content_image/371093.jpg content_image/371095.jpg
frequency process is characterized by a second-order susceptibility GLYPH<31> (2) via P ( !$_{1}$ + !$_{2}$ ) = GLYPH<31> (2) : F$_{1}$ ( !$_{1}$ ) F$_{2}$ ( !$_{2}$ ). In the above example of helix, the susceptibility may be written as a form of three dyadic vectors, GLYPH<31> (2) = A ( ZYX GLYPH<0> ZXY ), i.e., a rank-3 tensor. Thus the chiral sum-frequency susceptibility provides a measurement of the chirality of a physical object. If otherwise measured in linear optics, the e GLYPH<11> ect of the molecular chirality relies on the small magnetic moment of the molecules, and in turn on the small photon momentum of the probe light, similar to the case of linear optical e GLYPH<11> ects of spin currents [20]. 

For a systematic symmetry analysis, we consider a spin current with both longitudinal and transverse components. We define the Z -axis as the current direction and the X -axis as the spin direction of the transverse component. The spin current can be written as a rank-2 tensor J = J$_{X}$ XZ + J$_{Z}$ ZZ , in a form of dyadic vectors, in which the left / right vector is the spin / current direction and J$_{Z}$$_{=}$$_{X}$ is the longitudinal / transverse amplitude. Above all, the spin current breaks the inversion symmetry, satisfying the symmetry properties required by a second-order optical process [25]. 

In general, the sum-frequency susceptibility tensor has 27 independent terms, GLYPH<31> (2) = GLYPH<31>$_{XXX}$ XXX + GLYPH<31>$_{XXY}$ XXY + GLYPH<1> GLYPH<1> GLYPH<1> + GLYPH<31>$_{ZZZ}$ ZZZ , but the symmetry properties of a spin current will set many terms to be zero or non-independent [25]. For a longitudinal spin current, only the chiral terms are non-zero. In a non-chiral term, at least one of the three directions X , Y and Z appears even times (twice or zero times). Consider GLYPH<31>$_{XXY}$ XXY for example. Under reflection by the Y - Z plane, the longitudinal spin current is reversed, but GLYPH<31>$_{XXY}$ XXY is unchanged, so this term must be zero. Similar arguments apply to other non-chiral terms. Also, the susceptibility must be anti-symmetric under reflection by any plane parallel to the Z -axis. With these constraints, the sum-frequency susceptibility of a longitudinal spin current can be written as 

<img src='content_image/371093.jpg'>

with only three independent parameters. As for a transverse spin current J$_{X}$ XZ , it changes its sign under reflection by the X - Z plane but is invariant under reflection by the X - Y or Y - Z plane [see Fig. 1 (d)], each non-zero term in the susceptibility must contain odd times of Y and even times of Z or X , so 

<img src='content_image/371095.jpg'>

