Phonon Polaritonsin Polar Dielectric Heterostructures

Im Fachbereich Physikder Freien Universität Berlin

eingereichte Dissertation

zur Erlangung des akademischen Grades desDoktors der Naturwissenschaften (Dr. rer. nat.)

von

Nikolai Christian Paßler

Berlin, Mai 2020

Fritz-Haber-Institutder Max-Planck-Gesellschaft

ii

Erster Gutachter:Prof. Dr. Martin WolfFritz-Haber-Institut der Max-Planck-Gesellschaft, BerlinAbteilung Physikalische Chemie

Zweiter Gutachter:Prof. Dr. Paul FumagalliFreie Universität BerlinFachbereich Physik

Arbeitsgruppenleiter:Dr. Alexander PaarmannFritz-Haber-Institut der Max-Planck-Gesellschaft, BerlinAbteilung Physikalische Chemie

Tag der Disputation: 16. September 2020

Selbstständigkeitserklärung

Hiermit versichere ich, die vorliegende Dissertation eigenständig und ausschließlich unter Verwen-dung der angegebenen Quellen und Hilfsmittel angefertigt zu haben. Die vorliegende Arbeit ist indieser oder anderer Form nicht als Prüfungsarbeit zur Begutachtung vorgelegt worden.

Berlin, 29. Mai 2020

Nikolai Christian Paßler

iv

ContentsPublications Forming this Thesis vii

1 Introduction 1

2 Theoretical Concepts 52.1 The Dielectric Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Drude Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Lorentz Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Phonon Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 Dispersion Relation of a Surface Phonon Polariton . . . . . . . . . . . . . . . . . 122.2.2 Phonon Polariton Modes in Layered Heterostructures . . . . . . . . . . . . . . . 14

2.3 Polariton-Enhanced Nonlinear Optical Response . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Second Harmonic Generation (SHG) . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.2 Field Enhancement of Phonon Polaritons Probed with SHG . . . . . . . . . . . 21

3 Experimental Approaches 253.1 Excitation of Phonon Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 The Otto Geometry Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 The FHI Free Electron Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Publications 374.1 Second Harmonic Generation from Critically Coupled Surface Phonon Polaritons . . 374.2 Generalized 4 × 4 matrix formalism for light propagation in anisotropic stratied

media: study of surface phonon polaritons in polar dielectric heterostructures . . . . 454.2.1 Generalized 4 × 4 matrix formalism for light propagation in anisotropic

stratied media: study of surface phonon polaritons in polar dielectric het-erostructures: erratum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Strong Coupling of Epsilon-Near-Zero Phonon Polaritons in Polar Dielectric Het-erostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.4 Second Harmonic Generation from Phononic Epsilon-Near-Zero Berreman Modes inUltrathin Polar Crystal Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5 Surface Polariton-Like s-Polarized Waveguide Modes in Switchable Dielectric ThinFilms on Polar Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.6 Layer-Resolved Absorption of Light in Arbitrarily Anisotropic Heterostructures . . . 97

5 Perspectives 111

6 Conclusion 115

List of Publications 117

Bibliography 126

Appendices 127

v

Contents

A Supporting Information to Publications 129A.1 Supporting Information: Second Harmonic Generation from Critically Coupled Sur-

face Phonon Polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129A.2 Supporting Information: Strong Coupling of Epsilon-Near-Zero Phonon Polaritons

in Polar Dielectric Heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135A.3 Supporting Information: Second Harmonic Generation from Phononic Epsilon-Near-

Zero Berreman Modes in Ultrathin Polar Crystal Films . . . . . . . . . . . . . . . . . . . 139A.4 Supporting Information: Surface Polariton-Like s-Polarized Waveguide Modes in

Switchable Dielectric Thin Films on Polar Crystals . . . . . . . . . . . . . . . . . . . . . 147

List of Figures 154

List of Tables 155

List of Acronyms 159

Acknowledgements 161

Abstract 163

Kurzfassung 165

vi

Publications Forming this ThesisIn the course of this doctoral thesis, various experimental and theoretical ndings resulted in thepublication of ten scientic articles in several journals including the Journal of the Optical Societyof America (JOSA B), ACS Photonics, Nano Letters, ACS Nano, Physical Review B (PRB), AppliedPhysics Letters (APL), and Advanced Optical Materials (AOM). In the following table, the six (oneof which including the associated erratum) of those articles that form this thesis are listed, and inchapter 4, the original publications are reprinted.

Table: List of publications forming this thesis with the author of this thesis (N.C.P.) as rst author.Title Authors Published Journal Page

Second Harmonic Generation from Criti-cally Coupled Surface Phonon Polaritons[I]

N. C. Passler, I. Razdolski, S. Gewinner,W. Schöllkopf, M. Wolf and A. Paarmann

April 11,2017

ACSPhotonics

37

Generalized 4 × 4 matrix formalism forlight propagation in anisotropic stratiedmedia: study of surface phonon polaritonsin polar dielectric heterostructures [II]

N. C. Passler and A. Paarmann September18, 2017

JOSA B 45

Generalized 4 × 4 matrix formalism forlight propagation in anisotropic stratiedmedia: study of surface phonon polaritonsin polar dielectric heterostructures: erra-tum [III]

N. C. Passler and A. Paarmann November1, 2019

JOSA B 59

Strong Coupling of Epsilon-Near-ZeroPhonon Polaritons in Polar Dielectric Het-erostructures [IV]

N. C. Passler, C. R. Gubbin, T. G. Folland,I. Razdolski, D. S. Katzer, D. F. Storm, M.Wolf, S. De Liberato, J. D. Caldwell andA. Paarmann

June 12,2018

NanoLetters

65

Second Harmonic Generation fromPhononic Epsilon-Near-Zero BerremanModes in Ultrathin Polar Crystal Films[V]

N. C. Passler, I. Razdolski, D. S. Katzer,D. F. Storm, J. D. Caldwell, M. Wolf andA. Paarmann

June 3,2019

ACSPhotonics

75

Surface Polariton-Like s-Polarized Waveg-uide Modes in Switchable Dielectric ThinFilms on Polar Crystals [VI]

N. C. Passler, A. Heßler, M. Wuttig, T.Taubner and A. Paarmann

September26, 2019

AOM 85

Layer-Resolved Absorption of Light inArbitrarily Anisotropic Heterostructures[VII]

N. C. Passler, M. Jeannin and A. Paar-mann

April 23,2020

PRB 97

vii

CHAPTER 1IntroductionIn the past decades, the scientic eld of nanophotonics has emerged, aiming at understandingand harnessing the extraordinary properties of light and light-matter-interaction on the nanoscale.Restricted by the diraction limit [1], conventional optical components like lenses or parabolicmirrors cannot focus light far below the wavelength. In nanophotonics, this limit is overcome bycoupling light into subwavelength structures, enabling light connement of more than 400 timessmaller than the wavelength [2–4].

Squeezing light into nanoscale objects bears two intriguing advantages: Firstly, compared to the eldamplitudes of a light wave propagating in free space, the local electric elds in a nanostructure areimmensely enhanced, boosting the light-matter interaction considerably. These enhanced electricelds enable a variety of applications, such as improved solar cell eciency [5], all-optical switching[6], or ultra-ecient sensing [7]. Secondly, subwavelength-conned light allows for miniaturizationof optoelectronic devices, enabling advances such as on-chip, high-speed photodetectors [8] orultra-fast miniature lasers [9] for all-optical integrated circuitry and communication.

For many years, the principle workhorse of nanophotonics has been the surface plasmon polariton(SPP), a surface excitation enabled by collective electron oscillations – a plasmon – supported inmetallic materials at infrared (IR) up to visible wavelengths. Eminent milestones in the history ofplasmonics employing the SPP have been the discovery of surface-enhanced raman scattering (SERS)[10] and plasmon-enhanced uorescence [11], and the development of modern plasmonic devicessuch as single photon transistors [12] or nanolasers [13].

Only recently, other fundamental excitations than the SPP have started to attract broad attentionin the nanophotonics community. A promising complementary candidate is the surface phononpolariton (SPhP), where the light is coupled to an optically active phonon mode in a polar crystallattice instead of a plasmon in a metal. Owing to signicantly longer lifetimes of optical phonons insingle crystals compared to plasmons in metals, SPhPs provide an attractive low-loss alternative fornanophotonic applications at IR and THz frequencies [14].

Another advantage of SPhPs is the wealth of supporting materials, including most III-V semiconductorcompounds, silicon carbide (SiC), several II-VI semiconductors, and various oxides such as magnesiumoxide, molybdenum trioxide or gallium(III) trioxide, covering a broad frequency range in the IR.Furthermore, most of these materials feature optical anisotropy due to an anisotropic crystal structure,leading to non-trivial optical responses and novel nanophotonic phenomena. A prominent exampleare hyperbolic phonon polaritons [15], which have been demonstrated to enable subdiractionimaging and hyperlensing [16].

On the other hand, SPhPs supported by a single material possess several limitations that hinder theapplication in nanophotonic technologies: The frequency window where the SPhP is accessible issmall, the mode lacks active tunability, the spatial connement and eld enhancements are moderate,and only p-polarized light can be used for the excitation.

1

Chapter 1 Introduction

In this work, these limitations of SPhPs on bulk crystals are approached and overcome by theemployment of layered heterostructures, built from various dierent materials. By investigatingseveral exemplary systems, it is shown that the combination of SPhP-supporting crystals [IV, V],metals [VII], and other dielectric materials [VI] enables unique control over the properties of theexcitable polariton modes. These ndings demonstrate that layered heterostructures provide aversatile tool box for designing nanophotonic devices with user-dened optical responses and novelfunctionalities.

Experimentally, the investigation of phonon polaritons in layered heterostructures is realized in ahome built prism coupling setup. The specic design of the setup allows for full control over theexcitation conditions, enabling systematic experimental access to the phonon polariton properties.As an excitation source, the IR free electron laser (FEL) at the Fritz Haber Insitute (FHI) is employed.The machine is characterized by high-power, high-resolution radiation output, constituting the idealsource for non-linear optical methods such as second harmonic generation (SHG) spectroscopy [17].

By combining the prism coupling setup with the FEL, the non-linear optical response of phononpolaritons at IR frequencies can be probed. Establishing this concept, this work comprises the rstobservation of SHG from propagating SPhPs [I]. Subsequent studies demonstrate the SHG fromstrongly coupled SPhPs [VIII] and the Berreman mode [V], a phonon polariton excitable in thinlms of deep sub-wavelength thickness. As is shown in these studies, the employment of SHGspectroscopy provides experimental access to the local electric eld enhancement of the excitedphonon polariton modes, being inaccessible by linear optical techniques.

Even though of central importance for scientic progress, experimental data can be elusive without aproper theoretical model. But in layered anisotropic media, the theoretical description of light-matterinteraction is challenging, because most phonon polariton materials feature optical anisotropy. Aknown approach to this challenge is the 4 × 4 transfer matrix formalism. However, most previ-ous publications on this subject consider only special cases, are numerically unstable, or lead todiscontinuous solutions.

In this work, a numerically stable, generalized framework for the treatment of light-matter interactionin arbitrarily anisotropic layered heterostructures is developed [II, VII] that is implemented in anopen-access computer program [18, 19].1 As is demonstrated in the various studies of this work, theformalism allows for thorough simulations and analysis of phonon polaritons and their propertiesin layered media, enabling both corroborative calculations for experimental ndings as well aspredictive studies of yet unexplored nanophotonic structures.

This work is structured as follows: In chapter 2, the theoretical concepts are delineated, starting withthe dielectric permittivity in section 2.1 as the fundamental quantity for the description of light-matterinteraction. In section 2.2, phonon polaritons are introduced by deriving the dispersion relation of abulk SPhP. Then, the concept of layered heterostructures is presented, providing an overview overthe dierent phonon polariton modes arising in the investigated sample structures. Section 2.3 closesthe theoretial concepts by explaining the process of SHG, and presenting a summary of the resultsobtained from the various SHG spectroscopy measurements.

1It is noted that the original formalism contains an error in the calculation of electric elds for the case of birefringence, thatis, for anisotropic media with non-zero o-diagonal elements of the dielectric permittivity tensor in the lab frame. This error wascorrected in an erratum [III], see page 59. None of the publications employing the formalism prior to the erratum are aected by theerror, since birefringence does not occur in the systems investigated in these works.

2

Chapter 3 is dedicated to the experimental concepts. Section 3.1 discusses the most common methodsfor the excitation of phonon polaritons, and section 3.2 presents the implementation of the methodemployed in this work, the prism coupling in the Otto geometry. The chapter is completed by adescription of the FHI FEL (section 3.3).

This work is a cumulative thesis. In chapter 4, the scientic articles that form this thesis are reprinted.Each publication is presented in a separate section, providing a short description of the content of thearticle, the context in the scope of this thesis, and the author contributions. A list of all publicationsoriginated in the course of this thesis can be found on page 117.

Chapter 5 discusses the perspectives of the techniques implemented in this work by outlining severalpossible pathways for future advances in the eld of nanophotonics. Finally, chapter 6 concludes thepresented work.

3

CHAPTER 2Theoretical ConceptsIn 1950, the Ukrainian physicist Tolpygo analyzed the interaction of dipole-carrying lattice vibrationswith an electromagnetic eld in the limit of long wavelengths [20], obtaining the rst descriptionof what later would be termed a phonon polariton. His analysis shows the splitting and bendingof the light dispersion at frequencies of the lattice vibration, resulting in two distinct dispersionbranches featuring an anti-crossing. These two bulk phonon polaritons (BPhPs) are the result ofstrong light-matter coupling, where the interaction between light and medium is so strong that anew, fully hybridized quasiparticle – the phonon polariton – is created, carrying the properties ofboth light and the lattice excitation.

In later years, dierent types of polaritons were discovered, depending on the dipole-carrying matterexcitation that couples to the light. These types include exciton polaritons [21] and intersubbandpolaritons [22] resulting from electronic band transitions, SPPs [23] and SPhPs [14] arising fromplasmon and phonon modes that propagate along an interface, and various other polariton typessuch as magnon polaritons [24] or Cooper pair polaritons [25].

Astonishingly, the optical response of any of the above mentioned polaritons can be fully describedby the classical wave-interpretation of light given by Maxwell’s equations [26], as long as thelight intensity is large enough, that is, high above the single photon limit1 [28]. Independent ofthe underlying quasiparticle that is coupling to the light, Maxwell’s equations allow for a precisecalculation of the electromagnetic elds emerging from the strong light-matter interaction, even insub-wavelength structures of heterogeneous material compositions. Thus, owing to their universalvalidity and wide application variety, Maxwell’s equations are up until today the standard frameworkfor the semi-classical theoretical description of the interaction of light with matter.

Building on the description of light as electromagnetic wave in terms of the Maxwell’s equations,this chapter starts with the concept of the dielectric permittivity for modeling optical resonances(section 2.1), discussing the Drude and the Lorentz model for the description of metallic and polardielectric materials. In the following section 2.2, phonon polaritons in various heterostructures arediscussed, providing an overview over the polariton modes that are investigated in the publications(chapter 4). Finally, section 2.3 focuses on the eld enhancement associated with phonon polaritonresonances and the resulting enhanced SHG, discussing the systems that were studied by means ofSHG spectroscopy.

1Quantum optics eects [27] are not relevant for this work

5

Chapter 2 Theoretical Concepts

2.1 The Dielectric Permiivity

The optical response of a material under the inuence of an external electric eld ~E is characterizedby the polarization ~P generated in the material. When the external electric eld reaches highintensities, ~P is a nonlinear, complicated function of ~E, giving rise to nonlinear optical eects (seesection 2.3 for more details). In many cases, however, a linear approximation is sucient to describethe optical response, and the material’s polarization ~P can be written as follows (in the dipoleapproximation) [26]:

~P = ε0(ε− 1) ~E, (2.1.1)

where ε is the dielectric permittivity tensor, fully describing the optical response of a medium:

ε =

εxx εxy εxzεyx εyy εyzεzx εzy εzz

. (2.1.2)

Hence, in principle, with the knowledge of the exact shape of ε, any light-matter interaction of themedium can be predicted. In practice, however, the dielectric permittivity of a material cannot bedetermined for all optical frequencies in full detail. Instead, either experimental techniques suchas ellipsometry are employed [29] in order to obtain the shape of the dielectric permittivity in acertain frequency range, or, a model is used that captures the dominant characteristics of the opticalresponse of the material.

This work investigates the light-matter interaction of heterostructures made of various materials withvery dierent optical responses, requiring both experimentally determined dielectric permittivitydata (such as for the glass thallium bromo-iodide (KRS-5)), and theoretical models. The two modelsthat are employed are the Drude model and the Lorentz model, which are presented in this section.

2.1.1 Drude Model

The Drude model is based on Drudes considerations about the movement of electrons in a metalpublished in 1900 [30]. In his classical approach, the electrons in a metal can move freely between thepositively charged ions, forming an electron gas, and collisions with ions abruptly alter the velocityof the electrons. An external electric eld accelerates the electrons, leading to an electric current.However, for increasing electric elds, the collision rate increases as well and thus the electronsfeel an increasing resistance. The Drude model hence provides a physical explanation for Ohm’slaw [31].

Most certainly, the advances since 1900 of the quantum mechanical understanding of electrons insolid state physics have shown the limits of the Drude model. Specically, in frequency ranges whereother features are observable, such as electronic band transitions, the Drude model fails to predictthe optical response. Many improvements of the basic Drude model have been proposed, achievingmuch more accurate descriptions [32–36]. Nonetheless, the basic model yields reasonable results forthe overall trend of the optical response of a metallic material, which is why it is still broadly usedfor many applications.

6

2.1 The Dielectric Permittivity

In order to obtain the dielectric permittivity of a Drude metal, the equation of motion of a freeelectron is considered:

m~u+mγ~u = q ~E, (2.1.3)

where m is the eective mass of the electron, ~u the position in space, γ the damping constant arisingfrom the collisions with the ions, q the electron charge, and ~E = ~E0e

iωt the incident electromagneticeld driving the electron motion with frequency ω. The solution of this dierential equation is

~u =q ~E

m(−ω2 + iωγ). (2.1.4)

For an ensemble of N electrons, the macroscopic polarization ~P amounts to

~P = Nq~u = ε0χ~E = ε0(ε− 1) ~E, (2.1.5)

where in the second step the linear approximation of the polarization was assumed (Eq. 2.1.1), withχ being the electric susceptibility, and ε0 the vacuum permittivity. Rearranging Equation 2.1.5 yieldsfor the dielectric permittivity ε:

ε(ω) = 1− ω2p

ω2 − iωγ (2.1.6)

with the plasma frequency ωp

ωp =

√Nq2

ε0m, (2.1.7)

where N is given in units of inverse cubic centimeters, describing the free carrier concentration.At the plasma frequency, the real part of the dielectric permittivity crosses zero from negative topositive, representing the upper frequency limit of supported SPP modes, and marking the epsilon-near-zero (ENZ) frequency of metallic materials (see the publications reprinted in section 4.3 and 4.4for further details on ENZ polaritons).

In general, the dielectric permittivity starts at a value Re (ε) > 1 in the limit of zero frequency, anddecreases for increasing frequency with each occurring resonance, such as phonon modes, plasmonsand electronic transitions. At large enough frequencies, where there are no more optical resonances,Re (ε) = 1, meaning that any material at large frequencies behaves equal to vacuum.

The Drude model describes the optical response of free electrons in a metallic material, and ωp isthe resonance frequency of the plasmon, that is, a collective, coherent oscillation of all electrons.For higher frequencies than the plasma frequency, however, any metal or semiconductor featuresmore resonances, such as electronic transitions from core levels. These resonances lead to a quasi-static contribution to the dielectric permittivity at lower frequencies, i.e. in the range of the plasmafrequency. In order to account for this contribution, the parameter ε∞ is introduced, yielding thefollowing dielectric permittivity of the Drude model:

7

Chapter 2 Theoretical Concepts

0 10 20 30 40

-5

0

5

10 Re(ε ) Data

Im(ε ) Data

Re(ε ) Fit

Im(ε ) FitAu

Au

Au

AuD

iele

ctri

c P

erm

itti

vit

y ε

Photon Energy (eV)

a100

50

0

-50

-100

-150Die

lect

ric

Per

mit

tiv

ity ε

b

0 1 2 3 4Photon Energy (eV)

Figure 2.1.1: Drude model of the dielectric permiivity for gold. a The experimentally obtained dielec-tric permiivity of gold (Au) [37] is ploed as dashed lines, exhibiting various features in the shown frequencyrange. The Drude model fied to the experimental data (solid lines) fails to reproduce most features, except forthe IR frequency range (red shaded regions). b The same experimental data is fied only in the IR, achievinggood agreement. Thus, for nanophotonic studies in the IR, as they are performed in this thesis, the Drudemodel is a valid approximation for the optical response of Au.

ε(ω) = ε∞

(1− ω2

p

ω2 − iωγ

). (2.1.8)

This model is adequate for all metallic materials, where the plasmon resonance dominates the opticalresponse. This is, for example, the case for certain doped semiconductors [38], two-dimensional (2D)electron gases in quantum wells [39], or heavy-fermion metals [40]. In particular, in a recentpublication (reprinted in section 4.6), the Drude model was successfully employed to describe theanisotropic optical response of a doped semiconductor multi-quantum well supporting intersubandpolaritons [VII].

Other materials, such as Au or silver, on the other hand, feature various additional resonances inclose proximity to the plasma frequency, and thus in this spectral range, the Drude model is not veryaccurate. In Fig. 2.1.1, the measured dielectric permittivity of Au is plotted as dashed lines, and theDrude model (Eq. 2.1.8) is tted to the experimental data (solid lines). Clearly, the Drude model failsto describe the optical response in the mayor part of the plotted spectral range (Fig. 2.1.1a). However,the IR spectral range (λ = 50 µm − 780 nm, marked by a red shade in Fig. 2.1.1) is the frequencyregion where phonon polaritons, intersubband polaritons and semiconductor plasmon polaritons arelocated, and in this region, the Drude model achieves good agreement with the experimental data(Fig. 2.1.1b). Therefore, in the IR, where most investigations in this thesis are performed, the Drudemodel is an applicable and practical approximation even for Au.

