The polarization of a medium is the vector field that represents the dipole moment per unit volume induced in the material when subjected to an electric field.
When a material is placed in an electric field, the charges inside the material (positive and negative) are displaced slightly, creating tiny electric dipoles. A dipole is simply a separation of positive and negative charges.
The polarization is a measure of how much the material becomes polarized due to these dipoles. It is a vector field because it can vary in direction and magnitude at different points in the material. The polarization vector, \(\vec P(\vec E)\), represents the dipole moment (strength of the dipole) per unit volume at a given point.
General form
General form of the polarization
For a general medium, the polarization can be written as:
$$P_i (E(\omega))={{\epsilon_0 \left( \chi^{(1)}_{ij} E_j(\omega) + \chi^{(2)}_{ijk} E_j(\omega) E_k(\omega) + \chi^{(3)}_{ijkl} E_j(\omega) E_k(\omega) E_l(\omega) + \dots \right)}}$$
This expression is write with implicit summation.
With:
\(\chi^{(n)}\): the \(n\)-order susceptibility
Polarization in a linear mediul
In a linear medium, the polarization is just:
$$P_i(E(\omega))=\epsilon_0\chi_{ij} E_j(\omega)$$
With:
\(\chi\) : the \(1^{st}\) order susceptibility, also called the linear susceptibility
If the medium is isotropic: \(\chi_{ij}=\chi\)
Transfert function
Polarization as the response of the medium
We can view the polarization \(P_j(t)\) as the response of the medium to the applied electric field \(E_k(t)\). In the linear case, this response is often described by a convolution integral (Produit de convolution): $${{P_j(t) = \epsilon_0 \int \chi_{jk}^{(1)}(t - \tau) E_k(\tau) \, d\tau}}$$
Where:
\(\epsilon_0\) is the permittivity of free space
\(\chi_{jk}^{(1)}\) is the linear susceptibility of the medium
