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## Main Question or Discussion Point

Hello,

I hope I got the section right ;). I orginally posted this in the physics section, but as the problem is more mathematical. It would be nice if someone knows the right direction.

I've stumbled upon a math problem while going through some physics and got stuck with some mathematical cosmetics (pp. 40). It is the substition of:

[tex] \vec{P_L}(\vec{r},t) = \frac{1}{2} \hat{x} \left(P_L \exp{(-i \omega_0 t)} + c.c.\right) [/tex]

into

[tex] \vec{P_L}(\vec{r},t) = \epsilon_0 \int_{-\infty}^{\infty} \chi^{(1)} (t-t') \cdot E(\vec{r},t') dt' [/tex]

According to the author of the book (Agrawal, 'Nonlinear Fiber Optics') this should result in:

[tex] P_L(\vec{r},t) = \epsilon_0 \int_{-\infty}^{\infty}\chi^{(1)}_{xx} (t-t') \cdot E(\vec{r},t') \exp{(i \omega_0 (t-t'))} dt' [/tex]

under the assumption that the tensor [itex] \chi [/itex] was diagonal, lumping [itex] \hat{x} [/itex] and [itex] \chi^{(1)} (t-t') [/itex] together makes sense. But what I don't get is how to integrate the exponentials into the integral. It looks like the shift theorem, but the sum of the two exponentials leaves me puzzled. Can anyone give me a hint?

Thank you very much in advance,

spookyfw

I hope I got the section right ;). I orginally posted this in the physics section, but as the problem is more mathematical. It would be nice if someone knows the right direction.

I've stumbled upon a math problem while going through some physics and got stuck with some mathematical cosmetics (pp. 40). It is the substition of:

[tex] \vec{P_L}(\vec{r},t) = \frac{1}{2} \hat{x} \left(P_L \exp{(-i \omega_0 t)} + c.c.\right) [/tex]

into

[tex] \vec{P_L}(\vec{r},t) = \epsilon_0 \int_{-\infty}^{\infty} \chi^{(1)} (t-t') \cdot E(\vec{r},t') dt' [/tex]

According to the author of the book (Agrawal, 'Nonlinear Fiber Optics') this should result in:

[tex] P_L(\vec{r},t) = \epsilon_0 \int_{-\infty}^{\infty}\chi^{(1)}_{xx} (t-t') \cdot E(\vec{r},t') \exp{(i \omega_0 (t-t'))} dt' [/tex]

under the assumption that the tensor [itex] \chi [/itex] was diagonal, lumping [itex] \hat{x} [/itex] and [itex] \chi^{(1)} (t-t') [/itex] together makes sense. But what I don't get is how to integrate the exponentials into the integral. It looks like the shift theorem, but the sum of the two exponentials leaves me puzzled. Can anyone give me a hint?

Thank you very much in advance,

spookyfw