates with a fixed speed

vwithout 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 proposelater 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 Figure1.1shows

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

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 spatialdimension

Scalar Wave Equation for One Spatial Dimension

1.1 Wave Equation 3

In what follows, we consider the most simple wave form, a sinusoidal wave, whichcan be written as

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

where φ is some phase argument, we express the sinusoidal wave in the form

f (x, t )= ReA e i(kx −ωt) e iδ

= ReA e˜ i(kx −ωt)

, A˜= Ae iδ

In the last expression we have introduced a complex amplitude ˜A, which is the

product of A and the phase factor e iδ It turns out that the use of complex wavesand of the real part operation (in order to get the physically meaningful part) is sosuccessful that throughout this book we will no longer explicitly indicate the realpart operation Then, a sinusoidal wave is of the form

Sinusoidal Wave in One Spatial Dimension

and take real part operation Re .

at end

We have dropped the tilde from the amplitude, which from now on is alwaysunderstood as a complex quantity To make this point clear again: physical wavesare always real, the complex notation is only adopted for simplicity and we assumeimplicitly 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),

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)

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 determinesthe propagation properties of waves

Fourier Transform. An important theorem in mathematics states that any tion (which is sufficiently well behaved) can be decomposed into sinusoidal-likewaves through

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

1.1 Wave Equation 5

book we will usually adopt the above definition, but will occasionally deviatefrom 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 whenusing 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 sinusoidalwave evolves according to Eq (1.4), and we get

because of the modified ω(k) function As a representative example we consider

for ˜f (k) a Gaussian centered around k0with a width of σ−1

0 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+ v g k − k0

+β

Here v g is the group velocity and β a dispersion parameter As explicitly worked

out in Exercise1.5, the integral of Eq (1.7) can be solved analytically for theGaussian and the approximated dispersion relation, and we obtain

Thus, the Gaussian wavepacket propagates with the group velocity v g, 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 toemphasize that most of our analysis not only applies to wave propagation in free

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

where A is the amplitude and k = k ˆn 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

1.1 Wave Equation 7

Fig 1.2 Plane wave in three dimensions The wavevector k = k ˆn has the length of the

wavenumber k = 2π/λ, and points in the wave propagation direction ˆn The lines perpendicular

to k indicate planes of constant phase

Finally, upon insertion of plane waves into the scalar wave equation of Eq (1.9) weget

k x2+ k2

y + k2

z−ω2

v2 = 0

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.5when discussing thesolutions of the wave equation in terms of in- and out-going spherical waves Onlythe latter ones, which correspond to positive frequencies, fulfill the requirement ofcausality and must be kept We thus find for the dispersion relation of waves in threedimensions

Dispersion Relation for Scalar Wave Equation (3D)

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 asolution of the wave equation The Fourier transform of Eq (1.11) is a specialcase of this, where we have decomposed the wave into particularly simple planewaves If we know how a single plane wave evolves in time, we can describe thetime evolution of a more complicated wave by decomposing it into such simpleplane waves This always works because plane waves form a complete basis, asstated by Fourier’s theorem

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

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 alwaysdecompose 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 fromthe superposition of these simple waves

Wave Equation. If we insert Eq (1.13) into the wave equation of Eq (1.9) we get

∇2+ k2

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 waveequation Indeed, we can define a wave as a solution of the generalized waveequation



∇2+ n2(ω)k2



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 definewaves as solutions of the inhomogeneous wave equation

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”

1Note that in physics one usually introduces the time-harmonic form e −iωt In engineering one

usually writes e j ωt , where j is the imaginary unit and the sign in the exponential is reversed in

comparison to the physics convention.

