Research On The Dirac-cone Dispersions Induced By Accidental Degeneracy In Photonic And Phononic Crystals

Photonic and phononic crystals are the artificial composite structures with periodic spatial structures and band gaps. By exploiting the accidental degeneracy of Bloch eigenstates, many types of linear dispersion relations called Dirac-cone dispersions can be realized at the Brillouin zone center of the two kinds of artificial crystals. Potential applications of these peculiar dispersion relations can be expected in diverse fields such as classical wave manipulation and energy flow control. In this dissertation, we present a study on the scientific problems including physical mechanism and application associated with the Dirac-cone dispersions induced by accidental degeneracy. Using a perturbation theory and effective medium theory combined with finite element simulation, we deeply and systemically studied the Dirac-like cones, double Dirac cones and semi-Dirac cones in photonic and phononic crystals, and discussed the relation of the Dirac-cone dispersions to zero-index medium. We mainly carried out the research work as described below:1. We extended the k ×p method in quantum systems to study the two-dimensional dielectric photonic crystals and developed a selection rule for Dirac-like cones, which is derived from the perturbation theory. The selection rule states that a non-zero, mode-coupling integral between the accidental degenerate Bloch eigenstates guarantees a Dirac-like cone, regardless of the type of the accidental degeneracy. Based on the group theory, we pointed out that the selection rule can also be determined from the spatial symmetry of the Bloch eigenstates even without computing the integral. Thus, the existence of Dirac-like cones can be quickly and conclusively predicted for various photonic crystals independent of electromagnetic wave polarization, lattice structure, and composition.2. By exploiting the accidental degeneracy of the doubly-degenerate dipolar and quadrupolar modes, we show that a two-dimensional dielectric photonic crystal can exhibit the double Dirac cone dispersion at the Brillouin zone center. Using the k ×p perturbation method and group theory, we accurately predicted the linear slopes of the double Dirac cone and demonstrate that the double cone is composed of two identical and overlapping Dirac cones, and the linearity of the dispersion is guaranteed by the spatial symmetry of the Bloch eigenstates. Numerical simulations including wave-front shaping, unidirectional transmission and perfect tunneling show that the corresponding photonic crystal structure can be characterized by a zero effective refractive index.3. Accidental degeneracy of two double-degenerate eigenstates in a two-dimensional phononic crystal leads to a double Dirac cone at Brillouin zone center. Using the k ×p perturbation theory and selection rule, we demonstrate that the linear behaviors of double Dirac cone is determined by the spatial symmetry of the accidental degenerate eigenstates, and a double Dirac cone is composed of two identical and overlapping Dirac cones, which is the same as the case in photonic crystals. We calculated the effective parameters of the phononic crystal near the double Dirac point by using a standard retrieval method. And the results shows that at the double Dirac point frequency, the slab of the phononic crystal can be mapped onto a slab of zero refractive index material. Total transmission without phase change and energy tunneling at the double Dirac point frequency are unambiguously demonstrated by two examples, and these phenomena are roust regardless of whether or not the shear wave modes inside the solid components are included.4. By exploiting the accidental degeneracy of a monopolar eigenstate and a dipolar eigenstate, we realized a semi-Dirac cone at the Brillouin zone center of a two-dimensional phononic crystal, which consists of a square array of rectangle rods. We resort to the k ×p perturbation method to analyze the behaviors of the dispersions near the semi-Dirac point and found that the dispersion relation is linear only along one direction. By using a standard retrieval method, we obtained the effective parameters near the semi-Dirac point and proved that at the semi-Dirac point frequency, the phononic crystal possesses the properties of the “double-zero” and “single-zero” zero-index mediums. The hybridized property can be used to realize the unidirectional transmission and wave-front shaping.