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Tilted Light Cone in Circuit, Microwave and Photonic Crystal Structures

Motavassel, Ahmad | 2024

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 58246 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Jafari, Akbar
  7. Abstract:
  8. Based on the theory of General Relativity, the presence of mass and energy leads to a modification of the space-time metric, causing curvature that deviates the trajectories of particles and light (electromagnetic waves) from those in flat space-time. These effects manifest in the dispersion relation that describes the motion of particles or their associated waves. Similarly, in crystalline and periodic structures, dispersion relations describe the properties of the lattice. Recent studies have investigated the impact of lattice structure and coupling parameters between its units on the space-time geometry induced by the lattice. While much of this research has focused on atomic crystalline structures, investigations into non-flat geometries, such as hyperbolic geometry, in microwave circuit networks have also been conducted. However, the study of bosonic structures, particularly circuit-based systems that explore space-time geometry effects, has been relatively limited. This research explores the feasibility of implementing tilted optical cones, which correspond to a non-trivial space-time metric, within periodic circuit networks and photonic crystals. In the first part of the study, it is analytically demonstrated that the combination of capacitors and inductors arranged in a honeycomb lattice to form resonator arrays, with asymmetric inductive coupling, can produce controllable tilting in the frequency spectrum, resulting in a tilted Dirac cone. The effects of this asymmetry on the position of Dirac points, their shifts within the Brillouin zone from the corners K and K’ toward each other, and the creation of a frequency gap are analyzed. Finally, the relationship of impedance between different points of the network and the density of states in the frequency spectrum is identified, providing a foundation for experimental measurements. Subsequently, based on the proposed circuit design, three superconducting microwave circuits with 731 resonators were designed and fabricated, featuring varying circuit parameters to produce different degrees of tilting. By measuring the scattering matrix for transmission between specific points on the edges of the circuits and extracting resonance frequencies through peak detection methods, the theoretical analyses were validated. This was achieved by comparing the measured density of states with calculations obtained from solving the eigenvalue problem for the finite circuit network. The results demonstrated high controllability in tailoring spectral features of circuit structures, specifically enabling the design of networks with adjustable tilting to investigate the effects of varying space-time geometry in such systems. In another part of the study, the circuit-based concepts were extended to two-dimensional photonic crystals. Using the plane wave expansion (PWE) method, numerical simulations were performed to calculate the dispersion relations of photonic crystals. It was shown that a honeycomb lattice with elliptical dielectric rods in its unit cell can achieve up to 15% tilting of optical cones, appearing near the K and K’ points. Removing one of these cones enables the study of the group velocity differences in opposing directions without interference from the second cone in future investigations. In the final phase, the topological concept of breaking inversion and time-reversal symmetries was employed to eliminate one of the two optical cones. This was achieved by introducing dielectric asymmetry in the unit cell rods and adding gyromagnetic rods. This design, which can be experimentally constructed and tested in future research, provides a platform to study space-time geometry effects not only in the geometric optics regime but also in the wave regime. This research establishes a foundation for further exploration of tilted optical cones and non-trivial space-time geometries in artificial periodic systems
  9. Keywords:
  10. Honeycomb Lattice ; Microwave Circuits ; Photonic Crystal ; Space-Time ; Light Cone ; Dirac Cone ; Tilted Dirac Cone

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