From Dirac Points To Topological Monopole Resonators

The family of quantum Hall effects shows a new world of topology in condensed matter physics,research on topological phases of matter and topological phase transitions based on band topology has formed an upsurge in the past two decades.As an important branch of mathematics,topology has occupied an important position in the research of condensed matter physics for a long time.For example,the early topological defect theory plays an important role in the research on the arrangement of nematic liquid crystals and ordered spin structures.Different from ordinary defects,topological defects are stable against local disorders and system-wide minor perturbations.That’s also the most significant phenomenon in topological band theory,with non-trivial bulk band topology,there are topologically protected states at the phase transition interfaces or boundaries,showing robust propagation or resonance.In recent decades,new topological systems with fascinating features keep springing up.The extension of topological band theory to artificial systems like photonic and acoustic crystals also made significant achievements.From the perspective of band structure topology,researchers were able to design functional devices with advantages beyond traditional ways.Against this background,the research works proposed in this thesis were mainly in two parts.First,the investigation of topological band structures.In this part,the 3D ideal Dirac points were achieved in acoustic crystals,and the intrinsic topological surface states of the 3D Dirac points were experimentally observed for the first time.The Dirac points play a central role in topological band theory,they are the critical points of topological phase transitions between many different topological phases.From the Dirac points,various fascinating topological band structures can be constructed.In the past,numerous research works have obtained the 3D Dirac points based on the band inversion mechanism.Dirac points in such systems were not prevented to be annihilated in pairs by symmetry-protected perturbations.That’s also the reason why no intrinsic topological surface states were observed in all those attempts.Based on the space group symmetry analysis,in the first part of this work,we designed and realized two 3D ideal Dirac points in acoustic crystal structures under the protection of space group#230 or#206 symmetries,which provides a preferred platform for many follow-up studies.Surface-state dispersion characterized by quad-helicoid structure was experimentally observed for the first time using surface acoustic field measurements.Furthermore,novel topological band structures with Z2 topological invariants,such as nodal rings and Weyl dipoles,are obtained through the symmetric perturbation designs of the crystal structures.Second,the application exploration of topological band theory.Resonators are widely used in lasers,sensors,optical combs,etc.It’s difficult for conventional resonators to acquire an arbitrary number of mode degeneracy and single-mode designs with large mode volumes.Recently,the Dirac-vortex topological cavities based on 2D topological point defect modes in topological photonic crystals present a breakthrough in cavity designs.From the new perspective of band topology,researchers realized that the widely commercial products of stacked one-dimensional photonic crystal resonators such as DFBs and VCSELs were essentially topological non-trivial,which corresponds to the 1D Jackiw-Rebbi model.The Dirac-vortex topological cavities are the twodimensional generalization of this model,described by the Jackiw-Rossi model,which achieved an optimized single-mode behavior with the free spectral range inversely proportional to the square root of the modal volume,or an arbitrary number of degenerate modes.The topological cavities are already applied for the design of surface-emitting lasers.In 3D,the earliest Jackie-Rebbi model has given predictions at the same time,but it has not been realized.The second part of this work follows the core concept of this theoretical model,the 3D topological point defect of Dirac-mass distributions in real space was realized by adding geometry modulations to the acoustic Dirac crystal.The topological resonate mode at the center of the band gap was given by the zero-energy solution of the Dirac equation under an efficient approximation.The topological cavity thus designed can support an arbitrary number of degenerate modes,or theoretically the best single-mode performance at the large mode volume limit.In addition,the symmetry-protected perturbations promised all possible gapped phases with different mass terms of the 3D Dirac points,which can be used for further research of topological edge states or corner states,etc.Periodic structures like photonic and acoustic crystals are of great significance in the study of topological band theory.First,the photons and phonons all belong to bosons,the integer spin endows the systems with obviously different topological properties.For example,the experimental observation of 3D Dirac points with quadhelicoid surface states in the first part of this work can only be achieved in such bosonic systems,but cannot be realized in electronic systems.Secondly,photonic and acoustic crystals show remarkable advantages in band structure manipulations,with more complete theoretical and numerical tools,and simpler experimental operations compared to electronic systems.That makes them suitable as simulations and supplements for electronic band structures to more difficult challenges.Such as the 3D Jackiw-Rebbi zero mode obtained by complex geometry structures’ modulations,in the second part of this work,are extremely difficult to be realized in electronic systems.What’s more,the research on photonic and acoustic crystals has important application significance,many successfully applicable designs have been born on photonics crystal systems,such as the Bragg mirrors,photonic crystal fibers,and the photonic crystal surface-emitting lasers.The topological monopole cavities designed in this work are also oriented to the applied requirements of single-mode resonators with large modal volumes,which connected the theoretical model verifications and practical applications.