This work aims, in a rather contemplative manner phased into three related parts, to discuss two concepts indispensable for Hegel's entire philosophy of life and history: language and memory. In Part I, I examine the triple relation between language and thinking, thinking and reality, and reality and language. I argue that language shares a logical structure with thinking and reality, and is itself the performative principle (or acteur) of creating reality, being itself the externalizing tool of the movement of thinking without which any development of the Spirit would not be possible. Part II targets the concept of memory and its function within thinking and action of self-consciousness. It is argued that Hegel's language functions as the modern concept of discourse in terms of its agency in reality. Memory is understood as fundamentally entangled with matter, or the material objectivity that calls in memory to be named, i.e. posited in language. Memory is an interiorizing principle, language is the exteriorizing principle, both deeply rooted in the so-called night of the self of each spirit. I also discuss the subject-objective relation against the background of memory, before moving onto Part III which generally tackles the process of the self-expression of the Spirit in history, the distinction...

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques.

In dieser Dissertation wird erstmalig ein theoretisches Fundament zur Beschneiung virtueller Szenen entwickelt. Das theoretische Fundament wird als analytisches Modell in Form einer Diffusionsgleichung formuliert. Aus dem analytischen Modell lässt sich eine Gruppe von Algorithmen zur Beschneiung virtueller Szenen ableiten. Eingehende Voruntersuchungen zur allgemeinen Modellierung natürlicher Phänomene in der Computergraphik sowie eine Klassifikation der bestehenden Literatur über mathematische Schneemodellierung bilden den Anfang der Arbeit. Aus der umfassenden Darstellung der Eigenschaften von Schnee, wie er in der Natur vorkommt, ergeben sich die Grundlagen für die Modellbildung. Die Modellbildung fußt auf den grundlegenden Ansätzen der klassischen Mechanik und der statistischen Physik. Für die Beschneiung auf visueller Skala erweist sich der Diffusionsprozess als geeignete Beschreibung. Mit der Beschreibung lassen sich diffusiv Schneeoberflächen erzeugen. Der konkrete computergraphische Wert des theoretischen Fundaments wird anhand zweier Implementierungen exemplarisch dargestellt, und zwar in der Distanzfeldmethode und der Diffusionskernmethode. Die Ergebnisse werden mithilfe dreidimensionaler Rauschtexturen und Alpha-Masken an den Rändern fotorealistisch visualisiert.:1. Einleitung 7 2. Zentrale Beiträge dieser Arbeit 11 3. Natürliche Phänomenmodellierung in der Computergraphik 13 3.1. Die Rolle der computergraphisch modellierten Naturphänomene in der Informatik . . . . 14 3.2. Repräsentationsformen natürlicher Phänomene in der Computergraphik . . . . 16 3.3. Modellierungsmethoden im Überblick . . . . . . . . . . . . . . . . . . . . 20 3.3.1. Bildbasierte Methoden . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.2. Diskretisierungsbasierte Methoden . . . . . . . . . . . . . . . . . . 21 3.3.3. Kontinuumsbasierte Methoden . . . . . . . . . . . . . . . . . . . . 22 3.3.4. Modellreduktionsmethoden . . . . . . . . . . . . . . . . . . . . . . 23 3.3.5. Interaktionsmethoden . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4. Klassifikation der natürlichen Phänomene in der Computergraphik . . . . 25 3.4.1. Statische Phänomene . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4.2. Dynamische Phänomene . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4.3. Zusammengesetzte Phänomene . . . . . . . . . . . . . . . . . . . . 30 3.4.4. Sonstige natürliche Phänomene . . . . . . . . . . . . . . . . . . . . 32 3.5. Schlussfolgerung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4. Schnee in der Natur 35 4.1. Entstehung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2. Niederschlag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3. Akkumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4. Metamorphose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.5. Computergraphisch modellierbare Aspekte . . . . . . . . . . . . . . . . . . 40 5. Vorarbeiten zur computergraphischen Schneemodellierung 45 5.1. Modellierung statischer Aspekte . . . . . . . . . . . . . . . . . . . . . . . . 45 5.1.1. Optische Schneeeigenschaften . . . . . . . . . . . . . . . . . . . . . 45 5.1.2. Geometrische Form der Schneeoberflächen . . . . . . . . . . . . . . 46 5.1.3. Schnee als makroskopische Landschaftstextur . . . . . . . . . . . . 48 5.2. Modellierung dynamischer Aspekte . . . . . . . . . . . . . . . . . . . . . . 