Microscopic origin of non-linear optical effect I know that a non linear optical medium is a medium in which the optical response for example polarisation vector varies as
\begin{equation}
   \vec{P}=\epsilon_0\chi^{(1)}\vec{E}+\epsilon_0\chi^{(2)}\vec{E}^2+.......
\end{equation}
But I am not aware of reason behind this equation. Can anyone please give an intutive picture on the microscopic origin of non linear optical effect and non linear medium?
Or it is enough to have a comparison between the microscopic origin of linear and non-linear medium.
 A: Imagine an electron in a parabolic effective potential
$$V(\vec{\bar{x}}) = \tfrac{1}{2}k |\vec{\bar{x}}|^2 \\
\vec{F}(\vec{\bar{x}}) = \vec{\nabla} V(\vec{\bar{x}}) = k \vec{\bar{x}}\text{.}$$
A force $-e \vec{E}$ will displace it from the equilibrium position to
$$\vec{\bar{x}} = \frac{-e \vec{E}}{k} \text{,}$$
hence the polarization is
$$\vec{P} = -e \vec{\bar{x}} = \frac{e^2}{k} \vec{E} \text{.}$$
In this particular case the polarization is proportional to the electric field strength. But if the effective potential is different from a parabola higher powers of $\vec{E}$ appear in the equation for the polarization, which is the case for all polarizable materials if the field is strong enough.
A: The equation is just a Taylor expansion of an arbitrary function: by plugging in different values for the coefficients, you can construct practically any function.  The real question should be "Why are the coefficients for the higher terms typically very small?"
The answer is that Nature seems to be well-behaved.  If we examine most processes over a relatively small range of values, they appear linear or quadratic.
When second-  or third-order terms become important, there can be any of several causes.  A common cause of optical nonlinearities is heating: an increased temperature changes density, loosens bonds, etc.  Another cause is that charges driven by an electromagnetic field can't be pulled too far from their "home" positions without running into limits, analogously to a stretched spring.
A: An intuitively simple picture would be to consider terms in this infinit series to represent one-photon two-photon... N-photon processes. Altough this can give quantitatively wrong result, it can be considered to be equivalent with the classical series expansion.
