Nonlinear optics is usually looked at through a classical lens for most purposes. I do a Taylor expansion of the material polarization in terms of light-field strengths and we can derive all the essential mathematics for second-harmonic generation, sub-harmonic generation etc. Another way to look at high-harmonic generation is through multi-photon physics. For instance, I can look at second-harmonic generation as two photon emission. Is this the correct way to understand this? Is the classical description of nonlinear optics analogous to multi-photon physics? Are they just two ways of describing the same thing? Or am I missing a subtle difference in the terminology here? This is more of a terminology than a physics question.