Group velocity dispersion on two pulses of different lengths

[StudentonaOdyssey]

Homework Statement

GER:
Zwei bandbreitenlimitierte Pulse mit derselben Mittenfrequenz durchtreten dasselbe transparente optische Bauelement mit einer Dicke von 1 cm und einer GVD von 50 fs2/mm. Der eine Puls ist beim Eintritt 10-mal so lang wie der andere. Welcher der beiden Pulse ist nach dem Durchtritt unmittelbar nach dem Element länger? Um wievielmal länger ist dieser Puls als der andere?
ENG:
Two bandwith-limited pulses with the same middle frequency pass through the same transparent optic component with a thickness of 1cm and a GVD (Group Velocity Dispersion) of 50 fs^2/mm. One pulse is 10-times as long as the other on impact. Which one of the pulses is longer directly after the component? How many times longer is this pulse than the other.

Relevant Equations

Bandwidth-Limit
$$\Delta \omega \cdot \Delta t >= C$$

Pulse length after propagation
$$\tau _p = \tau _0 \sqrt{1+\frac{z^2}{D^2}}$$

$$D= \frac{\tau_0^2}{2\cdot GVD}$$

$$\tau _{01} = 10 \tau _{01}$$
If I calculate $\frac{\tau_{p1}}{\tau_{p1}}$ and set z=d=1cm I do not know how to continue from there as I can't solve the equation without knowledge of τ0 for D.
$$\frac{\tau_{p1}}{\tau_{p1}} = \frac{\tau_{02} \cdot 10}{\tau_{02}} \sqrt{\frac{1+\frac{d^2 \cdot 4 \cdot GVD^2}{\tau _{02}^4 \cdot 10^4}}{{1+\frac{d^2 \cdot 4 \cdot GVD^2}{\tau _{02}^4}}}}$$

 

[guv]

isn't $\tau_{p1}/\tau_{p1} = 1$? are we confusing something here?

 

[StudentonaOdyssey]

Yes that is my mistake, it should have been

$$\frac{\tau _{p1}}{\tau _{p2}}$$
Same with the ratio

$$\tau _{01} = 10 \cdot \tau _{02}$$