Second harmonic light scattering from dielectric and metallic spherical nanoparticles

Second harmonic light scatteringfrom dielectric and metallic spherical

nanoparticles

Der Naturwissenschaftlichen Fakultätder Friedrich-Alexander-Universität Erlangen-Nürnberg

zur

Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von

Sarina Janine Wunderlichaus Nürnberg

Als Dissertation genehmigtvon der Naturwissenschaftlichen Fakultätder Friedrich-Alexander Universität Erlangen-Nürnberg

Tag der mündlichen Prüfung: 10. April 2014

Vorsitzender des Promotionsorgans: Prof. Dr. Johannes Barth

Gutachter: Prof. Dr. Ulf PeschelFriedrich-Alexander-Universityof Erlangen-Nuremberg

Prof. Dr. Sylvie RokeÉcole Polytechniquede Fédérale Lausanne

Abstract

Second harmonic generation (SHG) is a useful tool for the study of interfacesbecause the method is intrinsically surface sensitive. As an optical method,SHG can be applied in-situ and enables real-time monitoring of the formationof nanoparticles and chemical and physical processes at their surface. In theseexperiments, the SH-intensity is recorded as a function of the scattering an-gle for different polarizations of the incident fundamental harmonic and thescattered second harmonic light. It is characteristic for the surface structure,surface charge, the kind and orientation of adsorbed molecules and many otherparameters. To gain maximum information from SH-experiments, a theoreticalmodel is needed that allows systematic variation of each parameter and studyof the influence on the SH-intensities. Such a model can narrow the param-eter space that is able to reproduce the measured signal. Comparison withexperimental data allows to determine the physical origin of the SH-signal.

In this work, a model for second harmonic and sum frequency scatteringbased on Mie theory is developed. It has the unique feature that the nonlin-ear polarization is modeled as an ensemble of discrete dipoles that are excitedby the fundamental harmonic light and radiate at second harmonic frequency.Comparison to other existing models, especially the widely used Rayleigh-Gans-Debye-model, confirms the stability of the method. The model is then used toinvestigate SHG from malachite green molecules adsorbed to polystyrene parti-cles and to determine the value of the surface susceptibility. In a second study,the model is extended and applied to the experimentally observed SHG frompolyelectrolyte brush particles. Several possibilities for the origin of SHG areinvestigated and a signal from the inner and outer interface is found to be themost realistic. The work closes with the investigation of SHG from dielectric-metallic core-shell particles where plasmonic enhancement of the SH-intensityis observed. The origin of these resonances is studied and an application assensing method for the shell thickness in a chemical growth process suggested.

iii

Kurzfassung

Frequenzverdopplung (englisch second harmonic generation, SHG) ist ein nütz-liches Werkzeug um Grenzflächen zu untersuchen, da die Methode intrinsischoberflächenempfindlich ist. Als optische Methode kann SHG in-situ angewendetwerden und ermöglicht es, die Herstellung von Nanopartikeln sowie chemischeund physikalische Vorgänge an deren Oberfläche in Echtzeit zu überwachen.In derartigen Experimenten wird die frequenzverdoppelte Intensität als Funk-tion des Streuwinkels für verschiedene Polarisationen des einfallenden Lichtsbei der Grundfrequenz und der Frequenz des gestreuten Lichts der zweitenHarmonischen gemessen. Diese ist charakteristisch für die Oberflächenstruk-tur, Oberflächenladung, Art und Orientierung adsorbierter Moleküle und vieleweitere Parameter. Um maximale Information aus Frequenzverdopplungsexpe-rimenten ziehen zu können, wird eine theoretisches Modell benötigt, das es er-laubt, jeden Parameter systematisch zu verändern und seinen Einfluss auf diefrequenzverdoppelte Intensität zu untersuchen. Ein solches Modell kann denParameterraum, der das Experiment reproduziert, einschränken. Der Vergleichmit dem Experiment legt die physikalische Ursache der Frequenzverdopplungfest.

In dieser Arbeit wird ein Modell für Frequenzverdopplung und Summenfre-quenzerzeugung basierend auf Mie-Streuung entwickelt. Dieses Modell hat dieeinzigartige Eigenschaft, dass die nichtlineare Polarisation als Ensemble diskre-ter Dipole modelliert wird. Die Dipole werden durch die fundamentale Harmo-nische angeregt und strahlen Licht mit der Frequenz der zweiten Harmonischenab. Der Vergleich mit anderen existierenden Modellen, insbesondere mit demweit verbreiteten Rayleigh-Gans-Debye Modell bestätigen die Stabilität dieserMethode. Das Modell wird anschließend genutzt, um SHG von Malachitgrün-molekülen zu untersuchen, die an Polysterolpartikel adsorbiert sind, und umden Wert der Oberflächensuszeptibilität zu bestimmen. In einer zweiten Unter-suchung wird das Modell erweitert und wird auf die experimentell beobachteteFrequenzverdopplung an Polylektrolyt-Bürsten-Partikeln angewendet. Verschie-dene Möglichkeiten für den Ursprung der Frequenzverdopplung werden unter-sucht und es erweist sich als realistischste Annahme, dass das Signal sowohl ander inneren als auch der äußeren Grenzfläche erzeugt wird. Die Arbeit schließt

v

vi

mit der Untersuchung der Frequenzverdopplung an dielektrisch-metallischenKern/Schale-Partikel, an denen plasmonische Verstärkung der SH-Intensitätbeobachtet wird. Der Ursprung dieser Resonanzen wird untersucht und die An-wendung als Messmethode zur Bestimmung der Schalendicke im chemischenWachstumsprozess vorgeschlagen.

Contents

1. Introduction 1

2. Second harmonic and sum frequency generation 52.1. Maxwell’s equations and polarization . . . . . . . . . . . . . . . 5

2.1.1. Linear optics and the wave equation . . . . . . . . . . . 62.1.2. Nonlinear optics and surface sum frequency generation . 8

2.2. Surface susceptibility . . . . . . . . . . . . . . . . . . . . . . . . 10

3. Vector spherical harmonics 153.1. Vector spherical harmonics . . . . . . . . . . . . . . . . . . . . 153.2. Expansion of electromagnetic fields into vector spherical harmonics 20

3.2.1. Plane wave . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.2. Field emitted by an electric dipole . . . . . . . . . . . . 21

3.3. Addition theorems . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.1. Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.2. Translation . . . . . . . . . . . . . . . . . . . . . . . . . 23

4. Mie scattering 274.1. Continuity condition . . . . . . . . . . . . . . . . . . . . . . . . 274.2. Stratified sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3. Scattering of a dipole at a stratified sphere . . . . . . . . . . . 334.4. Numerical implementation . . . . . . . . . . . . . . . . . . . . . 35

5. Theory of second harmonic and sum frequency scattering from thesurface of spherical particles 415.1. Existing theoretical models for second harmonic and sum fre-

quency scattering from spherical particles . . . . . . . . . . . . 415.1.1. Nonlinear Mie theory . . . . . . . . . . . . . . . . . . . 41

5.1.1.1. Östling’s anharmonic oscillator model . . . . . 425.1.1.2. Pavlyukh’s nonlinear sheet model . . . . . . . 425.1.1.3. De Beer’s and Roke’s reciprocity theorem . . . 43

vii

viii Contents

5.1.1.4. Gonella’s and Dai’s model . . . . . . . . . . . 435.1.1.5. Brevet’s model for metallic nanoshells . . . . . 44

5.1.2. Exact solutions in the small particle limit . . . . . . . . 465.1.3. Rayleigh-Gans-Debye-theory . . . . . . . . . . . . . . . 47

5.2. Molecular Mie model . . . . . . . . . . . . . . . . . . . . . . . . 48

6. Experimental realization of second harmonic generation 576.1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 576.2. Estimation of experimental errors . . . . . . . . . . . . . . . . . 59

6.2.1. Distortion by the cuvette geometry . . . . . . . . . . . . 596.2.2. Particle size distribution . . . . . . . . . . . . . . . . . . 64

7. Second harmonic generation from dielectric particles 677.1. Scanning for χ2 components . . . . . . . . . . . . . . . . . . . . 677.2. Effect of the number of orders . . . . . . . . . . . . . . . . . . . 707.3. Comparison to Rayleigh-Gans-Debye-theory . . . . . . . . . . . 737.4. Second harmonic generation from a non-centrosymmetric distri-

bution of the nonlinear polarization . . . . . . . . . . . . . . . . 767.5. Determination of the susceptibility of Malachite Green . . . . . 787.6. Second Harmonic generation from polyelectrolyte brush particles 83

8. Second harmonic generation from nanoshell particles 918.1. Silica particles with a gold shell of increasing thickness . . . . . 918.2. Second harmonic generation from silica—silver nanoshells . . . 96

8.2.1. Resonances at the second harmonic frequency . . . . . . 988.2.2. Resonances at the fundamental harmonic frequency . . 1048.2.3. Sensing application . . . . . . . . . . . . . . . . . . . . . 108

9. Conclusion and outlook 111

A. Mathematical definitions 115A.1. Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A.1.1. Spherical Bessel functions . . . . . . . . . . . . . . . . . 115A.1.2. Riccati Bessel functions . . . . . . . . . . . . . . . . . . 117

A.2. Associated Legendre functions . . . . . . . . . . . . . . . . . . . 118A.2.1. Legendre Polynomials . . . . . . . . . . . . . . . . . . . 118A.2.2. Associated Legendre functions . . . . . . . . . . . . . . 118A.2.3. Modified Legendre functions . . . . . . . . . . . . . . . . 120

Contents ix

A.3. Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . 122A.3.1. Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . 122

125Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . 125List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

1. Introduction

We study our environment mainly by looking at it: we detect the spectral andspatial distribution of sunlight that has been scattered by objects. Beside thisintuitive approach, optics provides an excellent method of sensing for severalreasons. In contrast to other methods, like scanning and transmission electronmicroscopy and different spectroscopic methods it is applicable in situ and doesnot destroy the object or alter its properties. Optics is literally as fast as thespeed of light. Depending on the exposure time of the detector, which is neededto collect a sufficient number of photons, and the number of measurement pointsthat are needed, optical methods are usually still fast enough to monitor mostprocesses in real time.

Therefore light scattering has been used to study nanoparticles. Even thoughtheir size is smaller than the wavelength of light and therefore below the Abberesolution limit, light scattering provides information about size, shape, concen-tration, speed of propagation, order and many more properties of nanoparticles.The interaction of light with matter is unique, even on a sub-wavelength scale.

In this work, the interaction of light with one special kind of particles is stud-ied: nanoparticles of spherical symmetry and without periodic order. Thisincludes simple spheres of dielectric or metallic material, with or withoutmolecules attached to the surface, but also more complex systems of strati-fied substructure, like particles consisting of a dielectric core with a metallicshell.

nanoparticles are of special interest in the development of advanced materi-als, and many medical biotechnology applications. The surface-to-volume ratioof nanoparticles is extremely high compared to macroscopic objects. There-fore, the surface structure dominates the properties of a material that consistsof nanoparticles. Also the synthesis of nanoparticles is determined by chemi-cal processes at their surface. Consequently, knowledge about the surface ofnanoparticles is crucial.

In contrast to linear light scattering processes, nonlinear light scatteringprocesses of all even orders exhibit an intrinsic surface sensitivity. Secondharmonic generation and sum frequency generation are such processes. Sumfrequency generation requires higher experimental efforts, so that during the

1

2 1. Introduction

Figure 1.1.: Principal sketch of second harmonic light scattering (λinc. = 800 nm,λSH = 400 nm) from a sample with colloidal spherical nanoparticles with adsorbedmolecules. The nonlinear parameter χ(2) of the bulk and the solvent are zero. Thesecond harmonic light therefore arises only from the surface, where χ(2) �= 0 and thesymmetry of the statistically oriented nonlinear molecules is broken. The angulardistribution I(ϑ) of the second harmonic intensity contains information about thenanoparticles.

last three decades most experiments on planar surfaces and nanoparticles havebeen performed using second harmonic generation. In angle-resolved measure-ments of the second harmonic generation from colloidal nanoparticles, size andsurface structure of the materials affect the polarization, intensity and angulardistribution of the SH light (Fig. 1.1). To fully understand this correlation,a theoretical model is needed where parameter ranges can be scanned, fittedto the experimental results, and where theoretical results can be conclusivelycompared with experimental results.

This work starts with an introduction of second harmonic and sum frequencygeneration and a summary of the basic equations of optics that describe lightpropagation and light-matter interaction (Chapter 1). The model that is devel-oped here is based on Mie-theory, which solves the problem of scattering froma sphere by the use of a spherical coordinate system based on vector sphericalharmonics. Therefore chapter 3 introduces vector spherical harmonics, the pos-sibility to expand electromagnetic fields into vector spherical harmonics andaddition theorems that account for rotation and translation of fields. Follow-ing this, Mie theory is explained (Chapter 4). Standard Mie-theory describesscattering of a plane wave from a spherical particle, but can be extended to any

3

kind of incident field and stratified spheres. A special situation is scattering ofa dipole inside or outside a stratified sphere. This is later re-used in the Miemodel for second harmonic generation.

Several models to calculate second harmonic scattering from spherical nanopar-ticles have been developed (Chapter 5). Some take advantage of the small sizeof the particles and the low refractive index contrast between particles and liq-uid. These models can not be used for larger particles or metallic particles withan imaginary part of the refractive index. Other models involve exact Mie scat-tering and can be used for larger particles of arbitrary material. However, theylack the ability to model systems of stratified structure or they are restrictedto certain surface properties. The scope of this work is to develop a numericalmodel based on Mie scattering that describes both second harmonic and sumfrequency generation from spherical colloidal nanoparticles and adds additionaldegrees of freedom to the source of the second harmonic or sum frequency scat-tering. Also an inhomogeneous distribution of the nonlinear polarization at thesurface or a nonlinear polarization that is not restricted to the surface of theparticle can be calculated with the new model.

The simulation is compared to results from second harmonic scattering ex-periments on colloidal particles that were performed by the Institute of ParticleTechnology of the University of Erlangen-Nuremberg. The set-up and majorexperimental parameters are shown in chapter 6 together with an estimate oferrors introduced by the set-up.

In chapter 7 simulations and experimental results are discussed. The influ-ence of simulation parameters on the modeled second harmonic signal is studiedand the results of the nonlinear Mie model is compared to other existing models.Especially the comparison to the Rayleigh-Gans-Debye model for vanishing re-fractive index contrast is used to study the effect of a varying refractive index onthe second harmonic signal. Also, the effect of different surface susceptibilitieson the second harmonic signal is studied systematically and used to determinethe surface parameters of polystyrene with adsorbed malachite green. The spe-cific possibilities of the Mie model based on discrete dipoles is used for thesimulation of the second harmonic intensity from a non-centrosymmetric distri-bution of polarization. Also the modeling of particles with elongated moleculesattached to the surface takes advantage of this feature.

The second type of nanoparticles that are studied are dielectric particles witha metallic shell (Chapter 8). For certain shell thickness, these particles showan enhancement of the SH-intensity. The origin of this enhancement lies in theformation of surface plasmon polaritons by either the incident plane wave at

4 1. Introduction

fundamental harmonic frequency or the radiated second harmonic light. When-ever the effective wavelength of the surface plasmon polaritons is in resonancewith the circumference of the sphere, the SH intensity is enhanced, which pre-destines it for sensing applications.

The definition of Bessel functions and other mathematical properties arecollected in the appendix. This also contains a list of the abbreviations usedin this work, as well as the list of figures and tables.

2. Second harmonic and sum frequencygeneration

Sum frequency generation (SFG) describes the process, where two photons offrequencies ω1 and ω2 unite their energies to form a new photon of frequencyω1 + ω2. A special case of SFG is second harmonic generation (SHG), whereboth fundamental harmonic frequencies ω1 and ω2 are equal. The new photonhas twice the original frequency ωSH = 2ωFH. This nonlinear optical processneeds high intensities at the fundamental harmonic frequency. Second harmonicgeneration was first observed by Franken et al. [37] in 1961 for λFH = 694.3 nmin crystalline quartz shortly after the invention of the first laser.

The basis of this nonlinear phenomenon lies in Maxwell’s equations, thatare summarized in section 2.1 with their consequences for linear and nonlinearoptics. This section also introduces the nomenclature used in this work (see alsothe list of symbols and abbreviations on page 125). Section 2.1.2 also explainsthe surface sensitivity of SFG. Section 2.2 focuses on the main parameter forSHG and SFG: the susceptibility χ(2).

2.1. Maxwell’s equations and polarization

The following four differential equations form the basis of all modern optics.Derived and arranged by Maxwell [62] in 1861 the equations were grouped byHeaviside [44] and first referred to as “Maxwell’s equations” by Einstein [35]:

∇ · D = ρ , ∇ × E + ∂

∂tB = 0 ,

∇ · B = 0 , ∇ × H − ∂

∂tD = j .

The electric and magnetic fields E [V/m] and B [T = Vs/m2] describe thephysical forces on an electric charge F = qE or a charge that is moving withvelocity v: F = qv × B. Inside a material, the fields consist of two parts:

ε0E = D − P ,1

μ0B = H + M .

5

6 2. Second harmonic and sum frequency generation

The electric displacement field D [C/m2] and the magnetizing field H [A/m]are caused by external currents and charges, the polarization P [C/m2] andthe magnetization M [A/m] by currents and charges induced in the material.The magnetization vanishes at optical frequencies (M = 0). Usually, also noexternal charges (ρ = 0) nor currents (j = 0) are present in optics. Maxwell’sequations for the electric and magnetic fields are then:

ε0∇ · E = −∇ · P , ∇ × E + ∂

∂tB = 0 ,

∇ · B = 0 ,1

μ0∇ × B − ε0

∂

∂tE = ∂

∂tP .

The polarization P acts as a source term. An infinitesimal polarization elementlocated at the origin and oscillating with ω leads to the electric and magneticfields of an electric dipole, located at the origin (see Jackson [48], with j = ∂P

∂t

and the use of the vector potential A). Any distribution of P can therefore bemodeled as a density of electric dipoles.

The polarization describes the response of the material to the optical fields.Usually P is written as a Taylor expansion into ascending numbers of photoninteractions, where the first (linear) term involves only one photon interactionwithin the relaxation time of the material, the second (quadratic) term involvestwo photon interactions etcetera:

P = P (1)︸︷︷︸linear

+ P (2)︸︷︷︸quadratic

+ P (3)︸︷︷︸cubic

+ . . . . (2.1)

2.1.1. Linear optics and the wave equationIn linear optics or strictly linear media, only the first term in Eq. (2.1) isconsidered and the polarization assumed to be directly proportional to theelectric field. For monochromatic fields,

D = ε0εE , H = 1μ0μ

B ,

with the permittivity ε and the permeability μ = 1 as material parameters.In the case of linear optics, evaluating ∇×(∇×E) in an homogeneous mediumresults in the wave equation:

ΔE − εμε0μ0∂2

∂t2 E = 0 .

2.1. Maxwell’s equations and polarization 7

The simplest solution of the wave equation is the so called plane wave:

E(r, t) = E0ei(k·r−ω·t) .

This plane wave plays a special role in optics, because it is possible to restrictgeneral considerations to plane waves. The reason for this lies in the property ofFourier transforms: Any arbitrary electric field can be decomposed into planewaves with amplitudes E0(k, ω)

E(r, t) =∞∫

−∞

d3k

∞∫−∞

dωE0(k, ω)ei(k·r−ω·t) ,

and thus the solution for an arbitrary field can be composed from the well-known solution for plane waves. This is possible in linear optics, where thesums and multiples of fields that fulfill Maxwell’s equations also fulfill Maxwell’sequations.

The plane wave is propagating in the direction of k with spatial frequency|k| and temporal frequency ω. The propagation speed of the wavefronts is

v = |ω|Re(|k|) = 1

Re(√εμ)√ε0μ0= c

Re(√εμ) = c

nR.

In vacuum, where E = ε0D and H = 1μ0

B, the speed is given by the speedof light v = c = 1√

ε0μ0. In a medium the speed is reduced by a factor nR, the

real part of the refractive index n = √εμ. This affects the wavelength inside

the material, that is shorter than in vacuum:

λ = 2πkR

= 2πnR

ωc

= λ0

nR.

The imaginary part of n causes a reduction of the amplitude as a plane wavepropagates through a medium (k = kez):

E(z) = E0ei(kz−ωt) = E0ei( ωc

(nR+inI)z−ωt) =(

E0e− ωc

nIz)

ei( ωc

nRz−ωt) .


2.1.2. Nonlinear optics and surface sum frequency generationIn 1962, Armstrong et al. [3] found a successful explanation for the recent ex-perimental discovery of second harmonic generation, sum frequency generationand phase matching: In his semi-classical approach, the polarization of the ma-terial does not only depend linearly on the electric field but higher orders exist(compare Eq. (2.1)). For fixed frequencies, the i-th component of the m-thorder polarization, oscillating at frequency ω =

∑iωi, is given by the product

of all xi-th components of the electric fields with frequencies ωi, where a param-eter χ

(m)i,x1,...,xm

(−ω|, ωi, . . . , ωm) determines the contribution of the product tothe sum:

P(n)i (r, ω) = ε0

3∑x1,...,xm=1

χ(m)i,x1,...,xm

Ex1(r, ω1) · . . . · Exm(r, ωm) .

This is also known as dipole approximation in contrast to the nonlocal electricquadrupole assumption (see later in this chapter). Here, it is assumed, that Pobeys causality and does not depend on nonlocal interactions.

Nonlinear optics leads to various effects, like the generation of harmonics,parametric down conversion, self focusing, Kerr-effect, self phase modulation,cross phase modulation, four wave mixing and the formation of solitons (Boyd[8]).

This work focuses on second harmonic and sum frequency generation, wheretwo photons of frequencies ω1 and ω2 unite their energies and momenta toform a new photon at frequency ω1 + ω2. The relevant part of the nonlinearpolarization is the second, quadratic, term:

Pi(r, ω = ω1 + ω2) = ε0

3∑j,k=1

χ(2)i,j,k(−ω|ω1, ω2)Ej(r, ω1)Ek(r, ω2) . (2.2)

The susceptibility χ(2) is the primary parameter for this nonlinear process. χ(2)

is a tensor of rank three, but its 21 elements reduce to a much smaller number,when symmetry relations can be used. Especially, all components are zero,whenever a system is centrosymmetric. Any inversion of the coordinate systemwould then also invert E and P . However, as the polarization is proportionalto the square of the electric field, this leads to conflicts as long as χ(2) isnonzero. Yet, also for a centrosymmetric material, P (2) �= 0 is possible, whenthe symmetry of the system is broken. This is the case at the surface of thematerial, making SHG an intrinsically surface sensitive method. Such second

2.1. Maxwell’s equations and polarization 9

harmonic generation was observed (Bloembergen et al. [5], Lee et al. [55], Chenet al. [14]) and SHG and SFG have been widely used probing techniques insurface science (Shen [82], Heinz et al. [46]).

As an optical method, SHG has various advantages over other surface probingtechniques. The nonlinear process is intrinsically surface sensitive. The setupis very simple and the measurement can be performed in-situ. The opticalprocess is fast enough to monitor adsorption processes at the surface.

The typical experimental setup for surface SHG from planar interfaces is asfollows (Heinz et al. [47]): A laser beam is incident on the surface of a nonlinearmedium under an angle α. The reflected light consists of the fundamentalharmonic frequency ω, but also of second harmonic light at 2ω. The linearpolarization of the incident light is varied and the second harmonic intensityrecorded as a function of polarization angle ϑ to obtain information about thesymmetry structure of the surface.

During the last two decades, SHG was also applied to other systems thanmacroscopic planar interfaces. Of special interest is SHG from the surfacesof spheres, because the system is by itself centrosymmetric. First experimen-tal SHG from spherical nanoparticles was observed by Wang et al. [93] andMartorell et al. [61] in 1996 and 1995.

Although the surface sensitivity of SHG and SFG is intrinsically strong, therestill exists a very small SFG/SHG signal from the bulk of centrosymmetricmedia, first observed by Terhune et al. [89] as SHG from the centrosymmetriccalcite crystal. The existence of SHG from a system with inversion symmetryis explained by a nonlocal electric quadrupole transition (Epperlein et al. [36])

Pi(r, ω) = ε0

3∑j,k,l=1

Qijkl(−ω|ω1, ω2)Ej(r, ω1)∇kEl(r, ω2)

instead of the much stronger, but forbidden, electric dipole transition that wasgiven in Eq. (2.2). Often, the four independent components Qi,i,i,i, Qi,i,i,j,Qi,j,i,j , and Qi,j,j,i are labeled α, β, γ, and δ. To avoid a further ambiguity,this nomenclature is not used here. As this contribution is very small[36], itis omitted in this work, however some SHG models include the nonlocal bulkcontribution (see Sec. 5.1.2).


2.2. Surface susceptibility

The nonlinear susceptibility χ(2)ijk relates the fundamental harmonic fields along

different spatial axes to the nonlinear polarization. χ(2)zxy indicates, that two

electric fields in x- and y-direction lead to a polarization in z-direction:

Pz = ε0χ(2)zxyExEy.

The SI-unit of the bulk polarization is Cm/m3, indicating a volume dipoledensity. The respective unit of the bulk susceptibility is m/V. For surfacesecond harmonic generation, a dipole density per area is used. The unit of thesurface polarization is Cm/m2 and the respective surface susceptibility m2/V;often χ(2) is given in esu = cm2/statvolt = 4.191 69 × 10−6 m2/V. From nowon, only surface susceptibilities are used.

Figure 2.1.: Euler angles: The start coordinate system S is rotated about its z-axisby an angle α. The new coordinate system S’ is rotated about its y-axis by an angleβ. The resulting coordiante system S” is rotated about its z-axis by an angle γ andresults in the final coordinate system S”’.

On a molecular basis, instead of the susceptibility χ(2), the molecular hyper-polarizability β is used. It is the molecular equivalent of the susceptibility andgives the nonlinear response per molecule. The SI-unit of β is (Cm)3/J2. Toobtain the nonlinear polarization of a certain surface element, the molecular

2.2. Surface susceptibility 11

hyperpolarizability is multiplied with the density of molecules, ρMolecule:

Pi =∑j,k

βijk ρMoleculeEjEk .

