12, 12 − ( ) 1 2 2 3 i − e e 12, 12 + 12, 12 − 32,12 1 2 2 i + e e 1 2 6 i + e e 1 2 6 i − e e 1 2 2 i − e e 12, 12 + 32, 12 − 32, 32 − 32,32 12, 12 − 12, 12 + ( ) 1 2 2 3 i + e e (c) (b) (a) SO HH CB LH F E 1 1 , 2 2 ± 3 3 , 2 2 ± 3 1 , 2 2 ± 1 1 , 2 2 ± 3 , 2 m 1 , 2 m 1 , 2 m 32 E S () 1 2 ∗ + e e 1 2 i + e e v 12 E (d) 2 F 1 F P FIG. 2. (Color online) Models for microscopic calculation of the sum-frequency susceptibility. (a) The full eight-band model and the electron spin distribution for a pure spin current in a semiconductor. (b) and (c) Selection rules and relative dipole moments from the spin-3 / 2 and spin-1 / 2 valence bands to the conduction band, respectively. (d) A simplified model with the HH-LH splitting neglected. The spin states and selection rules for inter-band transitions are independent of the momentum. The transition energies to the Fermi surface from di GLYPH<11> erent valence bands are indicated. with seven independent parameters. The unique polarization-dependence of the second-order sus- ceptibility of a spin current can be used to distinguish its transverse and longitudinal components, and also to single out the spin-current signature from the e GLYPH<11> ects of the material background or a charge current [26]. To determine the independent parameters of the susceptibility in Eqns. (1) and (2), we per- formed microscopic calculation for a pure spin current in a bulk GaAs, using the standard per- turbation theory [25, 27] with an eight-band model [28]. We assumed that the pure spin current result from a non-equilibrium distribution of electrons in the conduction band, with a small por- tion of electrons near the Fermi surface having opposite spin polarizations for opposite velocities [Fig. 2 (a)] , under conditions similar to those in Ref. [7]. The optical interaction includes the inter-band transitions between the valence bands and the conduction band and the intra-band ac- celeration of electrons and holes. To avoid real absorption of light, the light frequencies were chosen such that the sum frequency is below the band gap. For the sake of simplicity, we ne- glected the anisotropy of the valence bands. We also adopted the free-particle approximation, 5