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There are plenty of linear and nonlinear optical processes, that can be divided into two groups: parametric and non-parametric (link). However, why are they called parametric (non-parametric)? What is the origin for this? Where is the dependence/independence on a parameter? What is the parameter, according to which this classification makes sense?

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As stated in the link to the wikipedia page that you provide, the definition of a parametric optical interaction with a medium is usually given as that which leaves the quantum state of the medium unchanged. This includes quasi-instantaneous perturbative nonlinear effects such as sum-frequency generation, self-phase modulation, etc. A non-parametric optical process is the opposite, i.e., the quantum state of the medium is changed by the process. An example would be Raman scattering, where the input photon energy is converted into a phonon and an output photon with a frequency which is detuned from the input photon frequency by an amount determined by the phonon energy.

In these interactions it is momentum which is being parameterized. Although the quantum state of the material is unchanged by a parametric processes, the material is required to fulfil the phase matching conditions which ensure momentum conservation. Non-parametric processes rely on other things (such as absorption and energy storage within the material) to circumvent phase matching conditions, and thus don't parameterize momentum*. This is indicated in the third section of the Wiki link that you posted: Parametric processes interact only with the real part of the refractive index, whereas non-parametric processes interact with the imaginary part as well, which determines absorption (a very simplistic description for this is given here).

*This doesn't mean that momentum isn't conserved during non-parametric processes, it's just that the non-parametric process doesn't rely on it.

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