8

2.1 The Dielectric Permittivity

2.1.2 Lorentz Model

After Drude had developed his model of free electrons, the Dutch physicist Lorentz extended theDrude model in order to describe electrons bound to an atom by adding a restorative force term. Thecorresponding equation of motion of the electrons in a harmonic potential driven by an externalelectric eld is:

m~u+mγ~u+mω20~u = q ~E, (2.1.9)

where ω0 is the resonance frequency of the particle in the harmonic potential. Analogous to theDrude model, the solution of this equation is

~u =q ~E

m (ω20 − ω2 + iωγ)

, (2.1.10)

and the dielectric permittivity can be formulated as

ε(ω) = ε∞ +Nq2

ε0m

1

ω20 − ω2 + iωγ

. (2.1.11)

Even though originally conceived for bound electrons, the Lorentz model can be applied to anycharged particle in a harmonic potential, and thus also allows to describe the optical response oflattice atoms in a crystal. A crystal with Nu.c. atoms in the primitive unit cell has 3 acoustic and3Nu.c.−3 optical modes. For the long-wavelength limit, that is, at the Brillouin zone center, in opticalphonons, the atoms describe an out of phase movement with specic periodicity. In the case of polarcrystals, where at least two atoms with dierent partial charge are present, the optical phononscreate a time-varying dipole, which can couple to light.

Depending on the vibration direction with respect to the propagation direction, a distinction is madebetween longitudinal optical (LO) and transverse optical (TO) modes. In polar crystals such as SiC,aluminium nitride (AlN) or gallium nitride (GaN), a gap between the two resonance frequenciesωTO and ωLO can be found at the zone center, which does not exist in non-polar crystals. This liftingof the degeneracy between the optical branches arises due to the macroscopic polarization of theLO mode, leading to a higher LO frequency than the TO frequency [41]. The region between bothfrequencies is called the reststrahlen (German for residual rays) band.

However, due to the orthogonality between propagation direction and polarization of light, only theTO mode is IR active, while the LO mode (propagation direction and polarization are parallel) cannotbe excited. Thus, in polar crystals, the resonance frequency in Eq. 2.1.9 and 2.1.11 is the frequency ofthe TO phonon, ω0 = ωTO.

Solving Eq. 2.1.11 at ωLO, where ε(ωLO) = 0 [42], yields for the dielectric permittivity of a polarcrystal:

ε(ω) = ε∞ω2 − ω2

LO − iγωω2 − ω2

TO − iγω. (2.1.12)

This model is used to describe the optical response of all polar crystals investigated in this thesis. InFig. 2.1.2, the principle dielectric permittivities of SiC, AlN, and GaN are plotted, which are the three

9

Chapter 2 Theoretical Concepts

500 600 700 800 900

-500

0

500

1000

1500D

iele

ctri

c P

erm

itti

vit

y ε

εAlN εAlN εSiC εSiCεGaN εGaN

Frequency ω (1/cm)

Re(ε)

Im(ε)

Figure 2.1.2: Lorentz model of the dielectric permiivity for 6H-SiC, AlN, and GaN. The materialparameters are given in table 2.1.1. All three materials are uniaxial crystals, featuring dierent ordinary(⊥) and extraordinary (‖) principle dielectric permiivities. The reststrahlen bands, where the real part ofε is negative, are highlighted with horizontal color bars, delimited by the respective TO and LO frequenciesωTO and ωLO.

polar crystals that are most predominantly employed. At the TO frequency, the resonance in eachmaterial is apparent as a peak in the imaginary part of ε (dashed lines). The real part (solid lines), onthe other hand, is negative in the reststrahlen band (marked by horizontal bars in Fig. 2.1.2), crossingzero at both the TO and LO frequency.

For anisotropic materials, the principle dielectric permittivities εii dier (Eq. 2.1.2) and thus, each εiiis dened by Eq. 2.1.12 with a unique set of material parameters. 6H-SiC, AlN, and GaN are uniaxialcrystals, featuring one extraordinary crystal axis (‖) along which the dielectric permittivity is dierentthan along the two ordinary axes (⊥). Crystals where the extraordinary axis is orthogonal to thesurface plane are called c-cut. In Fig. 2.1.2, the respective ordinary and extraordinary permittivities (ε‖and ε⊥) of all three crystals are shown. A summary of the material parameters is given in table 2.1.1.

Table 2.1.1: Material parameters of uniaxial 6H-SiC, AlN, and GaN. Parameters employed to model thedielectric permiivity of 6H-SiC, AlN, and GaN in all publications that were wrien in the course of this thesis(reprinted in chapter 4).

Parameter 6H-SiC [43] AlN [44] GaN [44]

ω‖TO 788 cm−1 611 cm−1 531.8 cm−1

ω⊥TO 797 cm−1 670.8 cm−1 558.8 cm−1

ω‖LO 964 cm−1 890 cm−1* 734 cm−1

ω⊥LO 970 cm−1 912 cm−1 741 cm−1

γ 3.75 cm−1 2.2 cm−1 4 cm−1

ε‖∞ 6.78 5.35 ε

‖∞= 4.35

ε⊥∞ 6.56 5.35 ε⊥∞= 4.16*Note that in contrast to the here reported literature value, ttingof experimental data has yielded a value of 900 cm−1, which isused throughout the thesis and in all publications.

10

2.1 The Dielectric Permittivity

Some materials, such as doped semiconductors, feature both free electrons and IR-active phonons.In that case, the optical response can be described by a combined Drude-Lorentz model with thedielectric permittivity taking the form

ε(ω) = ε∞

(1− ω2

p

ω2 − iωγe+ω2 − ω2

LO − iγpωω2 − ω2

TO − iγpω

), (2.1.13)

where γe and γp are the electron and the phonon damping, respectively.

Analogous to metals supporting SPPs below the plasma frequency, polar crystals support SPhPsmodes in the reststrahlen band, where Re (ε) is negative. The following section discusses phononpolaritons in polar crystals, describing the dierent types of polariton modes that arise in dielectricheterostructures, as investigated in the scientic articles published in the course of this thesis(reprinted in chapter 4).

11

Chapter 2 Theoretical Concepts

2.2 Phonon Polaritons

Phonon polaritons are excitations that arise under light-matter interaction with phonons in the strong-coupling limit, yielding hybridized modes with both phononic and electromagnetic characteristics.The simplest phonon polariton is the bulk phonon polariton (BPhP), which emerges when lighttravels through a bulk polar crystal at frequencies in proximity to the ωTO resonance frequency. Asoriginally observed by Tolpygo [20], around the reststrahlen band, the light dispersion splits up intotwo branches, featuring an avoided crossing (see Fig. 2.2.1, blue lines). The dispersion relation of theBPhP is obtained by plugging the permittivity of a polar crystal (Eq. 2.1.11) into the light dispersionof a plane wave [26]:

kBPhP =ω

cε =

ω

cε∞

ω2 − ω2LO − iγω

ω2 − ω2TO − iγω

. (2.2.1)

Besides the solution of plane waves traveling through a bulk material, at interfaces of materials withdierent dielectric permittivity, Maxwell’s equations yield the solution of a surface wave, the socalled surface polariton. Here, the matter excitation that couples to the light is a surface electric dipolewave with resonance frequency ωS, where the associated electric eld strength decays exponentiallywith increasing distance to the interface [45].

Depending on the charged particles that form the surface electric dipole wave, dierent types ofsurface polaritons arise, such as the surface plasmon polariton (SPP) for free electrons, or the surfacephonon polariton (SPhP) for lattice ions. In the following, without loss of generality for other types ofsurface polaritons, the dispersion of a SPhP will be derived. The second part of this section discussespolaritons in layered systems of more than one interface and material, summarizing the polaritonicmodes investigated in this thesis, such as thin-lm modes, the Berreman mode, volume-connedhyperbolic polaritons, and polariton-like waveguide modes.

2.2.1 Dispersion Relation of a Surface Phonon Polariton

The simplest system supporting a surface phonon polariton is a single interface of an isotropic polarcrystal adjacent to a dielectric material of constant positive dielectric permittivity. The interfacebetween the polar crystal (material P) and the adjacent dielectric (material D), described by thedielectric permittivities εP and εD, is dened to be the x-y-plane at z = 0, where the polar crystalextends into z < 0, and the dielectric material into the opposite half-space (z > 0).

The surface wave satisfying Maxwell’s equations is chosen to propagate along the x-direction, withits electric elds pointing in the x-z-plane (transverse magnetic), yielding the following explicitform:

~E = ~ED,P0 ei(ωt− kxx± k

D,Pz z), (2.2.2)

where the superscripts D and P stand for the dielectric (z > 0) and the polar crystal (z < 0), and the± sign is to be read such that the top sign corresponds to the z > 0 half-space, and the bottom sign toz < 0. The in-plane momentum kx is conserved at the interface, which is why it is dened material-independently. Plugging this surface wave into the three-dimensional wave equation ∆ ~E = εD,P

c2~E

12

2.2 Phonon Polaritons

0 0.5 1 1.5 2 2.5 30

500

1000

1500

0.5 1 1.5 2

800

850

900

950

Fre

quen

cy ω

(1/c

m)

(In-plane) Momentum k (1/μm) (In-plane) Momentum k (1/μm)

a b

Fre

quen

cy ω

(1/c

m)

Figure 2.2.1: Dispersion relations of BPhPs and SPhPs Light traveling in a material of constant epsilonfeatures a linear dispersion, such as the light line in vacuum (black solid line). The dashed and doed blacklines represent the limits for large and small frequencies, respectively, of the dispersion of light in a polarcrystal (blue line). At frequencies in proximity to the reststrahlen band, the BPhP arises, featuring an avoidedcrossing. Inside the reststrahlen band, the SPhP is supported at the surface of the polar crystal, rangingfrom ωTO up to the cut-o frequency ωS (shown in b). The modes are exemplified for a SiC crystal. Notethat the BPhP has a momentum that points along the propagation direction through the bulk crystal, andhence the absolute value is ploed, while the SPhP is ploed along its in-plane momentum, because it onlypropagates along the crystal surface. The inset in b sketches the exponentially decaying electric field of aSPhP propagating along the surface of a polar crystal of dielectric permiivity εP adjacent to a dielectricmaterial with permiivity εD.

leads to an expression for the out-of-plane momentum kz:

kD,Pz =

√ω2

c2εD,P − k2x. (2.2.3)

In order to get an actual surface wave, however, the ~E-eld should vanish with increasing distanceto z = 0. This is only achieved for complex kD,P

z , because then the z part in the exponent of Eq. 2.2.2becomes real and leads to an exponential decay in both z directions:

~E = ~ED,P0 ei(ωt− kxx) e∓κ

D,Pz, (2.2.4)

where

κD,P ≡ ikD,Pz (2.2.5)

is required to be real and positive.

From Maxwell’s equations, the following equation for a p-polarized plane wave can be derived [42]:

∂2Ex∂z2

+ω2

c2εEx −

∂2Ez∂x∂z

= 0. (2.2.6)

When Eq. 2.2.4 is plugged into this equation, the following relation between the eld components isobtained:

ED,P0z = ∓i kx

κD,PED,P0x , (2.2.7)

13

Chapter 2 Theoretical Concepts

and applying Maxwell’s boundary conditions for the displacement ~D (εDED0z = εPEP

0z) and theelectric eld ~E (ED

0x = EP0x) yields:

κD

εD = −κP

εP . (2.2.8)

κD, κP and εD are positive by denition. This leads to the condition that in order to support a SPhP,εP has to be negative. A polar crystal fullls this requirement in the reststrahlen band betweenωTO and ωLO. However, it can be shown that the exact condition reads εP < −εD [42], which leadsto a reduced upper frequency limit ωS < ωLO. Substituting κD,P (Eq. 2.2.5) into equation 2.2.8 nallygives the dispersion relation for a SPhP:

kx = kSPhP =ω

c

√εDεP

εD + εP . (2.2.9)

Note that the derivation is independent on the supporting material. Therefore, Eq. 2.2.9 is not onlyvalid for SPhPs, but holds for any type of surface polariton, for example the SPP supported on metalsor doped semiconductors. In Fig. 2.2.1, the SPhP dispersion is plotted together with the dispersionrelation for bulk phonon polaritons.

2.2.2 Phonon Polariton Modes in Layered Heterostructures

Recent advances in the eld of nanophotonics have shown that SPhPs oer great potential fortechnological applications, such as super-resolution imaging [16, 46, 47], highly ecient sensing[48], or enhanced nonlinear-optical conversion eciency [49–51].

A single SPhP supported by the surface of a bulk polar crystal as derived in the previous section,however, possesses several limitations that hinder the application in nanophotonic technologies:

(i) SPhPs are only supported in the small frequency window of the reststrahlen band of thesupporting polar crystal,

(ii) the dispersion relation is xed by the material parameters, and thus the SPhP lacks activetunability,

(iii) the spatial connement in lateral direction is only moderate, because SPhPs are accessible atwavelengths in the micrometer range, and the evanescent penetration depth scales with thewavelength,

(iv) the electric eld enhancement of a SPhP only reaches moderate values of up to 10 [I], which isrelatively low compared to the eld enhancements achieved in some plasmonic nanostructures[52–54], and

(v) SPhPs are only excitable by p-polarized light, limiting the versatility of polariton-based nanopho-tonics.

14

2.2 Phonon Polaritons

In the course of this thesis, all these limitations of SPhPs on bulk crystals have been approachedand treated. The key tool to overcome these restrictions has been the employment of layeredheterostructures, built from various dierent polar crystals, metals, and dielectric materials. Thissection provides an overview over the investigated layered systems (visualized in Fig. 2.2.2), thearising polariton modes, and their advantageous properties, referencing the published article wherethe respective system is discussed in detail.

Additionally to the experimental studies, in this thesis a theoretical framework has been developedthat allows to calculate the optical response of a multilayer structure of arbitrarily anisotropicmaterials, including the layer-resolved absorption, transmittance, and electric eld distribution. Thisframework has been used for all simulations of the investigated samples, and is described in detail intwo published articles [II, VII] (reprinted on pages 45 and 97). For the simulations in this section,the same framework is employed.

Thin-Film Polaritons One approach to modify the bulk SPhP (sketched in Fig. 2.2.2a) is to reducethe lm thickness d of the sample. For thicknesses in the range of the penetration depth of thebulk SPhP (for polar crystals such as SiC and AlN, this is the case for d . 1 µm [V]), an interactionbetween the two interfaces arises, resulting in two thin-lm polaritons (TFPs) (sketched in Fig. 2.2.2b).In terms of the dispersion, the single bulk SPhP dispersion (gray line in Fig. 2.2.2c for AlN) splitsup into the symmetric and antisymmetric TFP branch (green lines), where the splitting increasesfor decreasing lm thickness. Thus, the lm thickness acts as a tuning parameter for modifying thepolariton dispersion.

The dispersion of polaritons in a three-layer system can be determined by numerical evaluation ofthe following equation [55–57]:

1 +ε1kz3ε3kz1

= i tan (kz2d)

(ε2kz3ε3kz2

+ε1kz2ε2kz1

), (2.2.10)

where the indexes 1-3 refer to the three layers, and the out-of-plane momenta kz are given by Eq. 2.2.3.The dispersions shown in Fig. 2.2.2c,e have been calculated using Eq. 2.2.10.

ENZ Polaritons In the limit of ultra-thin lms (d < λ/100, with λ being the free-space wavelength)[57], the symmetric TFP dispersion is pushed completely against the LO phonon frequency, wherethe dielectric permittivity crosses zero (Fig. 2.2.2c). Therefore, this TFP is called an epsilon-near-zero (ENZ) polariton, featuring intriguing properties such as ultra-long wavelengths and immenseeld enhancements compared to the bulk SPhP (see the publication reprinted on page 65 [IV] forfurther details on ENZ polaritons).

Berreman Modes Most phonon polaritons have in-plane momenta larger than light in vacuum(black line in Fig. 2.2.2c), resulting in an evanescent mode that is guided at the interface withoutradiating into the far-eld. An exception is the Berreman mode, which arises in ultra-thin lms onthe left side of the light line, and therefore can be probed via free-space excitation. Similar to thesymmetric TFP, the dispersion of the Berreman mode in a polar crystal (red lines in Fig. 2.2.2c) liesin close proximity to the LO phonon frequency, and thus features ENZ characteristics such as an

15

Chapter 2 Theoretical Concepts

immense out-of-plane eld enhancement (see the publication reprinted on page 75 [V] for furtherdetails on Berreman modes).

Strongly Coupled Polaritons Polaritons are by denition modes arising due to the strong couplingof light with matter. Interestingly, polaritons can again couple strongly with other polaritons, formingnew hybrid polaritons with shared characteristics of the uncoupled modes.

A system where such a strong coupling can be observed is an ultra-thin AlN lm on a SiC substrate,as sketched in Fig. 2.2.2d, where the modes that couple are the SiC substrate SPhP (blue dispersionline in Fig. 2.2.2e) and the ENZ TFP of the AlN lm (green dispersion line). Together, these modesform two new hybridized strongly coupled polaritons with an avoided crossing of the two dispersionbranches (red lines), combining the properties of the uncoupled modes (see the publication reprintedon page 65 [IV] for further details on strongly coupled polaritons in layered heterostructures).

Notably, as an alternative to layered heterostructures, three-dimensional (3D) nanostructures canalso support strongly coupled polaritons. This has been demonstrated for SiC nanopillars on a SiCsubstrate, where the phonon polariton resonances localized in the nanopillars strongly couple to theSPhPs propagating on the substrate (see publication [VIII] for further details).

Actively Tunable, Refractive Index-Shied Polaritons The SPhP dispersion depends on the dielec-tric permittivity of the supporting crystal εP and of the adjacent medium εD (or, equivalently, therefractive index n =

√ε). Changing εD inuences the SPhP, resulting in a frequency shift of the

dispersion that can be exploited for sensing applications [48, 58–60].

A system that features refractive index-shifted (n-shifted) polaritons is depicted in Fig. 2.2.2f, wherea thin Ge3Sb2Te6 (GST) lm is placed on a SiC substrate. In Fig. 2.2.2g, the dispersion of the bareSiC SPhP (blue solid line) and the n-shifted SPhPs (purple and red solid lines) are plotted. GST is aphase-change material (PCM) which can be switched reversibly between its crystalline (c-GST) andamorphous (a-GST) phase, and due to the refractive index contrast between the two phases, twodierent n-shifted SPhPs are supported. By electrically or optically switching the GST phase, then-shifted SPhP dispersion can be actively tuned. (see the publication reprinted on page 85 [VI] forfurther details on actively tunable, n-shifted polaritons).

Polariton-Like Waveguide Modes In thin lms of high refractive index embedded in media oflower n, standing waves are supported that cannot freely propagate in the adjacent media due to therefractive index contrast, forming so-called waveguide modes (WMs). These modes can be either p- ors-polarized, and their dispersion is usually very steep compared to SPhPs. A thin GST lm supportsWMs, and when placed on a SiC substrate, the dispersion of the s-polarized WM is pushed upwardsinto the SiC reststrahlen band, resulting in a comparable dispersion as the SPhP (dashed lines inFig. 2.2.2g). The dispersion of WMs is given by the following equation (specied for the GST/SiCheterostructure) [61]:

ω

c

√εGST − k2 dGST − ΦSiC/GST − ΦGST/air = mπ, (2.2.11)

16

2.2Phonon

PolaritonsPhonon Polariton Engineering

xyzAlN

air SPhP

AlNair

air

AlNSiC

air

SPhP ENZ

thin-film polaritons

GSTSiC

SPhPair

WM

a

b

0.5 0.6 0.7 0.8 0.9 1.0

880

900

920

940

SiC SPhPAlN ENZ ModeStrongly Coupled

In-plane Momentum k (1/μm)

Modes

SiC

AlN

dAlN

Fre

quen

cy ω

(1/

cm)

0.4 0.6 0.8 1.0600

700

800

900

leaky evanescent

Berreman modeThin-film polaritonAlN SPhP

203001000

dAlN in nm symmetric

antisymmetric

AlN

AlN

enil thgil

In-plane Momentum k (1/μm)

c

d

e

d AlN = 20 nm

800

900

1000

bare SiCa-GSTc-GST

1.04 1.11 1.18 1.25 1.32In-plane Momentum k/k0

f

ik0 = ω/c

AlNSiC

GaN

AlNGaN

hPhP

...

h

In-plane Momentum k/k0

800

900

700

600

1.1 1.4 10 20 30 40

gType IType II

SiCp-pol SPhPs-pol Waveguide Mode

SiC

d GST ≈ 140 nm

dGST

air

Figure 2.2.2: Phonon polariton engineering using polar dielectric heterostructures. a Sketch of a SPhP propagating on an AlN bulk crystal.The electric field strength is illustrated in yellow, evanescently decaying in z-direction into both media. b In a thin AlN film, two thin-film polaritons (TFPs) withsymmetric and antisymmetric electric field distribution are supported. c Dispersion lines of leaky and evanescent phonon polaritons in bulk and thin-film AlN.On the right side of the light line, the bulk AlN SPhP (gray line) and the two TFPs disperse, whereas on the le side of the light line, the leaky Berreman modearises. Both Berreman mode and the symmetric ultra-thin film polariton are epsilon-near-zero (ENZ) modes, dispersing in close proximity to ωAlN

LO . d Sketch ofan AlN/SiC heterostructure, where the bulk SiC SPhP strongly couples to the ENZ TFP of the AlN thin film. e Dispersion of the strongly coupled polaritonmodes (red lines) featuring an avoided crossing. f Sketch of a heterostructure comprising a thin film of the reversibly switchable phase-change material GST ona SiC substrate. The system simultaneously supports p-polarized SPhPs and s-polarized waveguide modes (WMs), both being actively tunable by switching ofthe GST phase. g Dispersion lines of the SPhP (solid lines) and the WM (dashed lines) for amorphous (purple lines) and crystalline (red lines) GST. h Sketchof a crystalline hybrid (XH) comprising a polar dielectric superlaice of atomically thin AlN and GaN films on a SiC substrate, supporting volume-confinedhyperbolic phonon polaritons (hPhPs) of type I and type II. i Dispersion lines of the phonon polariton modes arising in the XH. At large in-plane momenta, thetypical mode progression of volume-confined hPhPs is observed (turquoise and red lines). Note that the dispersion lines in subfigures c and e were calculatedusing Eq. 2.2.10, yielding a momentum axis with absolute values. The dispersion lines in subfigures g and i, on the other hand, were obtained by transfer-matrixsimulations (for details see the publication reprinted on page 45 [II]), and here, the momentum axis is ploed in relative values. The light line in subfigures g andi is a vertical line located at k/k0 = 1.

