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The Dirac cones in photonic crystals have aroused much interest in the last few years. Annular photonic crystals have also been well studied for designing and controlling the band gap because they have more parameters than usual photonic crystal. In this paper, we study a two-dimensional square lattice dielectric annular photonic crystal to explore the formation of the photonic Dirac cone by the accidental degeneracy method. The theoretical tool is the plane wave expansion method. The results show that this system can provide a Dirac point in the center of the Brillouin-zone in the photonic band if both the outer radius and the inner radius of each scatterer are chosen to be appreciate values when the dielectric refractive index of the annular rod is fixed. For example, there is a Dirac point at the photonic normalized frequency f=0.438(c/a) when n=3.4, RO=0.42a, RI=0.305a, where f is the frequency, c is the light speed in vacuum, a is the lattice constant, n is the refractive index, RO is the outer radius, and RI is the inner radius. It is also found that within a confined region of outer radius RO(0.37aROa), when a Dirac point is realized in the annular photonic crystal (n>1.4), the inner radius RI and the outer radius RO obey a relation of RI=-1.104+8.167RO+(-11.439)RO2, which is unrelated to the refractive index n of the dielectric annular rod. If n is less than 1.4, this rule is not valid. At the same time, the normalized frequency at which the Dirac point is realized, decreases with increasing both refractive index n and outer radius RO. Especially, the curves of the relation between photonic frequency f and outer radius RO almost do not change their profiles but only be shifted up and down with changing the refractive index n. Based on this, we also design and predict the annular photonic crystal which provides a Dirac point. The goal is to obtain the other relative parameters (frequency f, outer radius RO and the inner radius RI) of the photonic crystal system if the refractive index n is fixed. The values of the prediction agree very well with the values obtained by the rigid theoretical calculation within a relative error of only 4%.
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Keywords:
- annular photonic crystals /
- the Dirac point /
- plane wave expansion method /
- accidental degeneracy
[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666
[2] Geim A K, Novoselov K S 2007 Nat. Mater. 6 183
[3] Liu Y H, He L, Shi Y L 2012 J. Opt. Soc. Am. B: Opt. Phys. 29 621
[4] Zhang X D 2008 Phys. Lett. A 372 3512
[5] Wang L G, Wang Z G, Zhang J X, Zhu S Y 2009 Opt. Lett. 34 1510
[6] Bittner S, Dietz B, Miski-Oglu M, Oria-Iriarte P, Richter A, Schafer F 2010 Phys. Rev. B 82 014301
[7] Katsnelson M I 2006 Eur. Phys. J. B 51 157
[8] Sepkhanov R A, Beenakker C W J 2008 Opt. Commun. 281 5267
[9] Guo H, Liu H G, Zhang X, Chen H J, Wang W X, Wang S K, Cui Y P 2013 Appl. Phys. Express 6 042003
[10] Huang X Q, Lai Y, Hang Z H, Zheng H H, Chan C T 2011 Nat. Mater. 10 582
[11] Sakoda K 2014 Int. J. Mod. Phys. B 28 1441008
[12] Kurt H, Citrin D S 2005 Opt. Express 13 10316
[13] Kurt H, Hao R, Chen Y, Feng J, Blair J, Gaillot D P, Summers C, Citrin D S, Zhou Z 2008 Opt. Lett. 33 1614
[14] Wu H, Citrin D S, Jiang L Y, Li X Y 2013 Appl. Phys. Lett. 102 141112
[15] Xia F, Yun M, Liu M, Liang J, Kong W, Tan H, Lü W 2013 J. Appl. Phys. 113 013109
[16] Yucel M B, Cicek A, Ulug B 2013 Photon. Nanostruct. 11 270
[17] Jiang L, Wu H, Li X 2013 J. Opt. Soc. Am. B: Opt. Phys. 30 1248
[18] Guo S, Albin S 2003 Opt. Express 11 167
[19] Joannopoulos J D, Johnson S G, Winn J N, Meade R D 2011 Photonic Crystals: Molding the Flow of Light (Princeton: Princeton University Press) pp20-21
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[1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666
[2] Geim A K, Novoselov K S 2007 Nat. Mater. 6 183
[3] Liu Y H, He L, Shi Y L 2012 J. Opt. Soc. Am. B: Opt. Phys. 29 621
[4] Zhang X D 2008 Phys. Lett. A 372 3512
[5] Wang L G, Wang Z G, Zhang J X, Zhu S Y 2009 Opt. Lett. 34 1510
[6] Bittner S, Dietz B, Miski-Oglu M, Oria-Iriarte P, Richter A, Schafer F 2010 Phys. Rev. B 82 014301
[7] Katsnelson M I 2006 Eur. Phys. J. B 51 157
[8] Sepkhanov R A, Beenakker C W J 2008 Opt. Commun. 281 5267
[9] Guo H, Liu H G, Zhang X, Chen H J, Wang W X, Wang S K, Cui Y P 2013 Appl. Phys. Express 6 042003
[10] Huang X Q, Lai Y, Hang Z H, Zheng H H, Chan C T 2011 Nat. Mater. 10 582
[11] Sakoda K 2014 Int. J. Mod. Phys. B 28 1441008
[12] Kurt H, Citrin D S 2005 Opt. Express 13 10316
[13] Kurt H, Hao R, Chen Y, Feng J, Blair J, Gaillot D P, Summers C, Citrin D S, Zhou Z 2008 Opt. Lett. 33 1614
[14] Wu H, Citrin D S, Jiang L Y, Li X Y 2013 Appl. Phys. Lett. 102 141112
[15] Xia F, Yun M, Liu M, Liang J, Kong W, Tan H, Lü W 2013 J. Appl. Phys. 113 013109
[16] Yucel M B, Cicek A, Ulug B 2013 Photon. Nanostruct. 11 270
[17] Jiang L, Wu H, Li X 2013 J. Opt. Soc. Am. B: Opt. Phys. 30 1248
[18] Guo S, Albin S 2003 Opt. Express 11 167
[19] Joannopoulos J D, Johnson S G, Winn J N, Meade R D 2011 Photonic Crystals: Molding the Flow of Light (Princeton: Princeton University Press) pp20-21
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