1.2 Evanescent Waves 9

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 thesituation quite easily

Plane Wave Decomposition. We start by decomposing the initial field f (x, y, 0)

in a plane wave basis using

However, we must additionally fulfill the constraint of the wave equation and

therefore cannot chose ω and k x , k y , k zindependently We can thus express one

variable, for instance, k z, in terms of the others and are led to

where we have explicitly indicated the dependence of k z on k x , k y From this

expression one observes that for z > 0 each plane wave acquires an additional

phase

˜

f (k x , k y )−→

z>0 e ik z z f (k˜ x , k y )

Here comes the problematic point Using the dispersion relation of Eq (1.3), the k z

component has to be computed from

k z= ±k2− k2− k2, k=ω

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 k2+ k2 ≤ k2the

z-component of the wavevector

of evanescent waves will not be overly surprised by this finding However, thoseunfamiliar with this concept should take their time to check carefully whether

we have done everything properly up to this point, or whether we have missedsomething important However, the only two ingredients in the derivation ofevanescent waves are the plane wave decomposition, which is based on Fourier’stheorem (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 ourwhole analysis) So obviously there is nothing wrong with evanescent waves, andthey are here to stay

In order to understand how these evanescent waves propagate, we insert theimaginary wavenumber into the plane wave ansatz to get

exp



i k x + k y y ± iκz= expi k x + k y 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 e κz for z < 0 Evanescent waves are better known in quantum

mechanics Figure1.4a shows a quantum mechanical particle that impinges on apotential barrier If the kinetic energy of the particle is smaller than the height of thebarrier, it is reflected However, part of the wave penetrates into the barrier whereits amplitude decays exponentially This is the analog to evanescent waves for thescalar 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 apropagating wave

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 inquantum mechanics but also in electrodynamics Panel (c) shows a prism where

an incoming light beam becomes reflected under the condition of total internalreflection Similarly to the situation shown in panel (a), the reflection is not abruptbut part of the light field penetrates to the air side of the prism where it decaysexponentially (evanescent wave) This penetration can be observed as the so-calledGoos–Hänchen phase shift a reflected wave suffers in comparison to an abruptlyreflected 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 thesecond prism, where it becomes converted again into a propagating light field.While the above examples are of somewhat limited use, evanescent waves play amore 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

Scalar Wave Propagation(z > 0)

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) showsthe 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 (k x , k y )

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

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 k2+ k2≤ k2 in arbitrary units

1.2 Evanescent Waves 13

kx

ky Evanescent waves

Propagating waves

Fig 1.6 Diffraction limit of light Only the inner part of the wavevector space corresponds to

propagating waves, the larger wavenumber components, which carry information about the fine

spatial details of f (x, y), correspond to evanescent waves which decay exponentially when moving away from z= 0 Far away from this plane only the propagating waves survive, and in an optical image reconstruction all high-resolution information is lost

We are now in the position to qualitatively discuss the diffraction limit of light Asshown in Fig.1.6, only wavevectors with a sufficiently small modulus correspond topropagating waves, whereas components with a larger wavenumber, which carry thehigh spatial resolution, correspond to evanescent waves which decay exponentially

when moving away from the plane z = 0 If we assume that the spatial resolution Δ

is approximately given by the largest wavenumber kmaxavailable, we get

Thus, the spatial resolution is approximately given by the wavelength λ (which

is determined by the frequency ω) In later parts of this book we will provide a more rigorous analysis for the diffraction limit of light, and will show that Δ≈ λ

2.However, it is gratifying to see that already this simple analysis gives a reasonableestimate To summarize, all high-resolution information of the wave is carried bythe evanescent wave components, which decay exponentially at larger distances and

no procedure whatsoever will bring them back

Figure1.7shows the wavelengths (bottom axis) and photon energies (top axis) forthe near-infrared, visible, and ultraviolet part of the electromagnetic spectrum Thevisible regime ranges from 380–750 nm, and correspondingly the diffraction limit

is in the micrometer rather than nanometer regime Thus, optics and nanoscience donot come naturally together! Nano optics is the science that tries to push optics tothe nanoscale despite these limitations

First, and most importantly, we have to realize that the diffraction limit is based

on fundamental laws of physics, most importantly the dispersion relation which isdeeply rooted in the fundamental wave equation From the dispersion relation wefind that there exist two types of waves, propagating and evanescent ones, and thedecay of the latter waves is responsible for the loss of resolution Using conventionaloptics it is not possible to resolve objects that are closer to each other than the

wavelength of light λ, and conversely we cannot focus light to spots that are smaller

in dimension than λ In order to overcome the diffraction limit of light we can

hardly compete with the fundamental laws of physics, thus we have to change therules of the game Nano optics has come up with a number of successful solutions,which will be discussed in detail in this book Figure1.8shows three representativeexamples

