Structured-Gaussian beams are shown to be fully and uniquely represented by a collection of points (or a constellation) on the surface of the modal Majorana sphere, providing a complete generalization of the modal Poincaré sphere to higher-order modes. The symmetries of this Majorana constellation translate into invariances to astigmatic transformations, giving way to continuous or quantized geometric phases. The experimental amenability of this system is shown by verifying the existence of both these symmetries and geometric phases.

中文翻译：

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马氏模态球和结构高斯光束的隐对称性

高斯光束被模态马约拉那球面上的点（或星座）集合完全唯一地表示，从而将模态庞加莱球面完全推广到更高阶模态。马约拉那星座的对称性转化为像散变换的不变性，让位于连续或量化的几何相位。通过验证这些对称性和几何相位的存在，表明了该系统的实验适应性。