17

Chapter 2 Theoretical Concepts

where ΦSiC/GST and ΦGST/air are the phase dierences upon reection at the substrate/lm (SiC/GST)and the lm/incident medium (GST/air) interfaces, k ≡ k/k0 = nair sin θ is the in-plane momentum(with k0 = ω/c), and dGST is the GST lm thickness.

Strikingly, in the GST/SiC heterostructure, the distinctive properties of the polar crystal substratemodify the WM such that its spatial connement and electric elds are enhanced, resulting in an s-polarized WM with polariton-like characteristics. Simultaneously supporting p-polarized SPhPs and s-polarized WMs, the GST/SiC system provides a versatile platform for actively tunable, omnipolarizednanophotonic applications (see the publication reprinted on page 85 [VI] for further details onpolariton-like WMs).

Volume-Confined Hyperbolic Polaritons A hyperbolic material is a material that features fre-quency regions where one (type II) or two (type I) principle dielectric permittivities have a negativereal part, while along the other axes, Re (ε) > 0. In those regions, hyperbolic phonon polari-tons (hPhPs) can be observed [15]. Some materials are hyperbolic by nature, such as hexagonalboron nitride (hBN) or quartz (SiO2), but alternatively, a hyperbolic material can be synthesizedby periodically stacking thin layers of dierent media [62–64]. In Fig. 2.2.2h, such an articialhyperbolic material is sketched, where multiple atomically thin layers of AlN and GaN thin lmsform a ∼ 650 nm thick superlattice, also called crystalline hybrid (XH) [65].

In thin lms of hyperbolic materials, such as the sketched XH, standing waves with discrete reso-nances are supported, giving rise to volume-conned hPhPs [16, 66, 67]. In Fig. 2.2.2i, the dispersionof all polaritons supported by the XH system is shown, featuring a rich variety of modes in the broadfrequency range of 550− 950 cm−1. For large in-plane momenta, the volume-conned hPhPs of typeI (turquoise lines) and type II (red lines) can be observed, featuring their characteristic higher-ordermode progression [68] (see publication [IX] for further details on hPhPs).

In summary, in only four dierent material systems comprising dierent polar dielectric materialslabs, a large variety of polariton modes arises, covering TFPs [V], Berreman modes [V], ENZpolaritons [IV], strong coupling [IV], refractive index-shifting [VI], WMs [VI] and volume-connedhPhPs [IX]. By studying these systems it is demonstrated that tailoring the material system allowsto overcome the limitations of a bulk SPhP for nanophotonic applications: phonon polaritons canbe actively tuned, their frequency window broadened, their spatial connement and electric eldsdramatically enhanced, and the polarization bottleneck of SPhPs circumvented by employing s-polarized, polariton-like WMs.

This thesis provides a broad overview over phonon polariton modes arising in layered heterostructures.In principle, the same physics can be observed in systems of 3D structure, as is exemplied forsubdiractional nanostructures in publication [VIII]. However, 3D structures entail a signicantlyhigher level of complexity, rendering the unambiguous assignment and prediction of polaritonmodes a formidable task. By focusing on well-dened systems consisting of layered media, thisthesis provides a thorough and robust benchmark of the physics of phonon polaritons in anisotropicheterostructures.

18

2.3 Polariton-Enhanced Nonlinear Optical Response

2.3 Polariton-Enhanced Nonlinear Optical Response

When light is compressed into sub-wavelength length scales, the local electric eld can be immenselyenhanced compared to a freely propagating wave. Such a eld enhancement naturally occurs forpolaritons, and can reach immense values when the modes experience additional spatial connementin nanoscale structures, as it is the case for TFPs or volume-conned hPhPs. These eld enhancementscan be experimentally probed by measuring the nonlinear optical response.

In this thesis, second harmonic generation (SHG), a second-order nonlinear optical eect, is employedfor the experimental investigation of the eld enhancements of various phonon polaritons in layeredheterostructures. Therefore, in the rst part, this section provides a brief introduction to SHG. In thesecond part, the calculated eld enhancement of the studied phonon polaritons and the resultingexperimental SHG yield is discussed. The theoretical calculations of the optical eld enhancementswere performed employing the 4× 4 transfer matrix formalism developed in the course of this thesis(for details see the publication reprinted on page 45 [II]).

2.3.1 Second Harmonic Generation (SHG)

The optical response of a material, that is, the created polarization ~P under an external electric eld~E, is approximately linear, as long as the incident intensity is weak enough (Eq. 2.1.1). For strongerincident elds, however, every material exhibits a nonlinear response, which is the driving principleof nonlinear optics. The electric polarization ~P is then a complicated function of ~E, and cannot bewritten down analytically any more. This is the origin of many nonlinear optical eects like SHG,sum frequency generation, dierence frequency generation, or electro-optical rectication – all ofthem second-order eects – and many others of higher order.

In the special case of plane waves and in the electric dipole approximation, the electric polarization~P can be expanded in a power series of ~E [69]:

~P (ω) = χ(1)(ω) · ~E(ω)

+ χ(2)(ω3 = ±ω1 ± ω2) : ~E(ω1) ~E(ω2)

+ ... ,

(2.3.1)

where χ = 1 − ε is the electric susceptibility, describing the optical response of a material underthe inuence of an external electric eld ~E. The nth coecient χ(n) of χ is a tensor of rank (n+ 1)and, in principle, allows to describe any nonlinear optical response of the material of any nth order.In practice, however, exact solutions of χ are not feasible even for the most simple systems, whichmakes nonlinear optics a timely, thriving eld of research.

In a very simple picture, the eect of SHG can be understood in terms of an energy diagram as athree-level process, which is depicted in Fig. 2.3.1a. Two incident photons at energy ~ωFEL from theincident light source, in this work a FEL, are needed in order to excite a system from the ground state|0〉 to the state |2〉, through the intermediate state |1〉, which lies at half the energy dierence between|0〉 and |2〉. The energy from |2〉 is then emitted as a photon at double frequency ωSHG = 2ωFEL,constituting the SHG signal. The measured SHG intensity ISHG is, most generally, given by

ISHG ∝ |χ(2)(2ω = ω + ω) ~E(ω)2|2, (2.3.2)

19

Chapter 2 Theoretical Concepts

wTO wTO⊥| | wzf

(2)

aca

|χ |(2)

caa

|χ |(2)

ccc

|χ |

(2)

|| (

norm

.)χ

800 850

1

-110

-210

-310

-410

-510900

wTO

1

-110

-210

-310

-410

(2)

aca|χ |

800 850 900Frequency ω (1/cm)Frequency ω (1/cm)

0

1

2

ωFEL

ωFEL ωSHG

a b c

3C-SiC 4H-SiC

Figure 2.3.1: Second harmonic generation and the second-order susceptibility of SiC. a Energy di-agram with three states, visualizing the process of SHG, where two incident photons at same frequencyωFEL are converted into the SH photon with double frequency ωSHG. Within the electric dipole approximation,this process occurs only in materials with broken inversion symmetry. In general, |1〉 and |2〉 can be virtualenergy levels. If the material exhibits a resonance at incident frequency, however, these states are real, andthe process is called resonant SHG. b and c Dispersion of the non-zero χ(2) elements in 3C-SiC and 4H-SiC,respectively. Additional to the resonances at the TO phonon frequencies, 4H-SiC features a resonance at thezone-folded (zf) phonon frequency, arising due to the supperlaice along the extraordinary crystal axis [70,71]. χ(2) dispersions are adapted from Ref. [70]

where χ(2) is the second-order susceptibility tensor, which has in total 27 coecients, and ~E2 is asymmetric tensor. Therefore, the resulting SHG signal can exhibit enhancements of two types: (i)originating from a χ(2) resonance, and (ii) from a resonance in the local elds at the sample site.For the case of polar crystals, the χ(2) dispersion only exhibits features at the IR-active phononresonances such as the TO phonon (as an example, Fig. 2.3.1b and c show the relevant χ(2) elementsof 3C-SiC and 4H-SiC, respectively). On the other hand, phonon polaritons feature strong local eldenhancements and therefore contribute signicantly to the second harmonic (SH) signal via the~E2 term, enabling polariton-enhanced SHG. In the following section, the eld enhancement of thephonon polariton modes studied in this work is discussed in detail.

Regarding the SHG signal from the bulk, the χ(2) coecients depend on the crystal symmetry, and forthe case of centrosymmetric materials (in the electric dipole approximation), they vanish completely(Tables for the non-vanishing coecients can be found in several text books [69, 72]). At the surfaceof a crystal, inversion symmetry is broken. Hence, even though a material does not exhibit SHG fromthe bulk, it will produce a surface SHG signal, which can be employed as a very surface-sensitiveprobing method [73–75]. In a material with broken inversion symmetry, on the other hand, thesurface signal is usually negligible [74], which is the case for all samples investigated in this work.

When the incident frequency ωFEL coincides with a resonance in the probed material, resonant SHGcan be observed [17, 73], accounting for a strong enhancement. While resonant enhancement of SHGfrom the bulk in non-centrosymmetric materials [17, 76, 77] as well as surface SHG from inversionsymmetric media [73, 78, 79] has been studied intensively, the process of resonant SHG arising fromphonon polaritons is mostly unexplored and a major component of this thesis. Notably, in thatregard, also sum frequency generation (SFG) spectroscopy and microscopy has recently emerged asa complementary tool [80].

20

2.3 Polariton-Enhanced Nonlinear Optical Response

Here, by investigating the SH response of bulk SPhPs (reprinted on page 4.1 [I]), strongly coupledphonon polaritons [VIII], and Berreman modes (reprinted on page 4.4 [V]), this thesis providesthorough experimental evidence of the nonlinear optical response of phonon polaritons in polardielectric heterostructures. In the following section, the eld enhancements of the phonon polaritonsresponsible for the observed SH signals are discussed.

2.3.2 Field Enhancement of Phonon Polaritons Probed with SHG

The strong connement of light as it occurs for phonon polaritons in nanoscale structures leadsto enhanced light-matter interaction, enabling nanophotonic applications such as highly ecientsensing [7, 81], all-optical switching [6, 82], and low-loss frequency conversion [83, 84]. Here, SHGspectroscopy [17] is employed as experimental method to probe the eld enhancement of phononpolaritons, while the transfer matrix formalism (see page 45 [II]) is employed as theoretical tool forcalculating the eld enhancement in layered media responsible for the SHG yield. In the following,the eld enhancement and the obtained SH yield of three systems are discussed, comprising a bareSiC crystal, a thin-lm AlN/SiC heterostructure, and SiC nanopillars on a SiC substrate.

SiC SPhP The SHG yield of evanescent SPhPs can be probed in the so-called Otto geometry (fordetails see chapter 3). In Fig. 2.3.2a, the experimental SHG spectrum from a SPhP propagating on abare SiC crystal obtained in the Otto geometry is shown. The spectrum features a strong, sharp peakat the SPhP resonance frequency (∼ 910 cm−1) that originates in the eld enhancement associatedwith the SPhP.

Fig. 2.3.2b plots the calculated electric eld distribution of the SiC SPhP as a function of z-position,that is, along the direction normal to the surface, and the incident frequency ω. Here, the in-planecomponent of the electric eld ~Ex is shown, clearly featuring a peak at the SPhP resonance frequency,localized at the air/SiC interface. These enhanced electric elds give rise to the observed peak in theSHG spectrum. The out-of-plane component ~Ez (not shown here) features a similar enhancement inair [I], while inside SiC, ~Ez is slightly suppressed [56] (for more details on the SHG from the SiCSPhP see the publication reprinted on page 37 [I]).

AlN/SiC Berreman Mode As has been discussed in section 2.2.2, a AlN/SiC heterostructure supportsa leaky Berreman mode dispersing at in-plane momenta smaller than light in vacuum, and stronglycoupled phonon polaritons arise on the evanescent side of the light line. In Fig. 2.3.2c, an SHGspectrum of the AlN/SiC system is shown, which was obtained under free-space excitation in orderto probe the leaky Berreman mode. The SHG spectrum features a pronounced peak at the Berremanmode resonance frequency (∼ 900 cm−1), arising from the eld enhancement associated with theBerreman mode. As is shown in Fig. 2.3.2d, the out-of-plane electric eld features an extreme spatialconnement localized in the thin AlN lm and reaching immense enhancement values of > 40.Interestingly, the in-plane elds ~Ex (not shown here) are featureless at the Berreman resonancefrequency [V]. Hence, the out-of-plane elds are the sole origin of the enhanced SH signal, observableas a peak in the SHG spectrum.

Note that the SHG peak of the Berreman mode lies in the wing of the SH signal at the SiC LO phononfrequency [70]. The two SH signals interfere constructively and destructively depending on thefrequency, resulting in the asymmetric peak shape of the Berreman mode SHG (for more details

21

Chapter

2TheoreticalC

oncepts

Frequency ω (1/cm)800 850 900 950 1000

Refl

ecta

nce

0

1

0

2

4

6

2ωωω

p

LSPhRSPhPSiC

800 850 900 950

0

I

(ar

b. u

n.)

SH

G

00

800 850 900 950Frequency ω (1/cm)

0

6

|E | |E |x x0/SiC

air

6 μ

m1

.5 μ

mz

AlNSiC SiC

Frequency ω (1/cm)

SiC

air

AlN

024

6 μ

m1.

5 μ

mz 20 nm

0

900880800 920 940Frequency ω (1/cm)

AlN

0

40

SiC

air

AlN20 nm

6 μ

m1.

5 μ

mz

a

b

c

d

e

f

g

|E | |E |x x0/|E | |E |z z0/

I

(ar

b. u

n.)

SH

G

I

(ar

b. u

n.)

SH

G

I

(ar

b. u

n.)

SH

G

SiCair

KRS5AlNSiC

AlN

KRS5air

SiC

Figure 2.3.2: Field enhancement of phonon polaritons probed with SHG spectroscopy. a SHG spectrum of the reststrahlen band of bareSiC measured in the Oo geometry (see section 3.2 for details) at an incident angle of θ ≈ 30 , featuring a resonance peak at the frequency of theSPhP [I]. b In-plane electric field distribution along the z-direction of the SiC SPhP at the air/SiC interface. At the SPhP resonance frequency, thelocal electric field is enhanced, reaching a maximum value of ∼ 6 at the interface. This field enhancement is the origin of the measured peak inthe SHG spectrum. c SHG spectrum under free-space excitation of an AlN/SiC heterostructure at θ ≈ 60 , featuring a resonance peak at thefrequency of the Berreman mode [V]. d Out-of-plane electric field distribution of the Berreman mode in the multilayer system. At the Berremanresonance frequency in close proximity to ωAlN

LO , the local electric field features an immense enhancement of up to ∼ 40, strongly localized in theAlN thin film. This field enhancement gives rise to the enhanced SHG signal shown in c. e SHG spectrum of the same AlN/SiC heterostructure asin c, measured in the Oo geometry at θ ≈ 29 . In the Oo geometry, the evanescently bound, strongly coupled phonon polaritons can be excited[IV], yielding two resonance peaks in proximity to ωAlN

LO [42]. f In-plane electric field distribution of the strongly coupled phonon polaritons,featuring two peaks in the field enhancement at the resonance frequencies of the two modes bound to the AlN layer, originating the measuredSHG peaks. g Reflectance and SHG spectrum of SiC nanopillars on a SiC substrate. The localized surface phonon resonance (LSPhR) of thenanopillars couples to the propagating SPhP, yielding two resonance dips in the reflectance, and two peaks in the SHG spectrum [VIII]. In bothstrongly coupled systems, the SHG yield of the two coupled modes is subject to a phase modulation induced by the background contribution ofthe SiC substrate. Figure adapted from publication [VIII].

22

2.3 Polariton-Enhanced Nonlinear Optical Response

on the eld enhancement and SH yield of the Berreman mode see the publication reprinted onpage 75 [V]).

AlN/SiC Strongly Coupled Phonon Polaritons When probing the AlN/SiC heterostructure in theOtto geometry, the strongly coupled phonon polaritons discussed in section 2.2.2 can be excited. AnSHG spectrum obtained for this conguration is shown in Fig. 2.3.2e, where two resonance peaksarise at the frequencies of the coupled modes. The origin of the dierent peak heights is still underinvestigation. As for the two previous systems, the origin of this enhanced SHG signal is the eldenhancement associated with the phonon polariton modes.

Fig. 2.3.2f shows the calculated in-plane electric eld distribution of the two strongly coupled phononpolaritons in the AlN/SiC heterostructure, featuring two resonances that are of similar shape as theeld distribution of the bare SiC SPhP (Fig. 2.3.2b). The eld distributions (both ~Ex and ~Ez [IV])of the two modes are of similar amplitude and shape, indicating full hybridization of the substrateSPhP and the ENZ mode at strong coupling conditions. For more details on the eld enhancement ofstrongly coupled phonon polaritons see the publication reprinted on page 65 [IV].

SiC Nanopillar Localized Phonon Polariton Resonances Periodic nanostructures of polar crystalswith 3D connement, such as nanopillars as sketched in Fig. 2.3.2g, support localized surface phononresonances (LSPhRs), which are tunable in frequency by the nanopillar diameterD and the periodicityp [3, 85, 86]. In the case of SiC nanopillars on a SiC substrate, a so-called monopole mode is supported,where the entire pillar is charged with one polarity, while the substrate carries the counter-charge.In a nanopillar conguration of large periodicity compared to D, as it is illustrated in Fig. 2.3.2g(D ≈ 1 µm, p = 5− 7 µm), the substrate supports a propagating SPhP that can couple strongly tothe monopole mode [87].

In Fig. 2.3.2g, one exemplary reectance spectrum (red line) of the investigated nanopillar system isshown, exhibiting two pronounced dips in the reststrahlen band at the resonance frequencies of theLSPhR (the nanopillar monopole mode) and the propagating SPhP, marked by dashed lines. The eldenhancement of both modes lead to a resonant SH yield, resulting in peaks in the SHG spectrum (blueline). For further details on the nanopillar modes, see Ref. [3, 85, 86]. Publication [VIII] discusses theSHG from the strongly coupled LSPhR and SPhP modes in detail.

Within this thesis, SHG is demonstrated to be a powerful experimental tool for probing the en-hanced local electric elds of phonon polaritons, ranging from single-crystal SPhPs over Berremanmodes to strongly coupled polaritons. Notably, the devised method provides the only direct experi-mental access to the eld enhancement of phonon polaritons in the IR. Furthermore, the developed4× 4 transfer matrix formalism (see page 45) is shown to be a key theoretical tool for calculatingthe eld enhancement of phonon polaritons in layered heterostructures of anisotropic materials.Thus, the investigations summarized in this section demonstrate the high level of insight into thelocal electric elds of phonon polaritons that is gained by the presented experimental and theoreticalmethods.

23

CHAPTER 3Experimental Approaches

Phonon polaritons are, except for the leaky Berreman mode, evanescent modes that cannot radiateinto the far-eld, and vice versa cannot be accessed by incident plane waves in a free-space excitationscheme. This aspect of being non-radiative renders phonon polaritons excellent excitations fornanophotonic applications that aim at harvesting, localizing and guiding light in sub-wavelengthnanostructures. On the other hand, the study of non-radiative modes also requires special excitationschemes, such as grating coupling, prism coupling, or scattering-type scanning near-eld opticalmicroscopy (s-SNOM).

The rst section of this chapter describes the most common experimental methods for the excitationof phonon polaritons. In this work, the Otto geometry for prism coupling has been implemented,and the details of the setup are provided in section 3.2. Finally, in section 3.3 the FEL at the FHI inBerlin is characterized, which was employed as a high intensity, mid-infrared (MIR) light source forall spectroscopy measurements performed in the course of this thesis.

3.1 Excitation of Phonon Polaritons

The reason for the non-radiative nature of phonon polaritons is their momentum-mismatch withplane waves propagating in vacuum. In order to be excitable, the dispersion must feature a crossingpoint with the dispersion of the incident light, but evanescent phonon polaritons disperse only onthe right hand side of the light line, see Fig. 2.2.1. The additional in-plane momentum required forthe excitation of phonon polaritons can be provided by evanescent waves or nanostructures, such asfor the evanescent wave generated under total internal reection, or the light scattered o a sharptip. In the following, four common excitation schemes are discussed and compared.

Grating Coupling One way to couple to surface polaritons is to cut a periodic line grating intothe sample surface, as has been demonstrated for SPPs propagating on at metallic surfaces [88, 89]and on metal tips [90, 91], and for SPhPs on a SiO2 crystal [92]. Alternatively, the grating can beevaporated onto the sample, as has recently been shown for gold gratings on hyperbolic metamaterials[63, 64]. By this means, the induced polarization in the sample receives a total momentum composedof the incoming in-plane momentum and an additional grating momentum, and thus can cross thepolariton dispersion. In Fig. 3.1.1a, the excitation of a surface polariton with a grating on a at sampleis sketched. The periodicity p of the grating denes the provided momentum qgrating, leading to thefollowing total momentum kgx [93]:

25

Chapter 3 Experimental Approaches

kgx = qx + qgrating =ω

csin θ + n

2π

p, (3.1.1)

where qx is the in-plane momentum of the incident light in air and n ∈ Z is the grating order. Forthe excitation of a surface polariton, the dispersion given by Eq. 3.1.1 has to cross the polaritondispersion, fullling the equation

ksurface polariton = kx. (3.1.2)

In Fig. 3.1.1b, the SPhP dispersion on a bare SiC crystal (green line) is plotted together with thedispersion given by Eq. 3.1.1 (red lines) for various incident angles θ and the grating orders n = 1 and−1. For n = −1, the total momentum kx is negative, meaning that a SPhP is excited that propagatesin the reverse in-plane direction of the incoming light.

