Two-photon absorption and 3rd order susceptibility I am referring to introduction of Section 12.5 Multiphoton Absorption and Multiphoton Ionization (Page no. 550 of Nonlinear Optics, Boyd-3rd edition) where it has been said that the two-photon absorption is proportional to $Im\chi^{(3)}(\omega=\omega+\omega-\omega)$ and three-photon absorption is proportional to $Im\chi^{(5)}$ as calculated in Chapter 3.
In section 3.2.5, the expression of the 3rd order polarizability implies a dependence of $E^3$. 
I have 2 questions:


*

*How to understand that the two-photon absorption $\propto \chi^{(3)}$ and resolve the above seeming contradiction as we know that two-photon absorption probability $\propto E^2$?

*We say that the two-photon absorption is very less likely compared to the one-photon absorption as its proportional to $E^2$ and the intensity dies away $\propto \frac{1}{r^2}$ in the transverse direction of laser beam. I think this is the spatial aspect of the two-photon absorption probability. If we consider the probability of two-photon absorption at a particular point,say the focal point of the laser beam, isn't the probability of two-photon absorption $M^{(2)}(\textbf{r},t) \propto E(t)^2$, higher order than the one-photon absorption $M^{(1)}(\textbf{r},t) \propto E(t)$?

 A: Two-photon absorption goes as E^4 (or I^2) and NOT E^2 as you seem to misunderstand. 
The third-order polarizability goes as E^3 and is perfectly in sync with this ideas since the light-matter interaction term goes as P.E (The P bring along the E^3 term and the additional E in the dot product bring along the fourth E to make that E^4). 
A: 

*In the framework we are discussing (perturbative regime), the usual small parameter is $E/E_{at} \ll 1$ where $E_{at}$ is some characteristic atomic field (say, $e/a_0^2$ (CGS), where $a_0$ is the Bohr radius). In this case, $\chi^{(3)} \propto (E/E_{at})^3$ is usually smaller than $\chi^{(1)} \propto (E/E_{at})$ that gives you 1p absorption. However, you should notice that absorption is a resonant process, i.e. at a certain field frequency (while the small parameter accounts for the field amplitude) the 2p absorption will be larger than 1p absorption. Simply saying, if the dye absorbs 400 nm light, the 2p absorption will be dominant for the 800 nm excitation laser, but will be still less than 1p absorption of the 400 nm laser. 