A coordinate transformation between the local coordinate system of the mole-cule and the external coordinate system converts the hyperpolarizability β intothe surface susceptibility χ(2). The orientation of the molecule (x′, y′, z′) withrespect to the external coordinate system (x, y, z) is given by the Euler anglesα, β and γ (see Fig. 2.1 and Sec. A.3.1 for the definition of Euler angles). Therespective rotation matrix correlates the vector in both coordinate systems:

v′ = M−1v , v = Mv′ .

This can be used to express the polarization P with respect to the outer co-ordinate system and to the molecule’s coordinate system and compare thecoefficients:

Pi =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

= P · ei = ε0

3∑j,k=1

χ(2)ijkEjEk

=(

MP ′)

· ei =3∑

l=1

Mi,lP′l

(2.3)

(2.4)

The second case, Eq.(2.4) is expanded:

Pi =3∑

l=1

Mi,lP′l =

3∑l=1

Mi,l

⎛⎝ρ

3∑j,k=1

βijkE′jE′

k

⎞⎠

=3∑

l=1

Mi,l

⎛⎜⎝ρ

3∑j,k=1

βijk

⎛⎝ 3∑

m=1

MmjEm

⎞⎠⎛⎝ 3∑

m=1

MmkEm

⎞⎠⎞⎟⎠ .

Here, Pi is expanded into a sum of EjEk. The elements of Pi in (2.3) and (2.4)are then compared and solved for the susceptibility as a function of the Eulerangles M(α, β, γ) and the hyperpolarizability βi,j,k:

χ(2)ijk = 1

ε0

3∑l,m,n=1

3∑o,p,q=1

Ml,o(α, β, γ) Mm,p(α, β, γ) Mn,q(α, β, γ) βo,p,q .(2.5)


Sokhan and Tildesley [84] have derived the same equation also including addi-tional fields due to the presence of surrounding molecules. If not all moleculesare oriented with the same Euler angles α, β, γ, the susceptibility βo,p,q is multi-plied with the respective density ρ(α, β, γ) and Eq. (2.5) is integrated over α, β,and γ. On a homogeneous surface, molecules can be expected to be on averagerotated statistically around the surface normal. To evaluate equation (2.5) for

this case, β and α are kept fixed and an integral over180°∫0°

dγ is performed. This

yields that only seven independent χ(2) components exist on a surface withoutdistinguished direction: χ

(2)1,1,3 = χ

(2)2,2,3, χ

(2)1,3,1 = χ

(2)2,3,2, χ

(2)3,1,1 = χ

(2)3,2,2, χ

(2)3,3,3,

χ(2)1,2,3 = −χ

(2)2,1,3, χ

(2)1,3,2 = −χ

(2)2,3,1, and χ

(2)3,1,2 = −χ

(2)3,2,1.

For SHG the last two indices are interchangeable because the two fundamen-tal harmonic fields are identical. Four independent components remain thatare labeled χ

(2)zzz, χ

(2)zxx, χ(2)xxz and χ

(2)zxy in this work. For fixed inclination

angle β and otherwise statistical rotation about surface normal and molecular

axis (360°∫0°

dα and180°∫0°

dγ) and ρ(α, β, γ) = 1 m−2 these components are (see also

Fig. 2.2):• χ

(2)zzz = χ

(2)3,3,3 =

= 1ε0

(sin β sin 2β

β1,1,3+β2,2,32 + sin2 β cos β

β3,1,1+β3,2,22

+β3,3,3 cos3 β)

• χ(2)zxx = χ

(2)3,1,1 = χ

(2)3,2,2 =

= 1ε0

cos β

(cos 2β

(β1,1,3+β2,2,3

4 + β3,1,1+β3,2,28 − β3,3,3

4

)− β1,1,3+β2,2,3

4 + 3(β3,1,1+β3,2,2)8 + β3,3,3

4

)• χ

(2)xxz = χ

(2)1,1,3 = χ

(2)1,3,1 = χ

(2)2,2,3 = χ

(2)2,3,2 =

= 1ε0

cos β

(cos 2β

(β1,1,3+β2,2,3

4 + β3,1,1+β3,2,28 − β3,3,3

4

)+ β1,1,3+β2,2,3

4 − β3,1,1+β3,2,28 + β3,3,3

4

)• χ

(2)xyz = χ

(2)1,2,3 = χ

(2)1,3,2 = −χ

(2)2,1,3 = −χ

(2)2,3,1 =

= 1ε0

(3 cos 2β + 1) β1,2,3−β2,1,38

The only chiral component is χxyz, it depends only on βx,y,z – chiral and non-chiral components do not mix under inclination of the molecule. The materials

2.2. Surface susceptibility 13

0 50

0

0.5

1

χ(2

) / ar

b.un

its

inclination angle β (deg)

βz´z´z´

=1

0 50

0

0.5

1

χ(2

) / ar

b.un

its


βz´x´x´

=1

0 50

0

0.5

1

χ(2

) / ar

b.un

its


βx´x´z´

=1

0 50

0

0.5

1

χ(2

) / ar

b.un

its


βx´y´z´

=1

χzzz

χzxx

χxxz

χxyz

Figure 2.2.: χ(2)ijk

calculated from a single βi′j′k′ -component for an increasing inclina-tion angle β of the molecular z′ axis with respect to the surface normal (z axis). In allplots a uniform distribution of the molecule axis is assumed and respective coefficientsof the molecular susceptibility are kept constant.

used in second harmonic generation experiments are generally assumed to showno chiral behavior. The chiral component χxyz is therefore neglected in mostof the following discussion. The other three independent components dependon the molecular hyperpolarizability βi,j,k and the inclination angle β.

Whereas the hyperpolarizability βi,j,k is used for molecules a different ap-proach exists for metal surfaces. Rudnick-Stern-parameters (Rudnick and Stern[78]) make use of a model, where electric currents parallel or perpendicular tothe surface give rise to SHG from metallic surfaces. These parameters are notused in this work.

3. Vector spherical harmonics

Vector spherical harmonics (VSHs) form a set of basis functions for electro-magnetic fields in systems with spherical symmetry and homogeneous electricconstant. Therefore they are used as a basis for all calculations in this work.Section 3.1 will describe their properties and origin and section 3.2 will givesome expansions of electromagnetic fields into VSHs such as plane waves andelectric dipoles. Several addition theorems exist and are needed to translate orrotate VSHs; they are presented in section 3.3.

3.1. Vector spherical harmonics

Electromagnetic waves that fulfill the wave equation in frequency space (com-pare Sec. 2.1.1)

ΔE(r) + k2E(r) = 0 , ΔH(r) + k2H(r) = 0 , (3.1)

can be expanded into a set of vector spherical harmonics (VSH). These VSHsare an extension of the scalar spherical harmonics ψ(r), which fulfill the scalarwave equation:

0 = ∇2ψ + k2ψ , (3.2)

0 = 1r2

∂

∂r

(r

∂

∂rψ

)+ 1

r2 sin ϑ

∂

∂ϑ

(sin ϑ

∂

∂ϑψ

)+ 1

r2 sin ϑ

∂2

∂ϕ2 ψ + k2ψ .

With a separation ansatz (Bohren and Huffman [6], Stratton [86]), ψ(r) can bedetermined to be:

ψ(j)m,n(r) = P m

n (cos ϑ)z(j)n (kr)eimϕ ,

with P mn (cos ϑ) the associated Legendre function of degree n and order m (see

Sec. A.2.2) and z(j)n (kr) the spherical Bessel functions of order n and kind j

(see Sec. A.1.1). Instead of kr, an effective radius ρ = kr is often used forbrevity.

15

16 3. Vector spherical harmonics

The vector spherical harmonics N(j)m,n(r), M

(j)m,n(r) and L

(j)m,n(r) are related

to ψ(j)m,n(r) by

L(j)m,n(r) = ∇ψ(j)

m,n(r) , (3.3)

M (j)m,n(r) = ∇ ×

(cψ(j)

m,n(r))

,

N (j)m,n(r) = 1

k∇ × M (j)

m,n(r) .

They fulfill the wave equation (3.1), if ψ fulfills the scalar wave equation (3.2).

The pilot vector c is chosen c = r, so that M fulfills the vector wave equa-tion in spherical coordinates. The vector spherical harmonics in spherical coor-dinates are then:

L(j)m,n =

⎧⎪⎪⎨⎪⎪⎩

kz′(j)n (kr) P m

n (cos ϑ) eimϕ er +k

z(j)n (kr)

kr

∂P mn

∂ϑ(cos ϑ) eimϕ eϑ +

k z(j)n (kr)

kr

P mn (cos ϑ)

sin ϑimeimϕ eϕ ,

(3.4)

M (j)m,n =

⎧⎪⎨⎪⎩

0 er +z

(j)n (kr) P m

n (cos ϑ)sin ϑ

imeimϕ eϑ +−z

(j)n (kr) ∂P m

n∂ϑ

(cos ϑ) eimϕ eϕ ,

N (j)m,n =

⎧⎪⎪⎨⎪⎪⎩

n(n + 1) z(j)n (kr)

krP m

n (cos ϑ) eimϕ er +[krz

(j)n (kr)]′kr

∂P mn

∂ϑ(cos ϑ) eimϕ eϑ +

[krz(j)n (kr)]′kr

P mn (cos ϑ)

sin ϑimeimϕ eϕ .

For n = 0 all vector spherical harmonics are constantly zero. M(j)m,n does not

have a radial field component, so it describes H-like electric field modes. E-likeelectric field modes are described by N

(j)m,n. L

(j)m,n is only needed, if the wave

function has nonzero divergence.

3.1. Vector spherical harmonics 17

kx

kz N0,1(1)

−20 −10 0 10 20

−20

−15

−10

−5

0

5

10

15

20

kx

ky

kx

kz M0,1(1)

−20 −10 0 10 20

−20

−15

−10

−5

0

5

10

15

20

kx

ky

kx

kz N0,2(1)

−20 −10 0 10 20

−20

−15

−10

−5

0

5

10

15

20

kx

ky

kx

kz M0,2(1)

−20 −10 0 10 20

−20

−15

−10

−5

0

5

10

15

20

kx

ky

kx

kz N0,3(1)

−20 −10 0 10 20

−20

−15

−10

−5

0

5

10

15

20

kx

ky

kx

kz M0,3(1)

−20 −10 0 10 20

−20

−15

−10

−5

0

5

10

15

20

kx

ky

Figure 3.1.: Vector spherical harmonics |N (1)0,1|2, |M (1)

0,1|2, |N (1)0,2|2, |M(1)

0,2|2, |N(1)0,3|2,

and |M(1)0,3|2 in the kx-kz-plane. The inset shows the same VSH in the kx-ky-plane.

The arrows denote the field vectors N(1)0,1, M

(1)0,1, N

(1)0,2, M

(1)0,2, N

(1)0,3, and M

(1)0,3. Al-

though the intensities of the M - and N -fields look similar, the orientation of thefields is perpendicular to each other. The field vectors of the N

(1)0,1-fields lie in the

kx-kz-plane, the vectors of the M(1)0,1-fields lie in the kx-ky-plane.


The orthogonality of the vector spherical harmonics,

2π∫0

dϕ

π∫0

dϑ sin ϑ Mm′,n′ (r) · Mm,n(r) = 0 (m �= m′, n �= n′) ,

2π∫0

dϕ

π∫0

dϑ sin ϑ N m′,n′(r) · Nm,n(r) = 0 (m �= m′, n �= n′) ,

2π∫0

dϕ

π∫0

dϑ sin ϑ Nm′,n′ (r) · Mm,n(r) = 0 ,

is used in Mie theory (see Sec. 4.1) to reduce the continuity conditions to alinear equation of the expansion coefficients.

A vector potential A, expanded into vector spherical harmonics

A = iω

∞∑n=0

n∑m=−n

cm,nL(j)m,n + am,nM (j)

m,n + bm,nN (j)m,n ,

results in the electric and magnetic fields

H = 1μ

∇ × A = − k

iωμ

∑n,m

am,nM (j)m,n + bm,nN (j)

m,n ,

E = iωμ

k2 ∇ × H = −∑n,m

am,nN (j)m,n + bm,nM (j)

m,n .

This means, that only the expansion coefficients am,n and bm,n and the kindj of Bessel functions used in the VSHs are needed to fully describe any electro-magnetic field.

Also other variants of the vector spherical harmonics can be found in lit-erature: Bohren and Huffman [6] and Stratton [86] expand the exponentialfunctions eimϕ and imeimϕ used in Eq. (3.4) in real and imaginary part, la-beled “even” and “odd”.

eimϕ = cos ϕ︸︷︷︸even

+i sin ϕ︸︷︷︸odd

imeimϕ = −m sin ϕ︸︷︷︸even

+i m cos ϕ︸︷︷︸odd

(3.5)

3.1. Vector spherical harmonics 19

This also divides the vector spherical harmonics into an even and an odd part:

L(j)m,n = L(j)

e,m,n + iL(j)o,m,n ,

M (j)m,n = M (j)

e,m,n + iM (j)o,m,n ,

N (j)m,n = N (j)

e,m,n + iN (j)o,m,n .

As this results in an expansion into four independent sets of vector sphericalharmonics, it is sufficient to sum only over positive m. One can easily con-vert the expansion coefficients in the notation with even and odd VSHs intoexpansion coefficients in the notation with positive and negative m and viceversa:

∞∑n=1

n∑m=0

ae,m,nN (j)e,m,n + ao,m,nN (j)

o,m,n + be,m,nM (j)e,m,n + bo,m,nM (j)

o,m,n

=∞∑

n=1

n∑m=−n

am,nN (j)m,n + bm,nM (j)

m,n

To fulfill this equation for any ϑ and ϕ, the summands for positive and negativem have to be the same:

E = ae,m,nN (j)e,m,n + ao,m,nN (j)

o,m,n + be,m,nM (j)e,m,n + bo,m,nM (j)

o,m,n

E = am,nN (j)m,n + a−m,nN

(j)−m,n + bm,nM (j)

m,n + b−m,nM(j)−m,n m ≥ 0

Leading to:

⎛⎜⎜⎝

ae,m,n

ao,m,n

be,m,n

bo,m,n

⎞⎟⎟⎠ =

⎛⎜⎜⎜⎜⎜⎝

1 P −mn

P mn

0 0

i −i P −mn

P mn

0 0

0 0 1 P−mn

P mn

0 0 i −i P −mn

P mn

⎞⎟⎟⎟⎟⎟⎠⎛⎜⎜⎝

am,n

a−m,n

bm,n

b−m,n

⎞⎟⎟⎠ m ≥ 0

⎛⎜⎜⎝

am,n

a−m,n

bm,n

b−m,n

⎞⎟⎟⎠ = 1

2

⎛⎜⎜⎜⎜⎝

1 −i 0 0P m

n

P −mn

i P mn

P −mn

0 00 0 1 −i0 0 P m

n

P −mn

i P mn

P −mn

⎞⎟⎟⎟⎟⎠⎛⎜⎜⎝

ae,m,n

ao,m,n

be,m,n

bo,m,n

⎞⎟⎟⎠


Chew and Wang [16, 94, 15] write the the scalar spherical harmonic as

ψ(j)m,n(r) = z(j)

n (kr)Ym,n(ϑ, ϕ) ,

using the spherical harmonic similar to the definition by Jackson [48]:

Ym,n(ϑ, ϕ) = (−1)m

√2n + 1

4π(n − m)!(n + m)!

P mn (cos ϑ)eimϕ .

This definition differs only by the prefactor from the definition used in thiswork. The vector spherical harmonics are defined in the same way (Eq. (3.3)),where the different prefactor propagates into the VSHs.

3.2. Expansion of electromagnetic fields into vector sphericalharmonics

3.2.1. Plane waveA plane wave, polarized parallel to x and traveling along z, can be written as(Bohren and Huffman [6]):

E(r) = E0eikr cos ϑex

= E0

∞∑n=1

in 2n + 1n(n + 1)M

(1)o,1,n − in+1 2n + 1

n(n + 1)N(1)e,1,n .

The respective expansion coefficients of even and odd VSHs are:

bo,m,n = E0in 2n + 1n(n + 1)

δm,1 , ae,m,n = −ibo,m,n , ao,m,n = be,m,n = 0 .

Written in the notation used in this work (see Eq. (3.5)):

am,n = −E0in+1 2n + 12n(n + 1)

(δm,1 + δm,−1

), (3.6)

bm,n = −E0in+1 2n + 12n(n + 1)

(δm,1 − δm,−1

).

Other polarizations or propagation directions can be obtained from that withthe help of addition theorems for rotation (see Sec. 3.3.1).

3.3. Addition theorems 21

3.2.2. Field emitted by an electric dipoleThe electric field of an electric dipole with dipole moment p is given by (Jackson[48])

E(r) = 14πε0

[k2( r

|r| × p) × r

|r|eikr

r

+

(3 r

|r|

(r

|r|p)

− p

)eikr

(1r3 − ik

r2

)⎤⎦ .

Expressing the field of a dipole parallel to z (p = pez) in spherical coordinates,yields that

Epz (r) = k3p

4πε0iN (3)

0,1 .

The respective dipoles in x- and y-direction are obtained with the help ofaddition theorems for rotation (see Sec. 3.3.1). They are a composition ofN

(3)−1,1 and N

(3)1,1, where the exact factors depend on the definition of the vector

spherical harmonics.

3.3. Addition theorems

Every electromagnetic field can be expanded into a set of vector spherical har-monics (Eq. (3.7)). Therefore, also a translated or rotated field can be writtenas an expansion into vector spherical harmonics, however, with different expan-sion coefficients (Eq. (3.8)):

E(r) =N∑

n=1

n∑m=−n

am,nN (j)m,n(r) + bm,nM (j)

m,n(r) , (3.7)

E′(r) = E(r′) =N∑

ν=1

ν∑μ=−ν

aμ,νN (j)μ,ν(r) + bμ,νM (j)

μ,ν(r) . (3.8)

Realizing that both equations describe the same electric field, only in differentcoordinate systems, one can derive addition theorems for coordinate transfor-mations, that convert the expansion coefficients with respect to one coordinatesystem into expansion coefficients with respect to any other coordinate system.


Addition theorems were published by Friedman and Russek [38] and gener-alized by Stein [85]. Further addition theorems are in Cruzan [18] (with a signerror corrected by Tsang and Kong [90]) and Danos and Maximon [21]. Chew[15] published recursion relations for the expansion coefficients for scalar spher-ical harmonics and modified them for vector spherical harmonics (Chew andWang [16]).

3.3.1. RotationAn electric field E(r) is given with respect to a coordinate system S1. Withrespect to a coordinate system S2, the same electric field is given by E(r′).S2 arises from S1 by a rotation about the Euler angles α, β, γ (see A.3.1):r′ = MEulerr. The transformation from E(r) to E(r′) is given by (Stein [85]):

E(r) =N∑

n=1

n∑m=−n


m,n(r) ,

E(r′) =N∑

n=1

n∑μ=−n

n∑m=−n

am,namμ,n︸︷︷︸

a′μ,n

N (j)μ,n(r′) +

∑m

bm,namμ,n︸︷︷︸

b′μ,n

M (j)μ,n(r′) ,

with

amμ,n(α, β, γ) = eiμαDm

μ,n(β)eimγ

and

Dmμ,n(β) = Am

μ,n

n−max(μ,m)∑σ=max(0,−μ−m)

Bmμ,n,σ

(cos β

2

)2σ+μ+m(sin β

2

)2n−2σ−μ−m

,

Amμ,n = (n − μ)!

(n − m)! ,

Bmμ,n,σ = (−1)n+m−σ

(n + m

n − μ − σ

)(n − m

σ

).

The factors Amμ,n and Bm

μ,n,σ are independent of the rotation angle and can bestored, so that the calculation process becomes faster. Numerical instabilitiescan arise, when the cosine or sine are close to zero, but the exponent is largeand negative.


Also recurrence relations exist (Mackowski [59]), that help to overcome thisnummerical instability. They start from m = 0

Dm=0μ,n = (n − μ)!

n!

n−max(μ,0)∑σ=max(0,−μ)

(−1)n−σ

(n

n − μ − σ

)(nσ

)(

cos β

2

)2σ+μ(sin β

2

)2n−2σ−μ

=

{(−1)μP −μ

n (cos β) β ≥ 0P −μ

n (cos β) β < 0,

with recurrence relations for m �→ m + 1 and m �→ m − 1:

Dm+1μ,n = Dm

μ−1,n cos2 β

2−Dm

μ ,n μ sin β

−Dmμ+1,n (n − μ)(n + μ + 1) sin2 β

2,

(n + m)(n − m + 1)Dm−1μ,n = −Dm

μ−1,n sin2 β

2−Dm

μ ,n μ sin β

+Dmμ+1,n (n − μ)(n + μ + 1) cos2 β

2.

3.3.2. Translation

An electric field E(r) is given with respect to a coordinate system S1. Withrespect to a coordinate system S2, the same electric field is given by E(r′). S2arises from S1 by a translation r′ = r + a.

In contrast to rotation, when translating fields, they are expanded into a


different set of VSHs depending on whether |r′| is smaller or larger than |r|:

E(r) =N∑

n=1

n∑m=−n


m,n(r)

E(r′) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

E(1)(r′) =∑∞

n=1∑n

m=−na

(j)m,nN

(1)m,n(r′) + b

(j)m,nM

(1)m,n(r′)

|r′| < |a|E(3)(r′) =

∑∞n=1∑n

m=−na

(1)m,nN

(j)m,n(r′) + b

(1)m,nM

(j)m,n(r′)

|r′| > |a|

For |r′| < |a|, VSHs with Bessel functions of kind j = 1 are needed to omitdivergence at the origin. For |r′| > |a|, VSHs with the same kind of Besselfunction j is used as the original field. The number of orders needed for con-vergence increases when |r′| approaches |a|, for |r′| = |a|, an infinite numberof orders is needed.

In this work, only translations with a = aez are needed. The expansioncoefficients are given by (Bruning and Lo [12]):

a(j)m,n =

∑n

am,νA(j)m,ν,n(|a|) + bm,νB(j)

m,ν,n(|a|) ,

b(j)m,n =

∑n

bm,νA(j)m,ν,n(|a|) + am,νB(j)

m,ν,n(|a|) ,

with

A(j)m,ν,n(z) = (−1)min−ν 2n + 1

2n(n + 1)n+ν∑

p=|n−ν|

ν(ν + 1) + n(n + 1) − p(p + 1)ip ap,m,n,νz(j)

p (|a|) ,

B(j)m,ν,n(z) = (−1)min−ν 2n + 1

2ν(ν + 1)

n+ν∑p=|n−ν|

−2im|a|ip ap,m,n,νz(j)

p (|a|) .

The term ap,m,n,ν depends on the Wigner 3-j symbol

ap,m,n,ν = (2p + 1)√

(ν + m)!(n − m)!(ν − m)!(n + m)!

(ν n p0 0 0

)(ν n pm −m 0

),


kx

kz E(r)

a

−20 −10 0 10 20

−20

−10

0

10

20kx

kz E(1)(r’)

a

−20 −10 0 10 20

−20

−10

0

10

20

kx

kz E(3)(r’)

a

−20 −10 0 10 20

−20

−10

0

10

20kx

kz E(r’)

−20 −10 0 10 20

−20

−10

0

10

20

Figure 3.2.: Translation of E(r) = N(3)0,1(r) (top left) about a distance a to the bottom.

Two functions are used, depending on |r′| for the translated fields: E(1)(r′) divergesfor |r′| > a (top right), E(3)(r′) diverges for |r′| < a (bottom left). The translatedfield is a combination of both (bottom right), that converges everywhere, except for|r′| ≈ a. The region, where E(1)(r′) does not converge becomes smaller with anincreasing number of orders. Here N = 10 is used.


and can be calculated using recurrence relations (the indices m, n, ν of ap,m,n,ν

are omitted for brevity, p runs from |n − ν| to n + ν in increments of 2):

0 = αp−3ap−4 − (αp−2 + αp−1 − 4m2)ap−2 + αpap ,

αp =((ν + n + 1) − p2) (p2 − (ν − n)2)

4p2 − 1,

aν+n = (2ν − 1)!!(2n − 1)!!(2ν + 2n − 1)!!

(ν + n)!(ν − m)!(ν + m)! ,

aν+n−2 =(2ν + 2n − 3)

(νn − m2(2ν + 2n − 1)

)(2ν − 1)(2n − 1)(ν + n)

aν+n .

4. Mie scattering

Colloids, especially their color effects, were subject to research in the firstdecades of the 20th century. Two theories emerged. On the on hand, the bluecolor of colloidal metals can be described with the theory of Raileigh scattering(Strutt [87]), but not its change to violet and red with particle size. On theother hand, Garnett [40] succeeded in explaining the red color of glass with goldnanoparticles, but failed to explain the blue color. He calculated an effectiverefractive index of the glass using the theory of Lorenz (Lorenz [57]) that onlydepends on the volume fraction of metal. Ehrenhaft [34] suggested resonanteffects, but only Mie [66] proposed a full theory for the scattering at spheressolving Maxwell’s equations in spherical coordinates. Debye [29] transformedthe equations into the modern form using Bessel functions.

The original formulation is based on a plane wave that impinges a sphere.Generalized Mie theory describes scattering of an arbitrary beam by a homo-geneous sphere. As Mie theory directly follows from generalized Mie theory,this more general method is described here (section 4.1). The approach canbe further generalized to a stratified sphere consisting of several layers (sec-tion 4.2). In contrast to the straightforward formulation of Mie theory, someadaptations are needed for numerical stability when implementing the formulasinto programming code. They are described in section 4.4.

4.1. Continuity condition

Electric and magnetic fields have to fulfill continuity conditions at the transitionbetween any two materials in oder to fulfill energy conservation (Born and Wolf[7]).

