Semi-Dirac dispersion relation in photonic crystals

A semi-Dirac cone refers to a peculiar type of dispersion relation that is linear along the symmetry line but quadratic in the perpendicular direction. Here, I demonstrate that a photonic crystal consisting of a square array of elliptical dielectric cylinders is able to produce this particular dispersion relation in the Brillouin zone center. A perturbation method is used to evaluate the linear slope and to affirm that the dispersion relation is a semi-Dirac type. Effective medium parameters calculated from a boundary effective medium theory not only explain the unexpected topological transition in the iso-frequency surfaces occurring at the semi-Dirac point, they also offer a perspective on the property at that point, where the photonic crystal behaves as a zero-refractive-index material along the symmetry axis but functions like at a photonic band edge in the perpendicular direction.


Introduction
Photonic crystals (PhCs), periodic structures that control photons in a way comparable to the way semiconductors control electrons, have inspired extensive study since their emergence in the late 1980's [1,2]. Details about these fascinating structures can be found in a recent monograph [3], and references therein. One key feature of a PhC is the photonic band gap, which prevents light from propagating in certain directions at specific frequencies. It is analogous to the electron band gap in semiconductors. Engineering the photonic band gap to create a large and complete band gap was the goal of much early research in this area [1,2,4] . On the other hand, engineering the photonic bang gap to achieve a type of gapless band structure, namely the Dirac and Dirac-like cone dispersion relation, has been the focus of much recent work [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. This dispersion relation is analogues to the Dirac cones in electron systems, where two linear bands touch so that there is no band gap. Owing to this special property, remarkable wave transport behaviors and interesting applications in electromagnetic waves have been reported [5][6][7][8][9][10][11][12][13][14][15][16]. To name a few, Haldane and Raghu proposed the possible realization of a directional optical wave guide in a PhC with breaking time-reversal symmetry [5] and described classical analogs of edge states in Quantum-Hall-Effect systems [6]. wave manipulation characteristics, such as beam shaping and cloaking of an object [9].
Very recently, a unique and unprecedented band structure was discovered in a VO 2 /TiO 2 structure: near a point Fermi surface in the two-dimensional Brillouin zone, the dispersion relation is linear along the symmetry line ((1,1) direction) but quadratic in the perpendicular direction [23]. The associated quasiparticles are therefore massless along one direction, like those in graphene, but effective-mass-like along the other. This dispersion relation is called a semi-Dirac cone and the associated point is called a semi-Dirac point [23]. It was reported that such a point is associated with the topological phase transition between a semi-metallic phase and a band insulator [24]. Because both band gaps and Dirac cones in electron systems have found their classical analogies in PhCs, a natural question to ask is: if the band structure with the special peculiarity of a semi-Dirac cone can be transcribed into PhCs? If the answer to this question is yes, it is possible to envisage various interesting consequences that could be attributed to this unique dispersion relation, such as super anisotropic wave transport behavior near the semi-Dirac point.
Here, I demonstrate that by employing accidental degeneracy in a two-dimensional PhC with anisotropic scatterers, it is indeed possible to achieve a semi-Dirac type dispersion relation in the Brillouin zone center, which is associated with a semi-Dirac point. A kp  method [21,22], based on perturbation, is exploited to confirm that the dispersion relation is linear in one symmetry axis (the X  direction) and quadratic in the perpendicular one (the Y  direction). Beyond that, the method shows that the linear slope decreases as k rotates away from the X  direction and eventually vanishes when k is in the Y  direction. A somewhat unexpected discovery is that the semi-Dirac point results in a topological transition [25] between a hyperbolic-like shape and an elliptic-like shape in the iso-frequency surfaces. An effective medium theory based on boundary

The Photonic Crystal System and the Semi-Dirac Dispersion Relation
The PhC considered in this study is a square array of elliptical cylinders with a dielectric  , where  is the angular frequency and c is the wave speed in air. This degenerated point is created by accidental degeneracy [9,[19][20][21][22] of a monopolar state (see Fig. 2 (b)) and a dipolar state (see Fig. 2(c)) at the  point, when the frequencies of these two states are deliberately tuned to be identical by adjusting the size or the material of the inclusion. Figures 1(b) and 1(c) show the band structure near the  point for smaller and larger dielectric cylinders, whose eccentricity is kept at 1. Therefore, the task is to solve the following 33  secular equation: where A  ( B  ) is the frequency of the eigenstates at Point A (B), and ij P represents the mode-coupling integrals among the three eigenstates whose profiles are plotted in Figs.
2(a)-2(c). Figure 2(a) shows the electric field distribution of the state located at Point B, where a dipolar state with its magnetic field parallel to the horizontal axis is seen. This state is denoted as 1  . Figures 2(b) and 2(c) exhibit the electric field map of the doubly degenerated states at Point A. Obviously seen are a monopolar state ( Fig.2(b)) labeled as 2  and a dipolar state with its magnetic field parallel to the vertical axis ( Fig. 2 , the solution to Eq. (1) is easily derived, which reads: where  denotes the angle between k and the X  direction, and the dimensionless frequency, 2 ac Along the Y  direction, this integration is zero because the coupling between 2  and 3  is extremely weak. Even though the accidental degeneracy is achieved, no linear dispersion is therefore found [22]. In fact, if we examine the eigenstates, it is not difficult to find that the underlying physics lies in 3  , the dipolar mode of the doubly degenerated state. As shown in Fig. 2(c), the magnetic field of this dipolar mode is polarized vertically, implying it is a longitudinal mode along the Y  direction but a transverse mode along the X  direction. In electromagnetic waves, the longitudinal branch is localized and almost does not couple to the incident wave and other branches [9,27]. Thus, a flat band associated with the longitudinal dipolar mode is found in the Y  direction. However, since 3  is transverse to the X  direction, it easily couples to the incident wave and other branches. The coupling between 2  and 3  is therefore strong in the X  direction and the linear dispersion relation is found.