48 5.2.1. Schneefall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2.2. Schneeschmelze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2.3. Lokale Schneeumlagerung und Kompaktifizierung . . . . . . . . . . 50 5.2.4. Bisher nicht modellierte Aspekte . . . . . . . . . . . . . . . . . . . 51 5.3. Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6. Physikalische Methoden zur Darstellung von Materialflüssen und Phasengrenzen 55 6.1. Mikroskopische Modelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.1.1. Formale Schneecharakterisierung mit einer Vielteilchen-Hamilton- Funktion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.1.2. Statistische Formulierung der Vielteilchenbeschreibung . . . . . . . 57 6.2. Makroskopische Modelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2.1. Schneeverteilung als globales Strahlungsgleichgewicht . . . . . . . 59 6.2.2. Lokale stochastische Darstellung als getriebene Oberfläche . . . . . 61 6.2.3. Oberflächenentwicklung als Reaktionsdiffusion . . . . . . . . . . . 63 6.3. Zusammenfassung und Schlussfolgerung . . . . . . . . . . . . . . . . . . . 65 7. Eigenschaften und Lösungen von Diffusionsgleichungen 67 7.1. Das physikalische Prinzip der Diffusion . . . . . . . . . . . . . . . . . . . . 67 7.1.1. Grundlagen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.1.2. Diffusion auf Höhenfeldern . . . . . . . . . . . . . . . . . . . . . . 68 7.2. Mathematische Behandlung linearer Diffusionsgleichungen . . . . . . . . . 70 7.2.1. Konstruktion von allgemeinen Lösungen mittels Fundamentallösung 70 7.3. Analytische Lösungsbeispiele . . . . . . . . . . . . . . . . . . . . . . . . . 72 8. Computergraphische Erzeugung von Schneeoberflächengeometrien 75 8.1. Faltung als Grundprinzip der diffusiven Schneedeckenerzeugung . . . . . . 76 8.2. Datenstrukturen zur Darstellung von Schneedecken . . . . . . . . . . . . . 79 8.3. Darstellung mittels Distanzfeld . . . . . . . . . . . . . . . . . . . . . . . . 81 8.3.1. Details der Distanzfeldmethode . . . . . . . . . . . . . . . . . . . . 81 8.3.2. Diskussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.4. Darstellung als Diffusionsprozess . . . . . . . . . . . . . . . . . . . . . . . 86 8.4.1. Modelldetails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8.4.2. Diskussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.5. Erweiterung für Überhänge und Schneebrücken . . . . . . . . . . . . . . . 96 8.5.1. Brückenerzeugung . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.5.2. Überhangsberechnung . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.5.3. Ergebnisse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.5.4. Diskussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.6. Vergleich und Verallgemeinerbarkeit der Schneemodellierungsansätze . . . 101 9. Visualisierung virtueller Schneeoberflächen 103 9.1. Schneeoberfläche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.2. Schneeränder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 9.3. Diskussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 10.Zusammenfassung und Schlussfolgerungen 107 10.1. Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 10.2. Schlussfolgerungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 A. Beobachtungssammlung natürlicher Schneeformen 109 A.1. Randprofile und Stützflächenabhängigkeit . . . . . . . . . . . . . . . . . . 109 A.2. Verdeckung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 A.3. Glättung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A.4. Innenränder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 A.5. Brücken und Überhänge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 A.6. Nicht modellierte Aspekte . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 B. Sammlung virtuell beschneiter Szenen 125 B.1. Distanzfeldmethode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 B.2. Diffusionskernmethode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 B.3. Brückenbildung und Überhänge . . . . . . . . . . . . . . . . . . . . . . . . 130 Literaturverzeichnis 131 Abbildungsverzeichnis 143 Veröffentlichungen 151 / In this dissertation for the first time a theoretical foundation is developed for snow accumulation in virtual scenes. The theoretical foundation is formulated in an analytical model as diffusion equation. The analytical model leads to a group of algorithms for virtual snow accumulation. Comprehensive investigations for the modelling of natural phenomena in computer graphics in general are used to develop a method classification scheme. Another classification is given for an overview over the aspects of snow in the real world. This allows an efficient presentation of related literature on snow modelling. A new approach of snow modelling is then drawn from first principles of classical mechanics and statistical physics. Diffusion processes provide an efficient theoretical framework for snow accumulation. The mathematical structure of diffusion equations is discussed and demonstrated to be adequate to snow modelling in visual scales. The value of the theoretical foundation for computer graphics is demonstrated with two exemplary implementations, a distance field method and the diffusion kernel method. Results are visualized with 3D noise textures and alpha masks near borders delivering photorealistic snow pictures.:1. Einleitung 7 2. Zentrale Beiträge dieser Arbeit 11 3. Natürliche Phänomenmodellierung in der Computergraphik 13 3.1. Die Rolle der computergraphisch modellierten Naturphänomene in der Informatik . . . . 