Abstract
In this paper, we numerically study the threefold accidental degeneracy conical dispersion (Driac-like cone) at the Brillouin zone center of the two-dimensional photonic crystals, which are composed of silicon pillars arranging in a triangular lattice. The effective permittivity and permeability near the Dirac-like point evolve from negative to positive by using the method of eigen-field averaging. Also, the isotropic behaviour of the Dirac-like cone is revealed by analysing the isofrequency contours. Moreover, we carry out numerical simulations including the reverse Snell’s law effect, negative Goos–Hänchen shifts and superfocusing lens to verify the negative refractive index characteristics of the designed structure. The proposed structure might find significant applications in the on-chip photonic interconnect and the photonic integrated circuit techniques.
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AbdelMalek, F., Belhadj, W., Haxha, S., Bouchriha, H.: Realization of a high coupling efficiency by employing a concave lens based on two-dimensional photonic crystals with a negative refractive index. J. Lightwave Technol. 25(10), 3168–3174 (2007). https://doi.org/10.1109/jlt.2007.904027
Berman, P.R.: Goos–Hanchen shift in negatively refractive media. Phys. Rev. E 66(6), 067603 (2002). https://doi.org/10.1103/PhysRevE.66.067603
Chen, C.W., Lin, W.C., Liao, L.S., Lin, Z.H., Chiang, H.P., Leung, P.T., Sijercic, E., Tse, W.S.: Optical temperature sensing based on the Goos–Hanchen effect. Appl. Opt. 46(22), 5347–5351 (2007). https://doi.org/10.1364/ao.46.005347
Chen, J., Wang, Y., Jia, B., Geng, T., Li, X., Feng, L., Qian, W., Liang, B., Zhang, X., Gu, M., Zhuang, S.: Observation of the inverse Doppler effect in negative-index materials at optical frequencies. Nat. Photonics 5(4), 239–242 (2011). https://doi.org/10.1038/nphoton.2011.17
Dong, J.W., Chang, M.L., Huang, X.Q., Hang, Z.H., Zhong, Z.C., Chen, W.J., Huang, Z.Y., Chan, C.T.: Conical dispersion and effective zero refractive index in photonic quasicrystals. Phys. Rev. Lett. (2015). https://doi.org/10.1103/physrevlett.114.163901
Dubois, M., Shi, C., Zhu, X., Wang, Y., Zhang, X.: Observation of acoustic Dirac-like cone and double zero refractive index. Nat. Commun. 8, 14871 (2017). https://doi.org/10.1038/ncomms14871
Foteinopoulou, S., Soukoulis, C.M.: Negative refraction and left-handed behavior in two-dimensional photonic crystals. Phys. Rev. B (2003). https://doi.org/10.1103/physrevb.67.235107
Hänchen, F.G.H.: Ein neuer und fundamental Versuch zur Totalreflexion. Ann. Phys. 1, 333–346 (1947)
He, X.T., Huang, Z.Z., Chang, M.L., Xu, S.Z., Zhao, F.L., Deng, S.Z., She, J.C., Dong, J.W.: Realization of zero-refractive-index lens with ultralow spherical aberration. ACS Photon. 3(12), 2262–2267 (2016). https://doi.org/10.1021/acsphotonics.6b00714
Huang, Z., Narimanov, E.E.: Optical imaging with photonic hyper-crystals: Veselago lens and beyond. J. Opt. 16(11), 114009 (2014). https://doi.org/10.1088/2040-8978/16/11/114009
Huang, X.Q., Lai, Y., Hang, Z.H., Zheng, H.H., Chan, C.T.: Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials. Nat. Mater. 10(8), 582–586 (2011). https://doi.org/10.1038/nmat3030
Husakou, A., Herrmann, J.: Superfocusing of optical beams below the diffraction limit by media with negative refraction. Phys. Status Solidi A 204(11), 3862–3877 (2007). https://doi.org/10.1002/pssa.200776402
Johnson, N.P., Tsakmakidis, K.L., Özbay, E., Kirby, E.I., Hamm, J., Ziolkowski, R.W., Zheludev, N.I., Hess, O.: Trapped rainbow storage of light in metamaterials. Nature 7711, 77111C (2010). https://doi.org/10.1117/12.855120
Li, Z.F., Zhao, R.K., Koschny, T., Kafesaki, M., Alici, K.B., Colak, E., Caglayan, H., Ozbay, E., Soukoulis, C.M.: Chiral metamaterials with negative refractive index based on four “U” split ring resonators. Appl. Phys. Lett. 97(8), 081901 (2010). https://doi.org/10.1063/1.3457448
Li, Y., Kita, S., Munoz, P., Reshef, O., Vulis, D.I., Yin, M., Loncar, M., Mazur, E.: On-chip zero-index metamaterials. Nat. Photonics 9(11), 738–745 (2015). https://doi.org/10.1038/nphoton.2015.198