For the Mie formalism, all fields are decomposed into vector spherical har-monics. For the incident fields (E, H inc), the scattered fields inside the sphere(E, Hsc,i) and the scattered fields outside the sphere (E, Hsc,o), the followingdecomposition is used (see chapter 3 for other conventions for vector spherical

27

28 4. Mie scattering

harmonics):

{EincH inc

}(r) =

{1

− ikoωμ

} ∞∑n=0

n∑m=−n

{ainc

m,n

bincm,n

}N jinc

m,n(r) +{

bincm,n

aincm,n

}M jinc

m,n(r) ,

{Esc,oHsc,o

}(r) =

{1

− ikoωμ

} ∞∑n=0

n∑m=−n

{asc,o

m,n

bsc,om,n

}N

jsc,om,n (r) +

{bsc,o

m,n

asc,om,n

}M

jsc,om,n (r) ,

{Esc,iHsc,i

}(r) =

{1

− ikiωμ

} ∞∑n=0

n∑m=−n

{asc,i

m,n

bsc,im,n

}N

jsc,im,n(r) +

{bsc,i

m,n

asc,im,n

}M

jsc,im,n(r) .

jsc,o = 3 is chosen (Hankel functions), to ensure a spherical wave in the farfield, jsc,i = 1 is needed to omit divergence at the origin.

At the surface of the sphere (r = R, independent of ϑ or ϕ), the tangentialcomponents of the fields have to be continuous:

Esc,iϑ,ϕ = Esc,oϑ,ϕ + Eincϑ,ϕ , Hsc,iϑ,ϕ = Hsc,oϑ,ϕ + Hincϑ,ϕ . (4.1)

The continuity of the normal components of D and B is not needed to calculatethe fields, but can be used to determine surface charge and surface current.

The fields are expanded into vector spherical harmonics. Using the RicattiBessel function ψ (see Sec. A.1.2), the tangential components are:

Eϑϕ

∝∑m,n

eimϕ

⎡⎣amn

ψ′n(ρ)ρ

{∂P m

n (cos ϑ)∂ϑ

P mn (cos ϑ)

sin ϑim

}+ bmn

ψn(ρ)ρ

{P m

n (cos ϑ)sin ϑ

im− ∂P m

n (cos ϑ)∂ϑ

}⎤⎦ ,

Hϑϕ

∝∑m,n

eimϕ

⎡⎣amnψn(ρ)

{P m

n (cos ϑ)sin ϑ

im− ∂P m

n (cos ϑ)∂ϑ

}+ bmnψ′

n(ρ)

{∂P m

n (cos ϑ)∂ϑ

P mn (cos ϑ)

sin ϑim

}⎤⎦ .

Multiplying Eq. (4.1) by e−im′ϕ (⇒ δm,m′ ) andP m′

n′ (cos ϑ)sin ϑ

(⇒ δn,n′ , separation

of N and M , see Eq. (A.4) and Eq. (A.5)) and integrating over2π∫0

π∫0

dϕdϑ sin ϑ

yields, that if Eq. (4.1) is satisfied, it is satisfied for the radial part of eachsummand individually, M and N do not couple. Four equations remain, one

4.2. Stratified sphere 29

for each the the N -part or M -part of E as well as of H:

aincm,n

(ψ

(jinc)n (ρo)

1ρo

ψ′(jinc)n (ρo)

)=

(ψ

(jsc,i)n (ρi) −ψ

(jsc,o)n (ρo)

1ρi

ψ′(jsc,i)n (ρi) − 1

ρoψ′(jsc,i)

n (ρo)

)(asc,i

m,n

asc,om,n

),

bincm,n

(1

ρoψ

(jinc)n (ρo)

ψ′(jinc)n (ρo)

)=

(1ρi

ψ(jsc,i)n (ρi) − 1

ρoψ

(jsc,o)n (ρo)

ψ′(jsc,i)n (ρi) −ψ′(jsc,o)

n (ρo)

)(bsc,i

m,n

bsc,om,n

).

The solution for asc,im,n, bsc,i

m,n, asc,om,n and bsc,o

m,n does only depend on the radialcomponent of the field expansion, ψn(ρ), the m-dependence arises solely fromthe expansion of the incident field:(

asc,im,n

asc,om,n

)= ainc

m,n ·(

ψjsc,in (ρi) −ψ

jsc,on (ρo)

1ρi

ψ′jsc,in (ρi) − 1

ρoψ′jsc,o

n (ρo)

)−1(ψjinc

n (ρo)1

ρoψ′jinc

n (ρo)

)︸︷︷︸

=

(asc,i

n

asc,on

),

(bsc,i

m,n

bsc,om,n

)= binc

m,n ·(

1ρo

ψjsc,in (ρi) − 1

ρiψ

jsc,on (ρo)

ψ′jsc,in (ρi) −ψ′jsc,o

n (ρo)

)−1(1

ρoψjinc

n (ρo)ψ′jinc

n (ρo)

)︸︷︷︸

=

(bsc,i

n

bsc,on

).

Therefore usually reduced expansion coefficients ascn , bsc

n are calculated in a firststep and in a second step asc

m,n, bscm,n are calculated. The expansion coefficients

of the incident field aincm,n and binc

m,n act as beam shape coefficients (Kerker [52]).

If the incident field is not restricted to the outside of the sphere, but insteadto the inside of the sphere, only the sign of the incident field expansion ψ(jinc)

changes.

4.2. Stratified sphere

For a sphere consisting of L layers (core: l = 1, outermost layer l = L, surround-ing l = L + 1), the continuity condition has to be fulfilled at every boundary.The material properties of the different layers are given by the angular wavenumbers kl. For every boundary with radius rl two effective radii for the inner


and the outer interface exist: ρil = klrl for the inner interface and ρo

l = rlkl+1for the outer interface. The fields in the layers l = 2 . . . L will in general be a

k1 k2 k3 k4 k5

r1

r2

r3r4

ρi2

ρo2

Figure 4.1.: Stratified sphere

superposition of vector spherical harmonics of kind j = 1 and j = 2, or of kindj = 1 and j = 3:{

Esc,lHsc,l

}(r) =

{− iklr

ωμ

} ∞∑n=0

n∑m=−n

{al

m,n

blm,n

}N (1)

m,n(r) +{

alm,n

blm,n

}N (3)

m,n(r)

+{

blm,n

alm,n

}M (1)

m,n(r) +{

blm,n

alm,n

}M (3)

m,n(r) .

The continuity conditions are formed in exactly the same way as for a homo-geneous sphere. Again, Ricatti Bessel functions ψ (kind j = 1) and χ (kindj = 3) are used (see Sec. A.1.2). The equations are formulated for the case,that the incident field originates outside the sphere. For the case of a field thatoriginates inside the core or inside one of the layers see Sec. 4.3.

• Inside the core of the stratified particle, am,n and dm,n vanish, becausethe respective vector spherical harmonics of kind j = 2 or j = 3 divergeat the origin. Only fields with j = 1 remain. For the boundary betweenthe core and the first layer (r = r1) the continuity condition is:

4.2. Stratified sphere 31

0 = Esc,1(r1) − Esc,2(r1) ,

0 = Hsc,1(r1) − Hsc,2(r1) .

(00

)=

(ψ(ρi

1) −ψ(ρo,1) −ξ(ρo

1)ψ′(ρi

1)ρi

1− ψ′(ρo

1)ρo

1− ξ′(ρo

1)ρo

1

)⎛⎝a1a2a2

⎞⎠ , (4.2)

(00

)=

(ψ(ρi

1)ρi

1− ψ(ρo

1)ρo

1− ξ(ρo

1)ρo

1ψ′(ρi

1) −ψ′(ρo1) −ξ′(ρo

1)

)⎛⎝b1b2b2

⎞⎠ . (4.3)

• All boundaries rl between the shell layer l and l + 1, with l = 2 . . . L − 1form in the same way:

0 = Esc,l(rl) − Esc,l+1(rl) ,

0 = Hsc,l(rl) − Hsc,l+1(rl) .

(00

)=

(ψ(ρi

l) ξ(ρil) −ψ(ρo

l ) −ξ(ρol )

ψ′(ρil)

ρil

ξ′(ρil)

ρil

− ψ′(ρol

)ρo

l− ξ′(ρo

l)

ρol

)⎛⎜⎜⎝al

al

al+1al+1

⎞⎟⎟⎠ , (4.4)

(00

)=

(ψ(ρi

l)

ρil

ξ(ρil)

ρil

− ψ(ρol

)ρo

l− ξ(ρo

l)

ρol

ψ′(ρil) ξ′(ρi

l) −ψ′(ρol ) −ξ′(ρo

l )

)⎛⎜⎜⎝bl

bl

bl+1bl+1

⎞⎟⎟⎠ . (4.5)

• The incident field is only defined in the outside of the system. At theboundary r = rL between the outermost layer and the surrounding, theincident field enters the continuity condition:

Einc(rL) = Esc,L(rL) − Esc,L+1(rL) ,

H inc(rL) = Hsc,L(rL) − Hsc,L+1(rL) .


(ξ(ρo

L)ξ′(ρo

L)

ρoL

)=

(ψ(ρi

L) ξ(ρiL) −ξ(ρo

L)ψ′(ρi

L)ρo

L

ξ′(ρiL)

ρoL

− ξ′(ρoL)

ρiL

)⎛⎝ aL

aL

aL+1

⎞⎠ , (4.6)

(ξ(ρo

L)ρo

L

ξ′(ρoL)

)=

(ψ(ρi

L)ρi

L

ξ(ρiL)

ρiL

− ξ(ρoL)

ρoL

ψ′(ρiL ξ′(ρi

L) −ξ′(ρoL)

)⎛⎝ bL

bL

bL+1

⎞⎠ . (4.7)

The equations for every order n can be combined in one 2L×2L matrix thatis used to determine the expansion coefficients of the scattered fields inside andoutside the stratified sphere:

hn = Mnvn , (4.8)

gn = Nnwn .

The structure of the matrices Mn and Nn and the vectors vn, wn, hn, and gn

are:

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

00...0000...

XX

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

X X X 0 0 0 0 · · ·X X X 0 0 0 0 · · ·

......

· · · X X X X 0 0 · · ·· · · X X X X 0 0 · · ·· · · 0 0 X X X X · · ·· · · 0 0 X X X X · · ·

......

· · · 0 0 0 0 X X X· · · 0 0 0 0 X X X

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

x1x2x2...

xl

xl

xl+1xl+1

...xL

xL

xL+1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

The radius r is constant within every row, k is constant within every columns.So each row is equivalent to one continuity equation at one of the boundaries.This method was first described by Aden and Kerker [1] in 1951.

The expansion coefficients aincm,n and binc

m,n of the incident field (beam shapecoefficients) are then used to convert the reduced expansion coefficients intothe actual expansion coefficients for the scattered fields.

4.3. Scattering of a dipole at a stratified sphere 33

4.3. Scattering of a dipole at a stratified sphere

If the scattering process originates from a field source inside any of the shells orthe core of the particle, only the left hand side of Eq. (4.8) has to be changed.In the above formulas, the incident field enters at the boundary r = rL. TheBessel functions of the incident field are therefore in the last two rows of vectorhn and gn. If the incident field is in the core of the particle, the field enters atthe continuity condition at r = r1. The Bessel functions shift to the first tworows of the vectors hn and gn.

rcrs

�(a)� (b)

�(c)

Figure 4.2.: Sphere consisting of a core (radius rc) and one shell (radius rs). Withthree different possible positions of a dipole.

If the incident field is given by a dipole, the situation is a bit more compli-cated. A dipole at a position rd is given by the translation (see Sec. 3.3.2) ofthe expansion of a dipole (see Sec. 3.2.2):

E(r) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

E(1)(r) =N∑

n=1

n∑m=−n

a(1)m,nN

(1)m,n(r) + b

(1)m,nM

(1)m,n(r) |r| < |rd|

E(3)(r) =N∑

n=1

n∑m=−n

a(3)m,nN

(3)m,n(r) + b

(3)m,nM

(3)m,n(r) |r| > |rd|

.

The expansion coefficients and the kind of Bessel functions used, depend on theposition in space with a discontinuity at the surface of a sphere that includesthe origin of the dipole. Outside the sphere, the dipole is given by a expansionE(1)(r), inside the sphere by E(3)(r) with Bessel functions of first and thirdkind. This discontinuity propagates to the expansion of the scattered fields.Three cases have to be considered: (a) the dipole is inside the core of theparticle, (b) the dipole is in one of the layers, and (c) the dipole is outside thesphere. For simplicity, a sphere consisting of a core and one layer is assumed.The radius of the core is given by rc, the radius of the shell by rs (see Fig. 4.2).Additionally, only the continuity conditions for the electric fields are discussed


in this section. Like in the previous section also the magnetic fields have to betaken into account.

For the first case, of a dipole inside the core of the particle, only the E(3)(r)-fields enter the continuity conditions:

E(3)ϑ,ϕ(rc) + Esc,core

ϑ,ϕ (rc) = Esc,shellϑ,ϕ (rc) ,

Esc,shellϑ,ϕ (rs) = Esc,out

ϑ,ϕ (rs) .

The fields in the individual regions are:

|rd| rc rs

�(a)Esc,core

+E(1)Esc,core

+E(3)Esc,shell Esc,out

Case (c), where the dipole is outside the sphere is similar. Only E(1)(r)enters the continuity conditions:

Esc,coreϑ,ϕ (rc) = Esc,shell

ϑ,ϕ (rc) ,

Esc,shellϑ,ϕ (rs) = Esc,out

ϑ,ϕ (rs) + E(1)ϑ,ϕ(rs) .


|rd|rc rs

�(c)Esc,coreEsc,shell Esc,out

+E(1)Esc,out

+E(3)

Only case (b) includes both dipole field expansions in the continuity condi-tions:

Esc,coreϑ,ϕ (rc) = Esc,shell

ϑ,ϕ (rc) + E(1)ϑ,ϕ(rc) ,

E(3)ϑ,ϕ(rs) + Esc,shell

ϑ,ϕ (rs) = Esc,outϑ,ϕ (rs) .


|rd|rc rs

�(b)Esc,coreEsc,shell

+E(1)Esc,shell

+E(3)Esc,out

4.4. Numerical implementation 35

4.4. Numerical implementation

The formulation of Mie scattering in sections 4.1 and 4.2 encounters severalnumerical problems. Most importantly, the recurrence relations for Bessel func-tions are numerically not stable. Therefore the matrices from Sec. 4.2 have tobe reformulated.

Instead of Bessel functions, ratios of Bessel functions are used:

R(j)n (ρ1, ρ2) = ψ

(j)n (ρ1)

ψ(j)n (ρ2)

, (4.9)

and for the derivatives

D(j)n (ρ) = ψ′(j)

n (ρ)ψ

(j)n (ρ)

= ddρ

log[ψ(j)(ρ)

]. (4.10)

This follows naturally, when the Bessel function ψ(j)(ρil) in the equation is

shifted to the expansion coefficients al and bl. For example, equation (4.4)becomes:

(ψ(ρi

l) ξ(ρil) −ψ(ρo

l ) −ξ(ρol )

ψ′(ρil)

ρil

ξ′(ρil)

ρil

− ψ′(ρol

)ρo

l− ξ′(ρo

l)

ρol

)⎛⎜⎜⎝al

al

al+1al+1

⎞⎟⎟⎠

=

⎛⎜⎝ 1 1 − ψ(ρo

l)

ψ(ρil+1) − ξ(ρo

l)

ξ(ρil+1)

ψ′(ρil)

ρilψ(ρi

l)

ξ′(ρil)

ρilξ(ρi

l) − ψ′(ρo

l)

ρol

ψ(ρil+1) − ξ′(ρo

l)

ρol

ξ(ρil+1)

⎞⎟⎠⎛⎜⎜⎝

ψ(ρil)al

ξ(ρil)al

ψ(ρil+1)al+1

ξ(ρil+1)al+1

⎞⎟⎟⎠

=

⎛⎝ 1 1 −R(1)(ρo

l , ρil+1) −R(3)(ρo

l , ρil+1)

D(1)(ρil)

ρil

D(3)(ρil)

ρil

− ψ′(ρol

)ρo

lψ(ρi

l+1)ψ(ρo

l)

ψ(ρol

) − ξ′(ρol

)ρo

lξ(ρi

l+1)ξ(ρo

l)

ξ(ρol

)

⎞⎠⎛⎜⎜⎝

····

⎞⎟⎟⎠


=

⎛⎝ 1 1 −R(1)(ρo

l , ρil+1) −R(3)(ρo

l , ρil+1)

D(1)(ρil)

ρil

D(3)(ρil)

ρil

− D(1)(ρol

)R(1)(ρil+1,ρo

l)

ρol

− D(3)(ρol

)R(3)(ρil+1,ρo

l)

ρol

⎞⎠⎛⎜⎜⎝

····

⎞⎟⎟⎠ .

The logarithmic derivatives (Eq. (4.10)) can be calculated by upward ordownward recurrence relations. Dave [22] found that the upward recurrenceis unstable for absorbing spheres but much faster than downward recurrence.Wiscombe [95] found an algorithm that decides a priory whether D

(j)n should

be calculated using upward or downward recurrence and implemented a fasterdownward recurrence relation by Lentz [56]. To take advantage of vector struc-ture, he proposed an estimate of the numbers of orders needed in Mie series.Here the recurrence relations given by Mackowski et al. [58] in equations 62–67are used:

D(1)n−1(z) = n

z− 1

D(1)n (z) + n

z

, D(1)Nmax+15(z) = 0 + 0i ,

D(3)n (z) = D(1)

n (z) + iψn(z)ξn(z)

, D(3)0 (z) = i ,

ψn(z)ξn(z) = ψn−1(z)ξn−1(z)(

n

z− D

(1)n−1(z)

)(n

z− D

(3)n−1(z)

),

ψ0(z)ξ0(z) = −ieiz sin z ,

ψn(ρL) = ψn−1(ρL)(

n

ρL− D

(1)n−1(ρL)

), ψ0(ρL) = sin(ρL) ,

ξn(ρL) = ξn−1(ρL)(

n

ρL− D

(3)n−1(ρL)

), ξ0(ρL) = sin(ρL) − i cos(ρL) .

The ratios of Bessel functions (Eq. (4.9)) were defined by Sitarski [83] andalso used by Du [32], who presented a quantitative criterion to decide whetherR

(j)n is calculated using upward or downward recurrence. This method has

been further expanded by Kai and Massoli [51], Wu et al. [96], Yang [100] andothers. In Eq. 65, Mackowski et al. [58] gives the following recurrence relation:

R(j)n (z1, z2) = R

(j)n−1(z1, z2)

D(j)n (z2) + n

z2

D(j)n (z1) + n

z1

.


Using this approach and multiplying some of the equations with nlnl+1rl tomove ρ to a different position yields the improved form for equations (4.4) and(4.5) for every layer rl with l = 2 . . . L − 1:(

kl+1 kl+1 −klR(1)n (ρo

l , ρil+1) · · ·

D(1)n (ρi

l) D(3)n (ρi

l) D(1)n (ρo

l )R(1)n (ρo

l , ρil+1) · · ·

· · · −klR(3)n (ρo

l , ρil+1)

· · · D(3)n (ρo

l )R(3)n (ρo

l , ρil+1)

)·

⎛⎜⎜⎝

an,lψn(ρil)

an,lξn(ρil)

an,l+1ψn(ρil+1)

an,l+1ξn(ρil+1)

⎞⎟⎟⎠ =

(00

),

(1 1 −R

(1)n (ρo

l , ρil+1) · · ·

kl+1D(1)n (ρi

l) kl+1D(3)n (ρi

l) −klD(1)n (ρo

l )R(1)n (ρo

l , ρil+1) · · ·

· · · −R(3)n (ρo

l , ρil+1)

· · · −klD(3)n (ρo

l )R(3)n (ρo

l , ρil+1)

)·

⎛⎜⎜⎝

bn,lψn(ρil)

bn,lξn(ρil)

bn,l+1ψn(ρil+1)

bn,l+1ξn(ρil+1)

⎞⎟⎟⎠ =

(00

).

The equations for the first boundary (Eq. (4.2) and (4.3)) are:(k2 −k1R

(1)n (ρo

1, ρi2) −k1R

(3)n (ρo

1, ρi2)

D(1)n (ρi

1) D(1)n (ρo

1)R(1)n (ρo

1, ρi2) D

(3)n (ρo

1)R(3)n (ρo

1, ρi2)

)

·

⎛⎝an,1ψn(ρi

1)an,2ψn(ρi

2)an,2ξn(ρi

2)

⎞⎠ =

⎛⎝0

00

⎞⎠ ,

(1 −R

(1)n (ρo

1, ρi2) −R

(3)n (ρo

1, ρi2)

k2D(1)n (ρi

1) −klD(1)n (ρo

1)R(1)n (ρo

1, ρi2) −klD

(3)n (ρo

1)R(3)n (ρo

1, ρi2)

)

·

⎛⎝bn,1ψn(ρi

1)bn,2ψn(ρi

2)bn,2ξn(ρi

2)

⎞⎠ =

⎛⎝0

00

⎞⎠ .


At the boundary r = rL equations (4.6) and (4.7) are modified to(kL+1 kL+1 −kLξn(ρo

L)D

(1)n (ρi

L) D(3)n (ρi

L) − ξn(ρoL

)−nξn(ρoL

)ρo

L

)

·

⎛⎝an,Lψn(ρi

L)an,Lξn(ρi

L)an,L+1

⎞⎠ =

(kL · ψn(ρo

L)ψn(ρo

L)−nψn(ρo

L)

ρoL

),

(1 1 −ξn(ρo

L)D

(1)n (ρi

L) D(3)n (ρi

L) −kLξn(ρo

L)−nξn(ρoL)

ρoL

)

·

⎛⎝bn,Lψn(ρi

L)bn,Lξn(ρi

L)bn,L+1

⎞⎠ =

(ψn(ρo

L)kL

ψn(ρoL

)−nψn(ρoL

)ρo

L

).

These formulation was derived by Sitarski [83].Mackowski et al. [58] took the approach one step further and replaced the

system of linear equations by single equations for the expansion scatteringcoefficients al

n, bln, al

n, and bln in every layer. This elegant solution also gives

additional numerical stability and needs less computation time. However, itis not straight forward to move the incident field from the layer L + 1 to anyshell layer or the core, which is needed to calculate second harmonic scatteringwhich originates not only from the outer interface but from the interfaces insidethe stratified particle as well. Therefore the equations used in this section andthe section before are used for the numerical implementation.

An additional aspect in the numerical implementation is the storage of resultsand intermediate results. All fields are fully described by their decompositioninto vector spherical harmonics

E =N∑

n=1

n∑m=−n

am,nNm,n + bm,nMm,n .

It is therefore sufficient to perform all calculations on the expansion coefficientsam,n, bm,n only. If they were saved as a two-dimensional array of size (2N, N −1), one half of the array would be empty, as |m| ≤ n. Saving am,n into aone-dimensional array of length N(N + 2) saves computer memory. The array


can be stored as {a−1,1, a0,1, a1,1, a−2,2, a−1,2, a0,2, a2,1, a2,2,. . .} generallyam,n = Ni with index i =

∑n

k=1(2k) + m = n(n + 1) + m.Fields of the same kind j can be added by adding the expansion coefficients

for every order m and n. Only for fields of different j, or when not the fields,but the intensities are added, the actual fields have to be calculated.

5. Theory of second harmonic and sumfrequency scattering from the surface ofspherical particles

The main aim of this work was to develop a model to calculate second harmonicand sum frequency scattering from colloidal spherical nanoparticles. The de-veloped model needs to be viewed in context to other existing models for SHGand SFG. Section 5.1 therefore gives a review of the models that existed beforeand were developed by other groups during this work. The existing models thatallow modeling of SHG from spherical nanoparticles can be divided in threegroups: nonlinear Mie models (sec. 5.1.1), exact solutions in the small particlelimit (sec. 5.1.2), and exact solutions in the vanishing refractive index contrastlimit (sec. 5.1.3). Section 5.2 presents the newly developed model.

5.1. Existing theoretical models for second harmonic and sumfrequency scattering from spherical particles

This sections aims to give a summary of existing theoretical models, that de-scribe sum frequency and second harmonic scattering from spherical particles.

5.1.1. Nonlinear Mie theoryUsing Mie-theory to calculate sum frequency scattering was already proposedby Agarwal and Jha [2] in 1982. There exist now several nonlinear Mie models.All models divide the second harmonic generation process into three steps:First, the incident field at fundamental harmonic wavelength is scattered. Inthe second step, the resulting electric field at the surface of the particle givesrise to a second harmonic polarization at the interface of the particle. In athird step, this second harmonic polarization causes the second harmonic lightthat is radiated away from the particle.

One difference between the models lies in the way, the nonlinear polarizationis calculated. Not all models include all components of the nonlinear surface

41

42 5. Theory of SHG and SFG from spherical particles

susceptibility, Östling et al. [69] only calculates a surface charge instead of thenonlinear polarization. This restriction to certain components of χ(2) limitsthe possible applications of the model.

The main difference, however, are the different ways how the nonlinear polar-ization is converted to a radiated second harmonic field. The different modelsuse different kind of continuity conditions or take advantage of different physi-cal processes.

In contrast to the newly developed model, all existing Mie models restrictthe source of the second harmonic light to the surface of the particle. A sourceaway from the particle’s surface, for example when oriented molecules nextto the surface give rise to a second harmonic response, is not included in themodel.

5.1.1.1. Östling’s anharmonic oscillator model

In 1993, Östling, Stampfli, and Bennemann [69] calculated the second harmonicfields from a metallic particle, using the surface charge. The surface charge iscaused by the fundamental harmonic fields as [see 69, Eq. 4]:

σ = 1ε0

Re(

1 − 1ε(ω)

)r

|r| · Eoute−iωt .

The source of second harmonic radiation is taken as the square of the linearsurface charge. This approach is very intuitive and probably fits well to metallicsurfaces. It is equivalent to a pure χzzz-component and does not include othercomponents of χ(2). Thus it is not possible to apply this model for example toMalachite Green on polystyrene, where χ

(2)zxx is dominating. To determine the

second harmonic fields, the following continuity conditions are used for σ:

(Dout − Din) · n = 4πσ2 , (Eout − Ein) × n = 0 .

5.1.1.2. Pavlyukh’s nonlinear sheet model

In 2004, Pavlyukh and Hübner [70] proposed a nonlinear Mie model with con-tinuity conditions [see 70, Eq. 16–19]

n ×(

E2ωout − E2ω

in

)= 0 , n ·

(D2ω

out − D2ωin

)= σ2ω ,

n ×(

H2ωout − H2ω

in

)= j2ω , n ·

(B2ω

out − B2ωin

)= 0 .

5.1. SHG and SFG from sphercial particles 43

where the surface charge and surface current are caused by the nonlinear po-larization

j2ω = −2iδωn ×(

P 2ω × n)

, σ2ω = n · P 2ω .