14 3.2. Repräsentationsformen natürlicher Phänomene in der Computergraphik . . . . 16 3.3. Modellierungsmethoden im Überblick . . . . . . . . . . . . . . . . . . . . 20 3.3.1. Bildbasierte Methoden . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.2. Diskretisierungsbasierte Methoden . . . . . . . . . . . . . . . . . . 21 3.3.3. Kontinuumsbasierte Methoden . . . . . . . . . . . . . . . . . . . . 22 3.3.4. Modellreduktionsmethoden . . . . . . . . . . . . . . . . . . . . . . 23 3.3.5. Interaktionsmethoden . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.4. Klassifikation der natürlichen Phänomene in der Computergraphik . . . . 25 3.4.1. Statische Phänomene . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.4.2. Dynamische Phänomene . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4.3. Zusammengesetzte Phänomene . . . . . . . . . . . . . . . . . . . . 30 3.4.4. Sonstige natürliche Phänomene . . . . . . . . . . . . . . . . . . . . 32 3.5. Schlussfolgerung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4. Schnee in der Natur 35 4.1. Entstehung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2. Niederschlag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3. Akkumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4. Metamorphose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.5. Computergraphisch modellierbare Aspekte . . . . . . . . . . . . . . . . . . 40 5. Vorarbeiten zur computergraphischen Schneemodellierung 45 5.1. Modellierung statischer Aspekte . . . . . . . . . . . . . . . . . . . . . . . . 45 5.1.1. Optische Schneeeigenschaften . . . . . . . . . . . . . . . . . . . . . 45 5.1.2. Geometrische Form der Schneeoberflächen . . . . . . . . . . . . . . 46 5.1.3. Schnee als makroskopische Landschaftstextur . . . . . . . . . . . . 48 5.2. Modellierung dynamischer Aspekte . . . . . . . . . . . . . . . . . . . . . . 48 5.2.1. Schneefall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2.2. Schneeschmelze . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2.3. Lokale Schneeumlagerung und Kompaktifizierung . . . . . . . . . . 50 5.2.4. Bisher nicht modellierte Aspekte . . . . . . . . . . . . . . . . . . . 51 5.3. Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6. Physikalische Methoden zur Darstellung von Materialflüssen und Phasengrenzen 55 6.1. Mikroskopische Modelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.1.1. Formale Schneecharakterisierung mit einer Vielteilchen-Hamilton- Funktion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.1.2. Statistische Formulierung der Vielteilchenbeschreibung . . . . . . . 57 6.2. Makroskopische Modelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.2.1. Schneeverteilung als globales Strahlungsgleichgewicht . . . . . . . 59 6.2.2. Lokale stochastische Darstellung als getriebene Oberfläche . . . . . 61 6.2.3. Oberflächenentwicklung als Reaktionsdiffusion . . . . . . . . . . . 63 6.3. Zusammenfassung und Schlussfolgerung . . . . . . . . . . . . . . . . . . . 65 7. Eigenschaften und Lösungen von Diffusionsgleichungen 67 7.1. Das physikalische Prinzip der Diffusion . . . . . . . . . . . . . . . . . . . . 67 7.1.1. Grundlagen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.1.2. Diffusion auf Höhenfeldern . . . . . . . . . . . . . . . . . . . . . . 68 7.2. Mathematische Behandlung linearer Diffusionsgleichungen . . . . . . . . . 70 7.2.1. Konstruktion von allgemeinen Lösungen mittels Fundamentallösung 70 7.3. Analytische Lösungsbeispiele . . . . . . . . . . . . . . . . . . . . . . . . . 72 8. Computergraphische Erzeugung von Schneeoberflächengeometrien 75 8.1. Faltung als Grundprinzip der diffusiven Schneedeckenerzeugung . . . . . . 76 8.2. Datenstrukturen zur Darstellung von Schneedecken . . . . . . . . . . . . . 79 8.3. Darstellung mittels Distanzfeld . . . . . . . . . . . . . . . . . . . . . . . . 81 8.3.1. Details der Distanzfeldmethode . . . . . . . . . . . . . . . . . . . . 81 8.3.2. Diskussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.4. Darstellung als Diffusionsprozess . . . . . . . . . . . . . . . . . . . . . . . 86 8.4.1. Modelldetails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8.4.2. Diskussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.5. Erweiterung für Überhänge und Schneebrücken . . . . . . . . . . . . . . . 