## Introduction to polarization Basic concept of polarization and magnetization have been introduced already in the [[Maxwell equation recap]] page. In this chapter, we provide a brief and general introduction on how this quantities are defined, and how it is like in material. A more detailed scrutiny will be provided in a later chapter. >[!Notice] >From a microscopic view, polarization of material is a tricky concept. See the poly-book for Solid State Physics and Chemistry II for more information and get an ideal of how it is defined. The equations describe the response of material under external field belong to constitutive equations. The general way of writing polarization and magnetization could be $\mathbf{P}=\varepsilon_{0}\chi_{e} \mathbf{E}$ and $\mathbf{M}=\chi_{m} \mathbf{M}$ Under current nonlinear optics discussion, the polarization would be the mainly focus. For the first order susceptibility, we have $\tilde{P}(t) = \varepsilon_0 \chi^{(1)} \tilde{E}(t)$ This $\chi^{(1)}$ is a second rank tensor, with the subscript, it should be $\chi^{(1)}_{ij}$. This will be discussed in [[Nonlinear susceptibility]]. >[!Notice] >We adopt the notation used in [[Nonlinear Optics]] (Page 1), the $\tilde{}$ denotes a quantity varies rapidly in time. Constant quantities, slowly varying quantities, and Fourier amplitudes are written without the tilde. And for higher order process, we have $\tilde{P}(t) = \varepsilon_0 \left[ \chi^{(1)} \tilde{E}(t) + \chi^{(2)} \tilde{E}^2(t) + \chi^{(3)} \tilde{E}^3(t) + \cdots \right] \equiv \tilde{P}^{(1)}(t) + \tilde{P}^{(2)}(t) + \tilde{P}^{(3)}(t) + \cdots $ $\chi^{(2)}$ is called second order nonlinear susceptibility. $\tilde{P}^{(2)}(t)$ is the second order polarization, $\tilde{P}^{(3)}(t)$ is the third order polarization. Here we write $\tilde{P}(t)$ and $\tilde{E}(t)$ as a scalar quantity for simplicity but one should know they are not. The $\chi^{(2)}$ is a 3rd rank tensor, and $\chi^{(3)}$ is a 4th rank tensor, etc. The reason why we do not write it, like in may other tensor quantity, as a compact form using $\mathbf{P}$, is these $\chi$ have different ranks. The compact notation does not really simplify them. To clearly show the directions, they are in many cases written as $\tilde{P}_{ijk...}(t)$ using subscript. But in some case, especially when discussing one specific order of polarization we still write it in $\mathbf{P}$ form. > [!Note] > Since the these nonlinearities are electronic origin, an intuitive idea is that they should be comparable to $\tilde{P}^{(1)}(t)$ under a field with amplitude ~$E_{at}$, characteristic atomic electric field strength, > $E_{at} = \frac{e}{4 \pi \varepsilon_0 a_0^2}$ > Numerically calculated value is $E_{at} = 5.14 \times 10^{11} \, \text{V/m}$. And $\chi^{(1)}$ is in the order of 1, so $\chi^{(2)}\cong \frac{\chi^{(1)}}{E_{at}} \cong10^{-12}\text{ m/V}$, and $\chi^{(3)}\cong \frac{\chi^{(1)}}{E_{at}^{2}} \cong10^{-24}\ \mathrm{ m^2/V^{2}}$. These turn out to be okay approximation. > These are relative small value, and strong field would be required for higher order $\chi$ characterization. Therefore, to see nonlinear effect, pulse laser or high-Q factor cavity are applied in may cases.