Nearfield Optics. In scanning nearfield optical microscopy (SNOM) an opticalfiber is brought into close vicinity of a nano object, see panel (a) Throughthe fiber tip, the evanescent nearfields of the nano object can be converted intopropagating photons, which are detected at the other end of the fiber By raster-

Photon Energy (eV)

200 400

600 800

1000 1200

1400

Wavelength (nm)

Fig 1.7 Wavelengths (bottom) and photon energies (top) for near-infrared (770–3000 nm), visible

(380–750 nm), and ultraviolet (10–380 nm) regime

1.3 The Realm of Nano Optics 15

SNOM tip

Fig 1.8 Three examples of how to bring optics to the nanoscale (a) In scanning nearfield

optical microscopy ( SNOM ) an optical fiber is brought in close vicinity to some nano object Through coupling to the fiber the evanescent nearfields of the object are converted to propagating photons, which can be detected at the end of the fiber By raster-scanning the fiber over the

specimen, one obtains spatial information about the nearfields (b) In localization microscopy

fluorescent molecules are attached to the nano object to be investigated In the far-field one can measure the location of these molecules with nanometer spatial resolution, and thus obtains

indirectly information with high spatial resolution about the nano object under investigation (c)

In plasmonics one or several metallic nanoparticles ( MNP s) act as nano antennas which convert far-field radiation to evanescent fields, which allows delivering light to the nanoscale

scanning the fiber over the specimen, one obtains information about the opticalnearfields with nanometer resolution

Localization Microscopy. While in conventional optics there is a priori noinformation available about the system to be studied, in localization microscopythe objects under study (typically biological systems such as cells) are decoratedwith fluorescent molecules In conventional far-field optics the positions of thefluorescing molecules can be measured with nanometer resolution, providedthat the optically active molecules are further apart than the light wavelength,which allows to obtain optical images of nano objects with nanometer resolution.Localization microscopy has been awarded the Nobel Prize in Chemistry 2014

Plasmonics. In plasmonics one exploits coherent electron charge oscillations atthe interface between a metal and a dielectric, so-called surface plasmons orparticle plasmons, to convert far-field radiation to plasmons These plasmonscome along with strongly localized evanescent waves, which allows deliveringlight at the nanoscale By a similar token, a quantum emitter located close to aplasmonic nanoparticle can use the particle as a nano antenna in order to emitlight much more efficiently This principle form the basis for techniques such assurface-enhanced fluorescence or surface-enhanced Raman scattering (SERS)

1.3.1 Summary of Book Chaps 2 11

This book gives an introduction to the theoretical concepts underlying nano opticsand plasmonics Chapters2 11form the first part of the book and deal with thedescription using the framework of classical electrodynamics, while Chaps.12–18are concerned with quantum aspects In all brevity, the contents of the chaptersforming the first part can be summarized as follows

Chapter 2 : Maxwell’s Equations in a Nutshell. We start by giving a short mary of Maxwell’s theory of electrodynamics Readers already familiar with thetopic can easily skip this chapter

sum-Chapter 3 : Angular Spectrum Representation. This chapter provides the toolsneeded for the theoretical description of optical imaging We show how tocompute the optical far-fields and how to theoretically describe the operation

of collection and imaging lenses

Chapter 4 : Symmetry and Forces. Here we investigate the symmetries lying Maxwell’s equations, in particular momentum conservation and opticalforces, energy conservation and optical cross sections, and orbital angularmomentum We also discuss the working principles of optical tweezers, forwhich one half of the Nobel Prize in Physics 2018 has been awarded

under-Chapter 5 : Green’s Functions. This chapter gives an introduction to the concept

of Green’s functions, which are of paramount importance in the field of nanooptics We derive the Green’s functions for the Helmholtz and vector waveequations, and obtain the respective representation formulas