In a reectance measurement, the excitation of a SPhP leads to a resonance dip in the reststrahlenband, where otherwise the reectance is 1 due to the negative real part of the dielectric permittivity.By measuring the reectance spectra and extracting the SPhP resonance frequencies for variousincident angles θ, the SPhP dispersion can be reconstructed.

Gratings allow for the excitation of surface polaritons in a free-space experiment, where the samplecan be directly illuminated without the need of additional equipment. On the other hand, (i) thefabrication of gratings requires precise lithographic methods that have to be employed for eachinvestigated sample, (ii) the 3D structure of a grating provides numerous scattering sites that perturbthe propagation of the surface polariton and modify its dispersion, and (iii) a precise theoreticaldescription is dicult to achieve for samples with high surface roughness. Therefore, while gratingsprovide a straight-forward way to probe surface polaritons on a proof-of-principle basis, they are oflimited use for a detailed, theory-supported investigation of polaritonic systems.

Prism Coupling in the Kretschmann Geometry Another way to couple to surface polaritons isto provide the lacking in-plane momentum by employing a prism made of a material with highrefractive index. The total in-plane momentum is then given by

kpx = qx√εP =

ω

csin θ√εP, (3.1.3)

where εP is the dielectric permittivity of the prism.

In a prism coupling excitation scheme, the surface polariton has to be excited at a sample surfaceadjacent to air or another material with a dielectric permittivity smaller than εP. In the so-calledKretschmann geometry [94], this is achieved by placing a thin lm of the polariton-active mediumonto the prism, and the surface polariton is excited at the opposite sample surface adjacent to air(sketched in Fig. 3.1.1c). Furthermore, the incident angle θ has to be larger than the critical angle oftotal internal reection of the prism material, in order to generate an evanescent wave that leaksinto the sample and is capable of coupling to the surface polariton.

The Kretschmann geometry is mainly employed for the investigation of SPPs, where thin metal lmsare evaporated onto the prism [94–98], whereas SPhPs are typically studied by dierent methods.

26

3.1Excitation

ofPhononPolaritons

a

xyzsample

air

SPhP

pθ

dgap

KRS5

sampleSPhP

air

θ

e

airsample

SPhP

Ω

gc

SiO2

SPPsample

θ

ds

0.5 1 1.5 2

800

850

900

950

1000

Fre

que

ncy ω

(1/

cm)

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SiC

SiC

SiCS

45°θ = 30° 70°

0

2

4

6

8

Phot

onE

nerg

y(e

V)

0 20 40 60 80 100 120In-plane Momentum k (1/μm)

AuS

Aup

60°θ = 50° 75°d f 1200

1100

Fre

que

ncy ω

(1/

cm)

h

1180

1160

1140

1120

1080

0 20 40 60 80 100In-plane Momentum k/k0

d = 7 nm15 nm

30 nmBulk

aira-GST

SiO2

d

Grating Coupling Kretschmann Geometry Otto Geometry s-SNOM

-2 -1 0 1 2

800

850

900

950

In-plane Momentum k (1/μm)

θ = 0° 60° 0° 60°

Fre

quen

cy ω

(1/

cm)

b

n = 1n = -1

... ...

Figure 3.1.1: Experimental methods for the excitation of surface polaritons. a Sketch of a grating edged into the sample surface, allowingto couple to surface polaritons via free-space radiation. b Dispersion of a SiC SPhP (green lines) either launched in the same in-plane direction(k > 0) as the incoming beam or counter-propagating (k < 0), and the dispersion of the grating light line kgx, see Eq. 3.1.1 (red lines), for variousincident angles θ and the grating orders n = 1 and −1. Excitation of the SPhP occurs at the crossing points of kgx and kSPhP. c Sketch of prismcoupling in the Kretschmann geometry, where the surface polariton is launched at the boom side of the sample film (yellow). d Dispersionof a Au SPP (green line) and the prism light lines kpx, see Eq. 3.1.3 (red lines), for various incident angles. e Illustration of the Oo geometry,implementing a coupling prism parallel to the sample leaving a variable air gap of size dgap. SPhPs are launched by evanescent waves leakingacross the air gap. f Dispersion of a SiC SPhP (green line) and prism light lines for a KRS-5 prism. g Sketch of the operation principle of s-SNOM.The incoming light is scaered o the oscillating tip of an atomic force microscope (AFM), thereby launching surface polaritons on the samplesurface. h Exemplary dispersion graph with experimental data (triangles) obtained by s-SNOM (figure adapted from Ref. [4]). Compared to theother excitation methods, s-SNOM allows to excite surface polaritons at large in-plane momenta.

27

Chapter 3 Experimental Approaches

In Fig. 3.1.1d, the dispersion of a SPP on Au (green line) is plotted together with the in-plane lightdispersion (Eq. 3.1.3) in a SiO2 prism for three dierent incident angles. Analogous to the gratingcoupling, varying θ allows to reconstruct the polariton dispersion, however only up to a limit(θ = 90 ) dened by the refractive index of the prism material.

Compared to the grating coupling, the Kretschmann geometry stands out by its structural robustnessand simplicity. Furthermore, the excitation of surface polaritons in the Kretschmann geometry iswell-dened and can be easily treated theoretically by means of a transfer matrix formalism. On theother hand, each sample and even each thickness of the same sample requires a new prism, whichis probably the reason why SPhPs are typically not studied in the Kretschmann geometry, becausesuitable prism materials in the IR are quite expensive (for example KRS-5 or diamond).

Prism Coupling in the Oo Geometry In 1968, three years before Kretschmann originated hiscoupling conguration, Otto proposed a prism coupling method to surface polaritons where a smallgap of size dgap is opened between prism and sample [23]. The Otto geometry is illustrated inFig. 3.1.1e and is implemented in this work (see section 3.2 for details on the setup).

The total in-plane momentum kx in the Otto geometry is the same as for the Kretschmann geometry,see Eq. 3.1.1, and the condition for excitation is given by Eq. 3.1.2. The dispersion of a bulk SiC SPhPand the light dispersion in a KRS-5 prism is plotted in Fig. 3.1.1f. As for grating coupling and theKretschmann geometry, the incident angle θ serves as a tuning parameter for reconstruction of theSPhP dispersion, and it has to be larger than the critical angle of total internal reection of the prismmaterial (∼ 25 for KRS-5 in the IR).

As for the Kretschmann geometry, a surface polariton in the Otto geometry is a three-layer mode,sensing the presence of the coupling prism. This leads to a strong dependence of the excitationeciency on dgap. For large distances, the exponentially decaying evanescent wave under totalinternal reection cannot excite the surface polariton eectively. For smaller dgap, the evanescentwave and the surface polariton overlap such that optimal coupling conditions are reached. At thiscritical distance dcrit, the reectance reaches its minimum and will be even zero for resonant wavevector matching, meaning that the incoming light is fully absorbed by the surface polariton. Byfurther reducing the gap size, the radiative loss of the surface polariton into the prism increases, andthe coupling eciency decreases (see the publication reprinted on page 37 [I] for further details onthe critical coupling behavior).

Compared to grating coupling and the Kretschmann geometry, the Otto geometry only requires asingle coupling prism, which can be utilized to study any bulk sample or layered heterostructureof arbitrary thickness. Furthermore, a unique experimental advantage of the Otto conguration isthe adjustability of the gap size dgap. However, the precise control of the gap size is challenging.Specically for SPPs, the gap sizes lie in the nanometer range, whereas SPhPs are accessible formicrometer gap sizes.

The setup developed in this work (see section 3.2) is one of the rst1 experimental realizations ofthe Otto geometry that allows for reproducible sub-micrometer control over the gap with direct,non-invasive read-out of the gap size. As is demonstrated in the publications originated during thecourse of this thesis, this unique setup2 enables the possibility to investigate the critical coupling

1to the best of my knowledge2In correspondence, PIKE Technologies has recently started the development of a comercialized Otto geometry setup

28

3.1 Excitation of Phonon Polaritons

behavior of SPhPs [I], study several modes of dierent coupling conditions in the same sample [IV,VI], and reconstruct the SPhP dispersion at critical coupling conditions at all frequencies [I, IV, X],being inaccessible by other coupling methods.

Compared to grating coupling and the Kretschmann geometry, the Otto geometry proves to bea versatile, reproducible, and robust tool for the investigation of SPhPs in the IR. However, themomentum range that can be probed is limited by the refractive index of the prism material, renderingthe Otto geometry only suitable for modes that propagate in proximity to the light line in vacuum.As is demonstrated in this work, the low-momentum region possesses a rich variety of polaritonmodes featuring numerous intriguing phenomena. Nonetheless, some modes disperse at largerin-plane momenta, and therefore are unreachable with the Otto geometry. These modes, such asvolume-conned hPhPs in hyperbolic thin lms, can be studied employing s-SNOM, as is discussedin the following.

Scaering-type Near-Field Optical Microscopy A sub-wavelength small object scatters an incom-ing plane wave into waves covering a continuum of in-plane momenta, allowing to couple to surfacepolaritons that disperse at momenta far o the light line in air. scattering-type scanning near-eldoptical microscopy (s-SNOM) employs the tip of an AFM as a scattering object (sketched in Fig. 3.1.1g),and by bringing the tip close to the sample surface, high-momentum surface polaritons are launchedin the sample. The tip oscillates at frequency Ω, and by locking to the higher-harmonics of theback-reected light, the scattered contributions o the tip can be extracted [99–101].

In order to obtain information about the launched SPhPs, the tip is scanned across an edge or spoton the sample, where the SPhPs are reected. The back-reected wave interferes with the launchedwave, forming a standing wave with a measurable intensity modulation, which allows to retrievethe wavelength and propagation length of the excited SPhP. This technique has been demonstratedfor various systems, including bare SiC [102, 103], hPhPs in hBN [16], and SPhPs in SiO2 modulatedby a thin-lm PCM [4]. The dispersion of the latter system is shown in Fig. 3.1.1h, where thetriangles indicate the experimental data obtained with s-SNOM, yielding in-plane momenta of up tok/k0 = 80.

s-SNOM is a powerful technique to study the local elds of surface polaritons in nanoscale structures.A complementary approach is the photothermal-induced resonance (PTIR) technique, where theabsorption of short laser pulses by the sample are detected with an AFM [104, 105]. Enabled by thesharp tip of the AFM, the spatial resolution of s-SNOM and PTIR far exceeds other coupling methods,and the local properties of polaritons can be mapped out by scanning the sample surface. On theother hand, due to the need of a back-reector on the sample surface and the presence of the tip asscattering object, theoretical modeling of the experimental data is cumbersome.

In conclusion, s-SNOM provides information about the high-momentum modes on the nanoscale,whereas the Otto geometry is better suited for the investigation of low-momentum surface polari-tons in macroscopically smooth layered heterostructures. Compared to grating coupling and theKretschmann geometry, the Otto geometry is more versatile, and provides deeper insight into thesurface polariton properties by controlling the gap size. This work implements the Otto geometry,focusing on phonon polaritons in layered polar dielectric heterostructures and their characteristicsin the low-momentum range. The experimental setup is presented in the following section.

29

Chapter 3 Experimental Approaches

3.2 The Oo Geometry Setup

As discussed in the previous section, the Otto geometry enables thorough investigations of SPhPsin layered heterostructures due to the tunability of in-plane momentum and excitation eciencyvia incidence angle and gap size, respectively. The experimental implementation together with thespectroscopic setup is illustrated in Fig. 3.2.1. The collimated incident FEL beam is focussed by ano-axis mirror (f = 646 mm) onto the prism. After passing through the Otto geometry, where thebeam is totally reected at the prism backside, the light is separated by a dichroic beam splitter(longwave pass at λ = 7 µm) for measuring the reectance and the generated SHG signal separately.In order to ensure reproducibility and accuracy of the detection for varying incident angles, thepick-up mirror that directs the beam onto the beam splitter is mounted on a motorized translationand rotation stage.

The Otto geometry comprises the sample and the prism with a small gap in between. In Fig. 3.2.2,the constructed Otto geometry system is shown in a real image. The gap size can be adjusted bymeans of three Newport motorized actuators with a minimum incremental motion of 0.1 µm, whichare arranged in a triangular shape. This assembly is attached to a two-axis mirror mount, allowingthe horizontal and vertical matching of the incident and reected beam. Finally, this is mounted on arotation stage for measuring at dierent incident angles θ, and on a x-y-z translation stage for neadjustment, i.e. altogether being a 9-axis system.

The triangular prism used in this work is made of KRS-5, with a basis size of 1′× 1′ and side angles of30 . Due to the high refractive index of KRS-5 (in the IR, nKRS-5 ≈ 2.4), a rotation of the entire Ottogeometry translates into a much smaller change of the incident angle θ inside the prism. Geometricbeam propagation calculations allow to convert the incident angle θ inside the prism into the externalincidence angle with respect to the prism side surface and vice versa. The prism geometry was chosensuch that for an incident wavelength of λ = 11 µm, the incident angle θ can be tuned in a range fromstill below the critical angle of total internal reection (∼ 25 for KRS-5) up to a maximally possibleangle of θ ≈ 50 , allowing to map the polariton dispersion from the light line up to a maximummomentum.

As sketched in Fig. 3.2.1 in green, a reective ber collimator directs the light of a Thorlabs tungstenNIR source onto the ground-o top of the KRS-5 prism, and collects the signal reected at the prismback side and the sample surface. The reected light is guided through a bifurcated ber bundleand analyzed in an Ocean Optics NIR spectrometer. Interference between the reected light fromthe prism back side and the sample leads to a modulation in the white-light spectrum, and Fouriertransformation of the spectrum allows to determine the gap size dgap. Note that exclusively the smallgap size between prism and sample is probed, because owing to the incoherence of the white-light,the setup is not sensible to larger interface distances such as the prism front and backside.

Prior to the SPhP excitation measurements, the modulation contrast of the white-light spectrumenables to optimize the parallelism between prism and sample. After establishing parallelism, dgap ismonitored continuously and simultaneously to the SPhP experiments. The dgap range that can beread out by the white-light interferometry setup lies between ∼ 100 nm and ∼ 60 µm. Roughness ofthe sample and the prism backside further restricts the lower limit, leading to typical minimum gapsizes of approximately 1 µm. The resulting accessible dgap range is perfectly suited for the excitationof SPhPs, since the typical critical gap sizes of optimal coupling conditions of phonon polaritons

30

3.2 The Otto Geometry Setup

Reflectancedetection

reflectancesplitterbeamReflectance

rotation stagetranslation and

Reflectancedetection

SHG

Reflectancerotation stage

translation and

whitelight

Otto geometry

ReflectanceFEL beam

NIRspectrometer

Reflectancebroadbandlight source

off-axis

mirrorfocusing

pick-up mirror on

Figure 3.2.1: Setup for reflectance and SHG measurements employing the Oo geometry. Dierent incidentangles θ can be measured by rotating the Oo assembly and moving and rotating the pick-up mirror to theposition, where the detection beam path is met again. Motorized actuators allow the adjustment of the gapsize between prism and sample. The prism top is ground o in order to couple in a collimated white-light beam.The reflected white-light signal carrying interferometric information is analyzed in a near-infrared (NIR)spectrometer, allowing to determine the gap size.

pick-up

white-light

mirror

Ottogeometry

Figure 3.2.2: Home-made and designed 9-axis Oo geometry construction with mounted KRS-5 prism. Threemotorized actuators allow to control the gap size on the sub-micrometer scale. Note that in the image, nosample is mounted and the gap is opened maximally for illustration purposes. From the le, the fiber-coupledwhite-light beam is directed onto the ground-o prism top.

31

Chapter 3 Experimental Approaches

lie in the range of ∼ 1 − 20 µm. For further details on the white-light interferometry setup, seepublication [X].

In summary, the constructed Otto geometry setup enables thorough experimental studies of phononpolaritons in any at sample, allowing to determine the linear and nonlinear optical response ofthe polariton modes, analyze their critical coupling conditions by controlling the gap size, and mapout the polariton dispersion by scanning the wavelength and incident angle. This renders the heredeveloped Otto geometry setup a highly versatile and exible experimental tool for the investigationof phonon polaritons in layered heterostructures, as is demonstrated in the publications reprinted inchapter 4.

32

3.3 The FHI Free Electron Laser

3.3 The FHI Free Electron Laser

The FHI FEL is a MIR oscillator FEL which has been installed in 2011 on the campus of the FritzHaber Insitute (FHI) and started operating in 2013 [106]. The machine generates spectrally sharp,frequency-tunable, high-intensity laser output in the MIR, constituting the ideal excitation sourcefor the investigation of phonon polaritons by means of reectance and SHG spectroscopy. In thefollowing, the FHI FEL is briey described.

The setup is illustrated in Fig. 3.3.1. After generation in a thermionic gridded electron gun, two linearaccelerators (linacs) speed up the electrons to energies in the range of 15 MeV to 50 MeV. Whilethe rst linac operates at a nominal electron energy of 20 MeV, the second can be tuned to boostor lessen the electron energy in the mentioned range. The electron gun generates pulsed electronbunches of about 200 pC at a repetition rate of 1 GHz. Before entering the linacs, these electronmicro pulses pass through a 1 GHz buncher, which reduces the micro pulse length to 1 ps to 5 ps.Each bunch can then be eectively accelerated in the linacs, which create a macro pulse structureof pulse lengths of 1 µs to 15 µs with a repetition rate of 10 Hz. In table 3.3.1, the most importantproperties of the accelerator system are listed.

Table 3.3.1: Specifications of the electron accelerator of the FHI FEL [106].

Parameter Value

Electron energy 15-50 MeVBunch charge 200 pC

Micro pulse length 1-5 psMicro pulse repetition rate 1 GHz

Macro pulse length 1-15 µsMacro pulse repetition rate 10 Hz

After guidance through a 90 curve by means of isochronous bends, the electron beam enters theMIR wedged-pole undulator which has a total length of 2 m and is composed of 50 periods with aperiod length of λu = 40 mm. The tunable undulator gap has a nominal minimum size of 16.5 mmand features the generation of radiation in the range of 3 µm to 50 µm. Up to date, the MIR undulatoris the only operating FEL arm. The construction of the longer, far-IR undulator depicted in Fig. 3.3.1is currently in preparation and will be initiated in the near future. This second undulator is designedto cover the wavelength range of 10 µm to 150 µm and will be synchronized with the MIR undulator,allowing for two-color experiments in the IR.

The undulator is enclosed by two gold-plated copper mirrors, forming the laser cavity of L0 = 5.4 mlength. Both mirrors are of concave spherical shape, with radii of curvature of 2.65 m and 3.51 m ofthe end mirror and the outcoupling mirror, respectively. Since hole outcoupling is employed, vemirrors with dierent hole diameters (0.75, 1, 1.5, 2.5, and 3.5 mm) are available in order to optimizethe lasing performance at all wavelengths. The end mirror is mounted on top of a precision translationstage, which can be used for a wavelength dependent cavity length tuning [106]. This adjusts thecavity length to a value of L0−∆λ, where ∆λ = qλ is the wavelength synchronized cavity detuning.In practice, the factor q is usually set to 0 < q < 5. By this means, the accessible wavelength rangeat a specic electron energy, that is, in a single wavelength scan, can be signicantly increased.

33

Chapter 3 Experimental Approaches

Figure 3.3.1: Sketch of the FHI FEL showing the electron acceleration segment and the two FEL systemswith dierent undulator length. In two linear accelerators Linac 1 and Linac 2, the electrons are speeded up toenergies up to 50MeV, and then are conducted into one of the FEL cavities, where the IR laser pulses aregenerated. The picture is taken from Ref. [106].

Figure 3.3.2: Pulse structure of the FEL. Each macro pulse (with ∼ 100ms time spacing) exhibits a bunchof micro pulses, separated by 1 ns with a duration of 1− 10 ps.

34

3.3 The FHI Free Electron Laser

Furthermore and more importantly, the emission bandwidth decreases with increasing q, whichenables a practical method of enhancing the spectral resolution. In table 3.3.2, the most importantproperties of the MIR FEL are listed.

Table 3.3.2: Specifications of the MIR FEL [106].

Parameter Value

Undulator length 2 mperiod length λu 40 mm

number of periods 50Cavity length L0 5.4 m

The pulse structure of the FEL output radiation is dominated by the input electron pulse structure,exhibiting macro pulses at ∼10 Hz repetition rate, each consisting of a bunch of micro pulses at1 GHz repetition rate. The resulting intensity distribution is sketched in Fig. 3.3.2.

In summary, the FEL provides laser radiation of high coherence, small spectral band width andtunable IR wavelength, enabling spectroscopic measurements of phonon polaritons with high spectralresolution and well-dened excitation conditions. Furthermore, the high power of the FHI FEL enablesthe possibility of employing SHG spectroscopy, rendering the machine a unique tool for studyingthe linear and non-linear optical response of phonon polaritons in the IR.

35

CHAPTER 4PublicationsIn this chapter, the six scientic articles forming this thesis are reprinted. The erratum of onepublication is also included. A list of these publications can be found on page vii, and a list of allpublications originated during the course of this thesis can be found on page 117.

4.1 Second Harmonic Generation from Critically CoupledSurface Phonon Polaritons

Nikolai Christian Passler, Ilya Razdolski, Sandy Gewinner, Wieland Schöllkopf, MartinWolf, and Alexander Paarmann

This publication (Passler et al., ACS Photonics 2017, 4, 1048-1053 [I]) reports on the rst SHG fromcritically coupled SPhPs on SiC excited in the Otto-type prism coupling geometry. The supportinginformation is reprinted in appendix A.1.

This is the rst paper that emerged from the Otto geometry implementation constructed in the courseof the thesis (section 3.2). Employing the full mechanical control over the prism-sample distance, thispublication explores the dependence of the excitation conditions and the optical eld enhancement ofSPhPs on the size of the air gap between prism and sample. The work establishes the Otto geometryas a tool for the investigation of surface polaritons in layered media at infrared wavelengths, pavingthe way for the other publications reporting on experiments that where conducted during thisthesis. Furthermore, the FEL is employed for the rst time for performing SHG spectroscopy onprism-coupled SPhPs.