Lu, Z.L., Prather, D.W.: Calculation of effective permittivity, permeability, and surface impedance of negative-refraction photonic crystals. Opt. Express 15(13), 8340–8345 (2007). https://doi.org/10.1364/oe.15.008340
Marques, R., Martel, J., Mesa, F., Medina, F.: Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides. Phys. Rev. Lett. 89(18), 183901 (2002). https://doi.org/10.1103/PhysRevLett.89.183901
Moitra, P., Yang, Y.M., Anderson, Z., Kravchenko, I.I., Briggs, D.P., Valentine, J.: Realization of an all-dielectric zero-index optical metamaterial. Nat. Photonics 7(10), 791–795 (2013). https://doi.org/10.1038/nphoton.2013.214
Notomi, M.: Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap. Phys. Rev. B 62(16), 10696–10705 (2000). https://doi.org/10.1103/PhysRevB.62.10696
Pendry, J.B.: Negative refraction makes a perfect lens. Phys. Rev. Lett. 85(18), 3966–3969 (2000). https://doi.org/10.1103/PhysRevLett.85.3966
Ramadurgam, S., Lin, T.G., Yang, C.: Tailoring optical and plasmon resonances in core-shell and core-multishell nanowires for visible range negative refraction and plasmonic light harvesting: a review. J. Mater. Sci. Technol. 31(6), 533–541 (2015). https://doi.org/10.1016/j.jmst.2015.01.004
Turduev, M., Hayran, Z., Kurt, H.: Focusing of light beyond the diffraction limit by randomly distributed graded index photonic medium. J. Appl. Phys. 120(24), 243102 (2016). https://doi.org/10.1063/1.4972980
Valentine, J., Zhang, S., Zentgraf, T., Zhang, X.: Development of bulk optical negative index fishnet metamaterials: achieving a low-loss and broadband response through coupling. Proc. IEEE 99(10), 1682–1690 (2011). https://doi.org/10.1109/jproc.2010.2094593
Veselago, V.G.: The electrodynamics of substances with simultaneously negative values of ε and μ. Sov. Phys. Usp. 92, 517–526 (1968)
Wang, L.-G., Zhu, S.-Y.: Large positive and negative Goos–Hänchen shifts from a weakly absorbing left-handed slab. J. Appl. Phys. 98(4), 043522 (2005). https://doi.org/10.1063/1.2034084
Wang, L.G., Chen, H., Zhu, S.Y.: Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials. Phys. Rev. B 70(24), 245102 (2004). https://doi.org/10.1103/PhysRevB.70.245102
Wang, J.R., Chen, X.D., Zhao, F.L., Dong, J.W.: Full polarization conical dispersion and zero-refractive-index in two-dimensional photonic hypercrystals. Sci. Rep. 6, 22739 (2016). https://doi.org/10.1038/srep22739
Wang, X.F., Ye, Q.B., Zhou, Y.J., Li, Y., Yu, H., Yang, J.Y., Jiang, X.Q.: Theoretical large positive and negative lateral beam shift of the metal cladding waveguide in the mid-infrared region. Appl. Opt. 57(16), 4714–4717 (2018). https://doi.org/10.1364/ao.57.004714
Yallapragada, V.J., Ravishankar, A.P., Mulay, G.L., Agarwal, G.S., Achanta, V.G.: Observation of giant Goos–Hanchen and angular shifts at designed metasurfaces. Sci. Rep. 6, 19319 (2016). https://doi.org/10.1038/srep19319
Yannopapas, V., Moroz, A.: Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges. J. Phys. Condens. Matter 17(25), 3717–3734 (2005). https://doi.org/10.1088/0953-8984/17/25/002
Zhang, S., Fan, W.J., Panoiu, N.C., Malloy, K.J., Osgood, R.M., Brueck, S.R.J.: Experimental demonstration of near-infrared negative-index metamaterials. Phys. Rev. Lett. 95(13), 137404 (2005). https://doi.org/10.1103/PhysRevLett.95.137404
Acknowledgements
The authors are grateful to the support by the Natural Science Fund of China under Grant No. 11774103, Project for Cultivating Postgraduates’ Innovative Ability in Scientific Research of Huaqiao University (17013082023), Quanzhou City Science & Technology Program of China under Grant No. 2018C003, Open Project of Fujian Key Laboratory of Semiconductor Materials and Applications under No. 2019001.
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Wang, Z., Pan, G., Qiu, W. et al. Refractive index from negative to positive with frequencies at the Dirac-like cone in a photonic crystal. Opt Quant Electron 51, 209 (2019). https://doi.org/10.1007/s11082-019-1925-6
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DOI: https://doi.org/10.1007/s11082-019-1925-6