Reciprocity is a fundamental principle of wave physics and directly relates to the symmetry in the transmission through a system when interchanging the input and output. The coherent transmission matrix (TM) is a convenient method to characterize wave transmission through general media. Here, we demonstrate the optical reciprocal nature of complex media by exploring their TM properties. We measured phase-corrected TMs of forward and round-trip propagation in a single polarization state through a looped 1 m-long step-index optical multimode fiber (MMF) to experimentally verify a transpose relationship between the forward and backward transmission. This symmetry impedes straightforward MMF calibration from proximal measurements of the round-trip TM. Furthermore, we show how focusing through the MMF with digital optical phase conjugation is compromised by system loss since time reversibility relies on power conservation. These insights may inform the development of new imaging techniques through complex media and coherent control of waves in photonic systems.

中文翻译：

---

复杂系统往返传输中的互易性对称性

互惠是波物理学的基本原理，并且在交换输入和输出时直接与通过系统的传输对称性有关。相干传输矩阵（TM）是表征通过一般介质的波传输的便捷方法。在这里，我们通过探索它们的TM特性来证明它们的光学互易性。我们测量了通过环形1 m长阶跃折射率多模光纤（MMF）在单偏振态下的正向和往返传播的相位校正TM，以实验验证向前和向后传输之间的转置关系。这种对称性阻碍了往返TM的近端测量的直接MMF校准。此外，我们展示了如何通过数字光相位共轭的MMF聚焦会因系统损耗而受到损害，因为时间可逆性依赖于节能。这些见解可以通过复杂的介质和光子系统中的波的相干控制来指导新的成像技术的发展。