The use of surface currents instead of only surface charges allows to includefurther components of the nonlinear susceptibility. The authors recognize threeindependent χ(2) components in their paper. The numerical solution, however,is restricted to χ

(2)zzz.

5.1.1.3. De Beer’s and Roke’s reciprocity theorem

Roke, Bonn, and Petukhov [77] proposed a reciprocity theorem to calculatenonlinear scattering from a spherical particle in 2004 [77].

This model also uses currents to describe the nonlinear response. The cur-rent source j1 is related to the nonlinear polarization, caused by the scatteredfundamental harmonic waves:

j1(r) = ∂P (r)∂t

, Pi = ε0∑

jk

χ(2)ijkEk1

j Ek2k .

The electric fields at second harmonic frequency that are caused by thiscurrent source are calculated using a reciprocity theorem (Fig. 5.1). Thus, thesecond harmonic fields generated by the current source j1(r) are calculated bycalculating the field inside a particle caused by a dipole with current source j2at a position r0 far away from the particle.

Placing the dipole for away from the particle is an additional trick in orderto be able to use standard Mie scattering algorithms, that only work for planewaves, not for dipole fields. The fields generated by j2 can be treated like planewaves if |r0| is much larger than the dimensions of the particle. Thus, wellknown Mie scattering algorithms for plane waves can be used, as implementedby de Beer and Roke [24] in 2009. Roke et al. [77] solved the problem in thesmall particle limit.

5.1.1.4. Gonella’s and Dai’s model

In 2011, Gonella and Dai [42] expanded the theory for small particles to afull Mie theory as suggested by Dadap et al. [20]. The underlying continuity


Figure 5.1.: The two situation, that are related in the reciprocity theorem: SituationI: The fundamental harmonic fields Ek1. . . Ekn cause a nonlinear polarization andthus the current source j1 which causes the sum frequency field E1 detected at r0.Situation II: A dipole point source with current source j2 causes a field E2 inside theparticle.Drawing based on Fig. 1 from Roke et al. [77] – Sylvie Roke, Mischa Bonn, and AndreiV. Petukhov: "‘Nonlinear optical scattering: The concept of effective susceptibility"’.Physical Review B, 70 (11) Copyright 2004 The American Physical Society.

condition, which relates the nonlinear polarization to the sum frequency fields,is [see 42, Eq. S12]:

ΔE‖(2ω) = − 4πε′(2ω)

∇‖P⊥(2ω) ,

ΔH‖(2ω) = 4πi2ω

cn × P (2ω) .

A comparison between a graph from 2012 and a plot with the same parame-ters using the our own code, shows good agreement (Fig. 5.2).

5.1.1.5. Brevet’s model for metallic nanoshells

Recently the group of Brevet developed a Mie-model for nanoshell particles,which can also be applied to solid particles (Butet et al. [13]). Like Goneallaand Dai, they use the polarization sheet model based on Heinz [45].

At the boundary between dielectric core and metallic shell, as well as theboundary between the metallic shell and the dielectric surrounding, the po-larizations are calculated from the scattered fields inside the metal, but thepolarizations are located in the dielectric.


0.5 1

30

210

60

240

90 270

120

300

150

330

180

0

0.5 1

30

210

60

240

90 270

120

300

150

330

180

0

χzzz

χzxx

χxxz

Figure 5.2.: Comparison of the results of Fig. 2 of Gonella et al. [43] (top) with the Miemodel developed in this work (bottom). Left: Polystyrene sphere with 2R = 88 nm.Right: Silver sphere with 2R = 80 nm diameter. Surrounding: water.Reprinted (adapted) with permission from "‘The Effect of Composition, Morphology,and Susceptibility on Nonlinear Light Scattering from Metallic and Dielectric Nanopar-ticles"’ by Grazia Gonella, Wei Gan, Bolei Xu, and Hai-Lung Dai. The Journal ofPhysical Chemistry Letters 2012 3 (19), 2877-2881. Copyright 2012 American Chem-ical Society.


5.1.2. Exact solutions in the small particle limitIn 1999, Dadap et al. [19] proposed a model for surface second harmonic gen-eration in the small particle limit. Expanding the resulting second harmonicfield or the generating vector potential into vector spherical harmonics, yieldsthat for a small sphere electric dipole and quadrupole terms dominate. Usingj = ∂P (2ω)

∂t, P = ε0χ(2)EE, and a plane wave that is undisturbed by the sphere,

the vector potential can be written as ([see 20, Eq. 4,6–7]):

A(r) = 1c

∫d3r′ eik1|r−r′|j(r′)

|r − r′|

≈ −ikeik1r

r

(p − n × m − ik1

6Q(n)

),

with n = r|r| , the electric dipole moment

p =∫

drP (r) ,

the magnetic dipole moment

m = − ik2

∫drr × P (r) ,

and the electric quadrupole moment

Q(r) =∫

dr′3(2(nr′)P (r′) + nP (r′)r′)− 2r′P (r′)n .

In 2004, Dadap et al. [20] expanded the model to include a bulk contributionterm ([see 20, Eq. 3]), that has also been used by Agarwal and Jha [2] in 1982to calculate SHG from the bulk of metallic nanoparticles in the small particlelimit:

P bulk = ε0γ∇(EE) + δ′(E∇)E .

This results in the second harmonic field ([see 19, Eq. 1,2]):

E(2ω) = keikr

ε(2ω)r

[r

|r| ×(

p − ik6

Q

)]× r

|r| ,


where p and Q depend on two effective nonlinear susceptibilities

χ1 = χ(2)zzz + 4χ(2)

zxx − 2χ(2)xxz + 5γ ,

χ2 = χ(2)zzz − χ(2)

zxx + χ(2)xxz .

5.1.3. Rayleigh-Gans-Debye-theory

Rayleigh-Debye-Gans-scattering is an approximation for the linear scatteringof particles of arbitrary shape, but with low refractive index contrast and smallsize. It was first proposed by Rayleigh [75] in 1881, expanded by Debye [30]in 1915 and reformulated by Gans [39] in 1925. In RGD-Theory, the volumeof the scatterer is discretized into elements, that behave like dipole Rayleighscatterers.

It is assumed, that the refractive index of the particle and the surroundingare so similar, that the incident wave is not disturbed by the presence of theparticle. The field inside the particle is assumed to be equal to the incident wave.The fields of all scatterers in the volume of the particle are added coherentlywith a phase difference δ and contribute to the total scattered field.

This approach of linear RGD scattering is expanded to nonlinear scatteringfrom the surface of particles. Source for the second harmonic fields is thenonlinear polarization:

P(2)i (r) = ε0

3∑jk=1

χ(2)ijkE1,j(r)E2,k(r) .

As the fundamental harmonic fields are assumed to be undistorted by the pres-ence of the scatterer, a plane wave is used (Ei = Eieikirei):

P(2)i = ε0

3∑jk=1

eiχ(2)ijkE1eik1re1.ejE2eik2re2.ek

= ε0E1E2ei(k1+k2)r

3∑ijk=1

eiχ(2)ijk(e1.ej)(e2.ek) .

Maxwell’s equations can then be used to calculate the electric displacement


field D caused by P (2) [11, p. 48]:

∇2DSH − ε0μ0

c2∂2

∂t2 DSH = −4π∇ × ∇ × P (2) ,(∇2 − k2

)DSH = −4π∇ × ∇ × P (2) .

For large distances, where P (2) = 0 and where one can assume spherical waves,the electric field becomes:

ESH = 1ε0

∇ × ∇ ×∫

dr′ eik|r−r′|

|r − r′| P (2)(r′) .

Using some simplifications for large distances, one arrives at [11, p. 48]:

ESH = − eikr

ε0r

∫dr′er × er × P (2)(r′)e−ier .r′

.

Using q = kSH − (k1 + k2):

ESH = E1E2k2SH

eikSHr

ε0rer × er ×

∫dr′eiqr′

3∑ijk=1

eiχ(2)ijk(e1.ej)(e2.ek) .

The RGD-Theory has been used by several groups to simulate second har-monic and sum frequency scattering from nano spheres. These include Mar-torell et al. [61], who restricted the solution to the χ

(2)zzz-component, Roke et al.

[76] and de Beer and Roke [23], who expanded the model to all four χ(2)-components. De Beer et al. [28] also generalized the concept to particles ofarbitrary shape. Viarbitskaya et al. [91] even went a step further by develop-ing a generalized nonlinear RGD theory, where also dispersion is taken intoaccount. The Wentzel-Kramers-Brillouin model still assumes propagation ofthe fundamental harmonic wave without refraction, but includes a phase factorproportional to the index contrast. This was also studied by Roke et al. [77].

5.2. Molecular Mie model

The existing RGD model (sec. 5.1.3) and the exact solution in the small parti-cle (sec. 5.1.2) limit lack a consistent description of the linear field propagationaround the spheres. They are therefore not suitable to quantitatively studythe nonlinear light scattering from larger nano- and microparticles. Mie theory

5.2. Molecular Mie model 49

offers such a complete consideration of the physical effects due to the pres-ence of the sphere (sec. 5.1). However, the first two models only includedthe χ

(2)zzz-component. The two models, that allow for all χ(2)-components were

only developed very recently. Additionally, the different nonlinear Mie modelsuse different continuity conditions to calculate the second harmonic intensityfrom the nonlinear polarization at the surface of the particle (see discussion inde Beer and Roke [24]). Furthermore, the four existing models are restricted tothe case, where the nonlinear polarization is defined only in a very thin layerat the surface of the sphere and spans the whole surface area.

A model is therefore needed, that allows for an exact solution of the fieldpropagation at the fundamental and the sum frequencies and at the same timeis flexible with respect to the distribution of the nonlinear polarization. Theunique feature of the molecular Mie model is the description of the nonlinearpolarization P (2) directly as a surface density of individual electric dipoles. Theelectromagnetic fields of the dipoles is scattered at the sphere using generalizedMie theory and the sum of all fields forms the sum frequency or second harmonicfields. In a more graphic image, nonlinear molecules that are the only sourcefor a sum frequency response, sit on the surface or in the surrounding of thesphere, are excited by the scattered plane wave, and radiate in a dipole emissionpattern at the sum frequency. The basic ingredients for this model have beendescribed in sections on vector spherical harmonics (sec. 3) and Mie scattering(sec. 4), so that this section can directly combine the different parts.

As a first step, the incident fields are prepared. To save calculation time,calculations are performed on the basis of the expansion coefficients whereverpossible and not on the actual electric fields. The expansion coefficients fora plane wave, polarized along the x-axis and traveling along the z-axis areknown from Eq. (3.6) (Fig. 5.3). Linear polarization along any other axis canbe generated using the rotation theorem (Sec. 3.3.1), circular and ellipticalpolarization by combining orthogonally linearly polarized plane waves with aphase difference. In the case of second harmonic generation, light with linearpolarization in x-direction or parallel to the scattering plane (p-in) and lightwith linear polarization in y-direction or perpendicular to the scattering plane(s-in) are needed. However, in this case, it is faster to perform the calculationonly for x-polarized light. The SH-intensity in the x − z-plane corresponds to“p-in”, the intensity in the y − z-plane to “s-in”. For sum-frequency generation,a second plane wave is needed, that is traveling at an angle ϑ to the z-axisin the x − z-plane. This wave can be generated using the rotation theorem(Sec. 3.3.1). The incident wave is expanded with a maximum number of orders


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0200

400

−400−2000

200400

−400

−200

0

200

400

X (nm)Y (nm)

Z (

nm)

plane wave: |E|20 0.5 1

Figure 5.3.: Step 1a: Plane Wave (∣∣Re(E)

∣∣2) propagating along the z-axis, polarizedparallel to the x-axis. The point-wise convergence of the fields is clearly visible in thisgraph as the contrast towards the edges of the picture diminishes. The number oforders is NFH = 6, which is high enough to ensure convergence at all rd (see Fig. 5.5)around the excited particle at x = y = z = 0.

NFH.The plane waves are then scattered by the sphere using Mie theory (Fig. 5.4).

As generalized Mie theory accepts arbitrary fields as input parameters, thismodel is applicable to any form of the fundamental harmonic fields. Alsofocused Gaussian beams or other shapes are possible fundamental harmonicfields, as long as there exists an expansion into vector spherical harmonics.

The fundamental harmonic fields are used to calculate the nonlinear polar-ization. Discrete positions rd are chosen (Fig. 5.5), where P (2) �= 0. In theclassical case, they are distributed at the surface of the spherical particle. Thedensity of these points has to be high enough that it can resemble the dipoledensity P (2) but low enough to ensure reasonable calculation speed. The num-ber of dipoles ND is usually chosen between 100 and 1000, depending on thesize of the sphere. It is generally impossible to distribute a number of points ona spherical surface with equal distance and symmetry with respect to the z-axis,except for special cases, like ND = 6 (Platonic solids). This offers a quick checkif the number of dipoles is chosen large enough: Usually, the scattered sumfrequency intensities lack symmetry with respect to the z-axis I(ϑ) = I(−ϑ),


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0200

400

−400−2000

200400

−400

−200

0

200

400

X (nm)Y (nm)

Z (

nm)

plane wave with sphere: Re(E)20 0.5 1

Figure 5.4.: Step 1b: Plane Wave (∣∣Re(E)

∣∣2) that has been scattered at the sphere.Refractive indices used are that of polystyrene (sphere) and water (surrounding). Thewavelength is λ = 800 nm. In comparison to Fig. 5.3, the fields close to the sphere arealtered.

Figure 5.5.: Step 2: Discretization of the area where P (2) �= 0 (here: at the surface ofa sphere with R = 100 nm) by choosing positions rd (d = 1 . . . 500).


if ND is too small.The actual fundamental harmonic fields EFH,1(rd) and EFH,2(rd) are cal-

culated from the expansion coefficients only for those coordinates rd. Themaximum number of orders is NFH:

EFH,sc,1(rd) =NFH∑n=1

n∑m=−n

aFH,sc,1m,n N (j)

m,n(rd) + bFH,sc,1m,n M (j)

m,n(rd) ,

EFH,sc,2(rd) =NFH∑n=1

n∑m=−n

aFH,sc,2m,n N (j)

m,n(rd) + bFH,sc,2m,n M (j)

m,n(rd) .

From the scattered fundamental harmonic fields EFH,sc,1, EFH,sc,2, the nonlin-ear polarization is calculated at every discrete position:

P(2)i (rd) = ε0

∑j,k

χ(2)ijk(rd)EFH,sc,1

j (rd)EFH,sc,2k (rd) .

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400

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200400

−400

−200

0

200

400

X (nm)Y (nm)

Z (

nm)

dipole: |E|20 0.5 1

Figure 5.6.: Step 4a: Dipole parallel to the z-axis at the origin.

The nonlinear polarization P (2)(rd) at every point rd is now resembled by adipole, with axis pointing in the direction of P (2)(rd) and strength proportional


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400

−400−2000

200400

−400

−200

0

200

400

X (nm)Y (nm)

Z (

nm)

translated dipole: |E|20 0.5 1

Figure 5.7.: Step 4b: Dipole parallel to the z-axis translated about a distance R =100 nm along the z-axis. The translation introduces a discontinuity at |r| = R, thatdoes, however, not affect the further calculation. The sphere is not yet included inthis step.

to the magnitude∣∣∣P (2)(rd)

∣∣∣. This dipole is then scattered at the sphere usingMie theory.

To save calculation time, a dipole pointing in the direction of P (2)(rd) iscomposed from three dipoles, pointing into the local x, y, and z direction:

P (rd) = P x(rd) + P y(rd) + P z(rd) .

So the following steps have to be performed three times—once for every dipoledirection.

The generation of a dipole at rd starts with a dipole at the origin (Fig. 5.6).The expansion coefficients can be found in Sec. 3.2.2. The dipoles are translatedalong the z-axis about a distance |rd| (Fig. 5.7). The respective expansioncoefficients of the translated dipoles are generated using the translation theorem(Sec. 3.3.2). The number of orders in the translation theorem is limited to NSF.Most experimental situations can be simulated by a layer of dipoles at a fixeddistance to the surface of the sphere, so that the translation process needs tobe done only once for the tree dipoles. For SHG from the surface of a solidparticle, there is only one distance |rd| = R. For a SHG from the interfaces ofa core-shell particle, two values of |rd| exist—the radius of the inner and the


outer interface. For SHG from chain-like molecules several values of |rd| haveto be considered.

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200400

−400

−200

0

200

400

X (nm)Y (nm)

Z (

nm)

translated dipole with sphere: |E|20 0.5 1

Figure 5.8.: Step 4c: Field of the dipole in Fig. 5.7, now scattered scattered at thesphere. Compared to Fig. 5.7, the dipole fields are more concentrated near the sphere.

The result are now three dipoles located at (x, y, z) = (0, 0, |rd|). In the nextstep, they are scattered at the sphere (Fig. 5.8). The expansion coefficients ofthe scattered dipoles are found using the generalized Mie theory with a dipoleas incident field (Sec. 4.3). Mie scattering is very resource costing, especially inrelation to the rotation theorem. Therefore, Mie scattering is only calculatedonce for each distance |rd| and each dipole orientation, and the scattered dipolesare then rotated to the actual position rd (Fig. 5.9). This is done, using therotation theorem (Sec. 3.3.1).

The expansion coefficients for the translated, scattered and rotated dipolesare then summed up to give the expansion coefficients of the sum frequencyfield:

aSF =∑

d

(P (2)

x (rd)aDip.,x,rd+ P (2)

y (rd)aDip.,y,rd+ P (2)

z (rd)aDip.,z,rd

),

bSF =∑

d

(P (2)

x (rd)bDip.,x,rd+ P (2)

y (rd)bDip.,y,rd+ P (2)

z (rd)bDip.,z,rd

).

To summarize, the simulation steps are given for second harmonic generation


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400

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200400

−400

−200

0

200

400

X (nm)Y (nm)

Z (

nm)

rotated dipole with sphere: |E|20 0.5 1

Figure 5.9.: Step 5: Dipole parallel to the z-axis translated about a distance R =100 nm along the z-axis that has been scattered at the sphere and after-wards rotatedabout 60° to a position rd.

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400

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200400

−400

−200

0

200

400

X (nm)Y (nm)

Z (

nm)

second harmonic: |E|20 0.5 1

Figure 5.10.: Step 6/7: second harmonic intensity as sum over all dipoles. Two lobesin the x-z plane (Ipp) and four lobes in the y-z plane (Isp) are visible. The surfacesusceptibility used for this simulation is χ

(2)zzz : χ

(2)zxx : χ

(2)xxz = 1 : 3 : 2.


as pseudo-code. a is always meant to represent both expansion coefficients am,n

and bm,n.

1. As a preparation the linear scattering of the plane wave is calculated:aFH = planeWave(λFH)aFH,sc = MieScattering(aFH)

2. Discrete positions ri, where P (2) �= 0, are chosen.

3. The actual fundamental harmonic field E(ri) at each position ri is cal-culated from aFH,sc and used to determine the polarization P (2):P

(2)xyz

(ri) =∑

j,kχ

(2)xyz

,j,kEj(ri)Ek(ri).

4. The polarization is resembled by dipoles. Therefore three dipoles, parallelto x, y and z, are generated for every distance |rd|:a

D,xyz

= dipole( 12 λFH,

xyz)

aD,tr,

xyz

= translation(aD,

xyz

, |ri|)a

D,sc,xyz

= MieScattering(aD,tr,

xyz

)

5. For every ri, the scattered dipoles are rotated to ri:a

D,sc,xyz

,i= rotation(a

D,sc,xyz

, ri)

6. The expansion coefficients for the second harmonic field is calculated asa sum over all ri:aSH =

∑i

(aD,sc,x,i · P

(2)x (ri) + aD,sc,y,i · P

(2)y (ri) + aD,sc,z,i · P

(2)z (ri)

).

7. Finally, the SH-intensities are calculated from aSH:ESH =

∑m,n

aSHm,nN

(3)m,n + bSH

m,nM(3)m,n.

6. Experimental realization of second harmonicgeneration

SHG scattering experiments at spherical nanoparticles were performed at theInstitute of Particle Technology (Chair Prof. Dr. Wolfgang Peukert) of theUniversity of Erlangen-Nuremberg by Dr. Ing. Lars-Ole Martinez-Tomalino [60],Dr. Ing. Benedikt Schürer [79], Dipl. Ing. Christian Sauerbeck and Dipl. Ing. Mi-chael Haderlein during their PhD- and Diploma-works. Section 6.1 shows thesetup used to measure angular scattering profiles. Some error sources and theirneed for correction are discussed in section 6.2.

6.1. Experimental setup

The setup used for the experiments that are compared to the simulations inthis work is in detail described in section 3.1.2 in Schürer [79]. A pulsed Ti-tan:sapphire laser system at wavelength λ = 800 nm (1.5 W, pulse duration80 fs, repetition rate 80 MHz, pulse energy 15 nJ) is used as a light source. Ahalf-wave plate is used to turn the linear polarization of the laser to be eitherparallel (p) or perpendicular (s) to the optical table, which is identical to thescattering plane. The laser is then focused by a lens with f = 50 mm into aquartz class cuvette. The typical beam waist inside the sample is 2w0 = 35 μmwith a peak intensity of approximately I = 20 GW/cm2 in the focus. The cu-vette is mounted on a computer controlled goniometer who’s arm can be movedfrom ϑ = −100° to ϑ = 140° with a minimum step size of Δϑ = 0.54°. TheSH signal is detected by a photomultiplier tube, that is mounted on the pivotarm. Spectral filters remove light at the FH frequency. Only light parallel (p)or perpendicular (s) to the scattering plane passes through a linear polarizingfilter. The respective intensities are given as Isp(ϑ) for light, where the incidentpolarization is perpendicular (s) to the scattering plane or optical table andthe detected SH-light is polarized parallel (p) to the scattering plane. LikewiseIpp(ϑ) is the intensity where both polarizers are parallel to the optical table.Theoretically also Iss(ϑ) and Ips(ϑ) exist, but are zero for centrosymmetric par-ticles. At each position, 25 consecutive measurements with an acquisition time

57

58 6. Experimental realization of second harmonic generation

Figure 6.1.: Experimental setup for angle resolved SHG measurements: HW: half-wave-plate, LP: long pass filter, FL: focusing lense, C: cuvette, AP: aperture, SP:short-pass-filter, BP: band-pass filter, PF: polarizing filter, PMT: photo multipliertube, PCU: photon counting unit.

of 200 ms are averaged. For a step size of Δϑ = 3.24°, a complete measurementof I(ϑ) takes 7 min.

To expand the setup for sum frequency generation, a second laser at a differ-ent wavelength is needed. Usually a visible and a mid-infrared laser beam areused with wavelengths λ1 = 800 nm and λ2 = 3280 nm—3570 nm. The anglebetween the two laser beams is often around 15° and both beams are parallelto the optical table. The SF-intensity is measured at a single detection angle,often ϑ = 60° as a function of the infrared wavelength λ2. The polarizationof the two fundamental harmonic light beams and the detected sum frequencyintensity are noted as a three letter subscript with the SF polarization first,the VIS polarization in the center and the IR polarization last. Compared tosecond harmonic generation, the order of input polarization and output polar-ization is reversed. Issp(λSF) is the SF-intensity polarized perpendicular to theoptical table if the FH beam at λ1 = 800 nm is also polarized perpendicular tothe table and the infrared-beam at λ2 is parallel to the optical table.

6.2. Estimation of experimental errors 59

6.2. Estimation of experimental errors

The simulation is based on the scattering of a plane wave at a single particle.The experimental situation differs from this assumption in several points: Thescattering originates from several particles that are distributed inside a volumein the glass cuvette. The scattered light from particles at different positions isdistorted by the curvature of the cuvette. In addition, the size of the particlesis not exactly fixed, but some distribution of the radii has to be assumed. Theeffects of this on the scattered SH light are investigated in this section.

Additional correction is performed on the measurement results for back-ground light from Hyper Rayleigh scattering (HRS) from the liquid. Thisis in detail described in Schürer [79].

6.2.1. Distortion by the cuvette geometry

z (mm)

x (m

m)

f(z) ∝ ∫ d r r I2(r,z) Θ(I(r,z)−Ithr

)

−1 0 1

−1.5

−1

−0.5

0

0.5

1

1.50

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5

−0.05

0

0.05

0 0.5 1 1.50

0.5

1

f(z)

Figure 6.2.: Intensity distribution of the Gaussian laser beam inside a cylindricalcuvette with 3.3 mm inner diameter. The inset at the bottom shows the intensitydistribution on a finer scale. The inset on the top shows the factor by which particlesat this position contribute to the overall signal (proportional to the intensity of theFH beam squared and integrated over x and y.

Inside the cuvette all particles inside the focal area contribute to the secondharmonic signal, the contribution being proportional to the squared intensity


of the fundamental harmonic. The focal volume is determined by the intensitydistribution of a Gaussian beam with wavelength λ and focal diameter 2w0:

I(r, z) = I0

(w0

w(z)

)2

exp

(−2(

r

w(z)

)2)

,

zR = πw20

λ, w(z) = w0

√1 +(

z

zR

)2

.

For parameters of the experiment (λ = 800 nm, 2w0 ≈ 35 μm, n = 1.3), the area,where I(r, z) > 0.1I0 is |z| < 3zR = 2.7 mm and r < w0

√ln(10)/2 = 19 μm

(see Fig. 6.2). Particles in this area contribute to the overall signal by a factorproportional to the square of the intensity at the position. Light from particlesthat are not situated directly at the center of the cuvette, will be refracted into adifferent direction, when passing through the cuvette (see Fig. 6.3). This affectsparticles that are translated in x or z direction, but not particles translatedin y because of the cylindrical symmetry. The focal area is very thin (seeFig. 6.2), so that only particles along the z-axis are considered in the followingcalculation. A particle at position z then contributes proportional a factor f(z)(upper inset in Fig. 6.2):

f(z) =∫

dr

∫dϕ r I2(r, z) .

Light from a particle at the position z emitted under an angle ϑ1 will bedetected under an angle ϑ2. Figure 6.3 shows the beam path for z > 0 and0° < ϑ1 < 90°. In the triangle with corners at the position of the particle, thecenter of the circle and the intersection with the inner circle, α1 and ω1 can bedetermined using the law of sines:

sin α1

z= sin(180° − ϑ1)

R,

ω1 = 180° − (180° − ϑ1) − α1 .

