# Group Velocity Dispersion Pulse Broadening Assignments

To demonstrate the influence of the group (GVD) velocity dispersion on pulse propagation in optical fibers in “linear” regime. The basic effects related to GVD are:

- GVD induced pulse broadening
- GVD induced pulse chirping
- Pulse compression

The equation, which describes the effect of GVD on optical pulse propagation neglecting the losses and nonlinearities, is [1]:

where is the propagation direction, is the time, is the electric field envelope, and

is the GVD parameter, defined as the second derivative of the fiber mode propagation constant with respect to frequency.

For an input pulse with a Gaussian shape,

the pulse width T_{0}(related to the pulse full width at half maximum by

T_{FWHM }≈ 1.665T_{0}) increases with z (the pulse broadens) according to [1]:

and, consequently, the peak power changes, due to GVD, are given by:

In Equation 3 and Equation 4, the quantity

is the dispersion length.

Its meaning is quite straightforward: after propagating a distance equal to L_{D}, the pulse broadens by a factor of √2.

To demonstrate this, we created the following simple project (Figure 1).

**Figure 1: GVD project layout**

We set the Bit rate equal to 40 Gb/s, which corresponds to bit duration of 25 ps. Using

the default value of 0.5 for “width” of the **Optical Gaussian Pulse Generator**, the

resulting FWHM of the pulse is 12.5 ps.

The T_{0} parameter is then

Using the value of β_{2} ≈ -20(ps)^{2} at 1.55μm for SMF, the dispersion length is:

In the** Optical Fiber** properties, we set the length of the fiber equal to this value, and

we disable all the effects except GVD (Figure 2).

We calculate the project and the obtained results are presented in Figure 3. We see that the pulse is broadened (the peak power decreases in accordance with Equation 4). The origin of pulse broadening can be understood be looking at the instant frequency of the pulse, namely the chirp.

**Figure 2: Optical Fiber properties**

This is shown in Figure 4, where the pulse chirp is plotted together with the pulse intensity. Whereas the input pulse is chirpless, the instantaneous frequency of the output pulse decreases from the leading to the trailing edge of the pulse. The reason for this is GVD. In the case of anomalous GVD (β_{2 }<0), the higher frequency (“blue-shifted”) components of the pulse travel faster than the lower frequency (or “red-shifted”) ones [1].

**Figure 3: Left plot – input pulse, right plot pulse at**

**Figure 4: Same as in Figure 3. The straight lines represent the chirp**

**Note:** The leading edge of the pulse is blue shifted and the trailing edge of the pulse is red-shifted. Because the “blue” and “red” spectral components tend to separate in time, this leads to pulse broadening. However, the pulse spectrum remains unchanged, as Figure 5 shows.

**Figure 5: Input (left plot) and output (right plot) spectra corresponding to Figure 3 and 4**

If the input pulse is frequency modulated (i.e. chirped), Equation 2 is replaced by:

and the expression for the dependence of the pulse width on z is [1]

The pulse broadens monotonically with z if β_{2}C > 0 , however, it goes through initial

narrowing when . In the latter case, the pulse width becomes minimum at

distance [1]:

and is given by

The peak power of the pulse in this case is:

P(z_{min}) = P_{0}(1 + C^{2})^{1/2}

Initial narrowing of the pulse for the case β_{2}C < 0 can be explained by noticing that

in this case the frequency modulation (or “chirp”) is such that the faster (“blue” in the

case of anomalous GVD) frequency components are in the trailing edge, and the

slower (or “red” in the case of anomalous GVD) in the leading edge of the pulse. As

the pulse propagates, the faster components will overtake the slower ones, leading to

pulse narrowing. At the same time, the dispersion induced chirp will compensate for

the initial one. At z = z_{min} , full compensation between both will occur. With further

propagation, the fast and the slow frequency components will tend to separate in time

from each other and, consequently, pulse broadening will be observed.

To demonstrate this, we use a chirped Gaussian pulse with the chirp parameter C = 2 (since β_{2 }<0 in our case) (Figure 6).

**Figure 6: Setting the chirp parameter to observe pulse compression**

Using Equation 7 and Equation 9, we obtain:

We change the length of the fiber to 1.125km and calculate the project.

The results for the output pulse shape and chirp are presented in Figure 7. It can be seen that an exact compensation between the dispersion induced and initial chirp occurs, and that the peak power of the pulse is 2.2mW, as given by Equation 9.

**Figure 7: Pulse shape and chirp at z=0 (left) and z=zmin (right)**

**Note**: There is exact compensation between the initial and the dispersion-induced chirp.

**Reference:**

[1]G. P. Agrawal Nonlinear Fiber Optics, Academic Press (2001).

## Abstract

The influence of group velocity dispersion (GVD) on the self-focusing of femtosecond laser pulses is investigated by numerically solving the extended nonlinear Schrödinger equation. By introducing the GVD length L_{GVD} into the semi-empirical, self-focusing formula proposed by Marburger, a revised one is proposed, which can not only well explain the influence of GVD on the collapse distance, but also is in good agreement with the numerical results, making the self-focusing formula applicable for more cases.

© 2015 Chinese Laser Press

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