光线庞加莱球法构建的结构光场及其传输特性研究

结构光场在光信息传输、显微成像以及微粒俘获中有重要作用.本文基于光线庞加莱球法结合不同花瓣数的梅花曲线构建了一类结构光束.根据光线庞加莱球法可计算这类光束在束腰面上的光强与相位分布，以及光束内外焦散线的分布.这些焦散线的特征表明梅花形结构光束具有无衍射与自修复的特性.进一步采用角谱衍射法和光线追迹研究了这类光束在空间中的传输特性.当梅花花瓣数为0时，该光束退化为拉盖尔-高斯光束；当花瓣数为1时，内焦散线汇集到两点，使光束具有无衍射特性.通过光线庞加莱球法获得其光线在空间传播的轨迹，直观地展示了光束被遮挡后的自修复特性.此外，本文还展示了花瓣数为5的结构光束，其内焦散线为十角星结构，该光束同样具有自修复特性.通过修改梅花曲线参数或选择其他庞加莱球面曲线可以构造更加复杂的结构光场.

Structured beam designed by ray-optical Poincaré sphere method and its propagation properties

Structured beam plays an important role in optical communication, microscopy and particle manipulations. Traditionally, structured beam can be obtained by solving Helmholtz wave equation. This method involves complex mathematical procedures, and the properties of solved light beam are obscure. It is worth noting that the structured beam can also be constructed by ray-optical Poincaré sphere method: this method is a rather intuitive and convenient for designing the structured beam with novel properties. This method also provides a ray-based way to study the propagation properties of structured beam. In this paper, the ray-optical Poincaré sphere method combined with plum-blossom curve is used to build a family of structured beams. The optical field distributions on beam waist, including intensity and phase, are calculated by the ray-optical Poincaré sphere method. The shape of inner and outer caustics of optical field are also detailed in order to demonstrate the self-healing or non-diffraction features of beams. By using angular spectrum diffraction, the free space evolutions of such structured beams are demonstrated. The results show that the structured beam turns to be the well-known Laguerre-Gaussian beam when the leaf number of plum-blossom curve is 0. While the leaf number equals 1, the structured beam has non-diffraction property, for its inner caustic concentrates onto two points. In geometrical optics sight, all light rays are tangent to the inner caustic, and the optical fields carried by rays interfere near the caustic, leading the beam to possess a self-healing capacity. The self-healing property is demonstrated in terms of rays. With the beam's propagating, rays which launch from the inner side of beam gradually reach the outer side of beam. On the contrary, the rays launching from the inner side of beam arrive at the outer side of beam. When the center of beam is blocked, the inner rays are also blocked. After propagating, outer side rays will reach the inner side, fill up the hole of beam, and recover the injury of optical field. Furthermore, we demonstrate the structured beam with a 5leave plum-blossom curve. In this case, the inner caustic of this beam turns into a decagonal star structure; our simulation results show that this beam has relatively strong self-healing capability. Theoretically, one can simply change the parameters of plum-blossom curve or choose other kind of Poincaré sphere curve to create more complex structured beams.