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This paper contains a detailed derivation of the photoinduced current densityat third order in the coupling between a semiconductor and a multifrequencyphoton field, starting from its standard textbook expression as a third ordertime integral of a triple commutator. Due to a major intrinsic problem linkedto this triple commutator, such a derivation has been made possible quiterecently only, thanks to the tools developed in the composite-boson many-bodytheory we have just constructed. The photoinduced current density is shown toultimately read in a compact form, in terms of the Pauli and Coulombscatterings for exciton-exciton interactions introduced in this theory.Representation in Shiva diagrams is also given to better grasp the physics ofthe various contributions.

Third order susceptibility: general formalism for photoinduced current density in semiconductors