Die Arbeit untersucht planare optische Mikrokavitäten, welche aus einer beidseitig von Multischichtspiegeln umgebenen Kavitätsschicht bestehen. Im Rahmen einer Transfermatrixbeschreibung für ebene Wellen wird ein genereller Ansatz zur Berechnung von optischen Kavitätsmoden von planaren Mikrokavitäten entwickelt, welche aus optisch beliebig anisotropen Medien bestehen. Die zugrunde liegende Modenbedingung kommt ohne vorherige Einschränkungen bezüglich der betrachteten Lichtpolarisation aus. Basierend auf diesem Ansatz werden numerische Modenberechnungen von Mikrokavitäten mit optisch uniaxialen Kavitätsschichten vorgenommen. Generell sind die Moden in einem solchen System elliptisch polarisiert, und zudem i.A. nicht orthogonal. Ein besonderes Phänomen stellen sogenannte exzeptionelle Punkte dar. Dies sind Richtungen, für welche Energie und Verbreiterung der zwei Kavitätsphotonmoden zugleich entarten. Die Moden werden an solchen Punkten zirkular ko-polarisiert, die Orientierung der linearen Modenpolarisation windet sich im Impulsraum um diese Punkte herum. Die Eigenschaften der anisotropen Mikrokavitäten und exzeptionellen Punkte sind charakteristisch für singuläre, biaxiale Optik. So entsprechen die exzeptionellen Punkte Richtungen sogenannter singulärer optischer Achsen der effektiv biaxialen Strukturen, und können als Entartung nicht-Hermitescher Operatoren beschrieben werden. Die experimentelle Realisierung wird am Beispiel ZnO-basierter Mikrokavitäten gezeigt und bestätigt die theoretischen Vorhersagen im Wesentlichen, wenngleich im Experiment keine komplett zirkular polarisierten Zustände an den Entartungspunkten beobachtet wurden.:0 Introduction 1 Theory I: Linear optics principles 1.1 Maxwell theory 1.1.1 Plane-wave ansatz 1.1.2 Light polarization 1.1.3 Crystal optics 1.1.4 The polariton concept 1.2 Matrix formalisms for planar structures 1.2.1 Transfer-matrix approach 1.2.2 Scattering, Jones and Müller matrices 2 Theory II: Planar optical microcavities 2.1 Fabry-Pérot resonators and photonic modes 2.2 Practical mode computation 2.3 Quasi-particle approach 3 Computation: Exceptional points in anisotropic microcavities 3.1 Numerical methods 3.2 Model and findings for anisotropic, dielectric microcavities 3.3 Classification and discussion 3.3.1 General characteristics of exceptional points in anisotropic microcavities 3.3.2 Polarization vortices and singular optics 3.3.3 Net topology of the system 3.3.4 Effective-medium approaches 3.3.5 Quasi-particle approaches 3.3.6 Other familiar systems and phenomena 3.4 Anisotropic exciton-polaritons 4 Experiment: ZnO-based planar microcavities 4.1 Microcavity samples 4.2 Experimental methods 4.3 Experimental results vs. theoretical computations 4.4 Summary and discussion 5 Conclusion A Appendix A.1 Determining optic axes A.2 Exceptional points A.3 Expressions in Gaußian CGS units A.4 Polarization patterns of isotropic microcavities Bibliography Symbols and Abbreviations Authored and co-authored publications directly related to this thesis Acknowledgments Curriculum Vitae / In this thesis, planar optical cavities are investigated. They consist of a cavity layer which is surrounded by multi-layer mirrors. Using a transfer matrix technique for planar structures, a general mode condition is developed, which allows computation of cavity-photon modes for planar microcavities, which consist of optically arbitrarily anisotropic media. With this approach, no prior restriction of the considered light polarization is required. Based on this formalism, numerical computations of planar microcavities with optically uniaxial cavity layers are performed. Generally, the cavity-photon modes in such systems obtain elliptic polarization. Furthermore, they are in general not orthogonal to each other. A particular phenomenon is the occurrence of so called exceptional points. Here, the two cavity-photon modes degenerate in energy and broadening simultaneously, and the modes become circularly co-polarized. In addition, the exceptional points are vortex centers in momentum space for the orientation of the linear polarization of the modes. With this, anisotropic planar microcavities show typical characteristics of singular as well as biaxial optics. The exceptional points can be regarded as singular optic axes of the effectively biaxial structures. They can be described by the degeneracy of non-Hermitian operators. Experimental implementation is demonstrated by ZnO-based microcavities. In general, experimental findings prove the theoretical predictions, albeit the degree of circular polarization does not approach 100% at the exceptional points.:0 Introduction 1 Theory I: Linear optics principles 1.1 Maxwell theory 1.1.1 Plane-wave ansatz 1.1.2 Light polarization 1.1.3 Crystal optics 1.1.4 The polariton concept 1.2 Matrix formalisms