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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

{\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 n_r(n_i) is the real(imaginary) component of the refractive index of the medium. The effects of a parametric process will affect only n_r, whereas a nonzero value of n_i 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:

Absorption
Inelastic scattering
- Raman scattering
- Brillouin scattering
Various optical emission processes

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:

Second-harmonic generation (SHG), or frequency doubling, generation of light with a doubled frequency (half the wavelength)
Third-harmonic generation (THG), generation of light with a tripled frequency (one-third the wavelength) (usually done in two steps: SHG followed by SFG of original and frequency-doubled waves)
High harmonic generation (HHG), generation of light with frequencies much greater than the original (typically 100 to 1000 times greater)
Sum-frequency generation (SFG), generation of light with a frequency that is the sum of two other frequencies (SHG is a special case of this)
Difference frequency generation (DFG), generation of light with a frequency that is the difference between two other frequencies
Optical parametric amplification (OPA), amplification of a signal input in the presence of a higher-frequency pump wave, at the same time generating an idler wave (can be considered as DFG)
Optical parametric oscillation (OPO), generation of a signal and idler wave using a parametric amplifier in a resonator (with no signal input)
Optical parametric generation (OPG), like parametric oscillation but without a resonator, using a very high gain instead
Spontaneous parametric down-conversion (SPDC), the amplification of the vacuum fluctuations in the low gain regime
Optical Kerr effect, intensity dependent refractive index
Self-focusing
Kerr-lens modelocking (KLM)
Self-phase modulation (SPM), a $\chi ^{(3)}$ effect
Optical solitons
Cross-phase modulation (XPM)
Four-wave mixing (FWM), can also arise from other nonlinearities
Cross-polarized wave generation (XPW), a $\chi ^{(3)}$ effect in which a wave with polarization vector perpendicular to the input is generated

Sample non-parametric nonlinear processes:

Stimulated Raman scattering
Raman amplification
Two-photon absorption, simultaneous absorption of two photons, transferring the energy to a single electron
Multiphoton absorption
Multiple photoionisation, near-simultaneous removal of many bound electrons by one photon

Notes

^ ^a ^b See Section Parametric versus Nonparametric Processes, Nonlinear Optics by Robert W. Boyd (3rd ed.), pp. 13-15.

References

Boyd, Robert (2008). Nonlinear Optics (3rd ed.). Academic Press. pp. 13–15. ISBN 978-0-12-369470-6.
Paschotta, Rüdiger, "Parametric Nonlinearities", Encyclopedia of Laser Physics and Technology

[Boyd_NLO-1] See Section Parametric versus Nonparametric Processes, Nonlinear Optics by Robert W. Boyd (3rd ed.), pp. 13-15.

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