Chapter 6 : Diffraction Limit and Beyond. Here we provide a thorough sion of the diffraction limit of light, and show how to overcome this limit usingscanning nearfield optical microscopy (SNOM) and localization microscopy, forwhich the Nobel Prize in Chemistry 2014 has been awarded

discus-Chapter 7 : Material Properties. This chapter marks the end of the part cerned with optical fields only We introduce generic models for the materialresponse, including the Drude–Lorentz and Drude models, and show how toobtain through suitable averaging over the microscopic properties the macro-scopic Maxwell’s equations

con-Chapter 8 : Stratified Media. The most simple hybrid system combiningMaxwell’s equations with materials are stratified media, consisting of stackedplanar layers of different materials We show that a novel type of excitation exists

at the interface between a metal and a dielectric, so-called surface plasmons, anddiscuss how these and related excitations can be tailored and exploited fornumerous applications

Chapter 9 : Particle Plasmons. Surface plasmons in restricted geometries ticles”) give rise to particle plasmons, which exhibit resonances and come alongwith strongly localized evanescent nearfields We investigate simple geometriesfor which analytic models are available, and develop general description schemesfor more complicated particle shapes

(“par-Exercises 17

Chapter 10 : Photonic Local Density of States. Through the combination ofplasmonic nanoparticles with quantum emitters it becomes possible tocompletely alter the optical properties of the emitters We introduce to surface-enhanced fluorescence and Raman scatterings (SERS), and discuss the basicprinciples underlying electron energy loss spectroscopy (EELS)

Chapter 11 : Computational Methods in Nano Optics. The final chapter of thefirst part deals with computational methods in nano optics and photonics, includ-ing finite difference time domain (FDTD), boundary element method (BEM), andfinite element method (FEM) simulation approaches

Exercises

Exercise 1.1 Consider a complex number z = x + iy and its complex conjugate

z∗= x − iy Express the real part Re(z) and imaginary part Im(z) in terms of z and

z∗.

Exercise 1.2 Show that a standing wave f (x, t) = A sin(kz) cos(kvt) satisfies the

wave equation, and express it as a sum of a waves traveling to the left and right

Exercise 1.3 Obtain the solution of the one-dimensional wave equation, Eq (1.1),

through separation of variables, f (x, t) = φ(x)ψ(t).

Exercise 1.4 Consider a Gaussian wavepacket of the form

Compute the Fourier transform, and use Eq (1.7) to get the wave at a later time t.

The following Gaussian integral might be useful:

2 k − k0

2

.

Interpret the final result:

(a) How does the wave propagate in time?

(b) How does the width of the wavepacket change as a function of time?

Exercise 1.6 The position x= 0 separates two media:

– for x < 0 the wave velocity is v1and

– for x > 0 the wave velocity is v2

A wave impinges from the left-hand side on the interface Part of the wave becomesreflected and part transmitted,

x <0: f (x, t) = e i(k1x −ωt) + R e i( −k1x −ωt)

x >0: f (x, t) = T e i(k2x −ωt) , where R and T are reflection and transmission coefficients.

(a) Use the dispersion relations to compute k1and k2

(b) Assume that the function and its derivative are continuous at x= 0 in order to

compute R, T

Exercise 1.7 Same as Exercise1.6but for two dimensions Consider a wave ansatz

with a given wavenumber k y (with k y < ω/v1)

x <0: f (x, y, t) = e i(k 1x x +k y y −ωt) + R e i( −k 1x x +k y y −ωt)

x >0: f (x, y, t) = T e i(k 2x x +k y y −ωt) .

(a) What kind of solution does this function describe?

(b) Use the dispersion relations to compute k 1x and k 2x

(c) Assume that the function and its derivative in x direction are continuous at x=

0 in order to compute R, T

(d) Discuss under which conditions the solutions for x > 0 have an evanescent

character

Exercise 1.8 In the fabrication of computer chips one nowadays uses photons with

an energy of about 10 eV Use the diffraction limit of light to estimate the smalleststructure sizes achievable, and compare with actual gate lengths of about 15 nm

