In this paper, based on the Euler–Lagrange equation, an ABCD matrix is constructed out to study the paraxial transmission of light on a constant Gaussian curvature surface (CGCS), which is the first time to our knowledge. Then, by using the method of matrix optics, we extend the CGCS matrix to a general transfer matrix which is suitable for a gently varying curvature. As a beam propagation example, based on the Collins integral and the derived ABCD matrix elements, an analytical propagation formula for the hollow Gaussian beams (HGBs) on the CGCS is deduced. The propagation characteristics of HGBs on a CGCS are illustrated graphically in detail, mainly including the change of dark spot size and splitting rays. Besides its propagation periodicity and diffraction properties, a criterion for convergence and divergence of the spot size is proposed. The area of the dark region of the HGBs can easily be controlled by proper choice of the beam parameters and the shape of CGCS. In addition, we also study the special propagation properties of the hollow beam with fractional order. Compared with propagation characteristics of HGBs in flat space, these novel propagation characteristics of HGBs on curved surface may further expand the application range of hollow beams.

在本文中，基于欧拉-拉格朗日方程，构造了ABCD矩阵来研究光在恒定高斯曲率表面 (CGCS) 上的近轴传输，这是我们所知的第一次。然后，通过使用矩阵光学的方法，我们将CGCS矩阵扩展为适用于曲率缓慢变化的通用传递矩阵。作为光束传播的例子，基于柯林斯积分和导出的ABCD矩阵元，推导出CGCS上空心高斯光束（HGBs）的解析传播公式。HGBs在CGCS上的传播特性用图形详细说明，主要包括暗点大小的变化和分裂光线。除了其传播周期和衍射特性外，还提出了光斑尺寸收敛和发散的标准。The area of​​ the dark region of the HGBs can easily be controlled by proper choice of the beam parameters and the shape of CGCS. 此外，我们还研究了分数阶空心光束的特殊传播特性。与HGBs在平坦空间中的传播特性相比，HGBs在曲面上的这些新颖的传播特性可能会进一步扩大空心光束的应用范围。