Abstract 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 (D_{x} and D_{y}) and refraction regions (Δ_{x} and Δ_{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 D_{x, y} < 0, D_{x, y} > 0 and D_{x, y}=0, respectively. And the wave experiences the positive, negative refractions, and non-deflection when the refraction region meets the conditions:Δ_{x, y} < 0, Δ_{x, y} > 0 and Δ_{x, y}=0, respectively. By matching the input wave vectors to the contour lines of D_{x}=0 and D_{y}=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 D_{y}=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 D_{x}=D_{y}=0. There are six intersections on the contour lines of D_{x}=0 and D_{y}=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 k_{x} > 0 and k_{y} > 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.

Fund:Project supported by the National Natural Science Foundation of China (Grant Nos. 61675168, 11634010) and the Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics (Grant No. U1630125).