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Recently, Chan and his collaborators reported that a crossing point of bands can be achieved at the Brillouin zone center in two-dimensional (2D) dielectric photonic crystals (PhCs) by accidental degeneracy of modes. At the crossing point, the accidental threefold degeneracy of modes generates a Dirac cone and an additional flat band (longitudinal mode) intersecting the Dirac cone. This is different from that of the Dirac point at the corner of the hexagonal Brillouin zone in which only Dirac cone exists. As a result, the crossing point at the Brillouin zone center is called a Dirac-like point. If the accidental degeneracy occurs by a monopole mode and two dipolar modes, the dielectric PhCs can be mapped to a zero-refractive-index system in which the effective permittivity and permeability are zero at the Dirac-like point from the effective medium theory. According to the Maxwell equations, if the permittivity and permeability are zero, the optical longitudinal modes can exist, in additional to the well-known transverse modes. The additional flat band at the Dirac-like point is closely connected with the longitudinal mode. For a homogeneous zero-index material (ZIM), the flat band is dispersionless and the longitudinal mode cannot couple with the external light. But in a finite-sized PhC, there is always some spatial dispersion, so the flat band is not perfectly dispersionless when it is away from the zone center. Therefore, if the wave source is a Gaussian beam with non-zero k-parallel components, the longitudinal mode can be excited. And the effective wavelength of ZIM is extremely large, leading to many scattering properties. However, in a PhC which behaves as if it had a zero refractive index, it is very interesting to show how the longitudinal mode influences the wave propagations in the PhC when the longitudinal mode is excited. In this paper, the effect of longitudinal mode on the transmission properties near the Dirac-like point of PhCs is investigated by numerical simulation. The alumina dielectric rods can be moved randomly in the structure to result in the disorder of the structure. Our results show that the transmission properties at the Dirac-like point are very different from those near the Dirac-like point, when the longitudinal mode is excited. At the Dirac-like point, the transmittance decreases with increasing disorder, as a result of the influence of the longitudinal mode, which is similar to the one in the pass band. Above the Dirac-like point without the disturbance of longitudinal mode, the transmittance is insensitive to the disorder in the structure, so that the structure may mimic a near-zero index materials and have a large effective wavelength. These results may further improve the understanding about the optical longitudinal mode and the zero refractive material.
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Keywords:
- photonic crystals /
- disorder /
- Dirac-like point
[1] Zhong K, Zhang H Y, Zhang Y P, Li X F, Wang P, Yao J Q 2007 Acta Phys. Sin. 56 7029 (in Chinese) [钟凯, 张会云, 张玉萍, 李喜福, 王鹏, 姚建铨 2007 物理学报 56 7029]
[2] Mei J, Wu Y, Chan C T, Zhang Z Q 2012 Phys. Rev. B 86 035141
[3] Sepkhanov R A, Bazaliy Ya B, Beenakker C W J 2007 Phys. Rev. A 75 063813
[4] Zandbergen Sander R, de Dood Michiel J A 2010 Phys. Rev. Lett. 104 043903
[5] Bittner S, Dietz B, Miski-Oglu M, Richter A 2012 Phys. Rev. B 85 064301
[6] Zhang X D 2008 Phys. Rev. Lett. 100 113903
[7] Bahat. -Treidel O, Peleg O, Grobman M, Shapira N, Segev M, Pereg-Barnea T 2010 Phys. Rev. Lett. 104 063901
[8] Poo Y, Wu R X, Lin Z F, Yang Y, Chan C T 2011 Phys. Rev. Lett. 106 093903
[9] Ouyang C F, Han D Z, Zhao F Y, Hu X H, Liu X H, Zi J 2012 J. Phys. : Condens. Matter 24 492203
[10] Kuhl U, Barkhofen S, Tudorovskiy T, Stöckmann H-J, Hossain T, de Forges de Parny L, Mortessagne F 2010 Phys. Rev. B 82 094308