Chapter 2

Maxwell’s Equations in a Nutshell

In this book I will assume that readers are already familiar with the basic concepts ofelectrodynamics Many textbooks can be found on the topic and in general they willall suffice as a suitable starting point Jackson’s “Classical Electrodynamics” [2]

is certainly among the most comprehensive accounts of the topic My favoritebook for teaching is the “Introduction to Electrodynamics” by Griffiths [1], andreaders familiar with this book will probably recognize some of his notations here(albeit not the famousr) In the following I briefly summarize the basic ideas ofelectrodynamics, however, without going too much into details

Electrostatics can be briefly summarized through Coulomb’s law that describes how

a particle with charge q1situated at position r1becomes attracted or repelled by a

second particle with charge q2situated at position r2,

4π ε0

q1q2

Here ε0 is the vacuum permittivity, which appears because of the SI unit system

under use, r12= r1− r2is the distance vector between the two charges, andˆr12is

the unit vector pointing in the direction of r12 Let me emphasize a few importantpoints about Coulomb’s law of Eq (2.1)

Symmetry. Coulomb’s law only depends on the relative distance vector r12 Forthis reason, it respects the homogeneity of space (no point of space is distin-guished with respect to any other one) and the isotropy of space (no direction inspace is distinguished with respect to another one) We will come back to thispoint in Chap.4 when discussing the symmetries of the electromagnetic fields

© Springer Nature Switzerland AG 2020

U Hohenester, Nano and Quantum Optics, Graduate Texts in Physics,

https://doi.org/10.1007/978-3-030-30504-8_2

19

We also note in passing that the 1/r2dependence of Coulomb’s law is the onlydistance dependence compatible with massless photons as force carriers of thefield [2].

Superposition. When two or more charged particles are present, the total forcecan be simply computed by adding the respective forces together,

This is the essence of the so-called superposition principle that has been tested

experimentally to the highest degree of precision [2], and which plays animportant role in the theory of electromagnetism

Charge Distribution. In many situations we do not want to deal with

point-like particles but with a continuous charge distribution ρ(r) Suppose that

many particles are present within a small volume element ΔV i and we areonly interested in the fields on sufficiently larger length scales We may then

group together the particles in small bunches Δq i and relate them to the charge

distribution ρ(r) via

Δq i ≈ ρ(r i )ΔV i

Although the limit ΔV → 0 is not meaningful for point-like particles, we can

still introduce a continuous charge distribution ρ(r), which is expected to vary smoothly as a function of r (see Chap.7for a more thorough discussion of such

an averaging procedure), and obtain instead of Eq (2.2) the expression

From Eqs (2.2) and (2.3) one observes that one can pull out the charge q1 fromthe sum or integral This can be done because of the superposition principle thatallows separating a many-body force into its mutual two-body forces In the field of

electrostatics it is then convenient to introduce an electric field E(r) defined through

Electric Field of Given Charge Distribution

4π ε0 ρ(r )

|r − r |2 ˆR d3r , (2.4)

2.1 The Concept of Fields 21

where ˆR is the unit vector pointing in the direction of r − r E(r) is a so-called

vector function which assigns to each space point r a vector The force acting on a

charge q1located at position r1can then be computed from

Thus, the electric field E(r) gives the force acting on a unit charge located at position r.

Up to this point the electric field has appeared as a completely auxiliary device

It provides us with the answer to the question of how the force on a charged particle

would be if a particle was there However, nothing hinders us to go one step further

and to give physical sense to E(r) In fact, the tremendous success of the theory of

electromagnetism is the identification of the electromagnetic fields as the central

objects of the theory As we will discuss in the remainder of this chapter, theelectromagnetic fields acquire a dynamics that is determined not only by the chargeand current distributions, which act as sources, but also by the electromagneticfields themselves In this way fields can become decoupled from their sources andpropagate through space independently

Before we will present the full set of Maxwell’s equations in Sect.2.2, we shouldtake a step back and ponder on the theory of vector fields In fact, the central objects

in electrodynamics are the electric E(r, t) and magnetic B(r, t) vector fields which

depend on both space and time coordinates Maxwell’s equations state how thesefields change in space and time, and the solutions have to be supplemented byappropriate boundary conditions The spatial variations are best described in terms

of the so-called “nabla” operator, which we will introduce next

Faraday’s Lines of Force

The concept of fields was originally brought up by Michael Faraday, shown inFig.2.1, an ingenious experimentalist with only limited mathematical skills

