To find the frequency spread of the pulse, we will use the time-bandwidth product, which is a principle in physics that relates the duration of a pulse to its frequency spread. For a Gaussian-shaped pulse, this relationship can be approximated as:

$\Delta \nu \cdot \Delta t \approx 0.44$

where:

• $\Delta \nu$ is the frequency spread (in Hz).
• $\Delta t$ is the pulse duration (in seconds).

Given:

• $\Delta t = 20$ femtoseconds = $20 \times 10^{-15}$ seconds.

Plugging in the values: $\Delta \nu \approx \frac{0.44}{\Delta t} = \frac{0.44}{20 \times 10^{-15} \, \text{s}}$ $\Delta \nu \approx 2.2 \times 10^{13} \, \text{Hz}$

Calculating the Frequency Spread

The frequency spread of the pulse is approximately $2.2 \times 10^{13}$ Hz.

Monochromaticity Check

To determine whether the pulse can be considered monochromatic, we can compare the frequency spread with the central frequency of the laser. The central frequency $\nu_0$ of the laser is given by the formula:

$\nu_0 = \frac{c}{\lambda}$

where:

• $c = 3 \times 10^8$ m/s (speed of light).
• $\lambda = 400$ nm = $400 \times 10^{-9}$ m.

Calculating the central frequency: $\nu_0 = \frac{3 \times 10^8 \, \text{m/s}}{400 \times 10^{-9} \, \text{m}} = 7.5 \times 10^{14} \, \text{Hz}$

Now, comparing the frequency spread $\Delta \nu$ to the central frequency $\nu_0$: $\frac{\Delta \nu}{\nu_0} = \frac{2.2 \times 10^{13}}{7.5 \times 10^{14}} \approx 0.029$

This ratio indicates that the frequency spread is about 2.9% of the central frequency.

Conclusion

The pulse cannot be strictly considered monochromatic because a significant frequency spread exists relative to the central frequency. Although it is still relatively narrow, the pulse has a sufficiently broad spectrum to impact its monochromatic assumption.

Would you like more details or have any questions about this topic?

Here are 5 related questions to expand your understanding:

1. What is the significance of the time-bandwidth product in pulse analysis?
2. How does the shape of the pulse affect its time-bandwidth product?
3. What happens to the frequency spread if the pulse duration is decreased?
4. How do ultra-short pulse lasers differ from continuous-wave lasers in their applications?
5. What are some practical applications of femtosecond laser pulses in technology and science?

Tip: Shorter pulse durations always lead to a broader frequency spread according to the uncertainty principle, which is crucial for designing lasers for specific applications.