(II) The Topological Transition: Anisotropic Effective Medium Theory
This interesting anisotropic dispersion is comparable to a semi-Dirac cone in the sense that dispersion is linear along the symmetry line but quadratic along the perpendicular direction. This property is seen again in Figs Surprisingly seen is that the topological transition [25] in the iso-frequency surface occurs at the semi-Dirac point.
Because the Bloch wave vector is very small near the Brillouin zone center, the unexpected topological transition at the semi-Dirac point can be understood from an effective medium perspective. Here I adopt a boundary effective medium approach that was originally developed for elastic waves in Ref. [28].  Figure 4(a) shows the results of the effective medium parameters evaluated by this boundary integration method using the eigenstates highlighted by the solid dots shown in Fig. 1(a). Figure 4(a) demonstrates that the effective permittivity calculated from eigenstates along the ΓX and ΓY directions are identical, because the electric field is a scalar for TE polarized waves. However, the permeability is anisotropic owing to the vector nature of the in-plane magnetic field. . The topological transition in the iso-frequency surface of the PhC can indeed be attributed to these effective medium properties. In fact, the iso-frequency surface of an anisotropic material is described as: (3) The signs of the material parameters determine the topology of the iso-frequency surface [25]. The waveguide has boundary walls that are perfect magnetic conductors (PMC).
Obviously, when the incident wave is propagating in the X  direction, the PhC exhibits the typical transmission property of a double-zero-refractive-index material [9], i.e., total transmission is supported without any phase change inside the material as shown in Figs. 4(c) and 4(d) for the real and imaginary parts of the electric field, respectively. However, when the PhC is illuminated by the same incident wave along the Y  direction, the amplitude of the electric field decays as the wave penetrates into the PhC, which is similar to the transmission property at the edge of a photonic band gap. This anisotropic transport feature provides evidence that the PhC has both the property of a "zero-indexmaterial" and a "photonic-band-gap" material at the semi-Dirac point.
The anisotropic transport property and the topological transition in the iso-frequency surfaces lead to interesting wave manipulation behaviors. In Interesting wave manipulation properties, such as beam splitting and directional beam shaping, have been demonstrated. Figure 1 (a) The band structure of a two-dimensional photonic crystal comprising a square array of elliptical cylinders with a relative dielectric constant, 12.5 s   , embedded in air. The lattice constant is a , which also serves as the length unit. The semi-minor axis, a r , is 0.188a , and the semi-major axis is 1.3 times that of the semiminor axis. A doubly-degenerated state in the Brillouin zone center is found near the dimensionless frequency, 0.540, marked as "A". In the vicinity of this point, the dispersion relation is linear along the X  direction and quadratic along the Y  direction, which is shown more clearly in Fig. 2(d). (b) and (c) Enlarged views of the band structure for smaller and larger elliptical cylinders. The doubly-degenerated state shown in (a) splits into two single states, marked as A1 and A2, where A1 corresponds to a dipolar state and A2 corresponds to monopolar state.    Fig. 1(a). Blue upper triangles and black squares represent the effective permittivity calculated by using eigenstates along the Y  and X  directions, respectively. They almost overlap indicating that the effective permittivity, eff  , is a scalar and does not depend on the direction. Red circles are eff y  , which cross zero at the same point as eff  and the point is a semi-Dirac point.

Figures
Green lower triangles represent eff x  , which cross zero at dimensionless frequency 0.487.
Note that both blue upper triangles and green lower triangles are missing from the frequency regime at 0.487 to 0.540, which corresponds to a band gap along the Y  direction. No eigenstates are thus available to evaluate the related effective medium parameters. (b)-(d) The electric field for a plane wave impinging on a PhC slab in a waveguide whose walls have PMC boundary conditions at the semi-Dirac frequency 0.540. (b) The real part of the electric field when the incident wave is along Y  direction. The amplitude decays as the wave penetrates into the sample. The imaginary part is orders of magnitude smaller than the real part, which is why it is not shown here.
(c) and (d) The real and imaginary parts of the electric field when the incident wave is along the X  direction. Both suggest that there is no phase change in the sample, which is a typical property of a double-zero-index material. . A similar pattern to that shown in (c) is found.