General first-order optical systems—represented by a four-dimensional real symplectic group—can be realized using thin lenses and free propagation transformations. It is shown that such systems can be realized using four units of free propagation transformation and nine thin rotated cylindrical lenses (or equivalently, four thin rotated astigmatic lenses and a thin rotated cylindrical lens). If these nine thin lenses placed in five transverse planes can be realized using spatial light modulators (SLMs), then our gadget realizes any general first-order optical system using five SLMs. It is also outlined that any general first-order optical system with finite symplectic matrix entries can always be realized using at least any one of the identities presented here, when a particular decomposition demands thin lenses with impractical focal length.

中文翻译：

---

使用九个任意焦距的薄圆柱透镜和四个单位的自由传播距离实现一般的一阶光学系统

通用的一阶光学系统（由一个四维实辛群表示）可以使用薄透镜和自由传播变换来实现。示出了可以使用四个单元的自由传播变换和九个薄旋转柱面透镜（或者等效地，四个薄旋转像散透镜和薄旋转柱面透镜）来实现这样的系统。如果可以使用空间光调制器（SLM）实现放置在五个横向平面中的这九个薄透镜，那么我们的小工具可以使用五个SLM实现任何一般的一阶光学系统。还概述了，当特定分解需要具有不切实际焦距的薄透镜时，任何具有有限辛矩阵项的通用一阶光学系统都可以始终使用此处介绍的至少任何一个身份来实现。