96 8.5.1. Brückenerzeugung . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.5.2. Überhangsberechnung . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.5.3. Ergebnisse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 8.5.4. Diskussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.6. Vergleich und Verallgemeinerbarkeit der Schneemodellierungsansätze . . . 101 9. Visualisierung virtueller Schneeoberflächen 103 9.1. Schneeoberfläche . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 9.2. Schneeränder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 9.3. Diskussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 10.Zusammenfassung und Schlussfolgerungen 107 10.1. Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 10.2. Schlussfolgerungen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 A. Beobachtungssammlung natürlicher Schneeformen 109 A.1. Randprofile und Stützflächenabhängigkeit . . . . . . . . . . . . . . . . . . 109 A.2. Verdeckung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 A.3. Glättung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 A.4. Innenränder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 A.5. Brücken und Überhänge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 A.6. Nicht modellierte Aspekte . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 B. Sammlung virtuell beschneiter Szenen 125 B.1. Distanzfeldmethode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 B.2. Diffusionskernmethode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 B.3. Brückenbildung und Überhänge . . . . . . . . . . . . . . . . . . . . . . . . 130 Literaturverzeichnis 131 Abbildungsverzeichnis 143 Veröffentlichungen 151

From a physical perspective biological cells consist of active soft matter that exist in a thermodynamic state far from equilibrium. Not only in muscles but also during cell proliferation, wound healing, embryonic development, and many other physiological tasks, generation of forces on the scale of whole cells is required. To date, cellular contractions have been ascribed to adhesion dependent processes such as myosin driven stress fiber formation and the development of focal adhesion complexes. In this thesis it is shown for the first time that contractions can occur independently of focal adhesions in single suspended cells. To measure mechanical properties of suspended cells the Optical Stretcher – a dualbeam laser trap – was used with phase contrast video microscopy which allowed to extract the deformation of the cell for every single frame. For fluorescence imaging confocal laser scanning microscopy was employed. The ratio of the fluorescence of a temperature sensitive and a temperature insensitive rhodamine dye was utilized to determine the temperatures inside the optical trap during and after Optical Stretching. The rise in temperature at a measuring power of 0.7W turned out to be enough to open a temperature sensitive ion channel transfected into an epithelial cell line. In this way a massive Ca2+ influx was triggered during the Optical Stretcher experiment. A new setup combining Optical Stretching and confocal laser scanning microscopy allowed fluorescence imaging of these Ca2+ signals while the cells were deformed by optically induced surface forces, showing that the Ca2+ influx could be manipulated with adequate drugs. This model system was then employed to investigate the influence of Ca2+ on the observed contractions, revealing that they are partially triggered by Ca2+. A phenomenological mathematical model based on the fundamental constitutive equation for linear viscoelastic materials extended by a term accounting for active contractions allowed to quantify the activity of the measured cells. The skewness and the median of the strain distributions were shown to depend on the activity of the cells. The introduced model reveals that even in measurements, that seemingly are describable by passive viscoelasticity, active contractililty might be superimposed. Ignoring this effect will lead to erroneous material properties and misinterpretation of the data. Taken together, the findings presented in this thesis demonstrate that active processes are an essential part of cellular mechanics and cells can contract even independently of adhesions. The results provide a method that allows to quantify active contractions of suspended cells. As the proposed model is not based on specific assumptions on force generating processes, it paves the way for a thorough investigation of different influences, such as cytoskeletal structures and intra-cellular signaling processes, to cellular contractions. The results present an important contribution for better mechanical classification of cells in future research with possible implications for medical diagnosis and therapy.