for planar structures 1.2.1 Transfer-matrix approach 1.2.2 Scattering, Jones and Müller matrices 2 Theory II: Planar optical microcavities 2.1 Fabry-Pérot resonators and photonic modes 2.2 Practical mode computation 2.3 Quasi-particle approach 3 Computation: Exceptional points in anisotropic microcavities 3.1 Numerical methods 3.2 Model and findings for anisotropic, dielectric microcavities 3.3 Classification and discussion 3.3.1 General characteristics of exceptional points in anisotropic microcavities 3.3.2 Polarization vortices and singular optics 3.3.3 Net topology of the system 3.3.4 Effective-medium approaches 3.3.5 Quasi-particle approaches 3.3.6 Other familiar systems and phenomena 3.4 Anisotropic exciton-polaritons 4 Experiment: ZnO-based planar microcavities 4.1 Microcavity samples 4.2 Experimental methods 4.3 Experimental results vs. theoretical computations 4.4 Summary and discussion 5 Conclusion A Appendix A.1 Determining optic axes A.2 Exceptional points A.3 Expressions in Gaußian CGS units A.4 Polarization patterns of isotropic microcavities Bibliography Symbols and Abbreviations Authored and co-authored publications directly related to this thesis Acknowledgments Curriculum Vitae

In this diploma thesis precursors of quantum phase transitions in finite many-body systems are studied. The main attention is paid to the mechanism, how nonanalytic behaviour of the ground state is generated for certain critical values of real control parameters. It is shown that nonanalytic behaviour of energy levels and eigenstates is closely connected with exceptional points of the hamiltonian, which are points in control parameter space extended into a complex domain where at least two eigenvalues and corresponding eigenvectors coincide. Differences in the distribution of exceptional points in the complex plane of control parameter for the first and second order phase transitions and also evolutions of the position of exceptional points with increasing particle number are discussed.

Dirac fermions provide a prototypical description of topological insulators and their gapless boundary states, which are predicted by the bulk-boundary correspondence. Motivated by the unusual physical properties of these states, we study them in two different Hermitian quantum systems. In non-Hermitian systems, we investigate the failure of the bulk-boundary correspondence and show that non-Hermitian topological invariants impact a system’s bulk response. First, we study electronic topological insulators in three dimensions with time-reversal symmetry. These can be characterized by a quantized magnetoelectric coefficient in the bulk, which, however, does not yield an experimentally observable response. We show that the signature response of a time-reversal-invariant topological insulator is a nonlinear magnetoelectric effect, which in the presence of a small electric field leads to the appearance of half-integer charges bound to a magnetic flux quantum. Next, we consider topological superconducting nanowires. These feature Majorana zero modes at their ends, which combine nonlocally into a single electronic state. An electron tunneling through such a state will be transmitted phase-coherently from one end of the wire to the other. We compute the transmission phase for nanowires with broken time-reversal symmetry and confirm that it is independent of the wire length. Turning to non-Hermitian systems, we consider planar optical microcavities with an anisotropic cavity material, which may feature topological degeneracies known as excep- tional points in their complex frequency spectrum. We present a quantitative method to extract an effective non-Hermitian Hamiltonian for the eigenmodes, and describe how a pair of exceptional points arises from a Dirac point due to the cavity loss. Finally, we investigate generalized topological invariants that can be defined for non- Hermitian systems, but which have no counterpart (i.e. vanish) in Hermitian systems, for example the so-called non-Hermitian winding number in one dimension. Contrary to Hermitian systems, the bulk-boundary correspondence breaks down: Comparing Green functions for periodic and open boundary conditions, we find that in general there is no correspondence between topological invariants computed for periodic boundary con- ditions, and boundary eigenstates observed for open boundary conditions. Instead, we prove that the non-Hermitian winding number in one dimension signals a topological phase transition in the bulk: It implies spatial growth of the bulk Green function, which we define as the response of a gapped system to an external perturbation on timescales where the induced excitations have not propagated to the boundary yet. Since periodic systems cannot accommodate such spatial growth, they differ from open ones.

