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# Nonlinear Polarization

Definition: the part of the light-induced electric polarization which depends nonlinearly on the electric field of the light

German: nichtlineare Polarisation

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When light propagates in a transparent medium (a dielectric), its electric field causes some amount of electric polarization in the medium, i.e. some density of electric dipole moment. (This must not be confused with the polarization of the light field, which is the direction of its electric field.) That polarization of the medium propagates together with the electromagnetic field in the form of a polarization wave, which has the same phase velocity as the driving field.

Whereas at low light intensities the electric polarization is proportional to the electric field strength, nonlinear contributions become important at high optical intensities, as they can e.g. be produced with lasers. Such nonlinear polarization waves are essential e.g. for nonlinear frequency conversion processes.

## Second-order Nonlinear Polarization

The second (lowest) order of nonlinear polarization can arise from a <$\chi^{(2)}$> nonlinearity which can occur only in nonlinear crystal materials with a non-centrosymmetric crystal structure. (Nonlinear effects at crystal surfaces are an exception.) The nonlinear polarization then has a component which depends quadratically on the electric field of an incident light wave. More precisely, the tensor nature of the nonlinear susceptibility needs to be considered:

$${P_i}(t) = {\varepsilon _0}\sum\limits_j {{\chi _{ij}}\;{E_j}} (t) + \sum\limits_{j,\;k} {{\varepsilon _0}\chi _{ijk}^{(2)}\;{E_j}(t)\;{E_k}} (t)$$

where <$P_i$> is the <$i$>-th Cartesian coordinate of the polarization, <$\chi^{(2)}$> is the nonlinear susceptibility, and <$E(t)$> is the optical electric field. More commonly, this is written as

$${P_i}(t) = {\varepsilon _0}\sum\limits_j {{\chi _{ij}}{E_j}} (t) + 2\sum\limits_{j,\;k} {{d_{ijk}}{E_j}(t){E_k}} (t)$$

with the nonlinear tensor <$d$>. Many tensor components can actually be zero for symmetry reasons, depending on the crystal class.

In general, the direction of the nonlinear polarization will not be the same as that the polarization direction of the input light.

The nonlinear polarization contains frequency components which are not present in the exciting beam(s). Light with such frequencies can then be generated in the medium (→ nonlinear frequency conversion). For example, if the input field is monochromatic, the nonlinear polarization also has a component with twice the input frequency (→ frequency doubling). As the polarization has the form of a nonlinear polarization wave, the frequency-doubled light is also radiated in the direction of the input beam. Other examples are sum and difference frequency generation, optical rectification, parametric amplification and oscillation.

## Third-order Nonlinear Polarization

The next higher order of nonlinear polarization can arise from a <$\chi^{(3)}$> nonlinearity, as it occurs in basically all media. This can give rise to various phenomena:

## Phase Matching

In most cases, the nonlinear mixing products can be efficiently accumulated over a greater length of crystal only if phase matching is achieved. Otherwise, the field amplitudes at the exit face, generated at different locations in the crystal, essentially cancel each other, and the apparent nonlinearity is weak. Some nonlinear effects, however, are either automatically phase-matched (e.g. self-phase modulation) or do not need phase matching (e.g. Raman scattering).

### Bibliography

  D. A. Kleinman, “Nonlinear dielectric polarization in optical media”, Phys. Rev. 126 (6), 1977 (1962); https://doi.org/10.1103/PhysRev.126.1977

## Questions and Comments from Users

2020-12-06

Can the second order nonlinear polarizability value be negative? If yes, what does it mean?

I don't think there is a problem with such a negative tensor component; it just means a negative sign of the induced polarization, referring to the chosen coordinate system.

2023-07-27

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