In an article by Basil Mahon [3], which appeared at the occasion of the150’th anniversary of Maxwell’s equation in Nature Photonics and which isabsolutely worth reading, the author explains how Faraday’s “lines of force,”

as he called them, were ahead of their time

Faraday’s thoughts were running along lines quite different from those of everyone else The general scientific opinion was still that electric and magnetic forces resulted from material bodies acting on one another at a distance with the intervening space playing only a passive role The Astronomer Royal, Sir George Biddell Airy, spoke for many when he described Faraday’s lines of force as “vague and varying” One can understand this view Action-at-a-distance gave exact formulae, whereas Faraday supplied none While they respected Faraday as a superb experimenter, most scientists thought him ill-equipped to theorize as he knew no mathematics.

(continued)

Conscious of such views, Faraday was circumspect when publishing his thoughts

on lines of force Only once, in 1846, did he venture into speculation A colleague, Charles Wheatstone, was due to speak at the Royal Institution about one of his inventions but took fright at the last minute Faraday decided to give the talk himself but ran out of things to say on the advertised topic well before the allotted hour was up Caught offguard he let his private thoughts escape and gave the audience

an astonishingly prescient outline of the electromagnetic theory of light All space,

he surmised, was filled with electric and magnetic lines of force which vibrated laterally when disturbed, and sent waves of energy along their lengths at a rapid but finite speed Light, he said, was probably one manifestation of these ‘ray-vibrations’.

We now know he was close to the truth, but to most of Faraday’s fellow scientists the ray-vibrations seemed like an absurd fantasy Even his supporters were embarrassed, and Faraday regretted letting his guard slip He had left his contemporaries behind, and it would take someone forty years his junior, a man of equal stature and complementary talents, to reveal Faraday’s true greatness The man was James Clerk Maxwell.

Gradient. Consider a scalar function f (x, y, z) that depends on all three spatial coordinates The total derivative of f becomes

where θ is the angle between ∇f and d, we observe that df becomes largest

when both vectors are parallel, this is for θ = 0 In other words, if we move inthe same direction as∇f the total change df is maximized Thus ∇f , which is

usually called the “gradient” of f , points into the direction where f (r) changes

most

2.1 The Concept of Fields 23

Fig 2.1 Two of the grounding fathers of electrodynamics Michael Faraday (left, 1791–1867) was

the first to envision the “lines of force” as the fundamental objects in electrodynamics, but he did not have the mathematical skills to develop a rigid mathematical theory This was accomplished by James Clerk Maxwell (right, 1831–1879) who first wrote down the equations that are now named after him

Divergence. The nabla operator can also act on a vector function F (r) either as

an inner product∇ · F (r), which is called the divergence, or as an outer product

∇ × F (r), which is called the curl Let us discuss first the divergence

∇ · F (r) = ∂F x (r)

∂x +∂F y (r)

∂y +∂F z (r)

To get some insight into the physical meaning of the divergence, we approximate

the derivatives by finite differences and assume for simplicity that F has no

dependence on z We then get

The arrow to the right gives the vector component F x at position x + Δ/2, and

the arrow to the left the component−F x at position x − Δ/2, with a similar

interpretation for the up and down directions By applying the plaquette to a

given vector field one obtains its divergence When we think about F as a fluid,

the divergence gives us information about the sources and sinks of the fluid

Fig 2.2 Application of (a,

b) divergence and (c, d) curl

plaquettes to vector fields (a)

(b)∇ · F = 0 for curl field,

∇ · F = 0 means that the fluid simply flows through a given square element, but

the in-flow equals the out-flow, whereas∇ · F > 0 means that more flows out

from the element than flows in, in accordance to a source within the square Asimilar interpretation in terms of a sink holds for∇ · F < 0 In Fig.2.2we showthe examples of point-like source fields and curl fields

Curl. We finally discuss the outer product∇ × F (r) which becomes

2.1 The Concept of Fields 25

of interest In terms of fluids, a curl can be interpreted as a vortex where one gains

or loses energy by making a round trip Figure2.2c, d shows examples of sourceand curl fields We will return to such finite difference approximations of the curl

in Chap.11when discussing finite difference time domain (FDTD) simulations

Usually we are dealing with problems where we are seeking for a vector field F (r)

within a given volume Ω Throughout this book we shall denote the boundary of a volume with ∂Ω The boundary ∂Ω can consist of several unconnected parts, and often the volume Ω fills the entire space such that ∂Ω is pushed towards infinity.