[11] Plotnik Y, Rechtsman M C, Song D, Heinrich M, Zeuner J M, Nolte S, Lumer Y, Malkova N, Xu J, Szameit A, Chen Z, Segev M 2014 Nat. mater. 13 57
[12] Sepkhanov R A, Ossipov A, Beenakker C W J 2009 EPL 85 14005
[13] Wang X, Jiang H T, Yan C, Sun Y, Li Y H, Shi Y L, Chen H 2013 EPL 103 17003
[14] Bellec M, Kuhl U, Montambaux G, Mortessagne F 2013 Phys. Rev. Lett. 110 033902
[15] Rechtsman M C, Zeuner J M, Tünnermann A, Stefan Nolte, Segev M, Szamerit A 2013 Nat. photonics 7 153
[16] Huang X Q, Lai Y, Hang Z H, Zheng H H, Chan C T 2011 Nat. Mater. 10 582
[17] Sakoda K 2012 Opt. Express 20 25181
[18] Li Y, Wu Y, Chen X, Mei J 2013 Opt. Express 21 7699
[19] Chan C T, Hang Z H, Huang X 2012 Adv. in OptoElectron. 2012 313984
[20] Liu F M, Lai Y, Huang X Q, Chan C T 2011 Phys. Rev. B 84 224113
[21] Moitra P, Yang Y, Anderson Z, Kravchenko I I, Briggs D P, Valentine J 2013 Nat. photonics 7 791
[22] Zhao H, Shen Y F, Zhang Z J 2014 Acta Phys. Sin. 63 174204 (in Chinese) [赵浩, 沈义峰, 张中杰 2014 物理学报 63 174204]
[23] D'Aguanno G, Mattiucci N, Conti C, Bloemer M J 2013 Phys. Rev. B 87 085135
[24] Mattiucci N, Bloemer M J, D'Aguanno G 2013 Opt. Express 21 11862
[25] Wang X, Jiang H T, Yan C, Deng F S, Sun Y, Li Y H, Shi Y L, Chen H 2014 EPL 108 14002
[26] Yang Y B, Wang S F, Li X J, Wang Y C, Liang W 2010 Acta Phys. Sin. 59 5073 (in Chinese) [杨毅彪, 王拴锋, 李秀杰, 王云才, 梁伟 2010 物理学报 59 5073]
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[1] Zhong K, Zhang H Y, Zhang Y P, Li X F, Wang P, Yao J Q 2007 Acta Phys. Sin. 56 7029 (in Chinese) [钟凯, 张会云, 张玉萍, 李喜福, 王鹏, 姚建铨 2007 物理学报 56 7029]
[2] Mei J, Wu Y, Chan C T, Zhang Z Q 2012 Phys. Rev. B 86 035141
[3] Sepkhanov R A, Bazaliy Ya B, Beenakker C W J 2007 Phys. Rev. A 75 063813
[4] Zandbergen Sander R, de Dood Michiel J A 2010 Phys. Rev. Lett. 104 043903
[5] Bittner S, Dietz B, Miski-Oglu M, Richter A 2012 Phys. Rev. B 85 064301
[6] Zhang X D 2008 Phys. Rev. Lett. 100 113903
[7] Bahat. -Treidel O, Peleg O, Grobman M, Shapira N, Segev M, Pereg-Barnea T 2010 Phys. Rev. Lett. 104 063901
[8] Poo Y, Wu R X, Lin Z F, Yang Y, Chan C T 2011 Phys. Rev. Lett. 106 093903
[9] Ouyang C F, Han D Z, Zhao F Y, Hu X H, Liu X H, Zi J 2012 J. Phys. : Condens. Matter 24 492203
[10] Kuhl U, Barkhofen S, Tudorovskiy T, Stöckmann H-J, Hossain T, de Forges de Parny L, Mortessagne F 2010 Phys. Rev. B 82 094308
[11] Plotnik Y, Rechtsman M C, Song D, Heinrich M, Zeuner J M, Nolte S, Lumer Y, Malkova N, Xu J, Szameit A, Chen Z, Segev M 2014 Nat. mater. 13 57
[12] Sepkhanov R A, Ossipov A, Beenakker C W J 2009 EPL 85 14005
[13] Wang X, Jiang H T, Yan C, Sun Y, Li Y H, Shi Y L, Chen H 2013 EPL 103 17003
[14] Bellec M, Kuhl U, Montambaux G, Mortessagne F 2013 Phys. Rev. Lett. 110 033902
[15] Rechtsman M C, Zeuner J M, Tünnermann A, Stefan Nolte, Segev M, Szamerit A 2013 Nat. photonics 7 153
[16] Huang X Q, Lai Y, Hang Z H, Zheng H H, Chan C T 2011 Nat. Mater. 10 582
[17] Sakoda K 2012 Opt. Express 20 25181
[18] Li Y, Wu Y, Chen X, Mei J 2013 Opt. Express 21 7699
[19] Chan C T, Hang Z H, Huang X 2012 Adv. in OptoElectron. 2012 313984
[20] Liu F M, Lai Y, Huang X Q, Chan C T 2011 Phys. Rev. B 84 224113
[21] Moitra P, Yang Y, Anderson Z, Kravchenko I I, Briggs D P, Valentine J 2013 Nat. photonics 7 791
[22] Zhao H, Shen Y F, Zhang Z J 2014 Acta Phys. Sin. 63 174204 (in Chinese) [赵浩, 沈义峰, 张中杰 2014 物理学报 63 174204]
[23] D'Aguanno G, Mattiucci N, Conti C, Bloemer M J 2013 Phys. Rev. B 87 085135
[24] Mattiucci N, Bloemer M J, D'Aguanno G 2013 Opt. Express 21 11862
[25] Wang X, Jiang H T, Yan C, Deng F S, Sun Y, Li Y H, Shi Y L, Chen H 2014 EPL 108 14002
[26] Yang Y B, Wang S F, Li X J, Wang Y C, Liang W 2010 Acta Phys. Sin. 59 5073 (in Chinese) [杨毅彪, 王拴锋, 李秀杰, 王云才, 梁伟 2010 物理学报 59 5073]
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