Deux stratégies sont présentées à utiliser des effets de couplage de modes pour l’amplification de l’atténuation du son dans les conduits acoustiques. La première est de coupler le mode incident propagatif avec un mode localisé, aussi appelé résonance de Fano. Cette stratégie est présentée et validée dans un système conduit-cavité et un guide d’onde partiellement traité en paroi avec un matériau à réaction locale. La méthode “R-matrix” est introduite pour résoudre le problème de propagation d’onde. Une annulation de la transmission se produit quand un mode piégé (qui est formé par les interférences de deux modes voisins) est excité dans le système ouvert. Ce phénomène est aussi lié au croisement évité des valeurs propres et à un point exceptionnel. Dans la seconde stratégie, un réseau d’inclusions rigides périodiques est intégré dans une couche poreuse pour améliorer l’atténuation du son à basse fréquence. Le couplage de modes est du à la présence de ces inclusions. Le théorème de Floquet-Bloch est proposé pour analyser l’atténuation du son dans un guide d’onde périodique en 2D. Un croisement de l’atténuation de deux ondes de Bloch est observé. Ce phénomène est utilisé pour expliquer le pic de pertes en transmission observé expérimentalement et numériquement dans un guide 3D partiellement traitée par un matériau poreux avec des inclusions périodiques. Enfin, le comportement acoustique d’un liner purement réactif dans un conduit rectangulaire avec et sans écoulement est étudié. Dans une certaine gamme de fréquence, aucune onde ne peut se propager à contre sens de l’écoulement. Par analyse des différent modes à l’aide de la relation de dispersion, il est démontré que le son peut être ralenti et même arrêté. / Two strategies are presented to use the mode coupling effects to enhance sound attenuation in acoustic ducts. The strategy is to couple the incoming propagative mode with the localized mode, which is also called Fano resonance. This strategy is presented and validated in a duct-cavity system and a waveguide partially lined with a locally reacting material. The R-matrix method is introduced to solve the propagation problems. A zero in the transmission is present, due to the excitation of a trapped mode which is formed by the interferences of two neighboured modes. It is also linked to the avoided crossing of the eigenvalues and exceptional point. In the second strategy, a set of periodic rigid inclusions are embedded in a porous lining to enhance sound attenuation at low frequencies. The mode coupling is due to the presence of the embedded inclusions. Floquet - Bloch theorem is proposed to investigate the attenuation in a 2D periodic waveguide. Crossing is observed between the mode attenuations of two Bloch waves. The most important and interesting figure is that near the frequency where the crossing appears, an attenuation peak is observed. This phenomenon can be used to explain the transmission loss peak observed numerically and experimentally in a 3D waveguide partially lined by a porous material embedded with periodic inclusions. Finally, the acoustical behaviours of a purely reacting liner in a duct in absence and presence of flow are investigated. The results exhibit an unusual acoustical behaviour : for a certain range of frequencies, no wave can propagate against the flow. a negative group velocity is found, and it is demonstrated that the sound can be slowed down and even stopped.

This thesis theoretically and experimentally studies the characteristics of integrated microresonators (MRs) built by passive (no gain) and non-magnetic materials and characterized by both Hermitian and non-Hermitian Hamiltonians. In particular, I have studied three different microresonators: a typical Microring Resonator (MR), a Taiji Microresonator (TJMR), which consists of a microresonator with an embedded S-shaped waveguide, and a new geometry called the Infinity-Loop Microresonator (ILMR), which is characterized by a microresonator shaped like the infinity symbol coupled at two points to the bus waveguide. To get an accurate picture of the three devices, they were modeled using both the transfer matrix method and the temporal coupled mode theory. Neglecting propagation losses, the MR is described by a Hermitian Hamiltonian, while the TJMR and the ILMR are described by a non-Hermitian one. An important difference between Hermitian and non-Hermitian systems concerns their degeneracies. Hermitian degeneracies are called Diabolic Points (DPs) and are characterized by coincident eigenvalues and mutually orthogonal eigenvectors. In contrast, non-Hermitian degeneracies are called Exceptional Points (EPs). At the EP, both the eigenvalues and the eigenvectors coalesce. The MR is at a DP instead, and the TJMR and the ILMR are at an EP. Since the TJMR and ILMR are at an EP, they have interesting features such as the possibility of being unidirectional reflectors. Here, it is shown experimentally how in the case of the TJMR this degeneracy can also be used to break Lorentz reciprocity in the nonlinear regime (high incident laser powers), discussing the effect of the Fabry-Perot of the bus waveguide facets. The effect of backscattering, mainly due to the waveguide surface-wall roughness, on the microresonators is also studied. This phenomenon induces simultaneous excitation of the clockwise and counterclockwise modes, leading to eigenvalue splitting. This splitting makes the use of typical quality factor estimation methods unfeasible. To overcome this problem and mitigate the negative effects of backscattering, a new experimental technique called interferometric excitation is introduced. This technique involves coherent excitation of the microresonator from both sides of the bus waveguide, allowing selective excitation of a single supermode. By adjusting the relative phase and amplitude between the excitation fields, the splitting in the transmission spectrum can be eliminated, resulting in improved quality factors and eigenvalue measurements. It is shown that this interferometric technique can be exploited under both stationary and dynamic conditions of time evolution. The thesis also investigates the sensing performance of the three microresonators as a function of a backscattering perturbation, which could be caused, for example, by the presence of a molecule or particle near the microresonator waveguide. It is shown that the ILMR has better performance in terms of responsivity and sensitivity than the other two microresonators. In fact, it has both the enhanced sensitivity due to the square root dependence of the splitting on the perturbation (characteristic of EPs) and the ability to completely eliminate the region of insensitivity as the backscattering perturbation approaches zero, which is present in both the other two microresonators. To validate the models used, they were compared with experimental measurements both in the linear regime and, for TJMR, also in the nonlinear regime, with excellent agreement.