Ym. Shi et al., GENERALIZED STOKES-MUELLER FORMALISM FOR 2-PHOTON ABSORPTION, FREQUENCY-DOUBLING, AND HYPER-RAMAN SCATTERING, Physical review. A, 49(3), 1994, pp. 1999-2015
We consider processes quadratic in incident intensity, in which two id
entical photons are annihilated simultaneously. We construct a formali
sm for these processes that is maximally analogous to the Stokes-Muell
er formalism for linear processes. The two-photon properties of the in
cident beam require nine Stokes-like parameters for their description;
the Mueller-like matrix which gives a complete description of frequen
cy doubling or hyper-Raman scattering is of size 4 X 9. Of the 36 inde
pendent observable matrix elements, 12 are activated by pure real elas
tic dipole susceptibilities, four by an imaginary component in the ele
ctric dipole susceptibility (absorption), nine by handedness alone, tw
o by extramultipole mechanisms alone, and nine require both handedness
and extramultipole mechanisms.