What is the Fourier transform of second order polarization?

[Niles]

Homework Statement

Hi

I am trying to solve problem 2.1 in these notes (it is on the very first page): http://qis.ucalgary.ca/quantech/673/notes/chapter_two.pdf . The problem tells me to show that the Fourier transform of

$$P^{(2)} (t) = \varepsilon _0 \frac{1}{{(2\pi )^2 }}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\chi ^{(2)} (t_1 ,t_2 )E(t - t_1 )E(t - t_2 )dt_1 dt_2 } }$$

is given by

$$P^{(2)}(\omega ) = \varepsilon _0 \int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\chi ^{(2)} (\omega _1 ,\omega _2 )\delta (\omega - \omega _1 - \omega _2 )E(\omega _1 )E(\omega _2 )d\omega _1 d\omega _2 } }$$

The Attempt at a Solution

OK, so I know that the FT of a convolution yields a product and that the FT of a delta-function is a constant. I am not sure whether to

1) tack on a factor of exp(-iωt) and integrate over t on both sides OR
2) write every factor in terms of its FT, and then compare terms in the end

I choose option 2. Here goes

$$\int_{ - \infty }^\infty {P(\omega )e^{ - i\omega t} } = \varepsilon _0 \frac{1}{{(2\pi )^2 }}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {\chi ^{(2)} (\omega _1 ,\omega _2 )e^{ - i\omega _1 t_1 - i\omega _2 t_2 } E(\omega _1 )e^{ - i\omega _1 (t - t_1 )} E(\omega _2 )e^{ - i\omega _2 (t - t_2 )} d\omega _1 d\omega _2 dt_1 dt_2 } }$$

But this makes t₁ and t₂ go out in the exponentials, which is no good. But if I go back to option 1, I can't see how I would ever end up with a delta-function in ω-ω₁-ω₂.

I would really appreciate a hint.Niles.

 

[Niles]

No one has a hint? In principle this seems straightforward, but I simply cannot crack this nut.