To calculate the maximum data rate (or bandwidth) of an optical fiber, we need to consider several factors related to the fiber's physical and optical properties.

Key parameters given:

• Core radius ($r$): $25 \, \mu m = 25 \times 10^{-6} \, m$
• Refractive index of the core ($n_1$): 1.47
• Refractive index of the cladding ($n_2$): 1.45
• Fiber length: 10 km = $10 \times 10^3 \, m$

Step 1: Numerical Aperture (NA)

The numerical aperture (NA) of the optical fiber can be calculated using the equation:

$NA = \sqrt{n_1^2 - n_2^2}$

Substituting the values for $n_1$ and $n_2$:

$NA = \sqrt{(1.47)^2 - (1.45)^2}$

Step 2: Calculate V-number (Normalized Frequency)

The normalized frequency or V-number is given by:

$V = \frac{2 \pi r}{\lambda} \cdot NA$

where:

• $r$ is the core radius
• $\lambda$ is the wavelength of light (usually for optical fibers, $\lambda$ is around 850 nm or $1.3 \, \mu m$ for single-mode fiber)

Assume $\lambda = 850 \, nm = 850 \times 10^{-9} \, m$.

Step 3: Modal Capacity

For a multi-mode fiber, the number of modes $M$ supported is approximated by:

$M \approx \frac{V^2}{2}$

Step 4: Maximum Data Rate

The bandwidth-distance product of an optical fiber can be approximated by the empirical formula:

$\text{Bandwidth} \times \text{Distance} = \frac{c}{M \cdot L}$

where:

• $c$ is the speed of light in the fiber
• $M$ is the number of modes
• $L$ is the fiber length (in km)

Finally, the maximum data rate (in Mbps) can be found by solving for Bandwidth.

I'll perform these calculations step by step.### Results:

1. Numerical Aperture (NA): 0.2417
2. V-number (Normalized Frequency): 44.66
3. Number of Modes (M): 997.21

Using these results, the maximum data rate of the optical fiber is approximately 0.0205 Mbps.

This value indicates that the fiber's capacity to transmit data in a multimode fiber is limited by the number of modes and the length of the fiber.

Would you like further details or clarifications on any of the steps?

Here are 5 related questions:

1. How does the choice of wavelength affect the fiber's data rate?
2. What role does core radius play in determining the number of modes in an optical fiber?
3. How can the refractive indices $n_1$ and $n_2$ be optimized for better performance?
4. What is the difference in data rate between multimode and single-mode fibers?
5. How would reducing the fiber length impact the maximum data rate?

Tip: For long-distance communication, single-mode fibers are preferred due to their higher bandwidth-distance product.