Spectromicroscopy techniques with fast electrons can quantitatively measure the optical response of excitations with unrivalled spatial resolution. However, owing to their inherently scalar nature, electron waves cannot access the polarization-related quantities. Despite promising attempts based on the conversion of concepts originating from singular optics (such as vortex beams), the definition of an optical polarization analogue for fast electrons has remained an open question. Here we establish such an analogue using the dipole transition vector of the electron between two well-chosen singular wave states. We show that electron energy loss spectroscopy allows the direct measurement of the polarized electromagnetic local density of states. In particular, in the case of circular polarization, it directly measures the local optical spin density. This work establishes electron energy loss spectroscopy as a quantitative technique to tackle fundamental issues in nano-optics, such as super-chirality, local polarization of dark excitations or polarization singularities at the nanoscale.

具有快速电子的光谱显微镜技术可以以无与伦比的空间分辨率定量测量激发的光学响应。然而，由于其固有的标量性质，电子波无法访问与极化相关的量。尽管基于奇异光学（例如涡旋光束）的概念转换的有希望的尝试，但快速电子的光学偏振模拟的定义仍然是一个悬而未决的问题。在这里，我们使用两个精心选择的奇异波态之间的电子偶极跃迁矢量建立了这样的模拟。我们表明，电子能量损失光谱允许直接测量极化电磁局部状态密度。特别是在圆偏振的情况下，它直接测量局部光学自旋密度。