Author contributions

A.P. originated the concept and devised the SHG method and the Otto-conguration experiments.The manuscript was written by N.C.P. and A.P., with all authors assisting in the proof-readingand preparation for nal submission. N.C.P. constructed the Otto-conguration setup. The SHGmeasurements were performed by N.C.P., I.R., and A.P., and the results were analyzed by N.C.P. andA.P. S.G. and W.S. operated the FEL. Computations of the linear and non-linear optical response werecompleted by A.P. and N.C.P. The experimental design and project management were provided byM.W. and A.P.

go to list of publications

37

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44

4.2 4 × 4 transfer matrix formalism

4.2 Generalized 4 × 4 matrix formalism for light propagationin anisotropic stratified media: study of surface phononpolaritons in polar dielectric heterostructures

Nikolai Christian Passler and Alexander Paarmann

This publication (Passler and Paarmann, JOSA B 2017, 34, 2128-2139 [II]) presents a theoreticalframework for the calculation of light-matter interaction in arbitrarily anisotropic multilayer systems.Compared to previous work [107–111], the here developed formalism is numerically stable, free fromdiscontinuities, and handles media with any isotropic or anisotropic permittivity tensor. An onlineversion of the computer code programmed in Matlab [18] and Python [112] has been published asopen source.

Most of the experimental studies conducted during this thesis investigate stratied sample systemswith anisotropic materials. For all of them, the algorithm developed in this publication has beenemployed for predicting, analyzing and simulating the optical response of the sample, as well astting the experimental spectra with exceptional accuracy.

The broad applicability and user friendliness of the formalism resulted in frequent use by manyresearch groups, which eventually led to the discovery of an error in the calculation of the electriceld distributions for birefringence, that is, for media with non-zero o-diagonal elements of thedielectric permittivity tensor in the lab frame. The correction of the error was published in an erratumin 2019 [III], which is reprinted on page 59. All publications prior to this erratum investigate isotropicor anisotropic media with diagonal permittivity tensor, and thus are not aected by the error in theoriginal formalism.

Author contributions

N.C.P. and A.P. originated the concept. The manuscript was written by N.C.P. and A.P. N.C.P.devised the fundamental framework of the formalism, and N.C.P. and A.P. developed the nalalgorithm. Computations of the example systems were completed by A.P. and N.C.P. The projectmanagement was provided by A.P.

go to list of publications

45

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4.2 4 × 4 transfer matrix formalism

4.2.1 Generalized 4 × 4 matrix formalism for light propagation in anisotropic stratifiedmedia: study of surface phonon polaritons in polar dielectric heterostructures:erratum

Nikolai Christian Passler and Alexander Paarmann

This erratum (Passler and Paarmann, JOSA B 2019, 36, 3246-3248 [III]) presents corrections to theoriginally published transfer matrix formalism [II] (reprinted on page 45), allowing for the correctcalculation of the electric eld distribution for birefringence, that is, for anisotropic media withnon-zero o-diagonal elements of the dielectric permittivity tensor in the lab frame. All publicationsprior to this erratum investigate isotropic or anisotropic media with diagonal permittivity tensor,and thus are not aected by the error in the original formalism. The corrections are included in theonline version of the open-source computer code programmed in Matlab [18] and Python [112].

Furthermore, revising the original work and correcting the equations triggered the development of anextended formalism, which is capable of calculating the layer-resolved absorption and transmittancein a multilayer system. This work has been published most recently [VII] and is reprinted onpage 97.

Author contributions

N.C.P. and A.P. originated the concept. The manuscript was written by N.C.P. and A.P. N.C.P. andA.P. developed the corrections to the algorithm. The project management was provided by A.P.

go to list of publications

59

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Chapter 4 Publications

62

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4.2 4 × 4 transfer matrix formalism

63

4.3 Strong Coupling of ENZ Phonon Polaritons

4.3 Strong Coupling of Epsilon-Near-Zero Phonon Polaritonsin Polar Dielectric Heterostructures

Nikolai Christian Passler, Christopher R. Gubbin, Thomas Graeme Folland, Ilya Razdolski,D. Scott Katzer, David F. Storm, Martin Wolf, Simone De Liberato, Joshua D. Caldwell,

and Alexander Paarmann

This publication (Passler et al., Nano Letters 2018, 18, 4285-4292 [IV]) reports on the rst observationof the full hybridization of an ENZ thin-lm polariton with a bulk SPhP. The experimental resultswere obtained employing the Otto type prism coupling setup (section 3.2), where varying the incidentangle allows for reconstruction of the strongly coupled, anti-crossing polariton dispersions. Thesupporting information is reprinted in appendix A.2.

This work has been the rst in the scope of this thesis in experimentally investigating polaritonmodes in stratied heterostructures, and was the trigger for the subsequent study of ENZ Berremanmodes that are observable via free-space excitation in the same sample system [V] (reprinted onpage 75). The transfer matrix formalism [II] (reprinted on page 45) is used as theoretical tool for thesimulation of the measured optical response.

Author contributions

A.P. conceived the project. D.S.K. and D.F.S. performed the sample growth. N.C.P., A.P. and I.R.conducted the experiment and analyzed the results. N.C.P. and A.P. wrote the manuscript, withall authors assisting in the proof-reading and preparation for nal submission. C.G. and S.D.L.developed the analytical strong coupling model. Computations of the linear optical response in theOtto geometry were completed by A.P. and N.C.P. T.F. carried out the simulations of the near-eldscattering at a nanoparticle. The project management was provided by J.D.C. and A.P.

go to list of publications

65

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4.4 SHG from Phononic ENZ Berreman Modes

4.4 Second Harmonic Generation from PhononicEpsilon-Near-Zero Berreman Modes in Ultrathin PolarCrystal Films

Nikolai Christian Passler, Ilya Razdolski, D. Scott Katzer, D. F. Storm, Joshua D. Caldwell,Martin Wolf, and Alexander Paarmann

This publication (Passler et al., ACS Photonics 2019, 6, 1365-1371 [V]) reports on the rst experimentalobservation of enhanced SHG arising from the Berreman mode in a subwavelength-thin AlN lm,which can be excited at the AlN LO frequency where the dielectric function crosses zero. The originof the high SHG yield is the immense eld enhancement at ENZ frequencies in ultra-thin lms, andthe work provides a thorough analysis of the thickness dependence of the eld enhancement. Thesupporting information is reprinted in appendix A.3.

The samples studied in this work comprise an ultra-thin AlN lm on a SiC substrate, which are thesame samples where the strong coupling of ENZ polaritons was observed [IV]. Complementary tothe evanescent polariton modes that were investigated previously in the Otto geometry setup [IV],this work focuses on the optical response under free-space excitation, revealing the Berreman modeas a radiative virtual polaritonic mode. The theoretical calculations of the eld enhancement wereperformed using the transfer matrix formalism that has been published previously [II] (reprinted onpage 45).

Author contributions

A.P. originated the concept. D.S.K. and D.F.S. performed the sample growth. The manuscript waswritten by N.C.P. and I.R., with all authors assisting in the proof-reading and preparation for nalsubmission. The SHG measurements were performed by N.C.P., and the results were analyzed byN.C.P., I.R. and A.P. Calculations of the optical response were completed by N.C.P. The experimentaldesign and project management were provided by M.W., J.D.C. and A.P.

go to list of publications

75

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4.4 SHG from Phononic ENZ Berreman Modes

83

4.5 Surface Polariton-Like s-Polarized Waveguide Modes

4.5 Surface Polariton-Like s-Polarized Waveguide Modes inSwitchable Dielectric Thin Films on Polar Crystals

Nikolai Christian Passler, Andreas Heßler, Matthias Wuttig, Thomas Taubner, andAlexander Paarmann

This publication (Passler et al., AOM 2020, 8, 1901056 [VI]) presents a model system consisting of aPCM on a polar crystal substrate that supports simultaneously p-polarized SPhPs and s-polarized,polariton-like waveguide modes. By switching the crystal phase of the PCM, the work demonstratesvolatile control over both modes with a switching contrast that far exceeds typical values in the eldof modulated nanophotonics [113–116]. The concept is visualized in the cover picture of the specialissue Polaritons in Nanomaterials reprinted on the following page [117]. The supporting informationis reprinted in appendix A.4.

Most importantly, this work delineates a path to resolve two aspects of SPhPs that are disadvantageousfor nanophotonic applications: Firstly, introducing s-polarized waveguide modes with polariton-like properties provides a way to circumvent the polarization bottleneck of polaritons [118], andsecondly, employing a switchable PCM as waveguide core drastically improves the usually verylimited tunability of surface polaritons [119]. As such, the here presented approach extends theversatility of surface polaritons, providing a complementary building block for actively tunable,omnipolarized nanophotonic applications.

The experimental results for this work were obtained using the Otto geometry implementing thewhite-light interferometry [X], and the theoretical calculations are based on the transfer matrixformalism [II] (reprinted on page 45), which are the key experimental and theoretical tools developedin the course of this thesis.

Author contributions

A.P. and T.T. originated the concept. A.H. performed the sample growth. The manuscript was writtenby N.C.P., A.H. and A.P., with all authors assisting in the proof-reading and preparation for nalsubmission. The measurements were performed by N.C.P., and the results were analyzed by N.C.P.and A.P. The theoretical calculations were completed by N.C.P. The experimental design and projectmanagement were provided by T.T. and A.P.

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4.5 Surface Polariton-Like s-Polarized Waveguide Modes

95

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Chapter 4 Publications

96

4.6 Layer-Resolved Absorption in Anisotropic Heterostructures

4.6 Layer-Resolved Absorption of Light in ArbitrarilyAnisotropic Heterostructures

Nikolai Christian Passler, Mathieu Jeannin, and Alexander Paarmann

This publication (Passler et al., PRB 2020, 101, 165425 [VII]) presents a formalism to describe thelayer-resolved transmittance and absorption in layered heterostructures of arbitrarily anisotropic,birefringent, and absorbing media, capable of treating light of any polarization at oblique incidenceonto any number of layers. An online version of the computer code programmed in Matlab [19] andPython [120] has been published as open source.

The idea of this work emerged during the development of the erratum to the transfer matrix formalism[III] (reprinted on page 59). The resulting algorithm describes any linear light-matter interaction inlayered anisotropic media, and thus aims at a broad readership covering numerous research areasin nanophotonics and optics. The examples that are discussed in this publication are selected fromthree dierent research elds in order to highlight the generality of the formalism.

Author contributions

A.P. originated the concept. The manuscript was written by N.C.P. and M.J., with all authors assistingin the proof-reading and preparation for nal submission. N.C.P. devised the fundamental frameworkof the formalism, and N.C.P. and A.P. developed the nal algorithm. Design and computations of thenanophotonic device structures were completed by N.C.P., M.J. and A.P. The project managementwas provided by A.P.

go to list of publications

97

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CHAPTER 5PerspectivesIn recent years, the growing research interest in polaritons in nanostructures has triggered nu-merous advances in the eld of nanophotonics, including single-molecule sensing with SERS [7],plasmon-driven ultra-short electron point sources [91], highly directional thermal emission [121], orfemtosecond optical polarization switching with an ENZ-based perfect absorber [122]. Within thesedevelopments, phonon polaritons have emerged as a low-loss, versatile platform for nanophotonicapplications in the IR [14], enabling novel devices such as hyper-lenses based on hyperbolic materials[16, 68], phonon polariton-enhanced electroluminescence for ecient electronic cooling [123], andelectrically pumped phonon-polariton lasers [124].

As has been demonstrated in the previous chapter, the investigations performed in this thesis form partof these recent prosperous advances in nanophotonics, contributing to the development of moderndevices based on phonon polaritons in heterostructures of arbitrarily anisotropic materials. Based onthe presented advances, this chapter discusses the perspectives of the implemented techniques byoutlining several possible pathways for future developments in polariton-driven nanophotonics.

Hyperbolic materials have recently shifted into the spotlight of scientic interest. These materialsfeature a hyperbolic isofrequency surface, enabling a huge photonic density of states with arbitrarilylarge in-plane momenta [125]. These unusual properties lead to a variety of unique phenomena,such as sub-wavelength imaging [16, 46, 47] or slow light [126–128]. However, natural hyperbolicmaterials such as SiO2 or hBN only feature hyperbolicity in narrow spectral windows that are xed bythe intrinsic material parameters [129]. Alternatively, hyperbolic metamaterials (HMMs) composedof alternating ultra-thin layers of dierent materials oer a platform for engineering materials witha user-dened hyperbolic optical response [130], such as the XHs discussed in section 2.2.2 and inpublication [IX].

On a dierent note, doped semiconductors such as cadmium oxide (CdO) featuring plasmon polaritonshave recently been discussed in parallel with polar crystals supporting phonon polaritons. DopedCdO thin-lms support low-loss plasmonic modes in a broad spectral range in the IR, providing acomplementary polaritonic excitation at phonon polariton frequencies. In particular, Runnerstromet al. have demonstrated the strong coupling of a SPP to a plasmonic ENZ mode in a doped CdOthin lm [131], complementary to the strong coupling of a bulk SPhP to a thin-lm ENZ phononpolariton as discussed in the publication reprinted on page 65 [V]. By strongly coupling to an ENZmode, the resulting CdO plasmon polaritons combine long propagation distances with high spatialconnement, while featuring broad frequency tunability by controlling the electron concentrationvia photo-injection.

Doped semiconductors such as CdO also oer new routes in the eld of HMMs. Composed of alter-nating doped and undoped semiconductors, such semiconductor HMMs are, complementary to XHs,

111

Chapter 5 Perspectives

promising candiates for hyperbolic nanophotonic applications [62–64]. These metallic/dielectric mul-tilayer HMMs support so-called volume plasmon polaritons (VPPs), arising as a collective excitationcomposed of coupled SPPs supported in each metallic thin-lm [132–136]. Very recently, VPPs indoped semiconductor HMMs have been shown to feature high quality factors across a large frequencyrange in the IR [63], providing a potential platform for low-loss, IR optoelectronic devices.

The excitation of VPPs in doped semiconductor HMMs is typically realized via grating coupling[63, 64]. So far, however, these systems have only been investigated in the linear optical regime.Experiments investigating the non-linear optical response are currently in the planning stage andwill be performed in the near future, providing new insights for the thorough understanding of VPPsin HMMs and their potential for nanophotonic applications.

While HMMs oer more structural exibility than natural hyperbolic materials, the extraordinaryaxis in layered HMMs lies by design along the surface normal, while in-plane anisotropy cannot beachieved. Biaxial single crystals, on the other hand, feature three distinct dielectric permittivitiesalong the principal crystal axes, and thus can provide in-plane anisotropy with hyperbolic frequencybands for any crystal cut. An example of such a natural in-plane hyperbolic material is molybdenumtrioxide (MoO3). This material supports azimuth-dependent, hyperbolic phonon polaritons withnovel polaritonic properties, as has been highlighted in the publication reprinted on page 97 [VII].

One step beyond biaxial crystals, another intriguing material class are monoclinic crystals, which arecurrently on the verge of attracting broad attention in the nanophotonics community. In search ofnew materials for next-generation semiconductor power devices, β-gallium oxide (β-Ga2O3) has beendiscovered as a promising candidate, enabling applications as transparent electrodes [137] and high-performance power devices [138]. Only recently, the complicated dielectric permittivity tensor ofmonoclinic β-Ga2O3 has been described by Schubert et al., containing non-zero o-diagonal elementsthat do not vanish under orientation along the crystal axes [139]. In a subsequent study, couplingbetween the various LO phonon modes and free charge carriers in doped β-Ga2O3 was reported,forming so-called longitudinal phonon plasmon coupled modes with non-trivial directionality [140].

The phonon polariton modes supported by a monoclinic crystal such as β-Ga2O3, however, havenot yet been investigated1. The constructed Otto geometry setup enables such a study, and theexperiments on β-Ga2O3 are already programmed for the near future. Additionally, in order to accessthe dependence of the phonon polariton modes on azimuthal rotation, the Otto geometry setup hasbeen equipped with the possibility to rotate the sample while being mounted. Considering the richoptical response from IR spectroscopic ellipsometry [140], the planned pilot investigations on phononpolaritons in β-Ga2O3 bear great potential for the development of future nanophotonic devicesbased on monoclinic crystals, such as non-trivial azimuth dependency of the polariton dispersion, ordirection-dependent coupling to polaritons in adjacent media.

As has been demonstrated in the research conducted in this thesis, employing the developed transfermatrix formalism for the theoretical computation of the linear optical response, the electric elddistribution, and the layer-resolved absorption and transmittance enables thorough insight into theoptical properties of the experimentally investigated sample. The above mentioned experimentalperspectives employ layered systems of anisotropic materials, and therefore can be analogouslycorroborated by theoretical calculations. By scanning various parameters such as lm thicknesses,

1to the best of my knowledge

112

incident angle, azimuthal angle, and frequency, the optical response of the system can be systemati-cally analyzed, allowing to optimize the sample structure for the experiments, as well as predict theobservable polariton modes and the associated nanophotonic phenomena.

In general, the transfer matrix formalism allows to study all dielectric resonances – evanescent orradiative – ranging from electronic intersubband-transitions and IR-active phonons, over excitons andplasma resonances, up to interband transitions and core level excitations, covering any multilayersystem of materials with arbitrarily anisotropic permittivity. Therefore, going one step beyondcorroborating experimental data, the transfer matrix formalism resembles the ideal theoretical toolfor predictive studies of yet undiscovered nanophotonic materials and devices, as has been exemplarilydemonstrated in the most recent publication (reprinted on page 97 [VII]).

Another intriguing eld of research explores surface magnon polaritons (SMPs) in anisotropicantiferromagnets, where applying an external magnetic eld provides straight-forward, active controlover the supported magnon polaritons [141, 142]. Furthermore, hybrid plasmon-magnon polaritonsin graphene-antiferromagnet heterostructures have been demonstrated, opening new possibilitiesfor nanophotonic applications by combining the realms of photons, magnons, and plasmons [143].The optical response of magnetic resonances is dictated by the magnetic permeability µ, and foranisotropic antiferromagnets at terahertz frequencies, µ is a 3× 3 tensor with non-zero o-diagonalelements (analogous to the dielectric permittivity tensor, see Eq. 2.1.2). However, the transfer matrixformalism as it is derived in this work only handles materials with isotropic permeability µ. Therefore,incorporating the possibility to include media of fully anisotropic permeability would considerablyamplify the application spectrum of the formalism, allowing to investigate SMPs in antiferromagnetsas well as their interaction with SPPs and SPhPs in yet unexplored anisotropic heterostructures ofmagnetic, polar dielectric, and plasmonic material layers.

In the last three years in the course of this thesis, the combination of the Otto geometry and thetransfer matrix formalism has proven fruitful for the investigation of polaritons in anisotropiclayered heterostructures. As the above selection of perspectives demonstrates, this developedexperimental-theoretical tool box bears great potential for future investigations in modern polariton-based nanophotonics that is far from being exhausted.

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CHAPTER 6ConclusionThis thesis provides an extensive, detailed experimental and theoretical study of phonon polaritonsin layered heterostructures of arbitrarily anisotropic media. The ndings of various new fundamentalphenomena of phonon polaritons as well as the developed versatile experimental and theoreticaltools constitute a signicant contribution to the scientic eld, advancing the development of novelpolariton-based nanophotonic technologies.

As concept, this work promotes multilayer systems of various dierent materials as a platformfor designable polaritonic materials with user-dened characteristics and functionalities. Thisconcept is demonstrated on the basis of various heterostructures composed of SiC, AlN, GaN, GST,MoO3, molybdenum disulde (MoS2), Au, and hBN, revealing numerous novel phenomena such aspropagating ENZ modes, strongly coupled phonon polaritons, polariton-like s-polarized waveguidemodes, and azimuth-dependent hyperbolic phonon polaritons.

Experimentally, these systems were investigated by means of reectance and SHG spectroscopy inthe home-built Otto geometry setup, using the FHI FEL as excitation source. The ndings includethe rst SHG measurements of propagating SPhPs on a bulk SiC crystal, the rst experimentalproof of the immense optical eld enhancement of the Berreman mode in ultra-thin AlN lms,and the demonstration of the exceptional tuning capabilities of polaritons in a switchable GST/SiCheterostructure.

On the theoretical side, this work has developed a 4× 4 transfer matrix formalism for the descriptionof light-matter interaction in multilayer systems of arbitrarily anisotropic media, producing a versatile,robust framework that is implemented in an open-access computer program. The capabilities ofthe formalism are demonstrated by the accurate description of the experimental data as well as byvarious predictive theoretical studies, promoting yet unexplored nanophotonic device structures withnovel functionalities such as azimuth-dependent hyperbolic polariton tunneling or layer-selectiveexciton absorption.

As a summary, this work constitutes a comprehensive study of the physics of phonon polaritonsexcited with light in polar dielectric heterostructures. By implementing a new experimental andtheoretical methodology, novel phenomena of phonon polaritons in heterostructures and their uniqueinteraction nature were discovered, boosting the development of polariton-based technologies. Fur-thermore, this thesis lays out perspectives on how to use the developed experimental and theoreticalmethods for a number of future studies, bearing great potential to further advance the eld of infrarednanophotonics.

115

List of PublicationsIN. C. Passler, I. Razdolski, S. Gewinner, W. Schöllkopf, M. Wolf, and A. Paarmann, “Second-HarmonicGeneration from Critically Coupled Surface Phonon Polaritons”, ACS Photonics 4, 1048–1053 (2017).

IIN. C. Passler and A. Paarmann, “Generalized 4 x 4 matrix formalism for light propagation in anisotropicstratied media: study of surface phonon polaritons in polar dielectric heterostructures”, Journal of theOptical Society of America B 34, 2128 (2017).

IIIN. C. Passler and A. Paarmann, “Generalized 4 x 4 matrix formalism for light propagation in anisotropicstratied media: study of surface phonon polaritons in polar dielectric heterostructures: erratum”, Journalof the Optical Society of America B 36, 3246 (2019).

IVN. C. Passler, C. R. Gubbin, T. G. Folland, I. Razdolski, D. S. Katzer, D. F. Storm, M. Wolf, S. De Liberato, J. D.Caldwell, and A. Paarmann, “Strong Coupling of Epsilon-Near-Zero Phonon Polaritons in Polar DielectricHeterostructures”, Nano Letters 18, 4285–4292 (2018).