Robust states exist at the interfaces between topologically trivial and nontrivial phases of matter. These boundary states are expression of the nontrivial bulk properties through a connection dubbed the bulk-boundary correspondence. Whether the bulk is topological or not is determined by the value of a topological invariant. This quantity is defined with respect to symmetries and dimensionality of the system, such that it takes only quantized values. For static topological phases that are realized in ground-states of isolated, time-independent systems, the topological invariant is related to the properties of the Hamiltonian operator. In contrast, Floquet topological phases that are realized in open systems with periodical pumping of energy are topologically characterized with a unitary Floquet operator i.e., the time-evolution operator over the entire period. Topological phases of matter can be distinguished by the dimensionality of robust boundary states with respect to the protecting bulk. This dissertation concerns recently discovered higher-order topological phases where the difference between dimensionalities of bulk and boundary states is larger than one. Using analytical and numerical single-particle techniques, we focus on instances where static higher-order topology can be understood with insights from the mature field of Floquet topology. Namely, even though static systems do not admit a Floquet description, we find examples of higher-order systems to which certain unitary operators can be attributed. The understanding of topological characteristics of these systems is therefore conditioned by the knowledge on topological properties of unitary operators, among which the Floquet operator is well-known. The first half of this thesis concerns toy models of static higher-order topological phases that are topologically characterized in terms of unitary operators. We find that a class of these systems called quadrupole topological insulators exhibit a wider range of topological phases than known previously. In the second half of this dissertation, we study reflection matrices of higher-order topological phases and show that they can exhibit the same topological features as Floquet systems. Our findings suggest a new route to experimental realizations of Floquet systems, the one that avoids noise-induced decoherence inevitable in many other experimental setups.

This work investigates the initial growth of iron and iron oxides on Ag(001). Surface structure and morphology of both post deposition annealed Fe films (in UHV and O2 atmosphere) as well as reactive grown iron oxide films will be analyzed in detail by low energy electron diffraction (LEED) and scanning tunneling microscopy (STM). The stoichiometry at the surface of the iron oxide films will be determined by X-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES). The main focus of this work is to shed light on the question whether the growth of iron oxide films on Ag(001) is accompanied by the formation of strain reducing dislocation networks, or superstructures as found for other metal substrates in former studies. Here, we will distinguish between Fe films which were post deposition annealed in a thin O2 atmosphere and reactively grown iron oxide films.

Quasi-particles formed in lithium niobate after pulse exposure were investigated by transient absorption and photoluminescence spectroscopy as well as numerical simulations. This includes the formation process, the transport through the crystal, interim pinning on defects during the relaxation process, and the final recombination with deep centers. It was shown that the charge-transport through the crystal can be described by a hopping transport including different types of hops between regular or defective lattice sites, i.e., the transport includes a mixture of free and bound small polarons. Furthermore, the different types of hops connected with varying activation energies and their distribution are responsible for an altered temporal decay curve when changing the crystal composition or temperature. Additionally, it was shown that the hitherto accepted recombination model is insufficient to describe all transient absorption and luminescence effects in lithium niobate under certain experimental conditions, i.e., long-living absorption dynamics in the blue/UV spectral range do not follow the typical polaron dynamics and cannot be described under the assumption of charge compensation. However, similar decay characteristics between self-trapped excitons known from photoluminescence spectroscopy and the unexpected behavior of the transient absorption were found leading to a revised model. This includes, besides the known polaron relaxation and recombination branch, a significant role of self-trapped excitons and their pinning on defects (pinned STEs). Since the consideration of further absorption centers in the relaxation path after pulse exposure might result in misinterpretations of previously determined polaron absorption cross-sections and shapes, the necessity to perform a review became apparent. Therefore, a supercontinuum pump-probe experiment was designed and all measurements applied under the same experimental conditions (temperature, polarization) so that one can extract the absorption amplitudes of the single quasi-particles in a spectral range of 0.7-3.0eV. The detailed knowledge might be used to deconvolve the absorption spectra and transform them to number densities of the involved centers which enables one to obtain an easier insight into recombination and decay dynamics of small polarons and self-trapped excitons. As the hopping transport of quasi-particles and the concept of pinned STEs might be fundamental processes, a thorough understanding opens up the possibility of their exploitation in various materials. In particular, results presented herein are not only limited to lithium niobate and its applications; an extension to a wide range of further strongly polar crystals in both their microscopic processes and their use in industry can be considered.

