Spin currents, which carry information via spins in lieu of charges, play a key role in spin- tronics [1, 2]. Pure spin currents also signify the occurrence of some novel spin-related quantum phenomena such as the spin Hall e GLYPH<11> ect [3–10], the quantum spin Hall e GLYPH<11> ect and topological insula- tors [11–15]. Spin currents were previously observed via spin accumulation at stopping edges [7– 9, 16] or conversion to electrical signals [10, 17–19]. Direct and non-destructive measurement of pure spin currents where and while they flow [20, 21] is highly desired, but is very di GLYPH<14> cult because a pure spin current bears neither net charge current nor net magnetization. Noticing that a longitudinal spin current in which the spins point parallel or anti-parallel to the current is a chi- ral quantity, we envisaged that it can be probed by the chiral sum-frequency optical spectroscopy which was recently developed to detect molecular chirality [22–24]. By symmetry analysis in gen- eral cases and microscopic calculations in realistic models, we discovered that a pure spin current has sizable second-order optical susceptibility. This finding lays the foundation of direct, non- destructive measurement of spin currents by standard optical spectroscopy, facilitating application of spintronics [1, 2] and research on spin-related quantum phenomena [3–15]. As a basic principle of nature, a physical object is measurable only when it breaks certain fun- damental symmetries. Indeed, the probe must break the same symmetries as the object does, since the whole coupled system of an object and a probe has the fundamental symmetries. For example, in an Amp`ere meter, a “pure” charge current, which breaks the time reversal symmetry, is cou- pled to a microscopic current inside a magnet. Such symmetry consideration led to a scheme of detecting a pure spin current by a “photon spin current” carried by a polarized light beam [20]. A recent experiment [21] showing coupling between a spin current and a spin wave is a remark- able demonstration of the symmetry principle of measurement. Though as a direct probe of spin currents, the spin-wave technique [21] still requires special design and fabrication of magnetic nanostructures and the “photon spin current” probe [20] is limited by weak interaction since it involves the tiny light momentum, these previous works paved the way of searching methods of direct and non-destructive measurement of pure spin currents using symmetry analysis. Spin currents have peculiar symmetry properties owing to the characteristics of spins. Unlike a charge which is a scalar, a spin is a vector pointing to a certain direction. Physically, a spin is like a tiny magnet resulting from a quantized amount of current circulating about the spin direc- tion [Fig. 1 (a)]. Such physical nature makes a spin an unusual vector, namely, an axial vector. As illustrated in Fig. 1 (a), a spin reverses inside a parallel mirror and is unchanged inside a perpendic- ular mirror, in opposite to a polar vector. Spin-ups and spin-downs moving in opposite directions 2