Group delay with Gaussian pulse

[EmilyRuck]

Hello!
Starting from a gaussian waveform propagating in a dispersive medium, is it possible to obtain an expression for the waveform at a generic time $t$, when the dispersion is not negligible?

I know that a generic gaussian pulse (considered as an envelope of a carrier at frequency $k_c$) can be expressed for $t = 0$ as

$g(x) = e^{ - \frac{1}{2} \left( \frac{x - x_0}{\sigma_x} \right)^2} e^{i k_c (x - x_0)}$

and its Fourier transform is

$G(k) = \displaystyle \frac{\sigma_x}{\sqrt{2 \pi}} e^{- \frac{(k - k_c)^2 \sigma_x^2}{2}} e^{-i (k - k_c)x_0}$

Then, the pulse at a generic time $t$ can be obtained as

$G(x,t) = \displaystyle \int_{-\infty}^{+\infty} G(k) e^{i(kx - \omega(k) t)} dk$

with $\omega(k) = \omega(k_c) + (k - k_c) \displaystyle \left. \frac{d \omega}{dk} \right|_{k = k_c} + (k - k_c)^2 \left. \frac{d^2 \omega}{dk^2} \right|_{k = k_c}$.

The first derivative is known as the group velocity. Note that the second derivative is considered too.

I don't know how to proceed in order to obtain an explicit form for $G(x,t)$ which can show its evolution during time, showing the velocity of propagation of the pulse and its possible broadening. With a simple substitution of $\omega(k)$ and solution of the integral I did not get such an explicit form. Is there anyone who can help, or anyone who does know a site/document dealing with this topic?
Thank you anyway,

Emily

 

[vanhees71]

To neglect the 2nd higher derivatives of the dispersion relation is, of course, only an approximation. You can improve by keeping the 2nd order. Going beyond usually leads to cases, which cannot be integrated analytically anymore. See, e.g.,

http://nptel.ac.in/courses/112105165/lec8.pdf

Sometimes, it's necessary to consider the exact expression, e.g., in the interesting case of anomalous dispersion. There are famous historical papers on the subject by Sommerfeld and Brillouin. For a good introduction, see Sommerfeld, Lectures on theoretical physics, vol. 4 (Optics).

 

[EmilyRuck]

To neglect the 2nd higher derivatives of the dispersion relation is, of course, only an approximation.

Yes, of course, in fact in my post I wrote the 2nd derivative has to be considered.

You can improve by keeping the 2nd order. Going beyond usually leads to cases, which cannot be integrated analytically anymore.

Ok, I understand.

See, e.g.,

http://nptel.ac.in/courses/112105165/lec8.pdf

Thank you. Did you follow the whole procedure? I can't get how he obtains a Schrödinger equation describing the amplitude of the (8.7) (page 5).

Sometimes, it's necessary to consider the exact expression, e.g., in the interesting case of anomalous dispersion. There are famous historical papers on the subject by Sommerfeld and Brillouin. For a good introduction, see Sommerfeld, Lectures on theoretical physics, vol. 4 (Optics).

Thanks for this one too.

Emily

 

[vanhees71]

Concernig Eq. (8.5), isn't this just the substitution $k=k_0+\xi$ in (8.4) and then cancelling the factor $\exp(\mathrm{i} k_0 x)$ on both sides of the equations?

 

[EmilyRuck]

Yes, of course it is (as explained in the document): this is straightforward. Anyway I can't get how "the evolution of the amplitude $A(x,t)$ of the harmonic wave $e^{i (k_0 x - \omega (k_0) t)}$ in (8.7) is governed by the" Schrödinger equation (page 5).