Deriving the ∫dω1 integral in 2nd-order polarization

[Ngineer]

Homework Statement
Starting with the second order polarization in the time domain:

(1)

I am trying derive the frequency domain form:

(2)

Multiple sources give essentially the same formula with the same integral, I have obtained the particular ones in here from those lecture notes.

My issue is finding the origin of the ∫dω1 integral. After a day of attempts I still can't figure out how it comes into play.

The attempt at a solution
Attempt 1
I started by identifying the time-domain formula (equation 1) as a double convolution:

Which would map nicely to

But I did not get far as to deriving equation 2 from here.

Attempt 2
Another approach I have attempted is that recognizing the the desired frequency-domain form (equation 2) is very close to convolution with respect to ω=ω1+ω2:

However,
- Why would a convolution in the time domain map to a convolution in the frequency domain?
- The original formula (equation 2) does not have an integral with respect to ∫dω2.

Any help is incredibly appreciated.
Thank you!

 

[Cryo]

How is your Fourier transform defined?

Assuming it is
$E\left(t\right)=\frac{1}{2\pi}\int d\omega \exp\left(-i\omega t\right) \tilde{E}\left(\omega\right)$ Eq.(1)
$\tilde{E}\left(\omega\right)=\int dt \exp\left(i\omega t\right) E\left(t\right)$

Where $\tilde{\dots}$ denotes the frequency domain functions.

How about substituting Eq. (1) into your expressions, and then carrying out the integrals with respect to $t'$ and $t''$? In the process of doing this you will need to explain what does $\chi^{2}\left(\omega, \omega_1, \omega_2\right)$ mean. How does it relate to temporal version?

 

[Ngineer]

Thank you for your response, the definition of the Fourier transform is

The derivation checks out if it is assumed to be separable with respect to t' and t''

 

[Cryo]

The derivation checks out if it is assumed to be separable with respect to t' and t''

So does it settle your question then? If there are still problems can you show how far you got before getting stuck?

 

[Ngineer]

Yes. I have actually corresponded with the professor that posted the online lectures I linked in the original post, and he kindly answered in detail, so I marked this post as solved.

Thanks for your kind help.