content_image/369117.jpg content_image/369118.jpg content_image/369115.jpg content_image/369116.jpg
χ$_{i}$ ( t < 0) = 0. The Fourier transform of Eq. 3 for a purely ac field E ( t ) = E cos( ωt ) gives: 

<img src='content_image/369117.jpg'>

where the polarization P and the susceptibilities χ$_{i}$ are now taken in the frequency domain and the dots indicate again infinite sums involving higher order terms. The response P ( t ) to E ( t ) = E cos( ωt ) can thus be written 

<img src='content_image/369118.jpg'>

To obtain Eq. (5), we have used the fact that because χ$_{1}$ and χ$_{3}$ are real in the time domain, their Fourier transform verify χ ∗ $_{1}$( ω ) = χ$_{1}$ ( − ω ) and χ ∗ $_{3}$( ω$_{1}$, ω$_{2}$, ω$_{3}$ ) = χ$_{3}$ ( − ω$_{1}$, − ω$_{2}$, − ω$_{3}$ ) (the star denotes the complex conjugate), and the invariance of χ$_{3}$ by permutation of its argu- ments. For simplicity, we write χ$_{3}$ ( ω ) = χ$_{3}$ ( ω, ω, ω ) and χ ¯ $_{3}$( ω ) = χ$_{3}$ ( − ω, ω, ω ). Eq. (5) can be written 

<img src='content_image/369115.jpg'>

where the susceptibilities χ$_{i}$ are given as a function of their real and imaginary parts χ ′ i and χ ′′ $_{i}$. For practical applications, the modulii and arguments | χ$_{i}$ | and δ$_{i}$ are rather used: 

<img src='content_image/369116.jpg'>

We see in the rhs in Eqs 5-7 that the nonlinear susceptibility of interest, namely χ$_{3}$ 

