Details of a Fourier transform

Question

I'm currently reading this article about how coherence properties of light influence the dynamics of absorption in molecules. In the article, equation 6 describes a chaotic light source whose time autocorrelation function is given by$$\langle\bar{\epsilon}(t_1)\bar{\epsilon}^*(t_2)\rangle=E_0^2e^{i\omega_0(t_1-t_2)}e^{-|t_1-t_2|/{\tau_d}}.$$Moreover, the article has already determined that the magnitude of the off-diagonal elements of the molecular density matrix will be proportional to the double Fourier transform of this quantity to the frequency domain; that is, to the quantity$$\langle\epsilon(\omega_{ig})\epsilon^*(\omega_{jg})\rangle=\int_{-\infty}^\infty\int_{-\infty}^{\infty} e^{i\omega_{ig}t_1}e^{\omega_{jg}t_2}\langle\bar{\epsilon}(t_1)\bar{\epsilon}^*(t_2)\rangle dt_1dt_2.$$ Here, $E_0$ is the maximum electric field value, $\omega_0$ is the mean frequency, and $\omega_{ig}$ and $\omega_{jg}$ are the energies of the transition from the ground state $g$ to states $i$ and $j$, expressed in frequency units. Now, though, the article makes a jump and says that $\langle\epsilon(\omega_{ig})\epsilon^*(\omega_{jg})\rangle$ comes out to$$4\pi E_0^2\delta(\omega_{ig}-\omega_{jg})\times(1/\tau_d)/\left[(\omega_0-\omega_{ig})^2+(1/\tau_d)^2\right]$$At this point, I am completely lost - I simply don't see how you can use Fourier transforms to get between these two equations. I am especially surprised by the appearance of the Dirac delta-function and the apparent asymmetry between the $\omega_{ig}$ and $\omega_{jg}$ terms of the final expression. Could anyone offer me a more step-by-step explanation of what happened in between these two equations?

Answer 1

$$\langle\epsilon(\omega_{ig})\epsilon^*(\omega_{jg})\rangle=E_0^2\int_{-\infty}^\infty\int_{-\infty}^{\infty} \!\!dt_1dt_2~~e^{i\omega_{ig}t_1}e^{i\omega_{jg}t_2} ~~e^{i\omega_0(t_1-t_2)}e^{-|t_1-t_2|/{\tau_d}} .$$

Now change variables to "light-cone" ones, by rotating the axes by π/4, $$ \eta=t_1-t_2, ~~~~\xi={t_1+1_2 \over 2} , $$ and do the two decoupled integrals, $$ E_0^2\left (\int\!\!d\xi ~~e^{i(\omega_{ig} -\omega_{jg})\xi} \right ) \left(\int\!\!d\eta ~~ e^{i\left ({\omega_{ig} +\omega_{jg}\over 2} +\omega_0 \right )\eta}~~e^{-| \eta|/{\tau_d}}\right ) $$

You know what the FTs of the delta, δ, and Lorentzian functions are, no? Apply the delta function to the argument of the Lorentzian!

Answer 2

I'll use the kernel convention for the forward transform, $e^{-i\omega t}$, as this seems to get closer to the expected result.

$$\begin{align*} \langle\epsilon(\omega_{ig})\epsilon^*(\omega_{jg})\rangle &= \int_{-\infty}^\infty\int_{-\infty}^{\infty} E_0^2e^{i\omega_0(t_1-t_2)}e^{-|t_1-t_2|/{\tau_d}} e^{-i\omega_{ig}t_1}e^{-i\omega_{jg}t_2} ~dt_1~dt_2 \\ \\ &=E_0^2 \int_{-\infty}^\infty\int_{-\infty}^{\infty} e^{-|t_1-t_2|/{\tau_d}}e^{-i(\omega_{ig}-\omega_0)t_1} ~dt_1~e^{-i(\omega_{jg}+\omega_0)t_2} ~dt_2\\ \\ &=E_0^2 \int_{-\infty}^\infty\int_{-\infty}^{\infty} e^{-|u|/{\tau_d}}e^{-i(\omega_{ig}-\omega_0)(u+t_2)} ~du~e^{-i(\omega_{jg}+\omega_0)t_2} ~dt_2\\ \\ &=E_0^2 \int_{-\infty}^\infty\left[\int_{-\infty}^{\infty} e^{-|u|/{\tau_d}}e^{-i(\omega_{ig}-\omega_0)u} ~du\right]~e^{-i(\omega_{ig}+\omega_{jg})t_2} ~dt_2\\ \\ &=E_0^2 \int_{-\infty}^\infty\left[\int_{-\infty}^{0} e^{\left[\frac{1}{\tau_d}-i(\omega_{ig}-\omega_0)\right]u} ~du+\int_{0}^{\infty}e^{-\left[\frac{1}{\tau_d}+i(\omega_{ig}-\omega_0)\right]u} ~du\right]~e^{-i(\omega_{ig}+\omega_{jg})t_2} ~dt_2\\ \\ &=E_0^2 \int_{-\infty}^\infty\left[\dfrac{1}{\frac{1}{\tau_d}-i(\omega_{ig}-\omega_0)}+\dfrac{1}{\frac{1}{\tau_d}+i(\omega_{ig}-\omega_0)}\right]~e^{-i(\omega_{ig}+\omega_{jg})t_2} ~dt_2\\ \\ &=2E_0^2 \dfrac{\frac{1}{\tau_d}}{\frac{1}{\tau_d^2}+(\omega_{ig}-\omega_0)^2}\int_{-\infty}^\infty e^{-i(\omega_{ig}+\omega_{jg})t_2} ~dt_2\\ \\ &= 4\pi E_0^2 \delta(\omega_{ig}+\omega_{jg}) \dfrac{\frac{1}{\tau_d}}{\frac{1}{\tau_d^2}+(\omega_{ig}-\omega_0)^2} \\ \end{align*}$$

The argument of my $\delta()$ function has a sign difference compared to the expected result, so someone has made a sign mistake or typo somewhere.

And some observations on the result:

1. There is a Dirac delta "wall" along the line $\omega_{ig} = - \omega_{jg}$ in the 2D frequency domain
2. The peak "height" of this wall is at $\omega_{ig} = \omega_0$, so the coordinate $(\omega_0, -\omega_0)$
3. The "asymmetry" between $\omega_{ig}$ and $\omega_{jg}$ you observed in the result really isn't there. One can exchange $ \omega_{ig} \leftrightarrow \omega_{jg} $ and the result is fundamentally the same.