Generalized Stokes-Mueller formalism for two-photon absorption, frequency doubling, and hyper-Raman scattering

We consider processes quadratic in incident intensity, in which two identical photons are annihilated simultaneously. We construct a formalism for these processes that is maximally analogous to the Stokes-Mueller formalism for linear processes. The two-photon properties of the incident beam require nine Stokes-like parameters for their description; the Mueller-like matrix which gives a complete description of frequency doubling or hyper-Raman scattering is of size 4\ifmmode\times\else\texttimes\fi{}9. Of the 36 independent observable matrix elements, 12 are activated by pure real elastic dipole susceptibilities, four by an imaginary component in the electric dipole susceptibility (absorption), nine by handedness alone, two by extramultipole mechanisms alone, and nine require both handedness and extramultipole mechanisms.