The ray is refracted at the first interface:

n1 sin α1 = n2 sin β1 .

In the triangle with corners at the position of the center of the circle and theintersections with the inner and outer circle, α2 and ω2 can be determined


D

R + d

Rz

α1α2

ω1ω2

β1

β2ϑ1

ϑ2

Figure 6.3.: Geometry of the cuvette with inner radius R and thickness d. Light rayfrom a particle displaced from the center of the cuvette along the axis about a distancez emitted under an angle ϑ1 is detected under an angle ϑ2.

using the law of sines:

sin α2

R= sin(180° − β1)

R + d,

ω2 = 180° − (180° − β1) − α2 .

The ray is refracted a second time at the outer interface:

n2 sin α2 = n3 sin β2 .

The angle ϑ2 is:

ϑ2 = β2 + ω1 + ω2 = ϑ1 + β2 + β1 − α2 − α1 .

The beam is laterally displaced by a distance

D = (R + d) sin β2

from a beam coming from the center of the cuvette.The above equations hold for z > 0 and ϑ1 > 0 (see Fig. 6.3). For ϑ1 < 0,

ϑ2 < 0 the formula can be used with |ϑ1/2|. For negative z the situation isequivalent to ϑ1/2 �→ (180° − ϑ1,2) and z �→ −z.


−100 0 100−10

−5

0

5

10

scattering angle θ1 (deg)

devi

atio

n of

det

ecte

d an

gle

θ2−

θ1 (

deg)

z/R

−0.5

0

0.5

Figure 6.4.: Deviation of detected angle (ϑ2 − ϑ1) from original scattering angle (ϑ1)for varying particle position along the z-axis in a glass-cuvette of inner diameter2R = 3.3 mm and wall thickness d = 0.7 mm, filled with water.

The change of scattering angles ϑ1 to ϑ2 also results in a reshaping of theintensities. The values ϑ1 are usually equally spaced, whereas the values ϑ2 arenot. Because the energy contained in every angular element has to be conserved,this leads to a reshaping of the intensities. If the original intensity as a functionof ϑ1 is given by Iϑ1(ϑ1) = dI

dϑ1with equally spaced ϑ1. The subscript ϑ is

only used here and omitted everywhere else in this work, for better readability.The intensity as a function of ϑ2 will be Iϑ2 (ϑ2) = dI

dϑ2= dI

dϑ1dϑ1dϑ2

= Iϑ1Δϑ1Δϑ2

(see Fig. 6.5).When the aperture of the detector is smaller than the dimensions of the

cuvette, rays displaced by a distance D exceeding a certain value will not reachthe detector. Fig. 6.5 shows D for a particle at z = 0.9R, it is always smallerthan 80 % of the outer cuvette radius, that is 1.7 mm. For a typical apertureof radius 1.5 mm this means, that some of the light is blocked. Although all ofthese effects individually seem to alter I(ϑ) the SH profile, the total sum overthe whole cuvette shows, that the scattering signal is hardly affected (Fig. 6.6).


−100 0 1000

0.2

0.4

0.6

0.8

1

scattering angle θ (deg)

SH

inte

nsity

dI/dθ

1(θ

1), ∫ dθ=93

dI/dθ1(θ

2), ∫ dθ=103

dI/dθ2(θ

2), ∫ Idθ=93

D/(R+d)aperture

Figure 6.5.: Intensity of a single particle inside the cuvette (I1(ϑ1) = dIdϑ1

(ϑ1)), aftercorrection for the scattering angle (I1(ϑ2) = dI

dϑ1(ϑ2)) and after correction of the

intensity to account for conservation of energy (I2(ϑ2) = dIdϑ2

(ϑ2)) for a particlelocated at z = 0.9R in a glass cuvette of inner diameter 2R = 3.3 mm and wallthickness d = 0.7 mm, filled with water.

−150 −100 −50 0 50 100 1500

0.2

0.4

0.6

0.8

1


SH

inte

nsity

originalcorrected

Figure 6.6.: Scattered SH light from a single particle and from an ensemble of particlesinside the cuvette.


6.2.2. Particle size distributionFor increasing particle size, the scattering pattern usually shows more peaks atscattering angles ϑn

pp and ϑnsp and the peaks move to smaller angles. Figure 6.7

shows the angular scattering profile of a polystyrene sphere in water with apure χ

(2)zzz with a radius ranging from R = 50 nm to R = 200 nm. For an

increasing radius, the intensity of the second harmonic signal increases and thepeaks shift to smaller scattering angles. Additional peaks at larger scatteringangles evolve. This is shown in Fig. 6.8. For an increasing radius, the positionof the first maximum moves from 50° or 90° to less than 10°, when the radiusbecomes as large as R = 800 nm. At approximately R = 200 nm, a secondmaximum of the intensity Isp develops, more maxima in Ipp and Isp developfor larger spheres.

−150 −100 −50 0 50 100 1500

1

2

3

4x 10

−7

scattering angle θ (°)

SH

inte

nsity

(ar

b.u.

)

100

150

200

Figure 6.7.: Effect of particle size: angular SH scattering profile Ipp(ϑ) of a polystyrenesphere in water with pure χ

(2)zzz for a varying radius between R = 50 nm and R =

200 nm.

An additional feature, that changes with particle size, is the ratio betweenthe maximum intensity in pp and sp polarization: Imax

ppImax

sp(Fig. 6.9).

A particle size distribution of some percent or some nanometers around amean radius might therefore lead to a washed out SH intensity signal. Fig-ure 6.10 shows the angular scattering profile for a sphere of R = 200 nm toR = 250 nm. Figure 6.11 compares the scattering profiles Ipp(ϑ) and Isp(ϑ)for a sphere of mean radius R0 = 225 nm and for a mixture of spheres withradii varying from R = 200 nm to R = 250 nm. The individual simulationsare performed for a constant number of dipoles or positions rd. As P shouldbe proportional to the surface of the sphere, the intensity of each sphere is


200 400 600 8000

50

100

150

radius (nm)

posi

tion

of fi

rst m

axim

um (

°)

θpp1

θpp2

θpp3

θsp1

θsp2

θsp3

Figure 6.8.: Effect of particle size: position of the first, second and third maximum inpp and sp polarization for SHG from a polystyrene sphere in water with pure χ

(2)zzz

for a varying radius between R = 50 nm and R = 850 nm.

200 400 600 8003

3.5

4

4.5

5

5.5

6

radius (nm)

inte

nsity

rat

io I

max

pp/ I

max

sp

Imaxpp

/ Imaxsp

Imaxpp

0.5

1

1.5

2

2.5

3

3.5

4x 10

−7

inte

nsity

Im

axpp

(ar

b. u

)

Figure 6.9.: Effect of particle size: Intensity ratio Imaxpp /Imax

sp and maximum intensityImax

pp for SHG from a polystyrene sphere in water with pure χ(2)zzz for a varying radius

between R = 50 nm and R = 850 nm.


weighted with R2. Additionally a Gaussian distribution of the sizes of thesphere around R0 = 225 nm with half width at half maximum ΔR = 10 nm isassumed. The intensities from the individual simulations are therefore multi-plied with R2e− ln(2)(R−R0)2/ΔR2

and added to determine the mean signal fromsuch a mixture of spheres. It hardly deviates from the signal from the meansphere size R0 = 225 nm.

−150 −100 −50 0 50 100 1500

0.5

1

1.5x 10

−6


SH

inte

nsity

(ar

b.u.

)

200

210

220

230

240

250

Figure 6.10.: Angular SH scattering profile Ipp(ϑ) of a polystyrene sphere in waterwith pure χ

(2)zzz for a varying radius between R = 200 nm and R = 250 nm.

−150 −100 −50 0 50 100 1500

0.2

0.4

0.6

0.8

1


SH

inte

nsity

(ar

b.u.

)

I

pp(R

0)

Isp

(R0)

mean( Ipp

)

mean( Isp

)

Figure 6.11.: Effect of particle size distribution: angular SH scattering profile Ipp(ϑ)of a polystyrene sphere in water with pure χ

(2)zzz for a fixed radius of R0 = 225 nm

and mean SH intensity for a Gaussian size distribution ranging from R = 200 nm toR = 250 nm (FWHM 20 nm).

7. Second harmonic generation from dielectricparticles

7.1. Scanning for χ2 components

As already discussed in section 2.2, for SHG from a homogeneous surface, onlythree independent non-chiral components of the nonlinear surface susceptibilityexist: χ

(2)zzz, χ

(2)zxx and χ

(2)xxz. In general χ(2) can be complex. Far from resonances

we can assume the elements of χ(2) to be either positive or negative but notcomplex valued. This assumption is necessary to reduce the parameter space toa size that can be handled, and conforms to literature. The absolute intensityof the SH signal is proportional the absolute magnitude of χ(2). The angularshape, however, only depends on the ratio of these components, often given asχ2

zzz/χ2zxx and χ2

xxz/χ2zxx, both ranging from −∞ to +∞.

−100 −50 0 50 100 1500

10

20

30

40


SH

inte

nsity

(co

unts

/0.2

s)

Ippmax

Ispmax

−θpp1

θsp1

Ippsim(θ)

Ispsim(θ)

Ippexp(θ)

Ispexp(θ)

Figure 7.1.: Definition of the positions and intensities on the basis of a real spectrum.Position of the first maxima in sp- and pp-polarization (ϑ1

sp, ϑ1pp) and maxima of the

measured SH-intensities (Imaxpp , Imax

sp ).

The three parameters that can be extracted from the experimental scatter-ing profiles with highest accuracy are the angular position of the first maxi-mum in both polarizations ϑ1

pp and ϑ1sp and the ratio of the maximal inten-

sities in both polarizations Imaxpp /Imax

sp (Fig. 7.1). A good way, to find the

67

68 7. Second harmonic generation from dielectric particles

χ(2) that is able to explain the experimental scattering profile, is to plot ϑ1pp,

ϑ1sp, and Imax

pp /Imaxsp over the two-dimensional parameter space (χ(2)

zzz/χ(2)zxx and

χ(2)xxz/χ

(2)zxx) and mark all areas, that match the experimental result.

However, the range from −∞ to +∞ makes this parameter space inconve-nient to draw. Therefore a triangular parameter space is used, as it is commonpractice for color measurements in lighting applications or for ternary phasediagrams.

To make the formula easier to read, the following shorthand notation is usedin the next paragraphs: A = χ

(2)zxx, B = χ

(2)xxz, and C = χ

(2)zzz. For positive χ(2),

the ratio A : B : C is equal to the ratio a : b : c, when

a = A

|A| + |B| + |C| b = B

|A| + |B| + |C| c = C

|A| + |B| + |C| ,

with

a + b + c = 1 a, b, c ∈ [0, 1] .

a and b can be displayed as Cartesian coordinates, resulting in a triangle withthe lower left corners at (0, 0), indicating χ(2) = a (pure χ

(2)zxx), the lower right

corner at (1, 0), indicating χ(2) = b (pure χ(2)xxz) and the top corner at (0, 1),

indicating χ(2) = c (pure χ(2)zzz) (see Fig. 7.2). The edges of the triangle stand

for situations, where one of the three components is zero. For example, theedge connecting the a- and b-corners represents c = 0. Half way between thea- and b-corners, χ(2) consists of a = b, c = 0. The inner area of the trianglerepresents all situations, where all three components are nonzero, with eachcomponent becoming larger when getting closer to the respective corner.

a

b1

1(a, b, c) = (1, 0, 0)

(a, b, c) = (0, 0, 1)

(a, b, c) = (0, 1, 0)

Figure 7.2.: Triangle to display the relative magnitude of three variables.

When shearing the triangle, so that the top corner lies at ( 12 , 1), the triangle

7.1. Scanning for χ2 components 69

looks more symmetric. The coordinates (x, y) are then:

x = 12

− a − c

2a = 1

2− x − y

2

y = c b = 12 + x − y

2c = y

The components of χ(2) do not need to have the same algebraic sign. As theintensities do not depend on the absolute phase of the electric field, only thecases where one of the three components is negative needs to be consideredwithout lack of generality. These situations are represented in the four outertriangles with coordinates:

a < 0 x = 1 − c − b

2a = 1

2− x − y

2< 0

y = 1 − b b = 1 − y

c = 12

− x + y

2

b < 0 x = b

2+ c

2− 1

2a = 1 − y

y = 1 − a b = 12

+ x − y

2< 0

c = 12

+ x + y

2

c < 0 x = 12

− a + c

2a = 1

2− x + y

2

y = c b = 12

+ x + y

2c = y

Figure 7.4 shows the positions of the first maxima ϑ1pp and ϑ1

sp and theratio of the maxima Imax

pp /Imaxsp for a polystyrene sphere of radius R = 100 nm

immersed in water. The position of the first maxima vary between 15° and130° with a clear discontinuity, when a secondary maximum emerges at a lowerangle than the previous one. The intensity ratio Imax

pp /Imaxsp ranges over several

orders of magnitude from 0.025 to 532. Especially the intensity ratio thereforeis a very significant parameter for the determination of χ(2). If all three χ(2)-components have the same sign (center of the triangle), Imax

pp is always largerthan Imax

sp , with the largest ratio Imaxpp /Imax

sp for χ(2)zzz ≈ χ

(2)xxz and χ

(2)zxx ≈ 0.


x

y

(1, 0, 0) (0, 1, 0)

(0, 0, 1)(0, −1, 0) (−1, 0, 0)

(0, 0, −1)

1−1

−1

1

(+, −, +) (−, +, +)

(+, +, −)

(+, +, +)

Figure 7.3.: Triangle to visualize the ratio of three χ(2)-components. The relativesize of the three independent χ(2)-components at different coordinates is indicated as(χ(2)

zxx, χ(2)xxz, χ

(2)zzz) = (a, b, c).

Experiments performed with polystyrene beads (see also sec. 7.5) with radiusR = 100 nm with adsorbed Malachite Green molecules immersed in water showan intensity profile with first maxima at ϑ1

sp = 50.4° ± 4° and ϑ1pp = 67.3° ± 4°

and intensity ratio Imaxpp /Imax

sp = 0.61 ± 0.15 (see Sec. 7.5). Fig. 7.5 reproducesFig. 7.4 but all areas, where 63.3° ≤ ϑ1

pp ≤ 71.3°, 46.4° ≤ ϑ1sp ≤ 54.4° and

0.46 ≤ Imaxpp /Imax

sp ≤ 0.76 are fulfilled, are marked. The conditions are indi-vidually fulfilled for a wide range of χ(2)-components. However, only for avery small range of χ(2)-components, located near a pure χ

(2)zxx-contribution, all

three conditions are fulfilled simultaneously.

7.2. Effect of the number of orders

In the molecular Mie model, the fundamental harmonic and the second har-monic fields are expanded in a set of vector spherical harmonics up to orderNFH and NSH. It is crucial, that the number or orders is chosen high enough toensure conversion. In the simulation code, a test is used to determine NFH andNSH: For NFH, scattering of a plane wave is calculated and for every expansion

7.2. Effect of the number of orders 71

θ1pp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

θ

1sp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

pos.

of f

irst m

axim

umθ

1 pp, θ

1 sp (

deg)

0

20

40

60

80

Imax

pp/ Imax

sp χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

x

y

inte

nsity

rat

io I

max

pp/ I

max

sp

0.01

0.1

1

10

100

1000

Figure 7.4.: Position of the first maxima ϑ1pp and ϑ1

sp and the ratio of the maximaImax

pp /Imaxsp for a polystyrene sphere of radius R = 100 nm immersed in water.

order n, the fields Enr , En

ϑ and Enϕ are calculated on a 150 × 150 grid. This

grid forms a spherical surface with radius R = |rd|. As soon as

(|En

r − En−1r |

)+(|En

ϑ − En−1ϑ |

)+(|En

ϕ − En−1ϕ |

)(|En

r |)

+(|En

ϑ |)

+(|En

ϕ |) <

11000 ,

(x =

∑xn

n

)

that is, the fields changed less then 0.1 % in the last step, the expansion isstopped and NFH = n is used. For NSH the same procedure is performed, butthe fields of three dipoles, polarized along x, y, and z with position (x, y, z) =(0, 0, |rd|) are calculated on a spherical surface with R = 2 cm. This distanceis used to detect the SH-intensity in all simulations.

For a polystyrene sphere in water with R = 250 nm, this method yieldsNFH = 8 and NSH = 11. Figure 7.6 shows, that this estimate is rather cautious.Already for NFH = 4 and NSH = 7 the SH intensity have converged.


θ1pp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

θ

1sp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

pos.

of f

irst m

axim

umθ

1 pp, θ

1 sp (

deg)

0

20

40

60

80

Imax

pp/ Imax

sp χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

x

yin

tens

ity r

atio

Im

axpp

/ Im

axsp

0.01

0.1

1

10

100

1000χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

Figure 7.5.: Position of the first maxima ϑ1pp and ϑ1

sp and the ratio of the maximaImax

pp /Imaxsp for a polystyrene sphere of radius R = 100 nm immersed in water. The

first three graphs indicate the areas where ϑ1pp, ϑ1

sp, and Imaxpp /Imax

sp individually fitto the experiment. The last graph combines these areas (yellow) and shows, where allthree requirements are fulfilled simultaneously (dark red).

−150 −100 −50 0 50 100 1500

2

4

6x 10

−7


SH

inte

nsity

(ar

b.u.

)

8/114/73/62/41/1 (×5000)

Figure 7.6.: SH-intensity (λSH = 400 nm) Ipp(ϑ) for a polystyrene sphere in waterwith R = 250 nm for increasing number of expansion orders NFH/NSH from 1/1 to8/11. The intensity for NFH/NSH =1/1 is increased by a factor of 5000.

7.3. Comparison to Rayleigh-Gans-Debye-theory 73

7.3. Comparison to Rayleigh-Gans-Debye-theory

For a vanishing refractive index contrast, the Molecular Mie model should givethe same result as RGD theory (see Sec. 5.1.3). This can act as a validation testfor the molecular Mie model and also allows to study the effect of an increasingrefractive index contrast.

The RGD code used for this comparison is the one by de Beer and Roke[25]. The equations for Eppp

∧= Epp, Essp, Esps, and Epss∧= Esp (in SFG

and SHG notation) are given by equations 5—12 in [25]. Alternatively thefree simulation software NLS-Simulate [26] by de Beer and Roke can be used.Their susceptibility coefficients are defined slightly different than in section2.2. In NLS-Simulate, D = 1, ϕ = 0 and βccc = χ

(2)zzz, βaac = βaca = χ

(2)xxz,

and βcaa = χ(2)zxx have to be chosen. In the equations in [25], χ⊥,⊥,⊥ = χ

(2)zzz,

χ‖,⊥,‖ = χ(2)xxz, and χ⊥,‖,‖ = χ

(2)zxx are chosen.

H2O:H2O PS:H2O PS:PS40

50

60

70

80R=50nm

posi

tion

of m

axim

a (d

eg)

Ipp

/Isp

θpp

θsp

3.4

3.5

3.6

3.7

3.8

3.9

inte

nsity

rat

io

H2O:H2O PS:H2O PS:PS15

20

25

30R=250nm

posi

tion

of m

axim

a (d

eg)

5

5.2

5.4

5.6

inte

nsity

rat

io

Figure 7.7.: Effect of the refractive index contrast between sphere and surrounding onthe scattered SH light for a pure χ

(2)zzz-component.

If the materials of the sphere and the surrounding are chosen to be identical,no difference in Ipp(ϑ) and Isp(ϑ) were observed for the molecular Mie modeland the RGD model. As soon as a refractive index contrast is introduced,differences between RGD and Mie become visible (Figure 7.7). Starting fromthe RGD limit of a water sphere in water (nsurr. = nsphere = 1.3), the refractiveindex of the sphere is increased to nsphere = 1.6 (polystyrene sphere in water).Then the refractive index of the surrounding is also increased to nsurr. = 1.6(polystyrene sphere in polystyrene, RGD limit). As already shown by Jen et al.


−150 −100 −50 0 50 100 1500

0.2

0.4

0.6

0.8

1

Scattering angle θ (°)

norm

aliz

ed S

H in

tens

ity

H2O:H2O

PS:H2O

PS:PS

Figure 7.8.: SH-intensity for a varying refractive index contrast from water to polysty-rene. Compare also the right hand side of Fig. 7.7.

[49], even in RGD theory, the refractive index has a significant impact on theposition of the maxima ϑn

pp and ϑnsp. The position of the first maximum for a

polystyrene sphere in polystyrene is significantly lower than for a water spherein water. This is because the refractive index of polystyrene is higher, whicheffectively decreases the wavelength or increases the ratio R

λeff.. For larger

spheres, the maxima in the SH intensity are at smaller angles. This effect isapproximately linear, so that an intermediate refractive index could be chosento correct for this effect. However, this is not true for the intensity ratio Imax

ppImax

pp.

It is not possible, to predict the intensity ratio that a full Mie simulation wouldgive, from two RGD simulations using first the refractive index of the sphereand secondly the refractive index of the surrounding. Even for the small sphereof R = 50 nm, where the parameter to estimate the validity of the RGD model,2πR

∣∣∣n1n2

− 1∣∣∣ = 0.074 � 1, seems to be sufficiently small, the intensity ratio

depends noticeably on the changing refractive index ratio. For a larger sphereof R = 250 nm, the effect is even stronger. Figure 7.8 shows the SH intensityas a function of the scattering angle for the varying refractive index contrastshown in the right hand side of Fig. 7.7. The absolute intensity of the secondharmonic signal is highest for a full refractive index contrast due to the fieldenhancement caused by the sphere. The intensity of the RGD-limit with therefractive index of water is at 30 % of the maximum intensity, with the refractiveindex of polystyrene at 20 % of the maximum intensity. Fig. 7.8 shows, thatthe height of the second and third maximum of the normalized SH-intensity inrelation to the first maximum changes with the refractive index contrast.

7.3. Comparison to Rayleigh-Gans-Debye-theory 75

θ1pp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

θ

1sp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

pos.

of f

irst m

axim

umθ

1 pp, θ

1 sp (

deg)

0%

50%

100%

Imax

pp/ Imax

sp χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

inte

nsity

rat

io I

max

pp/ I

max

sp

0%

50%

100%

150%

200%

Figure 7.9.: Difference |XRGD−XMie|(XRGD+XMie)/2 between the results from RGD (XRGD) and

Mie (XMie) theory for polystyrene sphere in water with radius R = 100 nm. Coor-dinates (x, y) = (0, −0.36) and (0.16, 0.68) are indicated by a +; the correspondingangular scattering profiles are shown in Fig. 7.10.

This considerations were done for a polystyrene sphere in water with a pureχ

(2)zzz-component. Figure 7.9 shows a more detailed evaluation of the differences

between Mie theory and RGD theory all non-chiral χ(2)-components for SHG.The graphs show |XRGD−XMie|

(XRGD+XMie)/2 with X = ϑ1pp, X = ϑ1

sp, and X = Imaxpp /Imax

sp .

For a R = 100 nm polystyrene sphere in water (2πR∣∣∣n1

n2− 1∣∣∣ = 0.15), the

differences between RGD-theory and the Mie-model are low, for a pure χ(2)zzz-,

χ(2)zxx-, or χ

(2)xxz-component. There are, however, some specific values for χ(2),

for which the RGD solution for position of the first maximum deviates as muchas 150 % and the intensity ratio deviates as much as 200 % from full Mie theory.Two of these cases are shown in Fig 7.10.


−100 0 1000

0.2

0.4

0.6

0.8

1


SH

inte

nsity

(ar

b.u.

)

−100 0 1000

0.2

0.4

0.6

0.8

1


SH

inte

nsity

(ar

b.u.

)

I

ppMie

IspMie

IppRGD

IspRGD

Figure 7.10.: SH-intensity for a polystyrene sphere in water with radius R = 100 nm,calculated using Mie theory (solid lines) and RGD theory (dashed lines, using therefractive index of water).Left: χ

(2)zzz/χ

(2)zxx = 1.13, , χ

(2)xxz/χ

(2)zxx = 1. Right: χ

(2)zzz/χ

(2)zxx = 2.13, χ

(2)xxz/χ

(2)zxx = 0.

The corresponding coordinates in Fig. 7.9 are (x, y) = (0, −0.36) and (0.16, 0.68).

7.4. Second harmonic generation from a non-centrosymmetricdistribution of the nonlinear polarization

−100 0 1000

0.2

0.4

0.6

0.8

1x 10

−6


SH

inte

nsity

(ar

b.u.

)

∫ = 1.53 × 10−4 Ipp

(θ)

Isp

(θ)

Ips

(θ)

Iss

(θ)

−100 0 1000

1

2

3x 10

−7


SH

inte

nsity

(ar

b.u.

)

∫ = 1.04 × 10−4 Ipp

(θ)

Isp

(θ)

Ips

(θ)

Iss

(θ)

Figure 7.11.: SH-intensities for the scattering at a polystyrene sphere in water withR = 100 nm if the nonlinear polarization is concentrated on one half of the sphere(left, rd with d = 1 . . . 269) or covers the whole surface of the sphere (right, rd withd = 1 . . . 500). The intensities are given in arbitrary units but the intensities of bothplots refer to the same scaling factor and are therefore comparable.

The molecular Mie model offers the unique possibility to model spheres,where the nonlinear polarization is not distributed homogeneously over the

7.4. SHG from non centrosymmetric distribution of the nonlinear polarization77

surface of the sphere. However, it must be made sure, that this inhomogeneitydoes not affect the spherical symmetry with respect to the linear scattering pro-cess! A possible situation could be a nonlinear molecule like Malachite Green,that for a reason that does not affect the linear scattering properties, only ad-sorbs to part of the surface. Figure 7.11 shows the second harmonic intensitiesfor a polystyrene sphere in water, where rd is concentrated on one half of thesphere’s surface.

The simulation is done with an ensemble of rd, where all positions with z < 0are removed. 200 individual simulations are performed, where the positions rd

are rotated before the next step, so that the orientation of the hemisphereis not fixed with respect to the z-axis, which is the propagation direction ofthe fundamental harmonic laser beam. The intensities of the 200 simulationsare averaged. This corresponds to a incoherent superposition of the secondharmonic fields of 200 spherical particles in a cuvette with statistical orientationof the nonlinear-active hemisphere with respect to the laser beam.

