Group velocity dispersion on two pulses of different lengths

In summary, group velocity dispersion (GVD) is a phenomenon in optics that affects the velocity of a group of waves or pulses depending on their wavelength or frequency. This can lead to pulse broadening or distortion, especially for pulses of different lengths traveling together. The relationship between pulse length and GVD is that longer pulses are more susceptible to its effects. To compensate for GVD, dispersion compensating materials or pulse compression techniques can be used. Understanding GVD in two pulses of different lengths is important for fields such as optical communications and laser technology, where accurate detection and transmission of pulses is crucial.
  • #1
StudentonaOdyssey
7
1
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.
$$\frac{\tau_{p1}}{\tau_{p1}} = \frac{\tau_{02} \cdot 10}{\tau_{02}} \sqrt{\frac{1+\frac{d^2 \cdot 4 \cdot GVD^2}{\tau _{02}^4 \cdot 10^4}}{{1+\frac{d^2 \cdot 4 \cdot GVD^2}{\tau _{02}^4}}}}$$
 
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  • #2
isn't ##\tau_{p1}/\tau_{p1} = 1##? are we confusing something here?
 
  • #3
Yes that is my mistake, it should have been

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

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

1. What is group velocity dispersion (GVD)?

Group velocity dispersion is a phenomenon in which the group velocity (the velocity at which the overall shape of a pulse moves) of a light pulse changes with respect to its wavelength. This can lead to distortion and broadening of the pulse, which can affect the quality of optical communication signals.

2. How does GVD affect two pulses of different lengths?

In the case of two pulses of different lengths, GVD can cause the pulses to arrive at different times and with different shapes. This can result in a decrease in the quality and accuracy of data transmission, as the pulses may overlap and interfere with each other.

3. What factors can contribute to GVD?

GVD can be caused by various factors, such as the material properties of the medium through which the light is traveling, the spectral width of the light source, and the length of the optical path. In fiber optics, GVD is mainly influenced by the refractive index and dispersion properties of the fiber material.

4. How is GVD measured?

GVD is typically measured in units of ps/(nm·km) or fs/(nm·m), which represent the amount of time delay or dispersion per unit distance. It can be measured using specialized equipment, such as an optical spectrum analyzer or a frequency-resolved optical gating (FROG) device.

5. Can GVD be compensated for?

Yes, GVD can be compensated for by using dispersion compensation techniques, such as dispersion compensating fibers, chirped fiber Bragg gratings, and dispersion compensating modules. These methods aim to introduce an opposite amount of dispersion to cancel out the effects of GVD and maintain the integrity of the transmitted signals.

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