VN. C. Passler, I. Razdolski, D. S. Katzer, D. F. Storm, J. D. Caldwell, M. Wolf, and A. Paarmann, “SecondHarmonic Generation from Phononic Epsilon-Near-Zero Berreman Modes in Ultrathin Polar Crystal Films”,ACS Photonics 6, 1365–1371 (2019).

VIN. C. Passler, A. Heßler, M. Wuttig, T. Taubner, and A. Paarmann, “Surface Polariton-Like s-PolarizedWaveguide Modes in Switchable Dielectric Thin Films on Polar Crystals”, Advanced Optical Materials 8,1901056 (2020).

VIIN. C. Passler, M. Jeannin, and A. Paarmann, “Layer-resolved absorption of light in arbitrarily anisotropicheterostructures”, Physical Review B 101, 165425 (2020).

VIIII. Razdolski, N. C. Passler, C. R. Gubbin, C. J. Winta, R. Cernansky, F. Martini, A. Politi, S. A. Maier, M.Wolf, A. Paarmann, and S. De Liberato, “Second harmonic generation from strongly coupled localized andpropagating phonon-polariton modes”, Physical Review B 98, 125425 (2018).

IXD. C. Ratchford, C. J. Winta, I. Chatzakis, C. T. Ellis, N. C. Passler, J. Winterstein, P. Dev, I. Razdolski,J. R. Matson, J. R. Nolen, J. G. Tischler, I. Vurgaftman, M. B. Katz, N. Nepal, M. T. Hardy, J. A. Hachtel,J. C. Idrobo, T. L. Reinecke, A. J. Giles, D. S. Katzer, N. D. Bassim, R. M. Stroud, M. Wolf, A. Paarmann, andJ. D. Caldwell, “Controlling the Infrared Dielectric Function through Atomic-Scale Heterostructures”, ACSNano 13, 6730–6741 (2019).

XK. Pufahl, N. C. Passler, N. B. Grosse, M. Wolf, U. Woggon, and A. Paarmann, “Controlling nanoscaleair-gaps for critically coupled surface polaritons by means of non-invasive white-light interferometry”,Applied Physics Letters 113, 161103 (2018).

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126

Appendices

127

APPENDIX ASupporting Information to Publications

A.1 Supporting Information: Second Harmonic Generationfrom Critically Coupled Surface Phonon Polaritons

Nikolai Christian Passler, Ilya Razdolski, Sandy Gewinner, Wieland Schöllkopf, MartinWolf, and Alexander Paarmann

129

Supporting InformationSecond-Harmonic Generation from Critcally

Coupled Surface Phonon Polaritons

Nikolai Christian Passler, Ilya Razdolski, Sandy Gewinner, Wieland Schollkopf,Martin Wolf, and Alexander Paarmann∗

Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany

E-mail: alexander.paarmann@fhi-berlin.mpg.de

S1 Experimental

The experimental setup was schematicallyshown in Fig. 1 (a) of the main text. TheOtto-type prism coupling is realized by plac-ing a triangular prism (KRS5, 25mm high,25mm wide, angles of 30, 30, and 120, Ko-rth Kristalle GmbH) on a motorized mount infront of the sample. Details on the implem-tation are shown in Fig. S1. Three motorizedactuators (Newport TRA12PPD) are used topush (against springs) the prism away from thesample, which is mounted on a plate (25mm di-ameter) equipped with three mN force sensors(Honeywell FSS1500SSB). Each force sensorindicates the point of contact, allowing for par-allel alignment of the sample and the prismback surface. The motorized actuators allowfor continuous tuning of the air gap width d.

We note that macroscopic protrusions andnon-flatness of the sample can lead to strainbetween sample and prism, resulting in a lowerlimit of air gap width (typically ∼ 1.5 µm) andslight nonlinearities of the actual air gap widthand the actuator readback values, specificallyat close approach. Therefore, we fitted the val-ues of d from the reflectivity data, using theprocedure described in Sec. S3, the results ofwhich are in qualitative (but not quantitative)agreement with the actuator readback values.Gap width scans of the signals are typicallyperformed by starting at closest achievable ap-

proach, followed by defined steps (identical forall three actuators) away from the sample.

Given the large refractive index of KRS5 ofnKRS5 ≈ 2.4, external angles differ respectivelywhen going away from θext = 30, i.e., the in-ternal angle θ inside the prism is θ = 30 +arcsin(sin(θext − 30)/nKRS5). This arrange-ment allows for internal angles from below thetotal internal reflection cutoff at θcutoff ≈ 25

to about θ = 52, covering the majority of theSPhP dispersion. We note that using more con-venient semi-cylindrical prisms results in stronghorizontal focusing, increasing angular spreadof the beam and thereby much degraded mo-mentum resolution. Additionally, this horizon-tal focusing distorts the beam to an ellipticalshape making it harder to focus onto small de-tectors.

We use an infrared FEL1 as tunable, narrow-band, and p-polarized excitation source. Wefocus the beam mildly (f ≈ 60 cm, EdmundOptics) onto the sample surface to minimizeangular spread of the beam. Details on the IR-FEL are given elsewhere.1 In short, the electrongun is operated at a micropulse repetition rateof 1 GHz with an electron macropulse durationof 10 µs and a macropulse repetition rate of10 Hz. The electron energy is set to 31 MeV,allowing to tune the FEL output wavelengthbetween ∼ 7− 18 µm (∼ 1400− 550 cm−1) us-ing the motorized undulator gap, providing ps-pulses with typical full-width-at-half-maximum

1

Appendix A Supporting Information to Publications

130

Figure S1: Practical implementation of theOtto-type prism coupling to surface polari-tons. Three motorized actuators push (againstsprings) the prism mount away from the sam-ple. When pulled back, the point of contact isdetected using three mN force sensors, allowingfor parallel alignment of the prism and sam-ple surfaces and continuous scanning of the gapwidth d.

(FWHM) of < 5 cm−1.Reflected fundamental and SHG beams are

split using a 7 µm long-pass filter (LOT). Thereflectivity is detected with a home-built pyro-electric sensor (Eltec 420M7). The SHG signalis detected with a liquid nitrogen cooled mer-cury cadmium telluride/indium antimony sand-wich detector (Infrared Associates), after block-ing the fundamental light using 5 mm of MgF2

and a 7 µm short-pass filter (LOT). Addition-ally, intrinsic harmonics in the FEL beam areblocked prior to the setup using a 7 µm anda 9 µm long-pass filter (both LOT). We alsomeasure reference power on a single shot ba-sis to minimize the impact of shot-to-shot fluc-tuations of the FEL using the reflection off athin KBr plate placed in the beam prior to thesetup. This reference signal is also detected bya pyro-electric detector, and we normalize allsignals to this reference signal on a single-shotbasis, i.e. for each macropulse. The spectralresponse function of the setup is determined

by measuring the reflectivity for very large gapwidth d ≈ 40 µm, i.e. in total reflection acrossthe full spectral range, for each incidence an-gle. All spectra are normalized for these refer-ence data. Finally, we note that changing theincidence angle requires detection realignment,giving rise to an experimental uncertainty forthe signal magnitudes of 10-20% between dif-ferent incidence angles. This has no influenceon the reflectivity, since the reflectivity refer-ence is acquired for each incidence angle. How-ever, no such reference is available for the SHG,such that confident interpretation of the abso-lute SHG intensities between different incidenceangles is not given.

S2 Transfer Matrix Ap-

proach

For all calculations of reflectivity, local fields,and transmittance, a 4 × 4 - matrix formalismwas developed capable of treating any numberof layers, each described by a dielectric tensor ε.In contrast to other approaches,2–4 our formal-ism is numerically stable and handles isotropicand anisotropic materials simultaneously, in-cluding the substrate.

The multilayer surfaces are chosen to extendperpendicular to z and the incident wave im-pinges in the xz-plane with wavevector ~ki =ωc(ξ, 0, qi) in layer i. The x-component ξ =√εinc sin(θ) is conserved throughout the com-

plete multilayer system, and qi is the unit-free, layer-dependent z-component. In anymedium, Maxwell’s equations yield four inde-pendent plane-wave solutions characterized byqij (j = 1..4), which are determined as shownby Berreman.2 In order to avoid instability anddiscontinuities, the four solutions have to be or-dered unambiguously, which is done based onthe work of Li et al..5

In continuation, the treatment of singularitiesof Xu et al.6 is employed in order to generalizethe formalism to any isotropic or anisotropicmedium, yielding the electric field components~Eij = (γij1, γij2, γij3) of the four eigenmodes jin layer i. With these components, a matrixAi is formulated (see equation 24 in ref.6), rep-

2

A.1 SI: SHG from Critically Coupled SPhPs

131

FEL frequency ω (1/cm)800 900 1000 800 900 1000

0

0.2

0.4

0.6

0.8

1

Refl

ecti

vit

y

0

0.2

0.4

0.6

0.8

1

(a) (b)

(c) (d)

expsim

2.4 µm3.7 µm6.1 µm8.7 µm12.3µmθ = 26.8° θ = 28.8°

2.2 µm3.4 µm4.8 µm5.7 µm8.2 µm

2.2 µm2.8 µm3.9 µm4.9 µm6.1 µm

1.2 µm1.7 µm2.3 µm2.8 µm4.0 µmθ = 30.3° θ = 35.1°

Figure S2: Experimental (solid lines) and calculated (dot-dashed lines) reflectivity spectra of 6H-SiC in the Otto configuration, shown for four different incidence angles θ (a,b,c,d), each with a setof five selected air gap widths d (see legends).

resenting Maxwell’s boundary conditions at theinterface i − 1, i. The ~E-fields are then con-nected by the interface matrix Li as follows:

~Ei−1 = A−1i−1Ai

~Ei ≡ Li~Ei. (S1)

The propagation through layer i is describedby a diagonal matrix Pi with elements e−i

ωcqijdi ,

where j = 1..4 is the nth diagonal element. Thetransfer matrix ΓN of the complete multilayersystem reads

ΓN = L1P1L2P2...LNPNLN+1. (S2)

Finally, reflection and transmission coefficientsfor p-polarization are calculated in terms ofΓN components as shown by Yeh.3

S3 Fitting Procedure

The reflectivity data were fitted employing thematrix formalism, and convolving the resultsto account for finite broadening of the inci-

dence angle and FEL frequency, ∆θ = 0.2

and ∆ω = 4.4 cm−1, respectively. For eachincidence angle, the data were normalized by areference spectrum taken at large gap width d(≈ 40 µm). This accounts for the spectral re-sponse function of the setup, while additionallyslow FEL power drifts are accounted for in thefitting procedure by a multiplication factor a forevery individual spectrum (0.98 < a < 1.04).

The data sets of all four incidence angles θwere fitted globally, where θ, ∆θ, ∆ω, and the6H-SiC parameters such as TO and LO frequen-cies and damping were global fitting param-eters, while gap widths d and factors a wereadjusted individually. Thereby, ∆θ was fittedonly for the two smaller angles, and ∆ω for theother two. This is justified due to the shapeof the SPhP dispersion, being more sensitive tothe incidence angle but less to the frequency atsmaller θ, and vice versa for larger angles.

By fitting all four data sets in several itera-tions, the global parameters converged commonvalues. In a final run, all data sets were fitted

3

Appendix A Supporting Information to Publications

132

Normalized gap d/dcrit

0.5 1 1.5 2

SP

hP r

eso

nan

ce a

mpl

itu

de

0

0.2

0.4

0.6

0.8

1

40°

26°

26°

40° ReflectivitySHG (norm.)

θ = 30.5°θ = 35.7°

θ = 29.1°θ = 26.9°

Refl. expRefl. simSHG expSHG sim

Normalized gap d/dcrit

0.5 1 1.5 2

(a) (b)

Figure S3: (a) Experimental (symbols) and simulated (lines) SPhP resonance amplitudes in thereflectivity (circles, solid lines) and SHG (diamonds, dashed lines) as a function of the gap width dnormalized to the critical gap width dcrit of the respective incidence angle, and (b) simulated curvesfor a larger set of incidence angles.

again using averages of the global parameters,yielding the curves and parameters shown in fig.S2.

S4 Normalization to Criti-

cal Gap Width

The amplitude of the SPhP resonance (see Fig.3 (a,b) of the main text) depends on both theincidence angle θ and gap width d. We have nu-merically determined the critical gap width dcritas a function of θ, allowing for renormalizationof the gap width d to dcrit at corresponding in-cidence angle. This is illustrated in Fig. S3 (a)where we plot the experimental and simulatedamplitudes of the SPhP resonance in the reflec-tivity and SHG, respectively, against the nor-malized gap. For better clarity of the trends,we also show the simulated data for a larger setof incidence angles in (b). All spectra fall backonto a unversial curve; the reflectivity SPhPamplitude peaks at critical coupling conditions,while the SHG intensity drops to < 20% at dcritas compared to close gap due to inefficient SHGout-coupling in the Otto geometry. The trendin the SHG resonance amplitudes for increasingincidence angle is a result of multiple compet-ing effects: angular and spectral dispersion ofthe out-coupling efficiency, see section S5, spec-tral dispersion of the nonlinear suseptibility,7 aswell as increasing field enhancements (see Fig. 4

of the main text), as we move along in the SPhPdispersion with larger incidence angles.

S5 Calculation of the Second-

Harmonic Intensity

The calculations of the second-harmonic in-tensity are largely based on previous work ofSHG spectroscopy using free space excitation.7

In order to correctly describe the multilayer(KRS5/air/SiC) system, we needed to replacethe Fresnel transmission coefficients with ap-propriate expressions derived from the transfermatrix approach. Particular care is needed forthe transmission of the outgoing SHG, as willbe discussed below.

We first calculate the nonlinear polarization~PNL at 2ω generated in SiC, before projectingit onto the appropriate mode at 2ω responsiblefor the out-coupling of the SHG into the farfield. Therefore, we rewrite Eq. 2 of the maintext as:

I(2ω) ∝ |(Tb ~ESiC(2ω)) · ~PNL(2ω)/∆k|2 (S3)

with

~PNL(2ω) ∝ (χ(2)(−2ω;ω, ω) ~E2SiC(ω))|2. (S4)

We use the dispersion of the nonlinear suscep-tibility tensor χ(2)(−2ω;ω, ω) of 6H-SiC as ex-

4

A.1 SI: SHG from Critically Coupled SPhPs

133

tracted in our previous work.7,8 The local fields~E2SiC(ω) on the SiC side of the air–SiC interface

are calculated as follows.First, we evaluate the transfer matrix Γ for

a specific configuration of incidence angle, fre-quency and air gap width, employing the algo-rithm presented in Sec. S2. Then, the in-planefield component Ex,SiC can be directly calcu-lated from the transfer matrix Γ as the trans-mission coefficient for p-polarized light:3

Ex,SiC =Γ[3, 3]

Γ[1, 1]Γ[3, 3]− Γ[1, 3]Γ[3, 1](S5)

where Γ[i, j] denote the (i, j) components of the4 × 4 matrix (Eq. S2). The out-of-plane com-ponent is determined using Maxwell’s equation5×H = εdE

dt:

Ez,SiC = −qxεxqzεz

Ex,SiC , (S6)

with qx(z) and εx(z) the in-plane (out-of-plane)components of the wave vector and diagonal di-electric tensor of SiC, respectively. With thisinformation, Eq. S4 can now be evaluated.

The out-coupling of the SHG is accounted foras follows. First, we calculate the transfer ma-trix Γ at 2ω and, importantly, also 2k‖. Then,we invert Γ(2ω) to describe the backward prop-

agation. Finally we project ~PNL onto the re-spective mode propagating from SiC back intothe prism, and calculate the SHG intensity:

I(2ω) =1

|∆k|2∣∣(~ESiC(2ω) · ~PNL(2ω)

)×

×|Eprism(2ω)||ESiC(2ω)| |

2, (S7)

where Ex,SiC(2ω) ≡ 1 and Ez,SiC(2ω) is calcu-lated using the expression analogous to Eq. S6.The last factor represents the transmission co-efficient Tb for the second-harmonic field acrossthe air gap. Ex,prism(2ω) is evaluated analogousto Eq. S5, however, employing [Γ(2ω)]−1 as thetransfer matrix, and Ez,prism(2ω) is again cal-culated using Eq. S6.

References

(1) Schollkopf, W.; Gewinner, S.; Junkes, H.;Paarmann, A.; von Helden, G.; Bluem, H.;Todd, A. M. M. The new IR and THzFEL facility at the Fritz Haber Institutein Berlin. Proc. SPIE 2015, 9512, 95121L

(2) Berreman, D. W. Optics in Stratified andAnisotropic Media: 4x4-Matrix Formula-tion. Journal of the Optical Society ofAmerica 1972, 62, 502

(3) Yeh, P. Electromagnetic propagation inbirefringent layered media. Journal of theOptical Society of America 1979, 69, 742

(4) Lin-Chung, P. J.; Teitler, S. 4x4 Ma-trix formalisms for optics in stratifiedanisotropic media. Journal of the OpticalSociety of America A 1984, 1, 703

(5) Li, Z.-M.; Sullivan, B. T.; Parsons, R. R.Use of the 4x4 matrix method in the op-tics of multilayer magnetooptic recordingmedia. Applied Optics 1988, 27, 1334

(6) Xu, W.; Wood, L. T.; Golding, T. D.Optical degeneracies in anisotropic lay-ered media: Treatment of singularities ina 4x4 matrix formalism. Physical ReviewB 2000, 61, 1740–1743

(7) Paarmann, A.; Razdolski, I.; Gewin-ner, S.; Schollkopf, W.; Wolf, M. Effects ofcrystal anisotropy on optical phonon res-onances in midinfrared second harmonicresponse of SiC. Physical Review B 2016,94, 134312

(8) Paarmann, A.; Razdolski, I.; Mel-nikov, A.; Gewinner, S.; Schollkopf, W.;Wolf, M. Second harmonic generationspectroscopy in the Reststrahl bandof SiC using an infrared free-electronlaser. Applied Physics Letters 2015, 107,081101

5

Appendix A Supporting Information to Publications

134

A.2 SI: Strong Coupling of ENZ Phonon Polaritons

A.2 Supporting Information: Strong Coupling ofEpsilon-Near-Zero Phonon Polaritons in Polar DielectricHeterostructures

Nikolai Christian Passler, Christopher R. Gubbin, Thomas Graeme Folland, Ilya Razdolski,D. Scott Katzer, David F. Storm, Martin Wolf, Simone De Liberato, Joshua D. Caldwell,

and Alexander Paarmann

135

Supporting Information:Strong Coupling of Epsilon-Near-Zero Phonon Polaritons in Polar DielectricHeterostructures

Nikolai Christian Passler,1, a) Christopher R. Gubbin,2 Thomas Graeme Folland,3 Ilya Razdolski,1 D. ScottKatzer,4 David F. Storm,4 Martin Wolf,1 Simone De Liberato,2 Joshua D. Caldwell,3, 4 and AlexanderPaarmann1, b)1)Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6,14195 Berlin,Germany2)School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ,United Kingdom3)Vanderbilt Institute of Nanoscale Science and Engineering, 2301 Vanderbilt Place, PMB 350106, Nashville,TN 37235-01064)US Naval Research Laboratory, 4555 Overlook Avenue SW, Washington DC 20375,USA

(Dated: May 29, 2018)

S1. EXPERIMENTAL

The experimental setup is sketched in Fig. 2a of themain text. The Otto geometry is implemented usingthree motorized actuators (Newport TRA12PPD) to con-trol the position of the triangular coupling prism (KRS5,25mm high, 25mm wide, angles of 30, 30, and 120,with the 120 edge cut-off, Korth Kristalle GmbH) rela-tive to the sample. The motors push the prism againstsprings away from the sample and allow for continuoustuning of the air gap dgap. The 120 prism edge is cut-offparallel to the backside in order to perpendicularly cou-ple in the collimated light from a stabilized broadbandtungsten-halogen light source (Thorlabs SLS201L). Theback-reflected spectrum (registered with Ocean Optics,NIRQuest512) carries interferometric information aboutthe air gap, allowing to quantify dgap by means of themodulation period, and to achieve parallel alignment ofprism and sample by optimizing the contrast of the spec-tral modulations. The absolute read-out of dgap is lim-ited to maximally ∼ 60 μm due to the resolution of thespectrometer, while gap sizes below 1 μm could be mea-sured in principle, but are typically not feasible becauseof macroscopic protrusions and non-flatness of the sam-ple.

At external incidence angles θext away from 30,the high refractive index of KRS5 (nKRS5 ≈ 2.4)leads to strong refraction, resulting in an internalincidence angle θ inside the prism of θ = 30 +arcsin (sin (θext − 30) /nKRS5). In this configuration,angles from below the critical angle of total internal re-flection (θcrit ≈ 25) up to about 52 are accessible, cov-ering in-plane momenta across and far beyond the anti-crossing of the strongly coupled modes (52 correspondsto k‖/k0 = 1.86). Different in-plane momenta can be

a)Electronic mail: passler@fhi-berlin.mpg.deb)Electronic mail: alexander.paarmann@fhi-berlin.mpg.de

accessed via the incidence angle θ by rotating the entireOtto geometry, thus allowing for mapping out the com-plete dispersion curves experimentally.

As excitation source we employ a mid-infrared freeelectron laser (FEL) as tunable, narrowband, and p-polarized excitation source. Details on the FEL are givenelsewhere1. In short, the electron gun is operated at a mi-cropulse repetition rate of 1 GHz with a macropulse dura-tion of 10 μs and a repetition rate of 10 Hz. The electronenergy is set to 31 MeV, allowing to tune the FEL outputwavelength between ∼ 7 − 18 μm (∼ 1400 − 550 cm−1,covering the spectral ranges of the SiC and AlN rest-strahlen bands) using the motorized undulator gap, pro-viding ps-pulses with typical full-width-at-half-maximum(FWHM) of < 5 cm−1. The macropulse and micropulseenergies are ∼ 30 mJ and ∼ 10 μJ, respectively.