This work is dedicated to a comprehensive experimental study on the interaction of femtosecond laser pulses with the nonlinear optical medium lithium niobate. The nonlinear optical response in the nanosecond regime was already studied extensively with a variety of techniques, whereas femtosecond pulses were mainly used in transient absorption or transient grating experiments. Naturally, the temporal resolution of these measurements depends on the pulse duration, however, dynamics during the pulse excitation were barely investigated. The motivation of this work is to widen the limits of femtosecond spectroscopy, not only to temporally resolve faster nonlinear optical processes, but further to show a sensitivity to other coupling mechanisms between the pulses and the material. Especially, the role of transient, dynamic holographic gratings is investigated with a careful determination of the pulse duration, bandwidth and frequency chirp. A basis of this work is established in the first part by studying the material response via light-induced absorption before focusing on the main topic, the pulse interaction with elementary (holographic) gratings, both self-induced and static, in the second part. By this detailed study, several features of femtosecond laser pulses, holographic gratings and the ultrafast material response can be revealed: (i) grating recording is feasible even with pulses of different frequencies, provided that their pulse duration is sufficiently short, (ii) grating based pulse coupling causes a pronounced energy transfer even in a common pump-probe setup for transient absorption measurements with (non-)degenerated frequencies, (iii) beyond expectation, oscillations in the phonon frequency range become apparent in different measurements. The presented results point towards appropriate future experiments to obtain a more consistent, microscopic model for the ultrafast response of the crystal, involving the interplay between photo-generated polarons, self-induced gratings, and phonons.

Gegenstand der Untersuchungen dieser Arbeit ist die analytische und numerische Studie der plasmonischen Eigenschaften vorhanden in Silbernanodrähten von verschiedener horizontaler Geometrie aufgrund verschiedener Modelle der optischen Antwort der Leitungselektronen. Nach einer hierarchischen Anordnung von linearen Volumen-Materialmodellen, welche innerhalb der plasmonischen Literatur genutzt werden, untersuchen wir die Verwicklung von (nicht)lokaler und dispersiver Antwort mit geometrischen Parametern von Monomeren und Dimeren. Unsere analytischen Studien fokussieren sich auf einzelne zylindrische Drähte, wobei wir das Auftreten von Radius-abhängiger Dämpfung in lokalisierten Oberflächenplasmonen nachweisen, ähnlich dem Konzept der begrenzten mittleren freien Weglänge diskutiert von Kreibig und Mitarbeitern. Weiterhin wird ein Streuproblem mit transversaler Nichtlokalität und "No-slip"- Randbedingung gelöst, gefolgt von einer Diskussion einer Randbedingung, welche zwischen “No-Slip”- und “Slip”-Bedinung interpoliert. Aus numerischer Sicht wird die Streuung an abgerundeten und gleichseitigen dreieckigen und Bowtie-Drähten behandelt mit Fokus auf einer vollanalytischen Beschreibung der Eckenrundung mittels Bézier- Kurven. Dies enthüllt den Krümmungsradius als neuen geometrischen Parameter. Das Variieren der Lückenbreite und Eckenrundung führt zu Verstärkungsfaktoren, welche relevant für oberflächenverstärkte Raman-Streuung einzelner Moleküle sind, in ausgezeichneten räumlichen Bereichen abhängig von der Art der Resonanz. Innerhalb der Extinktionsspektren von dreieckigen und Bowtie-Drähten erscheint eine Sequenz von nichtlokalen Maxima. Diese Sequenz ist am sensitivsten in Bezug auf die Änderung der Krümmung. Die Identifikation der (Hybrid-)Resonanzen basiert auf simulierten Ladungsdichteverteilungen. / Subject of this thesis is the analytical and numerical study of the plasmonic properties present in silver nanowires of different horizontal geometries due to different models of optical response of conduction electrons. Following a hierarchical arrangement of linear bulk material models, used throughout the plasmonic literature, we investigate the intertwining of (non)local and dispersive response with geometrical parameters of monomers and dimers. Our analytical studies focus on single cylindrical wires, revealing the occurrence of