Fiber Optics & Waveguide Modes
Fiber Optics & Waveguide Modes
An optical fiber guides light by total internal reflection at the core-cladding interface: when the incidence angle exceeds the critical angle $\theta_c=\arcsin(n_2/n_1)$, light is completely reflected. The fiber\'s numerical aperture $\text{NA}=\sqrt{n_1^2-n_2^2}$ sets the acceptance cone.
The $V$-number $V=\frac{2\pi a}{\lambda}\text{NA}$ determines the mode count. Single-mode ($V\lesssim2.405$) eliminates intermodal dispersion, essential for long-haul communications; multimode ($V\gg2.405$) suits short links and sensors.
V-Number & Mode Cut-Off
$$V=\frac{2\pi a}{\lambda}\sqrt{n_1^2-n_2^2}.$$ Single-mode if $V<2.405$ (first zero of $J_0$).
Group-Velocity Dispersion
$$D=-\frac{\lambda}{c}\frac{d^2n_\text{eff}}{d\lambda^2}\;[\text{ps}/(\text{nm}\cdot\text{km})].$$ Pulse broadening: $\Delta\tau=|D|\,L\,\Delta\lambda$.
Example 1
Fiber: $a=4\,\mu\text{m}$, $n_1=1.468$, $n_2=1.464$, $\lambda=1310\,\text{nm}$. Compute $V$.
Solution: $\text{NA}=\sqrt{1.468^2-1.464^2}\approx0.108$; $V=2\pi\times4\times0.108/1.31\approx2.07$ — single mode.
Example 2
100-km fiber, $D=17\,\text{ps/(nm\cdot km)}$, source $\Delta\lambda=0.1\,\text{nm}$. Find pulse broadening.
Solution: $\Delta\tau=17\times100\times0.1=170\,\text{ps}$.
Practice
- Derive the critical angle from Snell\'s law.
- What limits bandwidth in a multimode fiber?
- How does dispersion-shifted fiber achieve $D\approx0$ at 1550 nm?
- Why is single-mode fiber preferred for long-haul links?
Show Answer Key
1. Snell's law: $n_1\sin\theta_1 = n_2\sin\theta_2$. At the critical angle, $\theta_2 = 90°$: $\sin\theta_c = n_2/n_1$ (for $n_1 > n_2$). For glass-air ($n_1=1.5$): $\theta_c = \sin^{-1}(1/1.5) \approx 41.8°$.
2. In multimode fiber, different modes travel at different group velocities (intermodal dispersion). A short input pulse spreads into a broad output pulse: $\Delta t \approx \frac{n_1 \Delta}{c} \cdot L$ where $\Delta = (n_1-n_2)/n_1$ is the relative index difference. This limits bandwidth to $B \sim 1/\Delta t$, typically $\sim$20 MHz·km for step-index multimode fiber.
3. Dispersion-shifted fiber (DSF) modifies the waveguide dispersion (by changing core diameter and refractive index profile) to cancel material dispersion at 1550 nm (the minimum-loss window). Total dispersion $D = D_{\text{mat}} + D_{\text{wg}} \approx 0$ at 1550 nm, maximizing bandwidth for single-channel systems.
4. Single-mode fiber eliminates intermodal dispersion entirely (only one mode propagates). This allows much higher bandwidth × distance products ($\sim$100 GHz·km vs. ~20 MHz·km for multimode). Dispersion is limited to chromatic dispersion (material + waveguide) and polarization mode dispersion, both manageable. Essential for long-haul telecom (>10 km).