The reflectivity signal is detected with a pyroelectricsensor (Eltec 420M7). A second sensor is employed tomeasure the power reflected off a thin KBr plate placedprior to the setup in the beam. This signal is used as ref-erence for normalization of all reflectivity signals, and ismeasured on a single shot basis in order to minimize theimpact of shot-to-shot fluctuations of the FEL. The spec-tral response function of the setup is determined by mea-suring the reflectivity at large gap widths dgap ≈ 40 μm,where the prism features total internal reflection acrossthe full spectral range. This is done for each incidenceangle individually, and all spectra are normalized to theirrespective response function.

S2. CRITICAL COUPLING CONDITIONS OF THESTRONGLY COUPLED MODES

The Otto geometry grants experimental control overthe excitation efficiency by means of the tunability ofthe air gap width dgap. At a certain critical gap dcrit,the excitation efficiency is optimal2, and we emphasizethat dcrit is significantly different for the two modes con-sidered (dENZ

crit < dSPhPcrit ). Therefore, in a single spec-

Appendix A Supporting Information to Publications

136

2

860 900 940

-1

1

600 700 800 900 1000

-200

0

200

400

600Re(ε )SiC

Im(ε )SiC

Re(ε )AlN

Im(ε )AlN

TOωAlNTOωSiC ωLO

AlN ωLOSiC

bulk SiC SPhP

bulk AlN SPhP

AlN thin film SPhPs

AlN ENZ

SωSiCωSAlN

Die

lect

ric

Per

mit

tivi

ty ε

Strong Coupling

a

Frequency ω (1/cm)1.04 1.11 1.18 1.321.25

In-plane momentum k /k0

Fre

quen

cy ω

(1/c

m)

700

750

800

850

900

0.5

0

1

Refl

ecti

vity

k SPhP AlN

k ENZ SiC

d = 10 nmSiCb

airultra-thin SiC

AlN

Figure S1. Frequency ranges of phonon polaritons in AlN and SiC. a The real (solid line) and imaginary (dashedline) part of the dielectric function ε of SiC (blue) and AlN (green) are shown. At the respective optically active TO phononfrequency ωTO, they feature a strong resonance, while at the LO frequencies ωLO, the real part of ε crosses zero, as illustratedin the inset. In a bulk material, a single SPhP is supported in the frequency range between ωTO and the cut-off frequency ωS

(blue and green color bars). In a thin AlN film (dAlN < λ/2), two thin film polaritons emerge with the asymmetric mode havinga dispersion at frequencies above ωS (orange color bar), see main text. In the ultra-thin limit of an AlN film (dAlN < λ/50), theasymmetric mode disperses close to ωAlN

LO where Re(εAlN ) ≈ 0, hence being an ENZ mode. The strong coupling demonstratedin the main text arises between the SiC substrate SPhP and the AlN ENZ mode. b Reflectivity map of the inverse structureof an ultrathin SiC film on top of an AlN substrate at frequencies around the TO frequency of SiC ωSiC

TO ≈ 796 cm−1. Thetheoretical dispersions of the asymmetric SiC ENZ mode (light blue line) and the substrate AlN SPhP (dark blue line) areplotted on top, exhibiting a crossing point analogous to the structure discussed in the main text. Interestingly, the dispersionin the combined structure also splits up into two branches, featuring an avoided crossing.

trum with fixed gap width, the two modes cannot beexcited equally efficiently, resulting in drastically differ-ent resonance dip depths. The data shown in Fig. 2bof the main text, however, were taken at in-plane mo-mentum corresponding to the strong coupling conditions.Note that the two reflectivity dips in each spectrum areof equal depth, evidencing the hybrid nature of the twomodes. The tunability and reproducibility of the gapin our setup is therefore a substantial tool for the mea-surement of polariton resonances, revealing informationabout their coupling efficiency.

S3. STRONGLY INTERACTING MODES INMATERIALS WITH OVERLAPPING RESTSTRAHLENBANDS

In Fig. S1a, the dielectric functions ε of AlN and SiCare shown. They feature a strong resonance at the respec-tive TO frequency ωTO due to its infrared activity, whileat the non-IR-active LO frequency ωLO, the real part ofε crosses zero. For the strong coupling to arise, it is nec-essary that the reststrahlen bands of the two materialsoverlap, i.e. in the case of an AlN film on SiC, that theωLO frequency of AlN is located inside the reststrahlenband of SiC. Only then, the ENZ mode supported by theultra-thin AlN film lies at frequencies where the SiC sub-strate supports a SPhP, allowing the modes to stronglyinteract, as demonstrated in the main text.

A similar scenario is obtained for the inverse structure

not discussed in the main text, i.e. an ultra-thin SiCfilm on top of AlN. The SiC film, as shown in Fig. 1a ofthe main text, supports not only the symmetric thin filmmode close to ωLO, but additionally an asymmetric thinfilm mode lying close to the TO frequency ωTO. In the in-verse structure, this lower mode disperses at frequencieswhere the AlN substrate supports a SPhP, and these twomodes can also strongly interact. In Fig. S1b, we show acalculated reflectivity map covering the dispersion rangeof the substrate AlN SPhP (theoretical curve plotted indark blue) and the SiC asymmetric thin film mode closeto ωSiC

TO = 796 cm−1 (light blue). In the coupled system,these two modes strongly interact and form an avoidedcrossing, resulting in two new dispersion branches.

We note that two distinctions have to be made com-paring this system with the structure discussed in themain text. First, the dielectric function of SiC close toωTO is highly dispersive, and therefore only slightly awayfrom ωTO the mode loses its ENZ character. Secondly,the material is strongly absorptive at ωTO and hence thethin-film mode is very likely lossy, hence limiting its prac-ticability. However, such strong coupling in the presenceof significant losses is out of the scope of this work.

S4. DEVIATION FROM STRONG COUPLING FORLARGER FILM THICKNESSES

As stated in the main text, for film thicknesses exceed-ing d & 50 nm, the interaction of the modes is not well-

A.2 SI: Strong Coupling of ENZ Phonon Polaritons

137

3

a b c

Frequency ω(1/cm)850 900 950

z P

osit

ion

1 μ

m11

0 nm

6.58

μm

air

AlN

SiC

0

2

4

6

|E |

|E

|

xx0

/

020406080

|Im(r )|p

020406080

|Im(r )|p

Hopfield model

900 1000 1100 1200800

850

900

950

Fre

quen

cy ω

(1/

cm)

In-plane momentum k (1/cm)

upper mode

lower mode

800

850

900

950

Fre

quen

cy ω

(1/

cm)

900 1000 1100 1200In-plane momentum k (1/cm)

d = 50 nmAlN d = 110 nmAlN

Figure S2. Deviation from strong coupling for larger film thicknesses. a,b Reflectivity maps of the AlN/SiC structurefrom the main text in the Otto geometry for dAlN = 50, 110 nm, respectively, featuring the two dispersion branches of theinteracting modes with avoided crossing. On top, the dispersion curves extracted from the strong coupling Hopfield modelare plotted (blue dashed lines), exhibiting increasing deviation from the numerical calculations for larger film thicknesses. cIn-plane (x) field enhancement Ex as a function of frequency and z-position plotted at the avoided crossing and for an AlN filmthickness of dAlN = 110 nm, where the layer thicknesses are not to scale with respect to each other. The prism lies adjacent toair at the top, but is not shown here. At the corresponding resonance frequency of the two modes, strong field enhancementscan be observed, which peak at the air/AlN and the AlN/SiC interface, demonstrating that the two modes are not hybridizedanymore.

described by our strong coupling model anymore. Thiscan be seen both in the mode dispersion predicted by Eq.2 of the main text and in the field profiles. However, whilethe latter originates from the approximation of the ENZmode dispersion, the former provides a physical limit ofstrong coupling.

The dispersion curves for d = 50, 110 nm are shownin Fig. S2a and b, respectively, plotted on top of nu-merically calculated reflectivity maps demonstrating agood agreement. However, the increasing deviation ofthe theoretically predicted dispersion (blue dashed lines)for larger film thicknesses is evident. The reason for thatis that in our model, the ENZ mode dispersion is assumedto be constant (900 cm−1), which is sufficiently accuratefor dAlN . 50 nm. However, for increasing film thicknessthe dispersion of the ENZ mode evolves into a thin filmmode, see Fig. 1a of the main text. The deviation of thepredicted mode dispersion for larger film thicknesses isthus given by the limitation of our model.

Nonetheless, for increasing film thicknesses, a bareAlN SPhP evolves from the lower mode, which is al-ready largely decoupled from the substrate SiC SPhP atd ≈ 110 nm. The upper mode, on the other hand, devel-ops towards a pure SiC/AlN interface phonon polariton(IPhP), spectrally confined by the ωAlN

LO frequency andthe SiC cut-off frequency ωSiC

S .

In order to show this disintegration into two distinctmodes and thus the breakdown of strong coupling fordAlN & 50 nm, we analyze the electric field distributionsat film thickness dAlN = 110 nm. In Fig. S2c, the nor-malized in-plane (x) electric field amplitude is plotted as

a function of frequency and z-position (perpendicular tothe interfaces) along the three media (air, AlN, and SiC).Clearly, the two modes appear as strong peaks in thefield enhancement at their corresponding resonance fre-quency. But even though the incidence angle was chosensuch that the modes are excited at the avoided crossing(θ = 28.5, k‖/k0 = 1.13), the field distributions of the

two modes differ strongly. The lower mode (∼ 865 cm−1)features the strongest Ex at the air/AlN interface, whilethe upper mode (∼ 915 cm−1) exhibits its maximum fieldenhancement at the SiC/AlN interface, clearly demon-strating that the hybridization of the strong coupling islost. This can be attributed to the fact that the lossesof AlN grow linearly and thus faster than g0 ∝

√d with

increasing film thickness d. Furthermore, the localizationof the modes at the two different interfaces stays the sameon both sides of the avoided crossing (not shown), em-phasizing their respective bare AlN SPhP and SiC/AlNIPhP nature across the entire dispersion. In fact, bothmodes feature a similar in-plane field enhancement whilethe extreme out-of-plane field enhancement is stronglyreduced (not shown), meaning that the ENZ character-istics observed in an ultrathin film (d < 50 nm) are notpresent anymore.

REFERENCES

1W. Schollkopf, S. Gewinner, H. Junkes, A. Paarmann, G. vonHelden, H. Bluem, and A. M. M. Todd, 9512, 95121L (2015).

2N. C. Passler, I. Razdolski, S. Gewinner, W. Schollkopf, M. Wolf,and A. Paarmann, ACS Photonics 4, 1048 (2017).

Appendix A Supporting Information to Publications

138

A.3 SI: SHG from Phononic ENZ Berreman Modes

A.3 Supporting Information: Second Harmonic Generationfrom Phononic Epsilon-Near-Zero Berreman Modes inUltrathin Polar Crystal Films

Nikolai Christian Passler, Ilya Razdolski, D. Scott Katzer, D. F. Storm, Joshua D. Caldwell,Martin Wolf, and Alexander Paarmann

139

Supporting Information:

Second Harmonic Generation from Phononic Epsilon-Near-Zero Berreman Modes in

Ultrathin Polar Crystal Films

Nikolai Christian Passler,1, a) I. Razdolski,1 D. Scott Katzer,2 D. F. Storm,2 Joshua D.

Caldwell,2, 3 Martin Wolf,1 and Alexander Paarmann1, b)

1)Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6,14195 Berlin,

Germany

2)US Naval Research Laboratory, 4555 Overlook Avenue SW, Washington DC 20375,

USA

3)Vanderbilt University, Institute of Nanoscale Science and Engineering,

2201 West End Ave, PMB 350106, Nashville, TN 37235-0106,

USA

(Dated: 24 April 2019)

a)Electronic mail: passler@fhi-berlin.mpg.deb)Electronic mail: alexander.paarmann@fhi-berlin.mpg.de

S1

Appendix A Supporting Information to Publications

140

0.4 0.6 0.8 1.0Inplane momentum k (1/μm)

600

650

700

750

800

850

900

950

Wav

enum

ber

ω(1

/cm

)

leaky evanescent

ωLOAlN

ωTOAlN

Berreman modeThin film polaritonSPhP

a

203001000

dAlN in nm

enil thgil

symmetric

antisymmetric

600

650

700

750

800

850

900

950

Wav

enum

ber

ω(1

/cm

)

leaky evanescent

symmetric

antisymmetric

203001000

dAlN in nm

ωLOAlN

ωTOAlN

Berreman modeThin film polaritonSPhP

enil thgil

0.4 0.8 1.2Inplane momentum k (1/μm)

0 1.4

ω real, k complexω complex, k realb

FIG. S1. Theoretical thin film polariton dispersions for either complex frequency or

in-plane momentum. Solving Eq. 1 of the main text for a complex frequency ω and real in-plane

momentum kx leads to the dispersions shown in a, while a real ω and complex kx yields the curves

in b. Depending on the chosen observable, one representation is better suited than the other1.

For instance, the complex frequencies representation is recommended for short pulse excitations,

while the complex wavevector representation should be used to analyze the local density of states

of a system2. For the here investigated air/AlN/SiC system, the complex ω representation (a)

is the adequate choice: Firstly, solving the system without damping and thus for real ω and real

kx, qualitatively, the same dispersion curves as shown in a are obtained3. Secondly, the leaky

Berreman mode resembles a virtual polariton mode, i.e. its fields are related to an energy flow

out of the system, which decreases with time4. As has been discussed by Kliewer and Fuchs4, the

correct choice for a standard virtual-mode treatment is the complex ω representation.

S2

A.3 SI: SHG from Phononic ENZ Berreman Modes

141

0 10 15 20 25 30-3

-2

-1

0

1

2

x (μm)

z (μ

m)

air

AlN

air

Re(E ) air

Re(E ) AlN

ω = 900.6 1/cm k = 0.2 1/μm

-3

-2

-1

0

1

2

x (μm)z

(μm

)

air

AlN

SiC

ω = 902.1 1/cm k = 0.2 1/μm

Re(E )air

Re(E )AlN

Re(E )SiC

a b

5 0 10 15 20 25 305

FIG. S2. Electric field distribution of the Berreman mode in a 1µm thick AlN slab. a

(b) Vector plots of the electric field distribution in an air/AlN/air (air/AlN/SiC) system of the

Berreman mode at an incidence angle of 21 (corresponding to an in-plane momentum of k =

0.2 µm−1 at ω = 900.6 cm−1 (ω = 902.1 cm−1), lying on the respective Berreman dispersion). The

fields of the Berreman mode (red arrows) perform a counterclockwise rotation in positive x-direction

characteristic for polaritonic modes. Please note that the real part is plotted in order to capture the

propagation characteristics. The imaginary part features the same behavior with a phase difference

of π/2. The absolute in-plane fields are conserved at the interfaces as required by Maxwells

boundary conditions, while the absolute out-of-plane fields experience a strong enhancement in

the AlN film. The film thickness was chosen to be 1 µm for illustration purposes. In thinner films,

the fields qualitatively exhibit the same distribution, but feature much more pronounced out-of-

plane field amplitudes. Interestingly, the substrate material (air in a, SiC in b) has no notable

effect on the field distribution of the Berreman mode in the AlN film.

S3

Appendix A Supporting Information to Publications

142

0.30 0.35 0.40 0.45 0.50 0.55 0.60890

900

910

920

930

940

950

Inplane momentum k (1/μm)

Wav

enum

berω

(1/c

m)

203001000

dAlN in nm air/AlN/airair/AlN/SiC

enil thgil

ωTOAlN

°58 = θ

0 1 2 3 4 5

900

910

920

930

940

950

AlN film thickness d (μm)AlN

Wav

enum

berω

(1/c

m)

numericaltheoretical

θ = 85°

900 910 920 930 9400.0

0.2

0.4

0.6

0.8

1.0

Refl

ecta

nce

Wavenumber ω (1/cm)

0.02 1.250.05 1.50.1 1.750.2 2.00.3 2.50.5 3.00.75 4.01.0 5.0

d (μm)AlN

a b

air/AlN/SiC

890

FIG. S3. Comparison of theoretically and numerically determined Berreman disper-

sion. a Berreman dispersions for three different AlN film thicknesses dAlN supported in an

air/AlN/air (red curves) and an air/AlN/SiC (green curves) system calculated with Eq. 1 from the

main text. The dispersion in a freestanding AlN is flatter for smaller in-plane momenta than for

the AlN on a SiC substrate, but exhibits a steeper upwards bending close to the light line. b Plot

of the frequencies of the Berreman dispersion as a function of dAlN in the air/AlN/SiC system at

an incidence angle of 85, calculated numerically with the transfer-matrix method (blue triangles)

and theoretically as shown in subfigure a (green circles), showing perfect agreement. The inset

shows the Berreman reflectance dip as a function of dAlN . The frequencies of the minima were

extracted to obtain the numerical data (blue triangles).

S4

A.3 SI: SHG from Phononic ENZ Berreman Modes

143

AlN film thickness d (μm)AlNWavenumber ω (1/cm)900 920 940 960 980 1000

0

1

2

3

4

|E z |

inS

iC

101005001000

dAlN (nm)

ωLOSiC

ba

isotropic SiCanisotropic SiC

c

d

e

0

20

40

60

0

0.4

0.8

1.2

020406080

|E |

|E

|

zz0

/|E

| |

E

|z

z0/

in AlN

in air

0 0.5 10

0.1

0.2

0.3

0.4900

910

920

0

20

40

60

ωB

erre

man

Re(ε

)

AlN

|E |

|E

|

zz0

/

in AlN

f

g

0 0.5 1

α(|E

|

)zre

fl. in air

00.20.40.60.8

|E

| |

E

|z

z0/

refl

.

FIG. S4. Out-of-plane field enhancement Ez at the interfaces of the air/AlN/SiC sys-

tem. a Ez field enhancement as a function of frequency in isotropic (dashed lines) and anisotropic (solid

lines) SiC for four different AlN film thicknesses dAlN at an incidence angle of 60. At the LO frequency

of anisotropic SiC ωSiCLO , the field enhancement reaches a maximum of ∼ 3.5 due to the zero-crossing of the

SiC dielectric function, while the peak at higher frequency arises from a resonance condition in the Fresnel

transmission coefficient (also observed in isotropic SiC)5. Both features are mainly independent of dAlN

because AlN is transparent in this frequency range. The Ez peaks give rise to the SHG signals reported in

Fig. 2a of the main text (compare also with5). b-d show the Ez field enhancement of the Berreman mode

at resonance frequency, the corresponding resonance frequency ωBerreman, and the real part of the dielectric

function of AlN at ωBerreman, respectively, as a function of AlN film thickness dAlN and at an incidence

angle of 60. Clearly, the Ez field enhancement (b) experiences a strong increase in the limit of ultrathin

films (dAlN < 200 nm), which can be attributed to the proximity of the Berreman resonance frequency (c)

to the ALN LO frequency at 900 cm−1, where the dielectric function (d) crosses zero. For thicker films, on

the other hand, the Berreman mode disperses at frequencies where Re(εAlN ) > 0, yielding a much weaker

field enhancement. It is thus seen that only for ultrathin films the Berreman mode, due to its ENZ nature,

exhibits the reported strong field enhancement. e-g show the Ez field enhancement at the air/AlN inter-

face in AlN and air, and the reflected wave together with its phase, respectively, as a function of AlN film

thickness dAlN at fixed frequency ω = 900 cm−1 and at an incidence angle of 60. Surprisingly, also at fixed

ENZ frequency, the field enhancement in AlN (e) decreases with increasing film thickness, which, following

Eq. 3 of the main text, happens because of a decreasing Ez field in air (f). The latter behavior arises

due to the following reasons. The total field in air is the difference of the incident and the reflected field,

Ez = Einc.z − Erefl.

z , where Einc.z ≡ 1 and its phase α(Einc.

z ) = 0. The reflected field Erefl.z has a minimum

value of ∼ 0.4 at dAlN = 30 nm due to the Berreman resonance. Its phase, however, approaches 90 in the

limit of a vanishing AlN film, but drops rapidly to values close to 0 for increasing film thickness. As a

consequence, for a phase difference of 0, the reflected field and the incoming field interfere destructively at

the air/AlN interface, leading to a small total field amplitude for increasing film thickness both in air and in

AlN. In the ultrathin film limit, on the other hand, the significant phase difference of incoming and reflected

wave leads to a sizable total field in air, and thus, in combination with the ENZ condition (see Eq. 3 of the

main text), to the strong Ez field enhancement inside the AlN film.

S5

Appendix A Supporting Information to Publications

144

0 200 400 600 800 10000.0

0.1

0.2

0.3

0.4

0.5

0.6

AlN film thickness d (μm)AlN

ω = ωTO,oAlN

|E |

|E

|

xx0

/|E

| |

E

|z

z0/

600 700 800 900 10000

2

4

6

8

10

12

14

TO

,eω

AlN

TO

,oω

AlN

LO

,eω

AlN

LO

,oω

AlN

TO

,eω

SiC

TO

,oω

SiC

LO

,eω

SiC

LO

,oω

SiC

SiCAlN

effe

ctiv

e S

HG

sou

rce

dep

th δ

(μ

m)

FEL wavenumber (1/cm)ω

a b

FIG. S5. Contributions to the SHG yield at the AlN TO frequency. a Effective SHG

source depth δ = 1/∆k for bulk SiC and bulk AlN calculated by means of the wavevector mismatch

∆k2 =∣∣∣~kSHG − ~k1,in − ~k2,in

∣∣∣2.6 At the ordinary TO frequency, δ features a minimum of ∼ 200 nm

whereas in the range of the LO frequencies, δ reaches maximum values of a couple of micrometer.

Thus, in a bulk sample, a lot more volume contributes to the SHG signal at the LO than at the TO,

whereas in a very thin film (d < 200 nm), the effective source depth is equal at both frequencies

and given by the film thickness. Therefore, in an ultrathin AlN film, the TO SHG signal is stronger

compared to the LO signal than for a bulk crystal. b In-plane Ex and out-of-plane Ez fields at

the ordinary AlN TO frequency (670.8 cm−1) as a function of AlN film thickness. As stated in the

main text, in a bulk crystal the field enhancement is strongly suppressed at the TO frequency6,

leading to a suppressed SHG yield despite the strong χ(2) resonance. In an ultrathin film, however,

this suppression is reduced, as can be seen by the sharp increase of the Ex field for dAlN < 100 nm.

