The pulse duration from an ultra-fast laser is 20 femtoseconds,
what is the frequency spread of that pulse? Assuming the nominal
wavelength of the laser is 400 nm can we consider the pulse to be
monochromatic in wavelength?

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.

The pulse duration from an ultra-fast laser is 20 femtoseconds, what is the frequency spread of that pulse? Assuming the nominal wavelength of the laser is 400 nm can we consider the pulse to be monochromatic in wavelength?

Calculate the frequency spread of a 20 femtosecond laser pulse with a wavelength of 400 nm, using the time-bandwidth product. This analysis explains the pulse's frequency spread and evaluates whether the pulse can be considered monochromatic. Suitable for undergraduate physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation