# Parametric process (optics)

A parametric process is an optical process in which light interacts with matter in such a way as to leave the quantum state of the material unchanged. As a direct consequence of this there can be no net transfer of energy, momentum, or angular momentum between the optical field and the physical system. In contrast a non-parametric process is a process in which any part of the quantum state of the system changes.[1]

## Temporal characteristics

Because a parametric process prohibits a net change in the energy state of the system, parametric processes are "instantaneous". For example, if an atom absorbs a photon with energy E, the atom's energy increases by ΔE = E, but as a parametric process, the quantum state cannot change and thus the elevated energy state must be a temporary virtual state. By the Heisenberg Uncertainty Principle we know that ΔEΔt~ħ/2, thus the lifetime of a parametric process is roughly Δt~ħ/2ΔE, which is appreciably small for any non-zero ΔE.[1]

## Parametric versus non-parametric processes

### Linear optics

In a linear optical system the dielectric polarization, P, responds linearly to the presence of an electric field, E, and thus we can write

${\displaystyle {\mathbf {P} }=\varepsilon _{0}\chi {\mathbf {E} }=(n_{r}+in_{i})^{2}{\mathbf {E} },}$

where ε0 is the electric constant, χ is the (complex) electric susceptibility, and nr(ni) is the real(imaginary) component of the refractive index of the medium. The effects of a parametric process will affect only nr, whereas a nonzero value of ni can only be caused by a non-parametric process.

Thus in linear optics a parametric process will act as a lossless dielectric with the following effects:

Alternatively, non-parametric processes often involve loss (or gain) and give rise to:

### Nonlinear optics

In a nonlinear media, the dielectric polarization P responds nonlinearly to the electric field E of the light. As a parametric process is in general coherent, many parametric nonlinear processes will depend on phase matching and will usually be polarization dependent.

Sample parametric nonlinear processes:

Sample non-parametric nonlinear processes: