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 (?x and ?y), 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 < 0, 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: ?x,y < 0, ?x,y > 0 and ?x,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 ?y = 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.%空间频率模式的光子带隙反映了光波在周期性结构中的线性传输特性.以这种线性传输特性为基础,研究了蜂巢光子晶格中光波的无衍射和反常折射.通过详细分析带隙结构第一通带上的衍射与折射特性,得出了光波发生反常衍射和折射的入射条件.匹配不同的入射条件,数值模拟了光波的无衍射传输和反常折射现象.结果表明:将入射光束的波矢设置在蜂巢晶格布里渊区中正常、反常衍射区的交界处,可使高斯光束沿x轴、y轴方向的衍射得到有效抑制;以多光束干涉场作为入射光场,可对蜂巢晶格进行模式匹配,激发第二布里渊区的传输模式;进一步将模式匹配后入射光场的波矢设置在反常折射区,可实现光波的反常折射.
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