Conical Dispersion and Effective Zero Refractive Index in Photonic Quasicrystals

Abstract

It is recognized that for a certain class of periodic photonic crystals, conical dispersion can be related to a zero-refractive index. It is not obvious whether such a notion can be extended to a noncrystalline system. We show that certain photonic quasicrystalline approximants have conical dispersions at the zone center with a triply degenerate state at the Dirac frequency, which is the necessary condition to qualify as a zerorefractive-index medium. The states in the conical dispersions are extended and have a nearly constant phase. Experimental characterizations of finite-sized samples show evidence that the photonic quasicrystals do behave as a near zero-refractive-index material around the Dirac frequency. [10]. However, the connection between conical dispersions at k ¼ 0 and zero-refractive index were built upon periodicity. The conical dispersion was obtained by tuning the system parameters of "one-atom-per-unit-cell" photonic crystals with a well-defined photonic band structure. Whether a conical dispersion can exist in a nonperiodic system is still an interesting and open question. Furthermore, the claim that a system behaves like a zero-index medium implicitly assumes that an effective medium description could be applied. While not explicitly stated, many coherent-potential-approximation-type effective medium theories, employed to map a Dirac cone to zero index [11], assume that each scatter resides in the same environment. Although this assumption of periodicity is not needed in the ω → 0, k → 0 limit, it is not immediately obvious that such effective medium description can be applied to nonperiodic systems if we consider effective parameters at k → 0 but at a finite frequency such as a Dirac point. Can a nonperiodic system behave operationally as if it has zero-refractive index?Photonic quasicrystal (PQC) is constructed by building blocks positioned on well-designed patterns but lacks translational symmetry [12][13][14][15][16][17][18][19][20][21][22][23]. Nonetheless, PQC can still have relatively sharp diffraction patterns due to longrange order. Such patterns confirm the existence of wave scattering and interference, providing similar functionalities as periodic counterparts, such as photonic band gaps [12][13][14][15], negative refraction [16], lasing [17][18][19], and nonlinear light propagations [20][21][22]. We will show theoretically and experimentally that some two-dimensional photonic quasicrystalline approximants can possess conical dispersion at k ¼ 0, and their finite-sized counterparts can behave like a zero-refractive-index medium as evidenced by different experimental measurements.In this Letter, we show the existence of conical dispersions and extended states with zero-refractive-index characteristics in some PQCs, and experimentally characterize these states. The extended states are close to the Dirac frequency, and form two cones intersecting at a "Dirac point." The eigenmodes have almost constant field intensity at each quasicrystalline site, regardless of the size and the bounda...

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