Mathematical Reformulation of Polarization Equation

[spookyfw]

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:
$$\vec{P_L}(\vec{r},t) = \frac{1}{2} \hat{x} \left(P_L \exp{(-i \omega_0 t)} + c.c.\right)$$
into
$$\vec{P_L}(\vec{r},t) = \epsilon_0 \int_{-\infty}^{\infty} \chi^{(1)} (t-t') \cdot E(\vec{r},t') dt'$$
According to the author of the book (Agrawal, 'Nonlinear Fiber Optics') this should result in:
$$P_L(\vec{r},t) = \epsilon_0 \int_{-\infty}^{\infty}\chi^{(1)}_{xx} (t-t') \cdot E(\vec{r},t') \exp{(i \omega_0 (t-t'))} dt'$$
under the assumption that the tensor $\chi$ was diagonal, lumping $\hat{x}$ and $\chi^{(1)} (t-t')$ 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

 

[jvicens]

Hello spookyfw,

Thank you for posting your question in this forum. I can understand the confusion you are facing in trying to integrate the exponentials into the integral. Let me try to guide you in the right direction.

Firstly, it is important to note that the integral you are dealing with is a convolution integral, which is a common operation in mathematics and physics. In order to solve this integral, you can use the convolution theorem, which states that the Fourier transform of a convolution is the product of the Fourier transforms of the individual functions.

In this case, the function you are integrating is the product of the time-domain function \chi^{(1)}(t-t') and the frequency-domain function \exp{(i \omega_0 (t-t'))}. So, to solve this integral, you can use the convolution theorem and take the Fourier transforms of both these functions. Then, you can multiply the Fourier transforms and take the inverse Fourier transform to find the solution.

I hope this helps to clarify the process for you. If you need further assistance, please don't hesitate to ask for more guidance. Good luck with your problem!