We are given the following:

• Input power ($P_{in}$): 2 mW
• Attenuation ($\alpha$): 0.2 dB/km
• Length of fiber: 18 km

To find the output power of the fiber optic system in dBm, we follow these steps:

Step 1: Calculate the total attenuation over the fiber length

$\text{Total attenuation} = \alpha \times \text{length} = 0.2 \, \text{dB/km} \times 18 \, \text{km} = 3.6 \, \text{dB}$

Step 2: Calculate the input power in dBm

The formula to convert power from mW to dBm is:

$P_{in \, (\text{dBm})} = 10 \log_{10} \left( P_{in} \right)$

Where $P_{in}$ is in mW. For $P_{in} = 2 \, \text{mW}$:

$P_{in \, (\text{dBm})} = 10 \log_{10} (2) \approx 3.01 \, \text{dBm}$

Step 3: Calculate the output power in dBm

To find the output power, subtract the total attenuation from the input power in dBm:

$P_{out \, (\text{dBm})} = P_{in \, (\text{dBm})} - \text{Total attenuation}$

$P_{out \, (\text{dBm})} = 3.01 \, \text{dBm} - 3.6 \, \text{dB} = -0.59 \, \text{dBm}$

Final Answer:

The output power of the fiber optic system is approximately -0.59 dBm.

Would you like more details or have any questions?

Here are some related questions to consider:

1. How do different fiber lengths affect the output power?
2. What is the relationship between input power and output power in optical systems?
3. How does changing the attenuation coefficient impact fiber optic performance?
4. How do you convert between dBm and milliwatts?
5. What factors contribute to attenuation in fiber optics?

Tip: Always remember that every 3 dB of attenuation reduces power by half in a logarithmic scale.