It is apparent, that the general rule Ips(ϑ) = 0 and Iss(ϑ) = 0, does not holdfor this system. The additional breaking of the symmetry leads to a nonzerosignal in ps- and ss-polarization. For ϑ = 0, pp- and sp-, as well as sp- andps-polarization are effectively identical from geometrical considerations. Thisis also seen in the SH-intensities, where Ipp(0) = Iss(0) and Ips(0) = Isp(0).

The second harmonic intensity is proportional to the surface area squaredor the number of dipoles ND squared. For the completely covered sphere,N full

D = 500 dipoles were used. For the partly covered sphere, all dipoleswith z < 0 were removed and only those with z ≤ 0 kept in the ensemble(Nhalf

D = 269) and then rotated. If SHG was independent from the geometry,one would expect the SH-signal from the half-covered sphere to be only a factor(

NhalfD

NfullD

)2

= 0.29 of the SH-signal of the completely covered sphere. The peak

intensity of the half-covered sphere is slightly higher than expected: Imax,halfpp

Imax,fullpp

=0.31. The graph also displays the integrated angle resolved intensities of all

polarizations: Iintegrated =180°∫

−180°dϑ(Ipp(ϑ) + Isp(ϑ) + Ips(ϑ) + Iss(ϑ)

). Here

Imax,halfintegrated

Imax,fullintegrated

= 0.68. This shows, that the additional symmetry breaking due to

the partly covered sphere leads to an increase in the second harmonic intensity.


7.5. Determination of the susceptibility of Malachite Green

Malachite green (MG, [(C6H5C(C6H4N(CH3)2)2]+), named after its brightgreen color, which is similar to the mineral malachite (Cu2CO3(OH)2), is anorganic dye molecule. Besides its use as a saturable absorber to generate ultra-short laser pulses, it has become a widely used probe molecule for the studyof surfaces with second harmonic generation because of its strong nonlinearresponse (Meech and Yoshihara [64, 65], Morgenthaler and Meech [68]). Thefirst SHG experiment on colloidal particles was performed for polystyrene par-ticles with adsorbed malachite green (Wang et al. [93] in 1996). However, thenonlinear surface susceptibility χ(2) remained unclear. Studies on planar andspherical surfaces found different values using different models (see table 7.1).

χ(2)zzz/χ

(2)zxx χ

(2)xxz/χ

(2)zxx (x, y) model

Kikteva et al. [53] 0.02 0.06 (−0.44, 0.019)Yang et al. [99] ∞ 0 (0, 1) RGDJen et al. [49] 0 0 (−0.5, 0) RGD

−2.2 −0.5 (−0.93, 0.14) RGDGonella and Dai [42] 0.43 −0.23 (−0.44, 0.40) MieWunderlich et al. [98] 0.14 −0.027 (−0.45, 0.14) Mie

Table 7.1.: Different values for the nonlinear susceptibility of Malachite Green retrievedfrom various experiments.

Kikteva et al. [53] measured the intensity of the second harmonic scatteredlight from a planar fused silica surface in air. Here the intensity of the secondharmonic radiation was measured for various polarizations as a function of theinput polarization angle. An analytical model was fitted to the data. Theyfound a dominant χ

(2)zxx at λFH = 800 nm and explain this with the electronic

structure of malachite green, where the S2 ← S1 transition is resonant at410 nm.

Yang et al. [99] studied colloidal polystyrene beads with 980 nm, 700 nm,and 510 nm diameter and performed the first angular-resolved scattering exper-iments. Their publication only shows Ipp(ϑ) and their RGD simulation with apure χ

(2)zzz. However, it seems they have accidentally plotted |ERGD

pp |4 instead of|ERGD

pp |2. Fig. 7.12 additionally shows |ERGDpp |2 (dashed line) using the formula

given in their analysis.

7.5. Determination of the susceptibility of Malachite Green 79

Figure 7.12.: Determination of χ(2) of MG by Yang et al. [99]: Ipp(ϑ) for particleswith diameter 980 nm (top), 700 nm (middle), and 510 nm (bottom) with their fits tothe RGD model. As they seemed to have accidentally plotted |ERGD

pp |4, the dashedline also shows |ERGD

pp |2, which agrees less with the measured data.Copyright 2001 by The American Physical Society

Jen et al. [49] recorded both polarizations Ipp(ϑ) and Isp(ϑ) for polystyrenespheres of 1053 nm, 202 nm, 88 nm, and 56 nm diameter at λFH = 840 nm. Thesignal from the smaller spheres was fitted using a RGD model with dominatingχ

(2)zxx. For the larger particle a dominating χ

(2)zzz was used and additionally the

refractive index in the RGD model was changed to that of polystyrene. Alater evaluation (Gonella and Dai [42]), using a Mie model, found χ

(2)zzz/χ

(2)zxx =

0.43, χ(2)xxz/χ

(2)zxx = −0.23. The data are shown as angular plots, which makes

it difficult to compare the experimental and simulated SH profiles in detail.Additionally it is not clear, whether the ratio of the maximal intensities in pp-and sp-polarization was taken into account.

The aim in this work was to determine the nonlinear susceptibility of mala-chite green. Three different particle diameters (200 nm, 535 nm, and 989 nm)were used to obtain different angular profiles and thus to increase the accuracy.Table 7.2 shows ϑ1

pp, ϑ1sp, and Imax

pp /Imaxsp for the three particle sizes. The right

hand side of Fig. 7.13 shows the SH-intensities as a function of the scattering an-gle measured in the experiment as well as the simulation with χ

(2)zzz/χ

(2)zxx = 0.09


χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

−150 −100 −50 0 50 100 1500

10

20

30

40

scattering angle θ (deg)S

H in

tens

ity (

coun

ts/0

.2s)

Ippsim(θ)

Ispsim(θ)

Ippexp(θ)

Ispexp(θ)

χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

−150 −100 −50 0 50 100 1500

20

40

60

80


SH

inte

nsity

(co

unts

/0.2

s)

Ippsim(θ)

Ispsim(θ)

Ippexp(θ)

Ispexp(θ)

χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

−150 −100 −50 0 50 100 1500

20

40

60


SH

inte

nsity

(co

unts

/0.2

s)

Ippsim(θ)

Ispsim(θ)

Ippexp(θ)

Ispexp(θ)

Figure 7.13.: Scattering profiles for polystyrene spheres with malachite green with di-ameter 200 nm, 535 nm, and 989 nm (from top to bottom) with fits to the Mie modelpresented in this work. The range of χ(2), that match ϑ1

pp, ϑ1sp, and Imax

pp /Imaxsp

(compare Tab. 7.1) are colored and a “x”-mark shows the value used for the fit(χ(2)

zzz/χ(2)zxx = 0.093 and χ

(2)xxz/χ

(2)zxx = −0.070; coordinate (−0.49, 0.14)). As a com-

parison, a small + mark shows the result of Gonella and Dai [42]

7.5. Determination of the susceptibility of Malachite Green 81

and χ(2)xxz/χ

(2)zxx = −0.07. In the left hand side, the triangular χ(2)-space is

shown and the regions that fulfill the values in Table 7.2 are colored. A largex mark shows the value used for the fitting (x, y) = (−0.49, 0.14)).

R ϑ1pp ϑ1

sp Imaxpp /Imax

sp

100 nm 67.3° ± 4° 50.4° ± 4° 0.61 ± 0.1267.5 nm 21.2° ± 2° 20.3° ± 2° 0.32 ± 0.1494.5 nm 11.0° ± 1° 10.7° ± 1° 0.25 ± 0.1

Table 7.2.: Experimental results for malachite green on polystyrene: Position of thefirst maxima ϑ1

pp and ϑ1sp, ratio of intensities Imax

pp /Imaxsp for three particle sizes.

The simulated SH intensities for χ(2)zzz/χ

(2)zxx = 0.09 fit very well to the mea-

sured data. In order to find the χ(2) that best fits to the experiment, the regionsmarked in the left hand side of Fig. 7.13 were studied in more detail. Fig. 7.13only shows what region of χ(2) reproduces ϑ1

pp, ϑ1sp, and Imax

pp /Imaxsp . However,

also the position of higher order maxima (ϑ2pp, ϑ2

sp, ϑ3sp) and the peak intensity

at these higher order maxima determine how well simulation and experimentagree. These additional parameters were taken into account.

In contrast to Yang et al. [99], our simulation is not restricted to χ(2)zzz. We

found the same χ(2) for all three particles sizes using the same material pa-rameters in contrast to Jen et al. [49]. We were not able to reproduce theresult of the Mie simulation of Gonella and Dai [42] (compare Fig. 7.14). Itis hard to compare their measurement to our measurement, because the angu-lar plot makes it difficult to see the exact position and intensity of the higherorder maxima. But it seems at least for their R = 101 nm, Ipp has the max-imum at a lower scattering angle. However, they use a different wavelength(λFH = 840 nm) and χ(2) might be frequency dependent. When comparing oursimulation using their χ(2) to our measurement, ϑ1

pp for the small sphere is notreproduced. For the larger spheres, the positions and intensities of the higherorder maxima do not fit. Our simulation results in a dominating χ

(2)zxx, which

agrees to the results of Kikteva et al. [53] for a planar interface.


−150 −100 −50 0 50 100 1500

10

20

30

40


SH

inte

nsity

(co

unts

/0.2

s)

Ippsim(θ)

Ispsim(θ)

Ippexp(θ)

Ispexp(θ)

−150 −100 −50 0 50 100 1500

20

40

60


SH

inte

nsity

(co

unts

/0.2

s)

Ippsim(θ)

Ispsim(θ)

Ippexp(θ)

Ispexp(θ)

Figure 7.14.: Scattering profiles for polystyrene spheres with malachite green withdiameter 200 nm (top) and 989 nm (bottom) with fits to the Mie model presented inthis work. For χ(2) the value determined by Gonella and Dai [42] was used.

7.6. Second Harmonic generation from polyelectrolyte brush particles 83

7.6. Second Harmonic generation from polyelectrolyte brushparticles

Experiments with polyelectrolyte brush particles offer the possibility to applythe nonlinear Mie model to a system where the origin of the second harmonicsignal is not restricted to a thin layer at the surface of the sphere (Schürer et al.[80]). These particles consist of a polystyrene core of 220 nm diameter withpolyelectrolyte chains with a maximum length of 100 nm grafted to the surface(Fig. 7.15). The charged chains repel each other and lead to the brush-like ge-ometry with radially stretched molecules. When adding salt to the suspension,the charges are shielded and the chains are no longer fully stretched so thatthe outer radius of the brush particle decreases.

RDLS

R

Figure 7.15.: Spherical polyelectrolyte brush particle consisting of a spherical polysty-rene core of radius R = 110 nm and linear polyelectrolyte chains of an approximatelylength of 100 nm that stretch out radially. The outer radius of the brush particle RDLSis accessible via dynamic light scattering.

The SH-intensities in pp- and sp-polarization are measured between ϑ = −90°and ϑ = 140° for four different concentrations of NaCl. Ipp(ϑ) is much strongerthan Isp(ϑ) and shows a maximum at ϑ1

pp ≈ 40° and perhaps a secondarymaximum above 100° (see Fig. 7.18). Table 7.3 summarizes the parameters ofthe SH-intensities and the outer radius of the brush particle, determined bydynamic light scattering.

To answer the question of the origin of the second harmonic signal, different


c(NaCl) ϑ1pp ϑ1

sp Imaxpp /Imax

sp RDLS

0.1 mM 31° ± 4° 35° ± 6° 9.1 ± 2 192 nm1 mM 37° ± 4° 47° ± 6° 9.1 ± 2 180 nm10 mM 38° ± 4° 47° ± 6° 8.3 ± 2 168 nm75 mM 38° ± 4° 50° ± 6° 7.7 ± 2 162 nm

Table 7.3.: Experimental results for polyelectrolyte brush particles: Position of thefirst maxima ϑ1

pp and ϑ1sp, ratio of intensities Imax

pp /Imaxsp , and outer radius of the

brush particles determined by dynamic light scattering for four different NaCl concen-trations.

Figure 7.16.: Geometry for the simulation of polyelectrolyte brush particles: Sphericalparticle of radius R with thin layer of nonzero nonlinear polarization P (2)(rd) �= 0 ata distance D1 (left) and an extended shell of P (2)(rd) �= 0 with thickness T (center)or two thin layers at distance Din = 0 and Dout = D2 (right).

geometries were considered (see Fig. 7.16). All simulations were performed fora polystyrene sphere with radius R = 110 nm radius in water. Because of thelow density, the refractive index of the polyelectrolyte chains is assumed to beidentical to water and any effect of the chains on the linear light scattering isneglected. The first picture in Fig. 7.16 shows a thin layer of P (2)(rd) with adistance |rd|−R = D1 to the surface of the sphere. This situation does not fit tothe geometry with the polyelectrolyte brushes, but gives an effective radius forthe origin of the second harmonic response. Additionally, the other geometriescan be superimposed from these simulations with little additional calculationeffort. The electric fields Epp(ϑ) and Esp(ϑ) of simulations for different D1 canbe added to determine Ipp(ϑ) and Isp(ϑ) in other cases.

This effective radius method is used to find matching χ(2)-components. Po-lystyrene sulfonate polyelectrolyte molecules do not show a second harmonicsignal at λFH = 800 nm (McAloney and Goh [63], Breit et al. [9]). This suggests,that water molecules, which are oriented due to charge effects, give rise to the


θ1pp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

θ

1sp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

pos.

of f

irst m

axim

umθ

1 pp, θ

1 sp (

deg)

0

20

40

60

80

Imax

pp/ Imax

sp χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

x

y

inte

nsity

rat

io I

max

pp/ I

max

sp

0.01

0.1

1

10

100

1000χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

R=110nm|r

d|=160nm

θ1pp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

θ

1sp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

pos.

of f

irst m

axim

umθ

1 pp, θ

1 sp (

deg)

0

20

40

60

80

Imax

pp/ Imax

sp χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

x

y

inte

nsity

rat

io I

max

pp/ I

max

sp

0.01

0.1

1

10

100

1000χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

R=110nm|r

d|=120nm

Figure 7.17.: Scan of χ(2) components for the Brush particles at salt concentrationc = 0.1 mM (top) and c = 10 mM (bottom). Respective fits of the SH-intensitiesusing χ2

zzz/χ2zxx = 3 and χ2

xxz/χ2zxx = 1 (position marked by a cross) are shown in

Fig. 7.18.


−50 0 50 1000

5

10


SH

inte

nsity

(co

unts

/0.2

s)

0.1mMD

1=50nm I

ppsim

Ispsim

Ippexp

Ispexp

−50 0 50 1000

5

10


SH

inte

nsity

(co

unts

/0.2

s)

1mMD

1=15nm

Ippsim

Ispsim

Ippexp

Ispexp

−50 0 50 1000

2

4

6

8


SH

inte

nsity

(co

unts

/0.2

s)

10mMD

1=10nm

Ippsim

Ispsim

Ippexp

Ispexp

−50 0 50 1000

2

4

6

8


SH

inte

nsity

(co

unts

/0.2

s)

75mMD

1=5nm

Ippsim

Ispsim

Ippexp

Ispexp

Figure 7.18.: SH-intensities in sp- and pp-polarization with fit to the model using asingle shell with χ2


xxz/χ2zxx = 1 in a distance D1.

second harmonic signal. The electric field Φ caused by the static charge at theinterface can interact with χ in the same way as the electric fields of light do. Φonly has a component perpendicular to the surface. Two electric fields from thefundamental harmonic light and the static field from the charged interface canlead to a pure χ(3)-effect. As de Beer et al. [27] have shown, a pure χ(3)-effectat a charged interface

P(2ω)i =

3∑j,k=1

χijklE(ω)j E

(ω)k Φl

has to obey the symmetry rule χ(3)zzzz = χ

(3)zxxz + χ

(3)xxzz + χ

(3)xzxz, or χ

(2)zzzz

χ(2)zxxz

=

1 + χ(2)xxzz

χ(2)zxxz

+ χ(2)xzxz

χ(2)zxxz

. As Φ is fixed with a field component only perpendicu-lar, there is no sum over l and the equation can be reduced to an effectiveχ(2) by omitting the last index. In fact, scanning for χ(2) components for|rd| = 120 nm . . . 200 nm, shows that χ2


xxz/χ2zxx = 1 give


a good agreement with the experiment (Fig. 7.17). This χ(2) was used for allsubsequent simulations.

150 200 250 3000

20

40

60

80

100

120

140

Dipole position |rd| (nm)

Pos

ition

of m

axim

a (d

eg)

Ipp

/Isp

θpp

θsp

6.5

7

7.5

8

8.5

9

Inte

nsity

rat

io

Figure 7.19.: Intensity ratio Imaxpp /Imax

sp and positions of first and second maximum ϑ1pp,

ϑ1sp, ϑ2

pp, and ϑ2sp for a polystyrene sphere in water with an increasing dipole position

|rd| = 110 nm . . . 300 nm. The radius of the sphere is kept constant at R = 110 nm(solid lines) or increased R = |rd| (dashed lines).

Possible regions, where water molecules might be oriented in large num-bers are the inner interface directly at the surface of the polystyrene core,the outer radius where the polyelectrolyte chains end, and distributed alongthe polyelectrolyte chains (see Fig. 7.16). Figure 7.19 shows the effect of thedistance D1 by comparing two situations: First, a polystyrene sphere in wa-ter of R = 110 nm radius with dipole position varying from |rd| = 110 nm(D1 = 0 nm) to |rd| = 300 nm (D1 = 190 nm). The intensity ratio as well asthe positions of the first and second maximum are plotted using solid lines.The dashed lines show the same values for a polystyrene sphere in water of in-creasing radius from R = 110 nm to R = 300 nm, where the dipoles are locateddirectly at the surface of the sphere (|rd| = R, D1 = 0 nm). The positions ofthe maxima are lower, if both the positions |rd| and the radius R are increased,as imposed to the situation, where only |rd| is increased. The intensity ratio,where R is increased shows an oscillation, indicating radii that resonate withthe wavelength.

Figure 7.20 compares the two situations shown in the center and on the


0 50 100 15020

40

60

80

100

120

140

160

Distance D2 or Thickness T (nm)

Pos

ition

of m

axim

a (d

eg)

Ipp

/Isp

θpp

θsp

6.5

7

7.5

8

8.5

9

Inte

nsity

rat

io

Figure 7.20.: Intensity ratio Imaxpp /Imax

sp and positions of first and second maximumϑ1

pp, ϑ1sp, ϑ2

pp, and ϑ2sp for a polystyrene sphere (R = 100 nm) in water. Solid lines:

Increasing thickness T of dipoles. Dashed Lines: Increasing distance D2 between twothin layers, where the first layer is directly at the surface of the sphere.

right hand side of Fig. 7.16: A thick layer with thickness T placed directly atthe surface of the sphere (solid lines) and two thin layers, one directly at thesurface of the sphere and the second in a distance D2 = T (dashed lines). Theintensity ratio and the position of the first maxima hardly deviate in these twocases. Only for larger scattering angles, the two scenarios deviate when D2 orT are larger than 40 nm. For example, for T = 100 nm, a secondary maximumin pp and sp polarization is visible that vanishes at around T = 130 nm. ForD2 = 100 nm a secondary maximum in pp-polarization exists and a secondarymaximum in sp-polarization evolves for D2 = 160 nm. However, the absoluteintensities at larger angles are very low, so that practically the differences arenegligible (see Fig. 7.21). In this simulation, the number of dipoles ND is keptconstant in every layer. This makes sense for the extended shell of thickness Tthat is formed by the polyelectrolyte chains. However, for the two layers withdistance D2, the outer interface is larger than the inner interface and differentcharges might lead to a different strength of the contribution. The ratio ofthe number of orders used for the inner and the outer interface, Nout

Ninis an

additional parameter in the model. The density of oriented water moleculesat the interface is proportional to ρ ∝ N/R2, so that the ratio of densities is


given as ρoutρin

= Nout/R2out

Nin/R2in

. If the two interfaces are identical, one would expectρoutρin

= 1, however, the different charge at the two interfaces and the differentroughness of the surfaces can lead to a different ratio.

−150 −100 −50 0 50 100 15010

−4

10−3

10−2

10−1

100

Scattering angle (deg)

Rel

ativ

e In

tens

ity

D2 = 100 nm

T = 100 nmD

1 = 40 nm

Figure 7.21.: Simulation of angle resolved SH-intensities for a R = 110 nm polystyrenesphere in water. rd is concentrated a) in two layers at distance 0 and D2 = 100 nm,b) in a thick layer of thickness T = 100 nm and c) in a thin layer at D1 = 150 nm.

Salt Geometry Thin layer Thick layer Two layersc(NaCl) RDLS TDLS D1 T D2

ρoutρin

0.1 mM 192 nm 82 nm 50 nm 105 nm 82 nm 0.461 mM 180 nm 70 nm 15 nm 30 nm 70 nm 0.0910 mM 168 nm 58 nm 10 nm 20 nm 58 nm 0.0675 mM 162 nm 52 nm 5 nm 10 nm 52 nm 0.05

Table 7.4.: Parameters for the three models (thick shell of thickness T , single thinlayer at distance D1, two layers at the core surface and in a distance D2, where theouter layer contributes with a factor ρout

ρin) that are needed to fit the experiment.

To summarize, all geometries are able to reproduce the measured SH-inten-sities. However, not all of them fit to the geometry of the particles in the sameway. Using a single layer at an effective distance, D1 has to be chosen 50 nm,15 nm, 5 nm, and 5 nm for the particles at 0.1 mM, 1 mM, 10 mM, and 75 mMrespectively (see Table 7.4). This does not match the geometry of the particles


with outer radius 192 nm, 180 nm, 168 nm, and 162 nm respectively. So theexplanation of the origin of the second harmonic signal from an effective radiusthat is much smaller than the outer radius of the brush particle is not very real-istic as the physical cause. Likewise, a thick shell of thickness T can reproducethe measured SH-intensities, but the shell thickness does not match the actualgeometry of the particles. It has to be much larger for the low salt concen-tration and shorter for the high salt concentration than the particle geometrymeasured by dynamic light scattering. Only the model with two thin shells,one at the core surface and the other in a distance D2 with a factor is ableto fit the measured SH-intensities and at the same time match the geometryof the particles. The contribution from the outer interface is lower than fromthe inner interface. Already for a slightly increasing salt concentration, whichleads to a shortening of the polyelectrolyte chains and presumably an raveledgeometry of the brush chains, this effect becomes even stronger with ρout

ρinbe-

tween 0.09 and 0.05. This most likely displays the undefined outer interfacewith low charges, where water molecules do not align in high concentration.

8. Second harmonic generation from nanoshellparticles

After SHG from dielectric particles has been discussed extensively, this chapterdeals with nanoshells: dielectric particles of radius R that are covered witha metallic shell of thickness D (Fig. 8.1). The light-matter interaction with ametallic material allows for the formation of surface plasmon polaritons: electro-magnetic waves trapped at the dielectric-metallic interface. In all simulations,the refractive indices for metals given by Johnson and Christy [50] are used.

DR

Figure 8.1.: Nanoshell particle, consisting of a dielectric core (radius R) and a metallicshell (thickness D).

8.1. Silica particles with a gold shell of increasing thickness

Experiments on silica particles (R = 150 nm) with a growing gold nanoshellshow that the SH-intensity increases during the growth process, after aboutone minute reaches a peak for a certain shell thickness or coverage of the par-ticle’s surface, and then decreases again (Figure 8.2). SEM-images of the par-ticles with different amounts of gold show that the growth of the gold shellis formed from seeds which grow individually and slowly form larger islands

91

92 8. Second harmonic generation from nanoshell particles

0 200 400 6000

0.2

0.4

0.6

0.8

1

time (s)

SH

−in

tens

ity I pp

(θ=

20°

) (a

rb.u

.)

0 5 100

0.5

1

shell thickness (nm)

I pp(θ

=20

°)

(arb

.u.)

Figure 8.2.: SH-intensity, measured under an angle of ϑ = 20° as a function of timeduring the growth process of a gold nanoshell on a silica particle. The inset shows asimulation of Ipp(20° for a silica particle with gold shell of increasing thickness.

that merge into a complete shell. Schürer [79] performed a FDTD simulation(Lumerical Solutions, Inc.) of the scattered fundamental harmonic fields. Inthis simulation, the particle’s geometry is built from a silica sphere with 2000gold hemispheres statistically distributed on the surface of the silica particle.The size of the gold hemispheres is varied between r = 2.5 nm and r = 17.5 nm.For certain configurations, strong field enhancement occurs in the nano gapsbetween the gold particles. The SH-intensity is assumed to by directly propor-tional to the fundamental harmonic field to the power of four, so that this fieldenhancement is expected to result in a strongly increased SH-intensity. Theaverage of the field at the surface of the particle, as simulated by FDTD, be-comes as high as 〈Esurface〉4

E4inc

= 108 for a hemisphere radius r = 12.5 nm. Whenthe gold islands close to a complete nanoshell and the nano gaps vanish, thefield enhancement at the surface is lowered. This explains the decrease of theSH-intensity.

However, the gold surfaces at the walls of these gaps are opposite of eachother and their surface normals point in opposite directions. The nonlinearpolarization vectors of both surfaces point in opposite directions and becausethey are so close, they should cancel each other. This contradiction, that ahigh intensity in the fundamental harmonic does not necessarily lead to a highsecond harmonic signal is not considered in this model.

A nonlinear Mie simulation with a shell of constant thickness also shows

8.1. Silica particles with a gold shell of increasing thickness 93

a strong enhancement of the SH-intensity. The inset into Fig. 8.2 shows thesimulated Ipp(ϑ = 20°) for an increasing thickness of the gold shell. The SH-in-tensity is strongly increased for a shell thickness of 7 nm to 8 nm. This shows,that the patchy surface structure gives rise to a very similar optical responseas a perfect shell.

−100 −50 0 50 100100

200

300

400

scattering angle (°)

inte

nsity

(co

unts

/0.2

s)

Ippexp.

Ipsexp.

Ippsim.

Ipssim.

Figure 8.3.: SH-intensities Ipp(ϑ) and Isp(ϑ) for a silica particle (R = 150 nm) withan approximately D = 30 nm thick gold shell.

