An optical fiber has n1
 = 1.48, n2
 = 1.46. and d = 9 km. What is the dispersion in ns?

To calculate the dispersion in an optical fiber, we first need to understand the parameters given:

- \( n_1 = 1.48 \) is the refractive index of the core.
- \( n_2 = 1.46 \) is the refractive index of the cladding.
- \( d = 9 \) km is the length of the optical fiber.

Dispersion is generally expressed in terms of time delay or spread per unit length and can be calculated using the difference in time taken by light to travel through the core and the cladding. The dispersion in this case is mainly due to the difference in group velocities between the two modes (core and cladding).

### Step-by-Step Calculation

1. **Speed of light in the medium**:
   The speed of light \( v \) in a medium is given by:
   \[
   v = \frac{c}{n}
   \]
   where \( c \) is the speed of light in a vacuum (\( 3 \times 10^8 \) m/s), and \( n \) is the refractive index of the medium.

   For the core:
   \[
   v_1 = \frac{3 \times 10^8 \text{ m/s}}{1.48} = 2.027 \times 10^8 \text{ m/s}
   \]

   For the cladding:
   \[
   v_2 = \frac{3 \times 10^8 \text{ m/s}}{1.46} = 2.055 \times 10^8 \text{ m/s}
   \]

2. **Time taken to travel 9 km**:
   Time \( t \) is calculated by:
   \[
   t = \frac{d}{v}
   \]
   
   For the core:
   \[
   t_1 = \frac{9 \times 10^3 \text{ m}}{2.027 \times 10^8 \text{ m/s}} = 4.44 \times 10^{-5} \text{ s} = 44.4 \text{ ns}
   \]

   For the cladding:
   \[
   t_2 = \frac{9 \times 10^3 \text{ m}}{2.055 \times 10^8 \text{ m/s}} = 4.38 \times 10^{-5} \text{ s} = 43.8 \text{ ns}
   \]

3. **Calculate the dispersion**:
   The dispersion \( \Delta t \) is the difference in time taken by the two modes:
   \[
   \Delta t = t_1 - t_2 = 44.4 \text{ ns} - 43.8 \text{ ns} = 0.6 \text{ ns}
   \]

### Final Answer
The dispersion in the optical fiber is **0.6 nanoseconds (ns)**.

Would you like any further details or explanations?

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Here are 5 related questions you might consider:

1. How does the difference in refractive index affect signal quality in an optical fiber?
2. What other factors contribute to dispersion in optical fibers?
3. How can dispersion be minimized in long-distance fiber optic communication?
4. What is the significance of group velocity in optical fibers?
5. How does modal dispersion differ from chromatic dispersion in optical fibers?

**Tip:** Dispersion in optical fibers can lead to signal degradation over long distances. Techniques like using single-mode fibers or dispersion-shifted fibers can help mitigate this effect.

An optical fiber has n1 = 1.48, n2 = 1.46. and d = 9 km. What is the dispersion in ns?

This problem calculates the dispersion in an optical fiber with a core refractive index of n1=1.48, a cladding refractive index of n2=1.46, and a fiber length of 9 km. The detailed step-by-step solution involves calculating time delay using refractive indices and determining the dispersion in nanoseconds. Suitable for college-level optics or fiber communications studies.

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