S6

A.3 SI: SHG from Phononic ENZ Berreman Modes

145

REFERENCES

1R. W. Alexander, G. S. Kovener, and R. J. Bell, “Dispersion Curves for Surface Electro-

magnetic Waves with Damping,” Physical Review Letters 32, 154–157 (1974).

2A. Archambault, T. V. Teperik, F. Marquier, and J. J. Greffet, “Surface plasmon Fourier

optics,” Physical Review B 79, 195414 (2009).

3K. L. Kliewer and R. Fuchs, “Optical Modes of Vibration in an Ionic Crystal Slab Including

Retardation. I. Nonradiative Region,” Physical Review 144, 495–503 (1966).

4K. L. Kliewer and R. Fuchs, “Optical Modes of Vibration in an Ionic Crystal Slab Including

Retardation. II. Radiative Region,” Physical Review 150, 573–588 (1966).

5A. Paarmann, I. Razdolski, A. Melnikov, S. Gewinner, W. Schollkopf, and M. Wolf, “Second

harmonic generation spectroscopy in the Reststrahl band of SiC using an infrared free-

electron laser,” Applied Physics Letters 107, 081101 (2015).

6A. Paarmann, I. Razdolski, S. Gewinner, W. Schollkopf, and M. Wolf, “Effects of crystal

anisotropy on optical phonon resonances in midinfrared second harmonic response of SiC,”

Physical Review B 94, 134312 (2016).

S7

Appendix A Supporting Information to Publications

146

A.4 SI: Surface Polariton-Like Waveguide Modes

A.4 Supporting Information: Surface Polariton-Likes-Polarized Waveguide Modes in Switchable DielectricThin Films on Polar Crystals

Nikolai Christian Passler, Andreas Heßler, Matthias Wuttig, Thomas Taubner, andAlexander Paarmann

147

Copyright WILEY-VCH Verlag GmbH & Co. KGaA, 69469 Weinheim, Germany, 2019.

Supporting Information

for Adv. Optical Mater., DOI: 10.1002/adom.201901056

Surface Polariton-Like s-Polarized Waveguide Modes in Switchable Dielectric Thin Films on Polar Crystals

Nikolai Christian Passler,* Andreas Heßler, Matthias Wuttig,Thomas Taubner, and Alexander Paarmann*

Appendix A Supporting Information to Publications

148

Supporting Information:

Surface Polariton-Like s-Polarized Waveguide Modes in Switchable Dielectric

Thin-Films on Polar Crystals

Nikolai Christian Passler,1, a) Andreas Heßler,2 Matthias Wuttig,2 Thomas Taubner,2

and Alexander Paarmann1, b)

1)Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, 14195 Berlin,

Germany

2)Institute of Physics (IA), RWTH Aachen University, 52056 Aachen,

Germany

(Dated: 21 August 2019)

a)Electronic mail: passler@fhi-berlin.mpg.deb)Electronic mail: alexander.paarmann@fhi-berlin.mpg.de

S1

A.4 SI: Surface Polariton-Like Waveguide Modes

149

150 200 250 300 350 4000

2

4

6

8

d (nm)aGST

TF

OM SPhP

Waveguide Mode

FIG. S1. Tuning figure of merit (TFOM) of the SPhP and the waveguide mode in

air/GST/SiC. TFOM = δω/FWHM as a function of GST film thickness daGST calculated from

reflectance simulations of the air/GST/SiC system in Otto geometry (with KRS5 prism as incident

medium) at an incident angle of 30 with the parameters given in the Methods section of the main

text. For daGST ≈ 150− 170 nm, the SPhP features a maximal TFOM, while the waveguide mode

reaches a maximal predicted TFOM of ∼ 9 at daGST ≈ 210− 230 nm. For both modes, the change

in the TFOM as a function of daGST primarily arises from the frequency shift δω (compare Figure

S3 c and f), which varies strongly as the modes shift to lower frequencies towards ωSiCTO .

S2

Appendix A Supporting Information to Publications

150

1 2 3 4 5

1000

1500

2000

2500

3000

1 2 3 4 50

500

1000

1500

2000

Wav

enum

ber

(1/c

m)

a

In-plane Momentum k/k0

Im(Φ

a-G

ST

/air

s)

00.20.40.60.8

Wav

enum

ber

(1/c

m)

b

ωTOSiC

ωLOSiC

300500800

d (nm)GST

Re(ε

)a-

GS

T

In-plane Momentum k/k0

air/a-GST/SiC air/c-GST/SiC

SiC

a-GST

air

Re(ε )SiC

ωSiCTO

Re(ε )a-GST

(i)

(ii)

(iii)

X

X

6

Re(ε

)c-

GS

T

FIG. S2. Theoretical dispersion of the m=0 order s-polarized waveguide mode in a-

GST and c-GST on SiC. a Imaginary part of the phase difference Im(Φsa-GST/air) upon reflection

at the a-GST/air interface calculated with Eq. 3 from the main text1. Where Im(Φsa-GST/air) > 0,

no waveguide modes can be supported, which is the case for k/k0 >√

Re(εa-GST). Therefore,

region (ii) is ruled out, even though the SiC/a-GST interface would allow a waveguide mode

(Im(ΦsSiC/a-GST) = 0), see Fig. 4a of the main text. Regions marked with an X are excluded because

there, Im(ΦsSiC/a-GST) > 0. Thus, the air/a-GST/SiC system only supports waveguide modes in

regions (i) and (iii). b Comparison of the dispersion of an m=0 s-polarized waveguide mode in an

air/a-GST/SiC (solid lines) and an air/c-GST/SiC system (dashed lines) for three different GST

film thicknesses. The upper momentum boundary of the dispersions for both systems is given by

the refractive index (n =√ε) of the respective GST phase (dashed black lines). The dispersions in

a-GST, which has neglectable absorption in the shown frequency range (Im(εa-GST) = 0), approach

the upper momentum boundary asymtotically, whereas the dispersions in c-GST experience a red-

shift in proximity to the upper momentum boundary due to significant absorption (Im(εc-GST) = 9).

For small in-plane momenta (1 < k/k0 < 2), however, which is the range studied in this work,

the dispersions in the two GST phases are qualitatively the same, where in c-GST the modes are

red-shifted compared to a-GST.

S3

A.4 SI: Surface Polariton-Like Waveguide Modes

151

0 1 2 3

0

2

4

6

8

800850900950

1000

a-GST film thickness (μm)

m=0m=1m=2

a

c

p-pols-pol

FE

max

ω

(1/c

m)

mod

e

Waveguide M

odes

SP

hP

Waveguide M

odes

100

5001000

5000104

δ (n

m)

b

a-GST

0 1 2 3800850900950

1000

c-GST film thickness (μm)

m=0m=1m=2

d

f

p-pols-pol

FE

max

ω

(1/c

m)

mod

e

100

5001000

5000104

δ (n

m)

e

c-GST

024 1 20

2000g

851 9851/cm 1/cm

0

2

4

6

8

024

h881

1/cm

0

2

4

6

8

1 20

9701/cm

0 500 1000 1500

FE

| Ex| /

| Ex

0|

FE

| Ey| /

| Ey

0|

z Position in a-GST (nm)

z Position in air and SiC (μm)

024

i 8971/cm

0

2

4

6

8

FE

| Ex| /

| Ex0

|

0

2

4

6

8

1 20

9631/cm

FE

| Ey| /

| Ey0

|

0.5 1 1.5 2 2.5 3z Position (μm)

0

0.5

1.0

1.5

2.0

2.5

3.0air SiCa-GST

m=3

FIG. S3. Higher-order waveguide modes in a-GST and c-GST on SiC. a,d Maximum

field enhancement FEmax for p- (blue curves) and s-polarization (red curves) as a function of a-

GST and c-GST film thickness dGST, respectively, at θ = 30. The respective penetration depths

δ into SiC are plotted in b,e, and c,f shows the corresponding mode frequencies ωmode. Above

certain dGST (marked by gray lines where the mode frequencies jump), however, the first-order

s- and p-polarized waveguide modes start being supported, appearing at the upper frequency

boundary. Noteworthily, the SPhP mode propagating at a single boundary smoothly fits into the

waveguide mode picture that dominates at larger dGST, and could even be seen as a continuation

of the p-polarized waveguide mode in the limit of vanishing film thicknesses. However, while all

higher-order p-polarized (m≥ 1) and all s-polarized waveguide modes are also supported in thin

films on dielectric materials, the SPhP requires the negative permittivity of the SiC substrate. In

summary, the reasons for a structure to support a SPhP or a waveguide mode are different, but

otherwise SPhPs and waveguide modes can be treated very similarly. The penetration depth δ

is a function of the SiC substrate permittivity, and is large for ε > 0 (above ωSiCLO = 970 cm−1)

while reaching a minimum of δ = 100 nm for mode frequencies close to ωSiCTO = 797 cm−1 (where

ε 0). For p-polarization, the field enhancement and the penetration depth experience a small,

sharp feature where the mode crosses the region between the LO ordinary (o) and extraordinary

(e) phonon frequencies ωSiCLO,o = 970 cm−1 and ωSiC

LO,e = 964 cm−1, being an effect of the SiC c-

cut uniaxial crystal anisotropy2. In comparison to a-GST, the high absorption of c-GST leads

to a significantly reduced field enhancement especially for the higher-order modes. Due to larger

internal losses, the critical coupling gap of c-GST (1.5 µm) where the modes are excited optimally

in the Otto geometry is smaller than for a-GST (3.7 µm). g-i Electric field profiles for p- and s-

polarized incident light across an air/a-GST/SiC system for a-GST films of 200, 1700, and 3200 nm

film thickness, respectively, excited in the Otto geometry. For da-GST = 1700 nm (h), the m = 1

waveguide modes for both polarizations are supported, featuring a node inside the a-GST film,

where for da-GST = 3200 nm (i), the m = 2 waveguide modes with two nodes inside the a-GST film

can be observed.

S4

Appendix A Supporting Information to Publications

152

REFERENCES

1P. K. Tien and R. Ulrich, “Theory of Prism-Film Coupler and Thin-Film Light Guides,”

Journal of the Optical Society of America 60, 1325 (1970).

2I. Razdolski, Y. Chen, A. J. Giles, S. Gewinner, W. Schollkopf, M. Hong, M. Wolf, V. Gi-

annini, J. D. Caldwell, S. A. Maier, and A. Paarmann, “Resonant Enhancement of Second-

Harmonic Generation in the Mid-Infrared Using Localized Surface Phonon Polaritons in

Subdiffractional Nanostructures,” Nano Letters 16, 6954–6959 (2016).

S5

A.4 SI: Surface Polariton-Like Waveguide Modes

153

List of Figures2.1.1 Drude model of the dielectric permittivity for gold . . . . . . . . . . . . . . . . . . . . . 82.1.2 Lorentz model of the dielectric permittivity for 6H-SiC, AlN, and GaN . . . . . . . . . 102.2.1 Dispersion relations of bulk phonon polaritons and the surface phonon polariton . . 132.2.2 Phonon polariton engineering using polar dielectric heterostructures . . . . . . . . . . 172.3.1 Second harmonic generation and the second-order susceptibility of SiC . . . . . . . . . 202.3.2 Field enhancement of phonon polaritons probed with second harmonic generation

spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Experimental methods for the optical excitation of surface polaritons . . . . . . . . . . 273.2.1 Otto geometry setup for reectance and SHG measurements . . . . . . . . . . . . . . . 313.2.2 Image of the home-made and designed 9-axis Otto geometry construction . . . . . . . 313.3.1 Sketch of the FHI FEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3.2 Pulse structure of the FHI FEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

155

List of Tables2.1.1 Material parameters of uniaxial 6H-SiC, AlN, and GaN . . . . . . . . . . . . . . . . . . . 10

3.3.1 Specications of the electron accelerator of the FEL at the FHI Berlin . . . . . . . . . . 333.3.2 Specications of the employed MIR FEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

157

List of Acronyms2D two-dimensional3D three-dimensional

AFM atomic force microscopeAlN aluminium nitrideAOM Advanced Optical MaterialsAPL Applied Physics LettersAu gold

β-Ga2O3 β-gallium oxideBPhP bulk phonon polariton

CdO cadmium oxide

ENZ epsilon-near-zero

FEL free electron laserFHI Fritz Haber Insitute

GaN gallium nitrideGST Ge3Sb2Te6

hBN hexagonal boron nitrideHMM hyperbolic metamaterialhPhP hyperbolic phonon polariton

IR infrared

JOSA B Journal of the Optical Society of America

KRS-5 thallium bromo-iodide

LO longitudinal opticalLSPhR localized surface phonon resonance

159

MIR mid-infraredMoO3 molybdenum trioxideMoS2 molybdenum disulde

n-shied refractive index-shiftedN.C.P. the author of this thesisNIR near-infrared

PCM phase-change materialPRB Physical Review BPTIR photothermal-induced resonance

s-SNOM scattering-type scanning near-eld optical microscopySERS surface-enhanced raman scatteringSFG sum frequency generationSH second harmonicSHG second harmonic generationSiC silicon carbideSiO2 quartzSMP surface magnon polaritonSPhP surface phonon polaritonSPP surface plasmon polariton

TFP thin-lm polaritonTO transverse optical

VPP volume plasmon polariton

WM waveguide mode

XH crystalline hybrid

160

AcknowledgementsI want to thank everybody involved in the development of this thesis. First of all, I thank Martin Wolfand Paul Fumagalli for supervising this work. As head of the PC Department at the FHI, Martin Wolfhas been very supportive and encouraging, and I thank him for nurturing a pleasant and fruitfulwork environment.

My special gratitude goes to Alexander Paarmann for being a particularly supportive mentor, whotriggered the ideas for this doctoral thesis, never hesitating to assist and encourage the ongoingprocess. I thank him deeply for cultivating the exceptionally cordial and positive working atmosphereand coherence in our Lattice Dynamics Group. To him goes a special thanks for attentively readingthe thesis and suggesting fruitful corrections.

I furthermore thank the Lattice Dynamics Group for the pleasant oce ambience, for the helpfuldiscussions and for the generally cheerful mood sprinkled with jokes and laughs. I highly appreciatethe sound mixture of responsibility and jollity, combining high productivity with interludes ofrening our frisbee skills in the late afternoon sun.

Another thanks goes to the Fritz Haber Institute for making this work possible in the rst place, andfor being such a marvelous place of work where I found a warm welcome and quickly felt at home.I also want to thank Wieland Schöllkopf and Sandy Gewinner for enabling the usage of the FreeElectron Laser, enduringly accompanying the numerous beamtimes until the early hours.

Moreover, I am very glad to have been part of the IMPRS Functional Interfaces in Physics andChemistry, and I want to thank everybody involved in running the school and organizing thestimulating meetings and lectures. I furthermore want to express my thanks to the IMPRS and thePC Department for facilitating my attendance at various inspiring conferences in Germany andabroad.

Furthermore, I want to deeply thank my parents and my sister for staying close and supportive,always believing in me and my choice to pursue a doctorate. Finally, my most profound gratitudegoes to my beloved wife Gabriela, for being the most beautiful and lovable life companion imaginable.This work would not have been possible without her, granting me the privilege of her unfailingencouragement, inspiring soulfulness and unlimited love.

161

AbstractThe eld of nanophotonics aims at understanding and harnessing light-matter interaction in structuresof dimensions far below the wavelength. By squeezing light into nanostructures, the local electricelds can be immensely enhanced, boosting the eciency of applications such as solar cells ormolecular sensing. Furthermore, nanophotonics facilitates the miniaturization of optical devices,pushing forward the development of modern communication technologies and all-optical integratedcircuitry.

The fundamental excitation driving nanophotonics is the surface polariton, arising in dierent typesdepending on the supporting material. A promising candidate for applications at infrared frequenciesis the surface phonon polariton (SPhP) supported by polar crystals. However, a SPhP on a single polarcrystal possesses several limitations that hinder the application in nanophotonic technologies.

This work implements layered heterostructures built from various dierent materials as a versatileplatform for phonon polariton nanophotonics, overcoming the limitations of a conventional SPhP. Bystudying a variety of dierent polar crystal heterostructures, novel polariton modes with intriguingcharacteristics are discovered, such as ultra-thin lm modes with immense eld enhancements,strongly coupled polaritons at epsilon-near-zero frequencies, and waveguide modes with polariton-like properties.

Furthermore, a new experimental and theoretical methodology is developed, enabling the systematic,extensive study of phonon polaritons in layered heterostructures of arbitrarily anisotropic media. Inthe conducted experiments, the phonon polaritons are excited via prism coupling in a home-builtOtto geometry, featuring full control over the excitation conditions with direct read-out of thecharacteristic air gap size between prism and sample. This unique setup allows to characterize apolariton dispersion at critical coupling conditions, yielding the discovery of new phonon polaritonmodes.

As an excitation source, the free electron laser at the Fritz Haber Institute is employed, allowing fornon-linear optical spectroscopy measurements at infrared phonon-polariton frequencies. The resultsinclude the rst observation of second harmonic generation from propagating SPhPs and from theultra-thin lm Berreman polariton, enabling experimental access to the eld enhancement of theexcited polariton mode.

The theoretical advances of this work comprise a transfer matrix formalism for the calculation of light-matter interaction in arbitrarily anisotropic layered heterostructures. The developed versatile androbust framework enables the simulation, analysis and prediction of polaritons and their properties inany multilayer system, constituting a signicant contribution to the eld of polaritonic nanophotonics.The formalism is implemented in an open-access computer program, facilitating future studiestowards phonon polariton-based technologies.

By discovering various new phenomena and implementing a new experimental and theoreticalmethodology with great potential for future investigations, this work constitutes a comprehensivestudy of phonon polaritons in polar dielectric heterostructures. Furthermore, this thesis lays outperspectives on how to use the developed experimental and theoretical methods for a number offuture studies, bearing great potential to further advance the eld of infrared nanophotonics.

163

KurzfassungDas Gebiet der Nanophotonik zielt darauf ab, die Wechselwirkung zwischen Licht und Materie inStrukturen mit Größenskalen weit unterhalb der Wellenlänge zu verstehen und zu nutzen. Durch dasEinbringen von Licht in Nanostrukturen können die lokalen elektrischen Felder immens überhöhtwerden, was die Ezienz von Anwendungen wie Solarzellen oder molekularer Sensorik verbessert.Darüber hinaus ermöglicht die Nanophotonik die Miniaturisierung optischer Bauelemente undtreibt so die Entwicklung moderner Kommunikationstechnologien und rein optischer integrierterSchaltkreise voran.

Die grundlegende Anregung, die die Nanophotonik vorantreibt, ist das Oberächenpolariton, das jenach Trägermaterial in verschiedenen Typen auftritt. Ein vielversprechender Kandidat für Anwendun-gen bei Infrarotfrequenzen ist das auf polaren Kristallen existierende Oberächenphononenpolariton(OPhP). Ein OPhP auf einem einzelnen polaren Kristall weist jedoch mehrere Einschränkungen auf,die die Anwendung in nanophotonischen Technologien behindern.

Diese Arbeit implementiert geschichtete Heterostrukturen aus verschiedenen Materialien als vielseit-ige Plattform für die Phononenpolaritonen-Nanophotonik und überwindet so die Einschränkungeneines herkömmlichen OPhP. Durch die Untersuchung einer Vielzahl verschiedener polarer Kristall-heterostrukturen werden neue Polaritonenmoden mit interessanten Eigenschaften entdeckt, wie etwaModen in ultradünnen Filmen mit immensen Feldverstärkungen, stark gekoppelte Polaritonen beiEpsilon-nahe-Null-Frequenzen und Wellenleitermoden mit polaritonenähnlichen Eigenschaften.

Darüber hinaus wird eine neue experimentelle und theoretische Methodik entwickelt, die die system-atische und umfassende Untersuchung von Phononenpolaritonen in geschichteten Heterostrukturenbeliebig anisotroper Medien ermöglicht. In den durchgeführten Experimenten werden die Phononen-polaritonen durch Prismenkopplung in einer selbstgebauten Otto-Geometrie angeregt, wobei dieAnregungsbedingungen vollständig gesteuert werden können und die charakteristische Luftspalt-größe zwischen Prisma und Probe direkt ausgelesen wird. Dieser einzigartige Aufbau ermöglicht dieCharakterisierung einer Polaritondispersion bei kritischen Kopplungsbedingungen, wodurch neuePhononenpolaritonenmoden entdeckt werden können.

Als Anregungsquelle wird der Freie-Elektronen-Laser am Fritz-Haber-Institut eingesetzt, der nicht-lineare optische Spektroskopiemessungen bei infraroten Phonon-Polariton-Frequenzen ermöglicht.Die Ergebnisse umfassen die erste Beobachtung der Erzeugung der zweiten Harmonischen vonpropagierenden OPhP und von dem Berreman-Polariton in ultradünnen Filmen, wodurch die experi-mentelle Einsicht in die Feldverstärkung der angeregten Polaritonenmode ermöglicht wird.

Die theoretischen Fortschritte dieser Arbeit umfassen einen Transfermatrix-Formalismus zur Berech-nung der Wechselwirkung zwischen Licht und Materie in beliebig anisotropen geschichteten Het-erostrukturen. Der entwickelte vielseitige und robuste Formalismus ermöglicht die Simulation,Analyse und Vorhersage von Polaritonen und ihren Eigenschaften in jedem beliebigen Mehrschicht-system, was einen wesentlichen Beitrag auf dem Gebiet der polaritonischen Nanophotonik darstellt.Der Formalismus ist in einem Open-Access-Computerprogramm implementiert, was zukünftigeStudien in Richtung Phonon-Polariton-basierten Technologien erleichtert.

Durch die Entdeckung verschiedener neuer Phänomene und die Implementierung einer neuenexperimentellen und theoretischen Methodik mit großem Potenzial für zukünftige Studien stellt dieseArbeit ein gründliches Werk über Phononenpolaritonen in polaren dielektrischen Heterostrukturendar. Darüber hinaus skizziert diese Arbeit Perspektiven für die Verwendung der entwickeltenexperimentellen und theoretischen Methoden für eine Reihe zukünftiger Studien, die ein großesPotenzial für die Weiterentwicklung des Bereichs der Infrarot-Nanophotonik bieten.

165