The angular scattering pattern of the silica particle with a thick gold shellof approximately 20 nm to 30 nm thickness is shown in Fig. 8.3. No background-signal was available for this measurement, so the y-axis starts at 100 counts/0.2 s.Subtracting a background-measurement would presumably make the peaksmore narrow. The solid lines show a fit to the nonlinear Mie model, usingχ

(2)zzz as surface susceptibility for the inner and outer interface and a shell thick-

ness of D = 30 nm.The nonlinear surface susceptibilities χ(2) of metals are not well investigated,

however, a dominating χ(2)zzz is most realistic (Butet et al. [13], Wang et al.

[92], Bachelier et al. [4]). For a silica sphere of radius (R = 100 nm) with a goldshell between 20 nm and 30 nm, the scattering profiles do not depend much onthe surface χ(2) (Figure 8.4, compared to 7.4 for a PS sphere of R = 100 nm).Here, χ

(2)ijk at the inner interface is the negative value of χ

(2)ijk at the outer

interface. In other words: χ(2)ijk is defined with respect to the surface normal of

the gold surface. The number of dipoles is chosen proportional to the surfacearea, so that the outer surface has a slightly stronger contribution than theinner interface.

The position of the first maximum in pp-polarization is between 20° and


θ1pp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

θ

1sp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

pos.

of f

irst m

axim

umθ

1 pp, θ

1 sp (

deg)

20

30

40

50

Imax

pp/ Imax

sp χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

inte

nsity

rat

io I

max

pp/ I

max

sp

0

2

4

6

Imax

pp χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

x

y

max

. int

ensi

ty I

max

pp (

arb.

u.)

D = 20 nm

0.5

1

1.5

θ1pp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

θ

1sp χ

zzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

pos.

of f

irst m

axim

umθ

1 pp, θ

1 sp (

deg)

20

30

40

50

Imax

pp/ Imax

sp χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

inte

nsity

rat

io I

max

pp/ I

max

sp

0

2

4

6

Imax

pp χzzz

−χzzz

χzxx

−χzxx

χxxz

−χxxz

x

y

max

. int

ensi

ty I

max

pp (

arb.

u.)

D = 30 nm

0.5

1

1.5

2

2.5

Figure 8.4.: Scanning for χ(2)-components for a silica-particle (R = 150 nm) with agold shell of D = 20 nm (top) or D = 30 nm (bottom).

8.1. Silica particles with a gold shell of increasing thickness 95

25°, the first maximum in sp-polarization lies between 40° and 50°. The ratiobetween maximum intensity in pp- and sp-polarization varies between 1 and 7.It is impossible to extract these parameters from the measurement (Fig. 8.3)with the precision needed. Remarkable is, that the maximum intensity in pp-polarization is largest for χ

(2)zzz. The graph shows, that the signal only originates

from the χ(2)zzz. As the ratio between the intensities for the two polarizations

does not vary much with χ(2)ijk and Imax

pp is always larger than Imaxsp , this holds

not only for Imaxpp but also for Imax

sp .


8.2. Second harmonic generation from silica—silver nanoshells

Gold has the disadvantage that the interband transition at roughly 500 nmwavelength makes it a bad candidate for plasmonic effects at the SH-wavelengthof 400 nm. Therefore, the study of plasmonic resonances in the SHG fromnanoshell particles is continued for silver, where the absorption is much lowerin the visible. For silver nanoshells strong plasmonic resonances at the funda-mental harmonic and the second harmonic frequency lead to an increase of thescattered second harmonic light (Wunderlich and Peschel [97]).

Sca

tterin

g an

gle

θ (

°)

Ipp

(θ)

5 10 15 20 25 30

−150

−100

−50

0

50

100

150

Inte

nsity

(ar

b. u

.)

0

0.2

0.4

0.6

0.8

1

Sca

tterin

g an

gle

θ (

°)

Shell Thickness (nm)

Isp

(θ)

5 10 15 20 25 30

−150

−100

−50

0

50

100

150

Inte

nsity

(ar

b. u

.)

0

0.2

0.4

0.6

0.8

1

Figure 8.5.: Ipp(ϑ) and Isp(ϑ) for SHG from a silica particle (R = 100 nm) with asilver shell (with zeros plasmon losses) of varying thickness.

The SH-intensities Ipp(ϑ) and Isp(ϑ) of a particle consisting of a silica coreand a silver shell show a very distinctive pattern for an increasing shell thickness

8.2. Second harmonic generation from silica—silver nanoshells 97

5 10 15 20 25 300

1

2

3

4

5

SH

Inte

nsity

(ar

b.u.

)

without plasmon lossesεshell

= − 4.4

Ipp

Isp

5 10 15 20 25 300

0.5

1

1.5


SH

Inte

nsity

(ar

b.u.

)

with plasmon lossesεshell

= − 4.4 + 0.2i

Ipp

Isp

Figure 8.6.: Mean SH-intensities 〈Ipp(ϑ)〉ϑ and 〈Isp(ϑ)〉ϑ from a silica particle (R =100 nm) with a silver shell of varying thickness.

from 0 nm to 30 nm (Figure 8.5). The intensity is strongly enhanced for 4 nm,7 nm, 18 nm, and 25 nm shell thickness (Fig. 8.6). The resonances are even morepronounced, when the imaginary part of the permittivity ε is artificially set tozero, i.e. the plasmon losses are suppressed. When plasmon losses are included,the field enhancement at resonance is lowered and resonances at larger shellthickness disappear (Fig. 8.6).

To determine the origin of the resonances, the SHG-process is decomposedinto its two steps: First, the nonlinear polarization is formed at the inner andouter interface of the core-shell particle. The nonlinear polarization is directlyproportional to the square of the scattered fundamental harmonic wave. Hence,plasmonic resonances at the fundamental harmonic frequency lead to a highersecond harmonic intensity.

The second step is the conversion of the nonlinear polarization at the parti-cle’s interface to a radiated second harmonic field. If the particle’s geometryis resonant with the generated second harmonic radiation, this also enhances


the second harmonic signal. Because this effect is more intuitive, section 8.2.1explains the origin and location of the resonances at the second harmonic fre-quency and section 8.2.2 focuses on the resonances at the fundamental harmonicfrequency. A possible application of SHG from nanoshell particles is monitoringof the growth process of a nanoshell as suggested in section 8.2.3.

8.2.1. Resonances at the second harmonic frequency

The nonlinear Mie model is based on the scattering of individual dipoles placednear the surface of the sphere. To study the plasmonic resonances at the secondharmonic frequency, the situation of a single dipole near a spherical particle isinvestigated.

This is not only a purely theoretical investigation but also corresponds toexperiments: Single molecule spectroscopy is often performed near metallic sur-faces in order to profit from the enhanced local fields also known from surfaceenhanced Raman scattering (SERS). The metallic tip of a near-field scanningoptical microscope can be used to enhance and detect radiation from singlemolecules (Gersen et al. [41]). The environment however, influences the molec-ular emission properties (Drexhage [31]). Kühn et al. [54] used a spherical goldnanoparticle as an optical nanoantenna to enhance the fluorescence signal. ACCD camera image of the single molecule shows that the emission pattern ishardly influenced by the nanoparticle, when the gold nanoparticle is movedover the molecule. However, no resonant geometry was used here. Taminiauet al. [88] used antennas with a fixed dipole moment that is resonant at theoptical frequency to control the angular profile of the fluorescence of singlemolecules. Pustovit and Shahbazyan [74] also modeled an ensemble of K = 30dipoles near a metallic nano-sphere and determined the decay rate and totalradiated energy.

Here, a dipole close to a nanoshell is studied. When placing a dipole closeto a nano sphere, in the case of a resonant situation, the emission pattern ofthe dipole is drastically influenced by the presence of the sphere and the fieldstrength enhanced by several orders of magnitude (Fig. 8.7). The emittingdipole is hardly visible and all radiation seems to emerge from the particle in astar-like pattern. When the dipole is perpendicular to the surface, a maximumis seen at the position of the dipole and at the opposite site. When the dipoleis parallel to the surface, maxima and minima change place. Obviously, thenear field of the dipole is imaged onto the opposite side of the sphere, showingthat a spherical particle has perfect imaging properties even on sub wavelength


RD

x (nm)

y (n

m)

−500 0 500

−500

0

500 Inte

nsity

|ED

ipol

e |2 (ar

b. u

.)

1e−3

1e−2

1e−1

x (nm)

y (n

m)

−500 0 500

−500

0

500 Inte

nsity

|ES

yste

m|2 (

arb.

u.)

1

1e1

1e2

1e3

x (nm)

y (n

m)

−500 0 500

−500

0

500 Inte

nsity

|ES

yste

mr

|2 (ar

b. u

.)

1

1e1

1e2

1e3

x (nm)

y (n

m)

−500 0 500

−500

0

500 Inte

nsity

|ES

yste

mθ

|2 (ar

b. u

.)

1

1e1

1e2

1e3

Figure 8.7.: Field distribution |EDipole|2 of a dipole of λ = 400 nm wavelength (topleft) and field distribution |ESystem|2 of the dipole modified by the prescence of asphere (top right). The lower images show the radial and tangential parts of |ESystem

r |2and |ESystem

ϑ|2 separately. The sphere has a silica core of radius R = 100 nm and a

silver shell, where plasmon losses are set to zero, of thickness D = 25 nm. The shellthickness is tuned for the resonance K = 5 with 10 lobes. The far-field is shown inFig. 8.7

x (cm)

y (c

m)

−2 0 2

−2

−1

0

1

2 Inte

nsity

|ES

yste

mr

|2 (ar

b. u

.)

1e−20

1e−19

1e−18

1e−17

1e−16

1e−15

x (cm)

y (c

m)

−2 0 2

−2

−1

0

1

2 Inte

nsity

|ES

yste

mθ

|2 (ar

b. u

.)

1e−9

1e−8

1e−7

Figure 8.8.: Field distribution of a dipole close to a silica-silver core-shell particle. Theparameters are as in Fig. 8.7, but the far-field is shown. The radial field componentsare slightly stronger close to the sphere (Fig. 8.7) but the tangential field componentsradiate better into the far-field.


scale.

Sca

tterin

g A

ngle

θ (

°) without plasmon losses

20 nm distance

10 20 30 40 50

−100

0

100

Inte

nsity

|Eθ|2 (

arb.

u.)

1e−1

1e0

1e1

1e2

1e3

1e4

1e5

10 20 30 40 5010

−3

10−1

101

103

Inte

nsity

⟨ |E

θ(θ

)|2 ⟩ θ

(ar

b.u.

)

without plasmon losses2 cm distance

Dipol at inner interfaceDipol at outer interface

10 20 30 40 500

1

2

3

4

5


Inte

nsity

⟨ |E

θ(θ

)|2 ⟩ θ

(ar

b.u.

)

with plasmon losses

2 cm distance

Dipol at inner interfaceDipol at outer interface

Figure 8.9.: Tangential component of the Intensity |Eϑ(ϑ)|2 of a dipole (λ = 400 nm),which is placed at the inner respectively outer interface of a sphere with silica core(R = 100 nm) and a silver shell (with and without damping) of variable thickness.Top: angle resolved intensity in 20 nm distance to the surface of the sphere. Middleand bottom: averaged intensity 〈|Eϑ(ϑ)|2〉ϑ in 2 cm distance without (ε400 nm

shell = −4.4)and with (ε400 nm

shell = −4.4 + 0.2 i) plasmon losses.

When changing the thickness of the shell from 0 nm to 50 nm, resonantenhancement of the intensity of the scattered dipole occurs in the far field(Fig. 8.9). For the case with plasmon losses, an enhanced field occurs forD = 25 nm, explaining one of the resonances in the SH-intensity (Fig. 8.6).For decreasing shell thickness, two more lobes arise at every resonance. Thissuggests a surface plasmon polariton running at the surface of the sphere. A


resonance with 2K lobes occurs, whenever the effective wavelength of the sur-face plasmon polariton in the multi-layered system fits to the circumference ofthe sphere: 2πR = KλSPP. Hence, resonances depend on both the diameter ofthe sphere and the effective wavelength of the surface plasmon polariton. Thelatter one is mainly determined by the thickness of the shell.

To test the hypothesis that resonances are caused by running surface plas-mon polaritons, a solid metallic particle is discussed shortly. If the sphere issufficiently large, so that the surface can be assumed to be flat, the effectivewavelength of the surface plasmon polaritons is solely determined by the ma-terial parameters of the metallic sphere and the surrounding. The wavelengthof a surface plasmon polariton running at the planar interface between a metal(εSphere) and a dielectric (εSurrounding) is known analytically as

λSPP = λDipole

√εSphere + εSurrounding

εSphere · εSurrounding.

scat

terin

g an

gle

θ (

°)

permittivity ε−4 −3.5 −3 −2.5 −2

−150

−100

−50

0

50

100

150

Figure 8.10.: Tangential component of the intensity |Eϑ(ϑ)|2 of a dipole (λ = 400 nm),which is placed at the surface of a metallic sphere (R = 100 nm) with varying permit-tivity εSphere. Resonances with 2K lobes occuring at certain values of the permittivityof a bulk metallic sphere.

Figure 8.10 shows the tangential component of the intensity |Eϑ(ϑ)|2 of adipole (λ = 400 nm), which is placed at the surface of a metallic sphere (R =100 nm) with varying permittivity εSphere. Resonances with 2K lobes occur atcertain values of the permittivity of a bulk metallic sphere. At each resonancestwo more lobes arise.


Resonance positions in fact occur almost exactly when the circumference ofthe sphere coincides with multiples of λSPP, even though the interface is highlycurved (Fig. 8.11).

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30−3.5

−3

−2.5

−2

K = number of lobes

perm

ittiv

ity ε

expected resonancesMie simulation

Figure 8.11.: Resonances with 2K lobes occuring at certain values of the permittivityof a bulk metallic sphere εSphere

K in comparison to predictions based on a simplifiedmodel where a planar surface plasmon polariton is assumed to run around the spherefor R = 100 nm and λDipole = 400 nm.

Not all of these resonances, seen in Fig. 8.9 for the core-shell-particle, radi-ate into the far field. For a smaller shell thickness, the plasmon wavelengthdecreases. But, surface plasmon polaritons are guided waves and do only in-teract with the farfield due to the curvature of the surface. The thinner theshell, the shorter is the surface plasmon polariton’s wavelength and the strongerthe guidance. Therefore resonances at small shell thickness can be excited bydipoles, but they are only visible close to the surface of the sphere and do notsignificantly contribute to an emission to the farfield (Fig. 8.12).


10 20 30 40 5010

−5

10−4

10−3

10−2

10−1

100

101

102

103

104

105


Inte

nsity

|Eθ|2 (

arb.

u.)

Detection distance

10 nm

100 nm

200 nm

300 nm

400 nm

Figure 8.12.: Intensity 〈|Eϑ(ϑ)|2〉ϑ of a dipole (λ = 400 nm), which is placed at theouter interface of a sphere with silica core (R = 100 nm) and a silver shell (withoutdamping) of variable thickness. The intensity is detected in a variable distance to thesurface of the sphere ranging from 10 nm to 400 nm.


8.2.2. Resonances at the fundamental harmonic frequencyResonances of a plane wave scattered from a spherical particle are known as Mieresonances. They are usually seen in the scattering cross sections. Prodan andNordlander [71], Prodan et al. [73], Prodan and Nordlander [72] have discussedthe optical response of nanoshells that can be calculated using Mie theory, andhave interpreted respective spectra as a result of hybridization of the resonancesof the simpler shapes—the cavity and the sphere.

0 10 20 30 40 5010

0

101

102

103

104

105


Fie

ld S

tren

gth

|Er|4 (

arb.

u)

without plasmon lossesεshell

= − 4.4

outer interfaceinner interface

0 10 20 30 40 500

50

100

150


Fie

ld S

tren

gth

|Er|4 (

arb.

u)

with plasmon lossesεshell

= − 4.4 + 0.2i

outer interfaceinner interface

Figure 8.13.: Plasmonic enhancement of the FH-fields at the inner and outer interfaceof a silica–silver core shell particle. |Er|4 is shown, because with a χ

(2)zzz surface

susceptibility the SHG-intensity will be proportional to |Er|4.

The electric fields at the inner and outer surface of the core-shell particleare enhanced for certain shell thicknesses (Fig. 8.13). As the SH-intensity isproportional to the forth power of the field at the interfaces, those resonancesare also seen in the SH signal. The figure shows only the radial component offield |Er|4. The tangential parts are not shown, but exhibit the same behaviorand are of the same order of magnitude. Strong field enhancement is seenfor a shell thickness of 2.4 nm, 3.0 nm, 4.0 nm, and 6.5 nm. The last two are


still clearly visible, when realistic plasmon losses are included into the model.These resonances at 4.0 nm and 6.5 nm shell thickness are responsible for theenhancement of the SH-intensity at this shell thickness (Fig. 8.6).

z (nm)

x (n

m)

−100 0 100

−150

−100

−50

0

50

100

150

Inte

nsity

|Ep.

w.

r|2 (

arb.

u.)

0

1

2

3

z (nm)

x (n

m)

−100 0 100

−150

−100

−50

0

50

100

150

Inte

nsity

|Ep.

w.

r|2 (

arb.

u.)

0

2

4

6

z (nm)

x (n

m)

−100 0 100

−150

−100

−50

0

50

100

150

Inte

nsity

|Ep.

w.

r|2 (

arb.

u.)

0

20

40

60

80

100

z (nm)

x (n

m)

−100 0 100

−150

−100

−50

0

50

100

150

Inte

nsity

|Ep.

w.

r|2 (

arb.

u.)

0

20

40

60

80

100

Figure 8.14.: Distribution of the radial component of the electric field of a scatteredplane wave around a silica sphere (R = 100 nm) with a growing silver shell (imaginarypart of ε set to zero). The shell thickness is 2.4 nm, 3.0 nm, 4.0 nm, and 6.5 nm, whichcorresponds to the resonant field enhancement in Fig. 8.13.

The spatial distribution of the radial component of the field shows that eachresonance is characterized by a pattern with 2K lobes, where the number oflobes decreases by two when the shell thickness increases from one resonantvalue to the next (Fig. 8.14). This is very similar to the field distributionseen in section 8.2.1 for the second harmonic. Also, the shell thickness, forwhich a resonant behavior is seen, are identical, no matter if a plane wave withλ = 800 nm excites the system or a dipole with λ = 800 nm, placed at the outerinterface as in section 8.2.1 (Fig. 8.15). Also the number of lobes is identicalat each resonant configuration (Fig. 8.16).

This comparison shows, that also the resonances at the fundamental har-monic frequency are caused by surface plasmon polaritons. The enhanced sec-ond harmonic intensity generated at the surface of a nanoshell particle are


0 2.5 5 7.5 10 12.5 1510

−210

−110

010

110

210

310

4

Inte

nsity

|Edi

p.r

|2 (ar

b.u.

)


Plane WaveDipole

0 2.5 5 7.5 10 12.5 1510

−2

100

102

104

106

108

Inte

nsity

|Ep.

w.

r|2 (

arb.

u.)

Figure 8.15.: Field enhandement at the surface of a silica sphere (R = 100 nm) witha growing silver shell as a function of shell thickness. The system is excited by aplane wave with λ = 800 nm (see also Fig. 8.13) or a dipole with λ = 800 nm, placedat the outer interface. Resonant field enhancement is seen at the same thickness,independent of the kind of excitation.

z (nm)

x (n

m)

−100 0 100

−150

−100

−50

0

50

100

150

Inte

nsity

|Edi

p.r

|2 (ar

b. u

.)

0

1

2

3

4

5x 10

6

z (nm)

x (n

m)

−100 0 100

−150

−100

−50

0

50

100

150In

tens

ity |E

dip.

r|2 (

arb.

u.)

0

2

4

6

8

10x 10

5

z (nm)

x (n

m)

−100 0 100

−150

−100

−50

0

50

100

150

Inte

nsity

|Edi

p.r

|2 (ar

b. u

.)

0

1

2

3

4

5x 10

5

z (nm)

x (n

m)

−100 0 100

−150

−100

−50

0

50

100

150

Inte

nsity

|Edi

p.r

|2 (ar

b. u

.)

0

1000

2000

3000

4000

5000

Figure 8.16.: Distribution of the radial component of the electric field of a scattereddipole, placed at the outer interface, around a silica sphere (R = 100 nm) with agrowing silver shell (imaginary part of ε set to zero). The shell thickness is 2.4 nm,3.0 nm, 4.0 nm, and 6.5 nm, which corresponds to the resonant field enhancement inFig. 8.15.


driven by the fundamental harmonic wavelength, the geometry of the particle(size and shell thickness), and the material parameters. Resonances at low shellthickness are caused by surface plasmon polaritons generated by the incidentplane wave at fundamental harmonic wavelength. Resonances at larger shellthickness by surface plasmon polaritons generated by the second harmonic light.This high sensitivity to the particle’s geometry makes SHG an excellent methodto study the growth of nanoshells in situ by matching the FH-wavelength to thedesired geometry and interrupt the formation of the nanoshell at a resonance.


8.2.3. Sensing applicationThe high sensitivity of SHG to the thickness of a metallic nanoshell arounda dielectric spherical particle makes this process a useful monitoring methodduring the growth process of a nanoshell. Using a fixed wavelength that is tunedto the desired shell thickness, the chemical process can be interrupted oncethe shell thickness is reached. The wavelength is chosen, so that an resonantenhancement of the SH-intensity is reached at a slightly lower shell thicknessthan the desired value. Once the measurement shows a peak and the SHintensity is decreasing again, the chemical process is interrupted.

600 650 700 750 800 8500

0.5

1

1.5

2

I pp(θ

=20

° ) (a

rb.u

.)

D= 10 nm

D= 7 nm

D= 5 nm

600 650 700 750 800 8500

0.1

0.2

Cex

tinct

ion (

μm2 )

FH−wavelength (nm)

Figure 8.17.: Second harmonic intensity Ipp(ϑ = 20°) as a function of the fundamentalharmonic wavelength for silica particles of R = 100 nm radius with a silver shell ofthickness D = 5 nm, 7 nm,.png -m2 and 10 nm. As a comparison, the linear extinc-tion spectrum is also shown. No wavelength-dependence of χ(2) is applied in thiscalculation.

In a similar manner, scanning the FH-wavelength can be used to determinethe shell thickness of a particle with known composition and core radius. Asthe SHG is dominated by the resonances at the fundamental harmonic, alsolinear scattering is sensitive to the thickness of the metallic shell. However,


the peaks in the SH-intensity have a much higher contrast and are narrowerthan the peaks in the linear extinction spectrum (Fig. 8.17). The high surfacesensitivity makes SHG superior to linear scattering.

9. Conclusion and outlook

To summarize, this work shows that second harmonic generation is a versatiletool for the study of surface properties of nanoparticles in aqueous suspension.Nanoparticles have attracted growing interest in recent years. They promisebenefits in several applications ranging from bio sensing, pharmacology, ultra-light composite materials, to catalysis, and many more. As nanoparticles areproduced by chemical synthesis, exact knowledge of their surface is essentialfor optimizing their production. SHG enables the characterization of particlesurfaces in-situ during the manufacturing process and of stuying the reactionsinvolving nanoparticles in real time.

The angle resolved SH-intensity for different polarization combinations de-pends strongly on the composition of the nanoparticles and their surroundings,the size of the nanoparticles, and the surface properties that are given by thesurface susceptibility. A detailed study of this influence, using numerical simu-lation techniques has been the interest of several groups during the last decade.The unique feature of the molecular Mie model introduced in this work is itsability to model complex systems like core-shell particles and particles, wherethe nonlinear polarization is not restricted to the surface. Comparison of thenewly developed molecular Mie model to existing other nonlinear Mie models,which were developed at the same time, has proven the high reliability of themodel. The influence of numerical parameters, such as number of orders andrefractive index contrast were studied.

The model allows a deeper understanding of data gained from second har-monic scattering experiments. The calculation costs are low. The results allowfor extracting fundamental properties of the studied system based on only afew assumptions. Thus, we were able to investigate the adsorption of malachitegreen to the surface of polystyrene particles and to determine the nonlinear sus-ceptibility which we found to be consistent with data from plane silica surfaces.We analyzed the geometry of polyelectrolyte brush particles that consist of apolystyrene core with attached polyelectrolyte chains. The chains are radiallystretched as they repel each other due to equal charging. The chains wereexperimentally observed to crumple when salt is added to the solution. Thesimulations showed that the second harmonic is generated by oriented water

111

112 9. Conclusion and outlook

molecules. These molecules do not align along the chains, as one might firstguess, but at the surface of the polystyrene core and at the outer interfaceformed by the ends of the chains. With increasing salt concentration, the outerinterface becomes less well defined and the density of oriented water moleculesat the outer interface is reduced drastically.

The molecular Mie model was also extended to nanoshell particles consistingof a dielectric core and a metallic shell. In-situ experiments on silica particleswith a growing gold shell showed an enhanced SH-intensity for specific coverageof the particles. The growth process starts with gold seeds that evolve intolarger gold clots and then islands that successively connect and form a closedshell. The simulations can reproduce the enhanced SH-intensity even thoughit assumes a closed gold shell of variable thickness. This shows that patchyparticles can influence light in a very similar way to a nanoshell.

Theoretical investigations of SHG from silver nanoshells where the plasmonlosses were artificially reduced gave deeper insight into the formation of res-onances. An enhanced SH-intensity is observed, when either the scatteredfundamental harmonic fields are enhanced close to the sphere (Mie resonances)or when the second harmonic, which is generated by the polarization at theinterface, is in resonance with the nanoshell. In both cases surface plasmonpolaritons are created and give rise to an enhanced SH signal when low integermultiples of their effective wavelength equal the circumference of the sphere.

The nonlinear Mie model is also applicable to sum frequency scattering fromspherical nanoparticles. Here, usually the fundamental harmonic wavelengthsλVIS = 800 nm along the z-axis and λIR = 3500 nm with an angle of 15° withrespect to the z-axis are used. Because of the asymmetric setup, the angle re-solved SF-intensities are no longer symmetric in ϑ (Fig. 9.1). The angle resolvedmeasurements that were planned are still to be experimentally implemented, sono comparison of theory and experiment was possible. The set-up more widelyused is one in which the wavelength dependence at a fixed angle is measured,rather than the angular dependence at a single wavelength. The wavelengthdependence comes from the pulsed fundamental harmonic at λIR = 3500 nm,that leads to a frequency broadening. To simulate the frequency dependence,detailed knowledge about the frequency dependence of the seven independentcomponents of the surface susceptibility χ

(2)i,j,k(−ωSFG|ωVIS, ωIR) is needed.

