## polarization, $$P$$

https://doi.org/10.1351/goldbook.P04712
The relevant material property that couples with the radiation field. May be called optical or dielectric polarization. Optical spectroscopies may be classified according to the dielectric polarization power-law dependence on the external electric field.
Notes:
1. Mathematically it is defined as the @E01929@ change per volume resulting from absorption of radiation of optical frequencies, defined as $$P = D - \epsilon_{0}E$$, where $$D$$ is the @E01930@, $$\varepsilon_{0}$$ the electric constant (vacuum @P04507@) , and $$E$$ the strength of the radiation electric field. A dielectric medium is characterized by the constitutive relation $$D = \epsilon_{0}\, \chi ^{(1)}$$ where $$\chi ^{(\text{1})} = \varepsilon_{r} - 1$$ is the linear 'susceptibility' for a transparent singly refracting medium. Depending on the molecular or atomic restoring force on the electron with respect to the displacement $$D$$, the field-induced motion of the electron can introduce other frequency components on the electron motion, and this in turn leads to non-linear optical effects.
2. The polarization component to the nth-order in the field is denoted as $$P^{(\text{n})}$$ Thus, the following equations apply,
$$P = P^{(\text{1})} + P_{\text{NL}}$$ and $$P_{\text{NL}} = P^{(\text{2})} + P^{(\text{3})} + \dots$$
$$P = \epsilon _{0}\left [ \chi _{e}^{(1)}E\: + (1/2)\chi _{e}^{(2)}E^{2}\: + (1/6)\chi _{e}^{(3)}E^{3} + \dots \right ] = P^{(1)} + P^{(2)} + P^{(3)} + \dots$$ where $$E^{\text{i}}$$ is the i-th component of the electric field strength and $$\chi _{\text{e}}^{(\text{n})}$$ is the usual 'susceptibility' $$\chi ^{(\text{1})} = \varepsilon_{r} - 1$$ in the absence of higher terms and $$P^{(\text{n})}$$ is the order of the field-induced polarization in the material.
In an anisotropic medium, $$\chi _{\text{e}}^{(1)}$$, $$\chi _{\text{e}}^{(2)}$$ and $$\chi _{\text{e}}^{(3)}$$ are the medium 'hyper-susceptibilities'; they are tensors of rank 2, 3, and 4, respectively.
Linear optical responses such as absorption, light @P04881@, reflection, and refraction, involving a weak incoming field, are related to $$P^{(\text{1})}$$. Non-linear techniques are connected to the non-linear polarization $$P_{\text{NL}}$$. Low order non-linear techniques, such as three-wave mixing, are related to the second order optical polarization $$P^{(\text{2})}$$. For a random @I03353@ medium (such as a liquid) or for a crystal with a centrosymmetric @U06562@, $$\chi _{\text{e}}^{(2)}$$ is zero by symmetry and then the lowest order non-linear techniques, as well as the higher order, are related to the third-order optical polarization, $$P^{(\text{3})}$$, and the corresponding hyper-susceptibility.
Source:
PAC, 2007, 79, 293. 'Glossary of terms used in photochemistry, 3rd edition (IUPAC Recommendations 2006)' on page 402 (https://doi.org/10.1351/pac200779030293)