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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.²

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

P = ε 0 χ E = ( n r + i n i ) 2 E , {\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:

• Refraction
• Diffraction
• Elastic scattering
  • Rayleigh scattering
  • Mie scattering

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

• Absorption
• Inelastic scattering
  • Raman scattering
  • Brillouin scattering
• Various optical emission processes
  • Photoluminescence
  • Fluorescence
  • Luminescence
  • Phosphorescence

Nonlinear optics

Main article: 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 χ ( 3 ) {\displaystyle \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 χ ( 3 ) {\displaystyle \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

See also

• Nonlinear optics

Notes

• Boyd, Robert (2008). "1.2.10 Parametric versus Nonparametric Processes". 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

References

1. See Boyd 2008, pp. 13–15 1.2.10 Parametric versus Nonparametric Processes - Boyd, Robert (2008). "1.2.10 Parametric versus Nonparametric Processes". Nonlinear Optics (3rd ed.). Academic Press. pp. 13–15. ISBN 978-0-12-369470-6. https://books.google.com/books?id=uoRUi1Yb7ooC&pg=PA13 ↩

2. See Boyd 2008, pp. 13–15 1.2.10 Parametric versus Nonparametric Processes - Boyd, Robert (2008). "1.2.10 Parametric versus Nonparametric Processes". Nonlinear Optics (3rd ed.). Academic Press. pp. 13–15. ISBN 978-0-12-369470-6. https://books.google.com/books?id=uoRUi1Yb7ooC&pg=PA13 ↩