For the future, it would be useful to expand the molecular Mie model to non-spherical particles. The T-Matrix method offers such a possibility (Mishchenkoet al. [67]). Like Mie theory it is based on the expansion into vector sphericalharmonics such that existing strategies can be adopted. The gold nanoshell

113

−100 0 1000

1

2

3

4


SF

−in

tens

ity (

arb.

u.)

χ

zzz(2)

Ippp

(θ)

Ipss

(θ)

Isps

(θ)

Issp

(θ)

−100 0 1000

1

2

3

4


SF

−in

tens

ity (

arb.

u.)

χ

zxx(2)

Ippp

(θ)

Ipss

(θ)

Isps

(θ)

Issp

(θ)

−100 0 1000

0.2

0.4

0.6

0.8

1


SF

−in

tens

ity (

arb.

u.)

χ

xxz(2) I

ppp(θ)

Ipss

(θ)

Isps

(θ)

Issp

(θ)

−100 0 1000

1

2

3

4


SF

−in

tens

ity (

arb.

u.)

χ

xzx(2)

Ippp

(θ)

Ipss

(θ)

Isps

(θ)

Issp

(θ)

Figure 9.1.: Sum frequency generation from polystyrene particles of R = 250 nm radiusin water. The SF-intensities are shown for the four independent non-chiral componentsof the surface susceptibility χ(2): χ

(2)zzz, χ

(2)zxx, χ

(2)xxz, and χ

(2)xzx. From the eight polar-

ization combinations, only the intensities Ippp(ϑ), Ipss(ϑ), Isps(ϑ), and Issp(ϑ) arenonzero.

particles from Chapter 8 consist of a very patchy structure of single gold seedsthat grow into larger islands. Using a commercial FDTD-software it was pos-sible to simulate the scattering of the fundamental harmonic plane wave butnot the generation of the second harmonic. The T-matrix method would allowthe simulation of a silica sphere covered with much smaller gold spheres andcalculate the SH-signal from this system. A comparison to the simulation of acomplete thin gold shell would yield further insight into the optical propertiesand surface properties of patchy particles with respect to perfect nanoshells.

A. Mathematical definitions

A.1. Bessel functions

Bessel functions Z(x), or cylinder functions, are solutions of the differentialequation

x2 d2Z

dx2 + xdZ

dx+ (x2 − n2)Z = 0 . (A.1)

The two linearly independent solutions for integer n are the Bessel function offirst kind Z

(1)n (x) = J(x) and the Bessel function of second kind Z

(2)n (x) = Y (x)

(Neumann function). These solutions can also be superimposed to the Hankelfunctions Z

(3)n = H

(1)n (x) = Jn(x)+iYn(x) and Z

(4)n = H

(2)n (x) = Jn(x)−iYn(x).

Bessel functions fulfill the recursion relation [see 10, Eq. 9.54a]

Zn−1(x) + Jn+1(x) = 2n

xZn(x) ,

12

[Zn−1(x) − Zn+1(x)] = [Zn(x)]′ .

A.1.1. Spherical Bessel functions

The spherical Bessel functions

zn(r) =√

π

2rZn+ 1

2(kr)

solve the differential equation

r2 d2z

dr2 + 2rdz

dr+ (k2r2 − n(n + 1))z = 0

which is equivalent to (A.1) for x �→ kr, Z(r) �→ √rz(r) and n �→ n + 1

2 .

115

116 A. Mathematical definitions

The first three Bessel functions are (Fig. A.1):

j0(x) = sin x

x, y0(x) = − cos x

x,

j1(x) = sin x

x2 − cos x

x, y1(x) = − cos x

x2 − sin x

x,

j2(x) =(

3x2 − 1

)sin x

x− 3 cos x

x2 , y2(x) =(

1 − 3x2

)cos x

x− 3 sin x

x2 .

0 5 10 15 20−0.5

0

0.5

1

ρ

j0(ρ)

j1(ρ)

j2(ρ)

0 5 10 15 20−1

−0.5

0

0.5

ρ

y0(ρ)

y1(ρ)

y2(ρ)

0i 5i 10i 15i 20i−100

−50

0

50

100

ρ

Re[j

0(ρ)]

Re[j1(ρ)]

Re[j2(ρ)]

Im[j0(ρ)]

Im[j1(ρ)]

Im[j2(ρ)]

0i 5i 10i 15i 20i−100

−50

0

50

100

ρ

Re[y

0(ρ)]

Re[y1(ρ)]

Re[y2(ρ)]

Im[y0(ρ)]

Im[y1(ρ)]

Im[y2(ρ)]

Figure A.1.: First three Bessel functions of first and second kind for real and imaginaryargument.

The numerical accuracy of the spherical Bessel functions can be checked by[see 81, Eq. 7]

jn(z)yn−1(z) − jn−1(z)yn(z) = 1z2 .

A.1. Bessel functions 117

A.1.2. Riccati Bessel functionsRiccati Bessel functions are a minor modification of spherical Bessel functionsthat offer a short hand notation in Mie scattering:

ψn(x) = xjn(x) , χn(x) = −xyn(x) , ξn(x) = xh(1)n (x) , ζn(x) = xh(2)

n (x) .

They fulfill the differential equation

x2 d2y

dx2 + [x2 − n(n + 1)]y = 0 .


A.2. Associated Legendre functions

A.2.1. Legendre Polynomials

Legendre polynomials are the solution to the Legendre differential equation

(1 − x2) d2

dx2 Pl(x) − 2xd

dxPl(x) + l(l + 1)Pl(x) = 0 .(

l ∈ N0 x ∈ [−1, 1])

The nth polynomial is a polynomial of rank n,

Pn(x) = 12nn!

dn

dxn

([x2 − 1]n

)= 1

2nn!

n∑ν=1

(−1)n−nu

(nν

)(2ν)!

(2ν − n)! x2ν−n ,

with special values P0(x) = 1, Pn(1) = 1, Pn(−x) = (−1)nPn(x).Legendre polynomials fulfill the recursion relation [see 10, Eq. 9.58]

(n + 1)Pn+1(x) = (2n + 1)xPn(x) − nPn−1(x) (n = 1, 2, . . .) ,

(x2 − 1) ddx

Pn(x) = nxPn(x) − nPn−1(x) .

Legendre polynomials form a set of orthogonal functions for a fixed m overthe interval −1 ≤ x ≤ 1:

1∫−1

dxPn(x)Pn′(x) = c[n(n + 1) − n′(n′ + 1)

]δnn′ .

A.2.2. Associated Legendre functions

Associated Legendre functions of degree n and order m are the solution to thegeneral Legendre equation

(1 − x2) d2

dx2 P ml − 2x

ddx

P ml +

(l[l + 1] − m2

1 − x2

)P m

l = 0 0 ≤ |m| ≤ n.(A.2)

A.2. Associated Legendre functions 119

For positive m

P ml (x) = (−1)m (1 − x2)m/2 dm

dxmPl(x) m ≥ 0

= (−1)m

2ll! (1 − x2)m/2 dl+m

dxl+m(x2 − 1)l . (A.3)

(−1)m is known as Condon-Shortley phase and is omitted by some authors(Bohren and Huffman [6], Courant and Hilbert [17]). Sometimes this is indi-cated by the position of the indices:

Plm(x) = (−1)mP ml (x) .

For negative m the literature is inconsistent. As the differential equation(A.2) does only depend on m2, P −m

l ∝ P mm should hold. If however, (A.3) is

expected to hold for positive and negative m, P −ml becomes

P −ml (x) = (−1)m (l − m)!

(l + m)!︸︷︷︸≤1

P ml (x) m ≥ 0 .

However, this induces an asymmetry as⟨|P m

l (x)|⟩

x=√

(l+m)!(l−m)!(2l+1) increases

with increasing n or m for m > 0 and decreases with increasing n or |m| form < 0. In Mie scattering this even propagates to the expansion coefficientsam,n and bm,n and should therefore be omitted. Instead

P −ml (x) = P m

l (x) m ≥ 0

can be used.The first associated Legendre functions are (Fig. A.2)

P 00 (cos ϑ) = 1 ,

P 01 (cos ϑ) = cos ϑ , P 1

1 (cos ϑ) = − sin ϑ ,

P 02 (cos ϑ) = 1

2 (3 cos2 ϑ − 1) , P 12 (cos ϑ) = −3 sin ϑ cos ϑ ,

P 22 (cos ϑ) = 3 sin2 ϑ .


−1 −0.5 0 0.5 1−2

−1

0

1

2

3

cosθ

P2−2

P2−1

P20

P21

P22

Figure A.2.: Associated Legendre functions of order n = 2

The associated Legendre functions form a set of orthogonal functions forfixed m on the interval −1 ≤ x ≤ 1 with weighting function 1

1∫−1

dxP mn (x)P m

n′ (x) = 22n + 1

(n + m)!(n − m)!

δnn′ ,

and with weighting function (1 − x2)−1

1∫−1

dxP mn (x)P m

n′ (x) 11 − x2 = (n + m)!

m(n − m)!δnn′ . (A.4)

Also the following relations hold [see 6, Eq. 4.24]:

m

π∫0

dϑ

(P m

n (cos ϑ)dP mn′ (cos ϑ)

dϑ+ P m

n′ (cos ϑ)dP mn (cos ϑ)

dϑ

)(A.5)

= m[P m

n P mn′]π

0 = 0 .

A.2.3. Modified Legendre functions

Bohren and Huffman [6] and Bruning and Lo [12] use modified associated Leg-

A.2. Associated Legendre functions 121

endre functions

πm,n(ϑ) = ∂P mn (cos ϑ)

∂ϑ,

τm,n(ϑ) = mP mn (cos ϑ)sin ϑ

,

which are numerically easier.


A.3. Spherical coordinates

The relation between Cartesian coordinates (x, y, z) and spherical coordinates(r, ϑ, ϕ) is defined as

x = r sin(ϑ) cos(ϕ) , y = r sin(ϑ) sin(ϕ) , z = r cos(ϑ) .

For (ϑ, ϕ) = (0, 0), er = ez, eϑ = ex and eϕ = ey.

ex

ey

ez

er

eϑ

eϕ

eϕ

x

y

z

ϑ

ϕ

r

Figure A.3.: Spherical Coordinates

Coordinate transformations are performed by:

axex + ayey + azez = arer + aϑeϑ + aϕeϕ ,⎛⎝ax

ay

az

⎞⎠ =

⎛⎝sin ϑ cos ϕ cos ϑ cos ϕ − sin ϕ

sin ϑ sin ϕ cos ϑ sin ϕ cos ϕcos ϑ − sin ϑ 0

⎞⎠⎛⎝ar

aϑ

aϕ

⎞⎠

⎛⎝ar

aϑ

aϕ

⎞⎠ , =

⎛⎝sin ϑ cos ϕ sin ϑ sin ϕ cos ϑ

cos ϑ cos ϕ cos ϑ sin ϕ − sin ϑ− sin ϕ cos ϕ 0

⎞⎠⎛⎝ax

ay

az

⎞⎠ .

A.3.1. Euler anglesEuler angles describe a rotation. They can be defined in several ways. Thisnotation follows Edmonds [33]: A right handed system is first rotated about

A.3. Spherical coordinates 123

α around the z-axis, then about β around the new y-axis and finally about γaround the new z-axis. This rotation is given by the rotation matrix

M =

⎛⎝cos α − sin α 0

sin α cos α 00 0 1

⎞⎠⎛⎝ cos β 0 sin β

0 1 0− sin β 0 cos β

⎞⎠⎛⎝cos γ − sin γ 0

sin γ cos γ 00 0 1

⎞⎠

=

⎛⎝−sαsγ + cαcβcγ −sαcγ − cαcβsγ cαsβ

cαsγ + sαcβcγ cαcγ − sαcβsγ sαsβ

−sβcγ sβsγ cβ

⎞⎠ ,

where sα is used as an abbreviation for sin α etc.

Figure A.4.: Euler angles: The start coordinate system (yellow) is rotated about itsz-axis by an angle α. The new coordinate system (orange) is rotated about its y-axisby an angle β. The resulting coordiante system (red) is rotated about its z-axis by anangle γ and results in the final coordinate system (black).

Euler angles are not unique. The rotation (α, β = 0, γ) is identical to(α′, β′ = 0, γ′ = α + γ − α′). The definition is also equivalent to a rotationaround a fixed coordinate system: The first rotation is about γ around thez-axis, the second about β around the y-axis and the last about α around thez-axis.

The rotation (α, β, γ) is reversed by the rotation (−γ, −β, −α).In this work the rotation from the position (ϑ = 0, ϕ = 0) at the surface

of a sphere to the position (ϑ, ϕ) is needed. This is given by the Euler angles(α = ϕ, β = ϑ, γ = 0).

Symbols and Abbreviations

χn(x) Riccati Bessel function of kind j = 2 30, 116ε permittivity [As/Vm] 6ε0 permittivity of free space [As/Vm] 5μ permeability [N/A2] 6μ0 permeability of free space [N/A2] 5ψ

(j)n (x) Riccati Bessel function of kind j 28, 116

ψn(x) Riccati Bessel function of kind j = 1 30, 116ψ

(j)m,n(r) scalar spherical harmonic 15

ρ effective radius ρ = kr 15ρ(r) volume or surface charge density [C/m3] or

[C/m2]5

ϑ scattering angle 57ϑn

pp position of the nth maximum of Ipp(ϑ) 67ϑn

sp position of the nth maximum of Isp(ϑ) 67ξn(x) Riccati Bessel function of kind j = 3 30, 116ζn(x) Riccati Bessel function of kind j = 4 116

A(r) vector potential 6, 18am,n expansion coefficient of electromagnetic fields

into vector spherical harmonics18

an reduced expansion coefficient 29

B(r) magnetic field [Vs/m2] 5bm,n expansion coefficient of electromagnetic fields

into vector spherical harmonics18

bn reduced expansion coefficient 29

c speed of light in vacuum 7

D(r) electric displacement field [C/m2 ] 5

125

126 Symbols and Abbreviations

E(r) electric field [V/m] 5

FH fundamental harmonic 5

H(1/2)n (x) Hankel function 115

H(r) magnetizing field [A/m] 5h

(1/2)n (x) spherical Hankel function 116

Ipp(ϑ) angle resolved SH intensity for input polariza-tion and detected polarization parallel to thescattering plane

57

Imaxpp maximum of Ipp(ϑ) 67

Ips(ϑ) angle resolved SH intensity for input polariza-tion parallel and detected polarization perpen-dicular to the scattering plane

57

Isp(ϑ) angle resolved SH intensity for input polariza-tion perpendicular and detected polarizationparallel to the scattering plane

57

Imaxsp maximum of Isp(ϑ) 67

Iss(ϑ) angle resolved SH intensity for input polariza-tion and detected polarization perpendicular tothe scattering plane

57

Jn(x) Bessel function of first kind 115j(r) electric current density [A/m2] 5j kind of VSH or Bessel function 15jn(x) spherical Bessel function of first kind 116

k angular wave number 7

L(j)m,n(r) vector spherical harmonic 15

M(r) magnetization [A/m] 5M

(j)m,n(r) vector spherical harmonic 15

m order of Legendre function or VSH 15MG malachite green 77

Symbols and Abbreviations 127

N(j)m,n(r) vector spherical harmonic 15

n degree of Legendre function, order of Besselfunction, order or degree of VSH

15

n refractive index 7ND number of discrete dipole positions rd, d =

1 . . . ND

50

NFH maximum number of orders n = 1 . . . NFH forthe expansion of the fundamental harmonicwaves

49

NSF maximum number of orders n = 1 . . . NSH forthe expansion of the sum frequency waves

53

P (r) polarization per volume [C/m2] or per surface[C/m]

5

rd discrete positions for P (2) 50

SFG sum frequency generation 5SH second harmonic 5SHG second harmonic generation 5

VSH vector spherical harmonic 15

Yn(x) Neumann function 115yn(x) spherical Bessel function of second kind 116

Z(j)n (x) Bessel function of kind j 115

z(j)n (x) spherical Bessel function of kind j 115

List of Tables

7.1. Different values for the nonlinear susceptibility of Malachite Greenretrieved from various experiments. . . . . . . . . . . . . . . . . 78

7.2. Experimental results for MG on PS particles. . . . . . . . . . . 817.3. Experimental results for polyelectrolyte brush particles. . . . . 847.4. Three simulations that fit to the experiments with polyelectrolyte

brush particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

129

List of Figures

1.1. Principal sketch of second harmonic light scattering. . . . . . . 2

2.1. Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2. χ

(2)ijk as a function of inclination angle β and molecular hyperpo-

larizability βi′j′k′ . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1. Vector spherical harmonics. . . . . . . . . . . . . . . . . . . . . 173.2. Addition theorem for translation electric fields. . . . . . . . . . 25

4.1. Stratified sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2. Different positions of a dipole at a core-shell-particle. . . . . . . 33

5.1. Reciprocity theorem for Roke’s nonlinear Mie modell. . . . . . 445.2. Comparison of the molecular Mie modell with Gonella et al. [43]. 455.3. Molecular Mie Model – Step 1a: Plane wave. . . . . . . . . . . 505.4. Molecular Mie Model – Step 1b: Scattered plane wave. . . . . . 515.5. Molecular Mie Model – Step 2: Positions rd. . . . . . . . . . . 515.6. Molecular Mie Model – Step 4a: Dipole at the origin. . . . . . . 525.7. Molecular Mie Model – Step 4b: Dipole translated about a dis-

tance |rd|. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.8. Molecular Mie Model – Step 4c: Dipole scattered at the sphere. 545.9. Molecular Mie Model – Step 5: Scattered dipole, rotated to rd. 555.10. Molecular Mie Model – Step 6/7: second harmonic intensity. . 55

6.1. Experimental setup for angle resolved SHG measurements . . . 586.2. Intensity a the Gaussian laser beam inside a cylindrical cuvette. 596.3. Refraction of a light beam when passing through the cuvette. . 616.4. Deviation of detected angle from original scattering angle due to

refraction in the glass cuvette. . . . . . . . . . . . . . . . . . . . 626.5. Correction of the second harmonic intensity of a single particle

due to refraction. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

131

132 List of Figures

6.6. Scattered SH light from a single particle and from an ensembleof particles inside the cuvette. . . . . . . . . . . . . . . . . . . . 63

6.7. Effect of particle size: angular scattering profile. . . . . . . . . 646.8. Effect of particle size: position of maxima . . . . . . . . . . . . 656.9. Effect of particle size: intensity ratio and maximum intensity . 656.10. Angular SH scattering profile Ipp(ϑ) of a polystyrene sphere in

water with pure χ(2)zzz for a varying radius between R = 200 nm

and R = 250 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . 666.11. Effect of particle size distribution . . . . . . . . . . . . . . . . . 66

7.1. Definition of ϑ1sp, ϑ1

pp, and Imaxpp . . . . . . . . . . . . . . . . . . 67

7.2. Triangle to visualize χ(2) . . . . . . . . . . . . . . . . . . . . . . 687.3. Triangle to visualize χ(2) . . . . . . . . . . . . . . . . . . . . . . 707.4. χ(2)-triangles for R = 100 nm PS in water. . . . . . . . . . . . . 717.5. χ(2)-triangles for R = 100 nm PS in water (Malachite green ex-

periments). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.6. Effect of the number of expansion orders on SH-intensity. . . . 727.7. Effect of the refractive index contrast between sphere and sur-

rounding: intensity ratio and position of maxima. . . . . . . . . 737.8. Effect of the refractive index contrast: angular scattering profile. 747.9. Comparison between RGD- and Mie-model for a R = 100 nm-

sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.10. Comparison between RGD- and Mie-model for a R = 100 nm-

sphere: angular scattering profiles. . . . . . . . . . . . . . . . . 767.11. SHG from a non-centrosymmetric distribution of the nonlinear

polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.12. Comparison of χ(2) of MG on PS to results by Yang et al. [99]. 797.13. χ(2)-triangles for polystyrene in water with (MG-experiments). 807.14. Comparison of measurement to the simulation of Gonella and

Dai [42] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.15. Polyelectrolyte brush particle. . . . . . . . . . . . . . . . . . . . 837.16. Geometry for the simulation of polyelectrolyte brush particles. 847.17. Scan of χ(2) components for the Brush particles. . . . . . . . . 857.18. Fit to the polyelectrolyte brush particle measurement using a

single shell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867.19. Effect of dipole distance on the angular scattering pattern. . . . 877.20. Effect of thickness vs. distance of polarization layer. . . . . . . 88

List of Figures 133

7.21. Three simulations that fit to the experiments with polyelectrolytebrush particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.1. Nanoshell particle, consisting of a dielectric core (radius R) anda metallic shell (thickness D). . . . . . . . . . . . . . . . . . . . 91

8.2. SH-intensity during shell growth. . . . . . . . . . . . . . . . . . 928.3. Angular scattering profiles for silica-gold core-shell-particle. . . 938.4. Scanning for χ(2)-components of a silica-gold core-shell-particle. 948.5. Angle-resolved SH-intensity for silica-silver core-shell-particles

during shell growth. . . . . . . . . . . . . . . . . . . . . . . . . 968.6. SH-intensity for silica-silver core-shell-particles during shell growth. 978.7. Field distribution of a dipole at a sphere: near-field. . . . . . . 998.8. Field distribution of a dipole at a sphere: far-field. . . . . . . . 998.9. Intensity of a dipole at a sphere for increasing shell thickness. . 1008.10. Intensity of a dipole at a sphere for increasing permittivity. . . 1018.11. Explanation of resonances with a surface plasmon polariton, . . 1028.12. Intensity of a dipole at a sphere for increasing shell thickness in

different detection distances. . . . . . . . . . . . . . . . . . . . . 1038.13. Intensity of a plane wave at a sphere for increasing shell thickness.1048.14. Intensity of a plane wave at a sphere for different shell thicknesses.1058.15. Comparison of plasmonic resonances excited by a dipole and a

plane wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068.16. Intensity of a dipole at a sphere for different shell thicknesses. . 1068.17. SHG as sensing application for the thickness of a nanoshell. . . 108

9.1. Sum frequency generation from polystyrene particles (R = 250 nm)in water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

A.1. Bessel functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 116A.2. Associated Legendre functions. . . . . . . . . . . . . . . . . . . 120A.3. Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . 122A.4. Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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[97] Sarina Wunderlich and Ulf Peschel. Plasmonic enhancement of secondharmonic generation on metal coated nanoparticles. Optics Express, 21(16):18611–18623, 2013.

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List of PublicationsJournals

1. S. Wunderlich, U. Peschel (2013): “Plasmonic enhancement of secondharmonic generation on metal coated nanoparticles.” Opt. Expr. 21(13)18611-18623

2. S. Wunderlich, B. Schürer, C. Sauerbeck, W. Peukert, U. Peschel (2011):“Molecular Mie Model for second harmonic generation and Sum Frequencygeneration” Phys. Rev. B 84(23) 235403

3. B. Schürer, M. Hoffmann , S. Wunderlich , L. Harnau , U. Peschel, M. Ballauff, W. Peukert (2011): “Second Harmonic Light scatteringfrom Spherical Polyelectrolyte Brushes” J. Phys. Chem. C 115 (37)18302–18309

4. B. Schürer, S. Wunderlich, C. Sauerbeck, U. Peschel, W. Peukert (2010):“Probing colloidal interfaces by angle-resolved second harmonic light scat-tering” Phys. Rev. B 82, 241404(R)

Conferences

1. C. Sauerbeck, B. Schürer, S. Wunderlich, B. Braunschweig, U. Peschel,W. Peukert: “Probing colloidal interfaces by angle-resolved second-har-monic light scattering”, Partec 2013 - International Congress on ParticleTechnology, Nuremberg, April 2013 (presentation)

2. S. Wunderlich, O. Zhuromskyy, B. Schürer, M. Haderlein, C. Sauer-beck, W. Peukert, U. Peschel: “Second Harmonic generation from Metal-lic and Dielectric Spherical nanoparticles”, CLEO: 2012 Laser Science toPhotonic Applications, San Jose, USA May 2012 (presentation)

3. S. Wunderlich, B. Schürer, W. Peukert, U. Peschel: “Second harmonicscattering from colloidal particles”, 13th International SAOT Workshopon Nonlinear Optics and Interfaces, Erlangen April 2011 (presentation)

4. B. Schürer, M. Hoffmann, S. Wunderlich, U. Peschel, M. Ballauff, W. Peu-kert: “Grenzflächencharakterisierung von elektrosterisch stabilisierten Par-tikelsystemen mittels optischer Frequenzverdopplung”, Fachausschüsse Par-tikelmesstechnik und Grenzflächenbestimmte Systeme und Prozesse, Claus-thal-Zellerfeld February 2011 (presentation)

146 Bibliography

5. B. Schürer, S. Wunderlich, U. Peschel, W. Peukert: “In-situ SurfaceCharacterization of Nano- and Microparticles by Optical second harmonicgeneration”, NSTI Nanotech 2010, Anaheim, USA June 2010 (presenta-tion)

6. S. Wunderlich, U. Peschel, B. Schürer, W. Peukert: “Investigating theSurface of Nanoparticles by second harmonic generation”, CLEO/QELS:2010 Laser Science to Photonic Applications, San Jose, USA Mai 2010(poster)

7. S. Wunderlich, B. Schürer, W. Peukert, U. Peschel: “Ab initio Simula-tion of second harmonic generation from the Surface of Nanoparticles”,World Conference of Particle Technology (WCPT), Nürnberg April 2010(presentation)

8. S. Wunderlich, B. Schürer, W. Peukert, U. Peschel: “Investigating theSurface of Nanoparticles by second harmonic generation”, SymposiumEngineering of Advanced Materials (EAM), Wildbad Kreuth November2009 (presentation)

9. S. Wunderlich, O. Zhuromskyy, U. Peschel: “Concentration and en-hancement of dipole radiation by nano-spheres”, Jahrestagung der Deut-schen Physikalischen Gesellschaft (DPG), Hamburg, March 2009 (poster)

Documents

Second harmonic light scattering from dielectric and metallic spherical nanoparticles