Hi I am currently reading through a semiclassical approach to nonlinear optics. I learned about effects of second-order nonlinear optics like second harmonic generation and three wave mixing. I understand three-wave mixing as a process in which you send two (or three) monochromatic beams through a nonlinear medium in which those beams "interact" (via the medium) if they are phase-matched, the net result being a third beam generated (and / or changes in the intensities of the individual beams). At the end of the treatment of second-order nonlinear optics that I'm reading, spontaneous parametric down-conversion is mentioned as an example. I don't understand how this is an example for second-order nonlinearity. In the description, a pump beam is sent into a nonlinear medium (note: just one monochromatic beam). What comes out of the crystal is the weakened pump beam, together with a family of idle and signal beams, arranged in a cone (since there are many solutions for those two beams to be phase-matched with the pump beam). How can this be an effect of second-order nonlinear optics? If I send one single monochromatic beam through a second-order nonlinear medium, the only effect is the generation of the second harmonic and a constand polarization, none of which correspond to idler and signal beam, right? And if this should be a three-wave mixing, there should be at least one more monochromatic beam entering the medium, right? So can anyone explain to me how this should be an example of second-order nonlinear optics? Thanks in advance.