See also Fig.2.3 Nevertheless, a boundary is always present and one has to be

careful about the behavior of F (r) at the boundaries An important theorem in

vector analysis states:

Helmholtz Theorem

A vector function F (r) is uniquely determined upon knowledge of the

following quantities:

∇ · F , ∇ × F , and F (r ∈ ∂Ω) at the boundary.

Below we will interpret Maxwell’s equations in light of the Helmholtz theorem

In many cases the boundary conditions will not be explicitly stated but will besomehow built into the solutions, such as outgoing waves at infinity or electric fieldsthat become zero at infinity

Volume Ω

Boundary Ω

Normal vector

Fig 2.3 Volumes and boundaries In solving Maxwell’s equation we often deal with dielectric or

metallic bodies We denote a given regions of space (volume) with Ω, the boundary of this volume with ∂Ω, and the vector perpendicular to the boundary (and pointing outwards of the volume)

withˆn

2.1.3 Gauss’ and Stokes’ Theorem

We will often employ two integral theorems The first one is Gauss’ theorem

which states that the integral of∇ · F over a volume Ω equals the (directed) flow

of the vector field through the boundary ∂Ω of the volume Here dS = ˆn dS,

whereˆn is the outer surface normal and dS denotes an infinitesimal surface element.

Figure2.4a gives a graphical interpretation of this theorem in terms of the previouslyintroduced plaquettes As the divergence measures the net difference between in-and out-flow in a given square element, the in and out fluxes of two neighborelements precisely cancel each other, and the only non-vanishing contributions arelocated at the boundary

The second theorem is Stokes’ theorem

which states that the integration of the curl of a vector function over an open

surface S equals the line integral of F (r) along the boundary ∂S of the surface.

Figure2.4b gives a graphical interpretation of this theorem in terms of the previouslyintroduced plaquettes The curl contributions cancel each other at the edges ofneighbor elements, such as , and the only non-vanishing contributions arelocated at the surface boundary

In which direction does the outer surface normal of dS = ˆn dS point? And in which direction goes d? In case of Gauss’ theorem ˆn points to the outside of the

volume If one wants to define the boundary differently, and we will do so in laterparts of the book, one has to be careful about this point Similarly, the direction of

Fig 2.4 Graphical representation of (a) Gauss’ and (b) Stokes’ theorem In the Gauss’ theorem

∇ · F is integrated over a volume, where F flows from one volume element to a neighbor one, and

the in and out fluxes compensate each other with the exception of boundary terms Similarly, when integrating the curl over an open boundary, the curl terms in neighbor elements cancel each other with the exception of boundary elements with no neighbors

2.2 Maxwell’s Equations 27

dS dictates the circulation of d according to the right-hand rule, which means that

if one points with the thumb of the right hand upwards (pointing in the direction of

ˆn) the other fingers point in the direction of d.

In electrodynamics we deal with time-dependent electric E(r, t) and magnetic

B(r, t) fields The force acting on a particle with charge q at position r and moving

with velocity v can be computed from the Lorentz force

F = q [E(r, t) + v × B(r, t)] (2.14)The electromagnetic fields themselves are created by charge and current distribu-

tions ρ(r, t) and J (r, t), respectively The dynamics of these fields is determined

by Maxwell’s equations which form the basis of the theory of electrodynamics,

field E(r, t), whereas the second equation (no name) and Ampere’s law determine the magnetic field B(r, t) Additionally we have to specify appropriate boundary

conditions Maxwell’s equations can be interpreted as follows:

Gauss’ Law. The sources and sinks of the electric fields are given by the chargedistribution

No Name. There exist no magnetic charges

Faraday’s Law. A time-dependent magnetic field induces an electric curl field.This equation is also called the induction law and describes in electric circuits the