radius-dependent damping of localized surface plasmons similar to the concept of limited-mean-free-path discussed by Kreibig and coworkers. Further, a scattering problem with transverse nonlocality and s no-slip condition is solved followed by a discussion of a boundary condition interpolating between the slip and no-slip conditions. On a numerical level, the scattering by rounded and equilateral triangular and bowtie nanowires is treated based on a full analytical description of the corner rounding via Bézier curves revealing the radius of curvature as a new geometrical degree of freedom. Tuning of gap size and corner rounding reveals enhancement factors relevant for surface-enhanced Raman scattering of single molecules in distinguished spatial domains dependent on the type of resonance. Within the extinction spectra a nonlocal peak sequence emerges. This sequence is most sensitive to curvature variations and arises in the triangular monomer and bowtie dimer. The identification of (hybrid) resonances is based on charge density simulations.

A new method, probability distribution of diffusivities (time scaled square displacements between succeeding video frames), was developed to analyze single molecule tracking (SMT) experiments. This method was then applied to SMT experiments on ultrathin liquid tetrakis(2-ethylhexoxy)silane (TEHOS) films on Si wafer with 100 nm thermally grown oxide, and on thin semectic liquid crystal films. Spatial maps of diffusivities from SMT experiments on 220 nm thick semectic liquid crystal films reveal structure related dynamics. The SMT experiments on ultrathin TEHOS films were complemented by fluorescence correlation spectroscopy (FCS). The observed strongly heterogeneous single molecule dynamics within those films can be explained by a three-layer model consisting of (i) dye molecules adsorbed to the substrate, (ii) slowly diffusing molecules in the laterally heterogeneous near-surface region of 1 - 2 molecular diameters, and (iii) freely diffusing dye molecules in the upper region of the film. FCS and SMT experiments reveal a strong influence of substrate heterogeneity on SM dynamics. Thereby chemisorption to substrate surface silanols plays an important role. Vertical mean first passage times (mfpt) in those films are below 1 µs. This appears as fast component in FCS autocorrelation curves, which further contain a contribution from lateral diffusion and from adsorption events. Therefore, the FCS curves are approximated by a tri-component function, which contains an exponential term related to the mfpt, the correlation function for translational diffusion and a stretched exponential term for the broad distribution of adsorption events. Lateral diffusion coefficients obtained by FCS on 10 nm thick TEHOS films, thereby, are effective diffusion coefficients from dye transients in the focal area. They strongly depend on the substrate heterogeneity. Variation of the frame times for the acquisition of SMT experiments in steps of 20 ms from 20 ms to 200 ms revealed a strong dependence of the corresponding probability distributions of diffusivities on time, in particular in the range between 20 ms and 100 ms. This points to average dwell times of the dye molecules in at least one type of the heterogeneous regions (e.g. on and above silanol clusters) in the range of few tens of milliseconds. Furthermore, time series of SM spectra from Nile Red in 25 nm thick poly-n-alkyl-methacrylate (PnAMA) films were studied. In analogy to translational diffusion, spectral diffusion (shifts in energetic positions of SM spectra) can be studied by probability distributions of spectral diffusivities, i.e. time scaled square energetic displacements. Simulations were run and analyzed to study contributions from noise and fitting uncertainty to spectral diffusion. Furthermore the effect of spectral jumps during acquisition of a SM spectrum was investigated. Probability distributions of spectral diffusivites of Nile Red probing vitreous PnAMA films reveal a two-level system. In contrast, such probability distributions obtained from Nile Red within a 25 nm thick poly-n-butylmethacrylate film around glass transition and in the melt state, display larger spectral jumps. Moreover, for longer alkyl side chains a solvent shift to higher energies is observed, which supports the idea of nanophase separation within those polymers.