Pulse Broadening Analysis
The light pulses travel through the optical fiber via different paths. The output pulses will remain distinguishable if the time broadening (dispersion) due to the path difference does not exceed the time separation between the pulses ($\Delta t$).
1. Travel Times:
Let $L$ be the length of the fiber and $\mu$ be the refractive index. The speed of light in the fiber is $v = c/\mu$.
- Axial Ray (Shortest Path): Travels straight along the axis. $$ t_{min} = \frac{L}{v} = \frac{L\mu}{c} $$
- Extreme Ray (Longest Path): Travels at an angle $\theta$ to the axis. It undergoes Total Internal Reflection at the core-cladding interface. The path length $L’$ is related to $L$ by $L = L’ \cos \theta$, so $L’ = L \sec \theta$. $$ t_{max} = \frac{L’}{v} = \frac{L \sec \theta}{v} = \frac{L\mu \sec \theta}{c} $$
2. Condition for Distinguishability:
The pulse broadening (dispersion) is the difference between the maximum and minimum travel times. To keep consecutive pulses distinguishable, this broadening must be less than or equal to the time gap between pulses ($\Delta t$).
$$ t_{max} – t_{min} \le \Delta t $$
3. Solving for Length L:
Substitute the expressions for $t_{max}$ and $t_{min}$:
$$ \frac{L\mu \sec \theta}{c} – \frac{L\mu}{c} \le \Delta t $$
$$ \frac{L\mu}{c} (\sec \theta – 1) \le \Delta t $$
Rearranging to solve for the maximum length $L$:
$$ L \le \frac{c \Delta t}{\mu (\sec \theta – 1)} $$