To calculate the dispersion in an optical fiber, we can use the following formula for material dispersion:

$\Delta t = \frac{d}{c} \times (n_1 - n_2)$

Where:

• $\Delta t$ is the dispersion in seconds (which we will convert to nanoseconds),
• $d$ is the distance the light travels in the fiber (9 km = 9000 m),
• $c$ is the speed of light in a vacuum ($3 \times 10^8 \, \text{m/s}$),
• $n_1$ is the refractive index of the core of the fiber (1.48),
• $n_2$ is the refractive index of the cladding (1.46).

Step-by-step calculation:

1. The difference in refractive indices: $n_1 - n_2 = 1.48 - 1.46 = 0.02$

2. Using the formula for the time delay or dispersion: $\Delta t = \frac{9000}{3 \times 10^8} \times 0.02$

3. Simplifying: $\Delta t = 3 \times 10^{-5} \times 0.02 = 6 \times 10^{-7} \, \text{seconds}$

4. Converting this to nanoseconds: $\Delta t = 6 \times 10^{-7} \, \text{seconds} = 0.6 \, \text{nanoseconds}$

Thus, the dispersion over the 9 km fiber is 0.6 ns.

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Would you like further details on the calculation? Here are 5 related questions for deeper understanding:

1. How does dispersion affect signal quality in optical fibers?
2. What are the types of dispersion in optical fibers besides material dispersion?
3. How do refractive indices affect the propagation speed of light in a medium?
4. What is the difference between step-index and graded-index fibers?
5. How can we minimize dispersion in optical communication systems?

Tip: Increasing the purity of the glass in optical fibers can reduce dispersion and improve signal transmission quality.