Where does the temporal stretch come from in Group Velocity Dispersion?

[chris_avfc]

Homework Statement

Our lecturer seemed to skip over how to get from the Group Velocity Dispersion to the actual temporal stretch of a pulse sent down an optical fibre, instead we were given just the two formula. I've been trying to work out where the temporal stretch comes from but can't work it out.

Homework Equations

The GVD in this case is given by
$$ GVD = \frac{dv_g}{dk} = -\frac{\omega}{n}\frac{d^2n}{dk^2}=-\frac{\omega}{n}\frac{\lambda^4}{4\pi^2}\frac{d^2n}{d\lambda^2}$$

The temporal stretch is given by
$$ \Delta\tau = -\frac{L}{c}(\lambda^2\frac{d^2n}{d\lambda^2}) \frac{\Delta\lambda}{\lambda} $$

The Attempt at a Solution

Have made very little progress, only got as far as getting the same factors as in the temporal stretch.

$$ GVD = -(\lambda^2\frac{d^2n}{d\lambda^2})(\frac{\lambda^2}{4\pi^2}\frac{\omega}{n})$$

 

[Goddar]

Hi. I'm not familiar with the subject but there seems to be something wrong with your GVD equation. Check out these pages for reference:
http://www.rp-photonics.com/group_velocity_dispersion.html
http://en.wikipedia.org/wiki/Dispersion_(optics)
It seems from there that the temporal stretch would be defined as: Δτ = –LD Δλ,
Where the dispersion D = –2πc(GVD)/λ²
Maybe somebody else can help you further with this...