Hollow higher-order cosh-Gaussian (HhCG) beam is a new mathematical model to describe the dark-hollow beam. An analytical expression for HhCG beams propagating through a paraxail ABCD optical system is derived and used to investigate its propagation properties in a quadratic index medium (QIM) and a Fractional Fourier transform (FRFT) optical system. By using the derived expressions, the properties of HhCG beams of both systems are graphically illustrated with numerical example. Several influence parameters on the evolution of HhCG in the both systems QIM and FRFT, respectively, are discussed in detail. Results show that the intensity distribution of HhCG evolves periodically during propagation in the QIM. In FRFT system, it is found that the normalized intensity distribution of HhCG beams plane depends not only on the fractional order but also on the beam order and the truncation parameter. The results obtained in this work are valuable to provide a convenient way for HhCG beams shaping, optical fibers, lenses, and other fields.

中文翻译：

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二次折射率介质和分数阶傅里叶变换中空心高阶cosh-Gaussian光束的传播特性

空心高阶cosh-Gaussian (HhCG)光束是一种新的描述暗空心光束的数学模型。推导出通过旁轴 ABCD 光学系统传播的 HhCG 光束的解析表达式，并用于研究其在二次折射率介质 (QIM) 和分数傅立叶变换 (FRFT) 光学系统中的传播特性。通过使用导出的表达式，两个系统的 HhCG 光束的特性都通过数值例子进行了图形说明。分别详细讨论了两个系统 QIM 和 FRFT 中 HhCG 演变的几个影响参数。结果表明，HhCG 的强度分布在 QIM 传播过程中周期性变化。在 FRFT 系统中，发现HhCG光束平面的归一化强度分布不仅取决于分数阶，而且还取决于光束阶和截断参数。这项工作获得的结果为 HhCG 光束整形、光纤、透镜等领域提供了一种方便的方法。