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Photonic band-gap of light wave in spatial frequency model depicts the linear propagation characteristics of the light wave in period structures, based on which the linear diffraction and refraction of light are defined. In this paper, we numerically study the non-diffraction propagation and anomalous refraction of light waves in honeycomb photonic lattices according to the diffraction relationship of the photonic band-gap.By calculating the photonic band-gap structure, the linear propagation characteristics in the first transmission band are analyzed. The first Brillouin zone of the honeycomb lattice can be divided into different diffraction (Dx and Dy) and refraction regions (Δx and Δy), according to the definitions of light diffraction and refraction along the x-and y-axis. Light wave can present normal, anomalous diffraction and even non-diffraction when the wave vector matches the regions of Dx, y Dx, y > 0 and Dx, y=0, respectively. And the wave experiences the positive, negative refractions, and non-deflection when the refraction region meets the conditions:Δx, y x, y > 0 and Δx, y=0, respectively.By matching the input wave vectors to the contour lines of Dx=0 and Dy=0, we can realize the non-diffraction propagation along the x-and y-axis, respectively. When the input wave vector is set to be (0, 0), the light wave experiences normal diffraction and beam size is broadened. When the wave vector matches the point where Dy=0, the diffraction in the y-axis is obviously suppressed. To totally restrain the beam diffraction, the wave vector is set to be at the point where Dx=Dy=0. There are six intersections on the contour lines of Dx=0 and Dy=0, and these intersections are named non-diffraction points.The refraction of light can be also controlled by adjusting the input wave vector. When the wave vector is located on the contours of Δy=0, light wave propagates along the x-axis, without shifting along the y-axis. To excite the negative refractions, we need to match the input light wave to the eigen modes of the lattice, and adjust the wave vector to the negative refraction regions. We set the input wave vector to be kx > 0 and ky > 0, so that the beam would be output in the first quadrant of the coordinate if refracted normally. The eigen modes are approximated by multi-wave superposition, and the wave vector is adjusted to different refraction regions. From the numerical results of the light propagations, it is clearly seen that the propagations of a good portion of light energy follow the preconceived negative refractions, and output field is in the fourth, third, second, and third quadrant, respectively. Notably, the light waves generated by multi-wave superposition not only contain the eigen modes we need, but also include other modes. As a result, there are also energy outputs arising from the undesired modes in the other quadrants.The above conclusions are expected to provide a reference for the optical mechanisms of graphene-like optical phenomena in honeycomb photonic lattices.
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Keywords:
- photonic lattice /
- photonic band-gap /
- anomalous diffraction /
- negative refraction
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[21] Song D, Liu S, Paltoglou V, Gallardo D, Tang L, Zhao J, Xu J, Efremidis N K, Chen Z 2015 D Mater. 2 034007
[22] Durnin J, Miceli J J, Eberly J H 1987 Phys. Rev. Lett. 58 1499
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[1] Eisenberg H S, Silberberg Y S 2000 Phys. Rev. Lett. 85 1863
[2] Pertsch T, Zentgraf T, Peschel U, Bräuer A, Lederer F 2002 Phys. Rev. Lett. 88 093901
[3] Rosberg C R, Neshev D N, Sukhorukov A A, Kivshar Y S, Krolikowski W 2005 Opt. Lett. 30 2293
[4] Liu S, Zhang P, Xiao F J, Yang D X, Zhao J L 2009 Sci. China G 52 747
[5] Zhang P, Liu S, Zhao J L, Lou C B, Xu J J, Chen Z G 2008 Opt. Lett. 33 878
[6] Liu S, Hu Y, Zhang P, Gan X T, Xiao F J, Lou C B, Song D H, Zhao J L, Xu J J, Chen Z G 2011 Opt. Lett. 36 1167
[7] Zhang P, Lou C B, Liu S, Zhao J L, Xu J J, Chen Z G 2010 Opt. Lett. 35 892
[8] Liu S, Hu Y, Zhang P, Gan X T, Lou C B, Song D H, Zhao J L, Xu J J, Chen Z G 2012 Opt. Lett. 37 2184
[9] Liu S, Hu Y, Zhang P, Gan X T, Lou C B, Song D H, Zhao J L, Xu J J, Chen Z G 2012 Appl. Phys. Lett. 100 061907
[10] Yeh P, Yariv A, Hong C 1977 J. Opt. Soc. Am. 67 423
[11] Trompeter H, Krolikowski W, Neshev D N, Desyatnikov A S, Sukhorukov A A, Kivshar Y S, Pertsch T, Peschel U, Lederer F 2006 Phys. Rev. Lett. 96 053903
[12] Liu S, Rao B J, Wang M R, Zhang P, Xiao F J, Gan X T, Zhao J L 2017 Opt. Express 25 7475
[13] Schwartz T, Bartal G, Fishman S, Segev M 2007 Nature 446 52
[14] Peleg O, Bartal G, Freedman B, Manela O, Segev M, Christodoulides D N 2007 Phys. Rev. Lett. 98 103901
[15] Bahat-Treidel O, Peleg O, Grobman M, Shapira N, Segev M, Pereg-Barnea T 2010 Phys. Rev. Lett. 104 063901
[16] Rechtsman M C, Plotnik Y, Zeuner J M, Song D, Chen Z, Szameit A, Segev M 2013 Phys. Rev. Lett. 111 103901
[17] Plotnik Y, Rechtsman M C, Song D, Heinrich M, Zeuner J M, Nolte S, Lumer Y, Malkova N, Xu J, Szameit A 2014 Nat. Mater. 13 57
[18] Rechtsman M C, Zeuner J M, Tnnermann A, Nolte S, Segev M, Szameit A 2013 Nat. Photon. 7 153
[19] Rechtsman M C, Zeuner J M, Plotnik Y, Lumer Y, Podolsky D, Dreisow F, Nolte S, Segev M, Szameit A 2013 Nature 496 196
[20] Song D, Paltoglou V, Liu S, Zhu Y, Gallardo D, Tang L, Xu J, Ablowitz M, Efremidis N K, Chen Z 2015 Nat. Commun. 6 6272
[21] Song D, Liu S, Paltoglou V, Gallardo D, Tang L, Zhao J, Xu J, Efremidis N K, Chen Z 2015 D Mater. 2 034007
[22] Durnin J, Miceli J J, Eberly J H 1987 Phys. Rev. Lett. 58 1499
[23] Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901
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