Propagation mechanism of light in fibre

Propagation mechanism of light in fibre

1. Two Views — Ray vs Wave

• Ray (geometric) model: Light rays zig-zag via TIR (step-index) or bend sinusoidally (graded) — intuitive for large $V$.
• Wave (mode) model: Fibre is cylindrical waveguide; Maxwell's equations with boundary at $r=a$ give discrete modes LP$_{lm}$ (like waveguide) with propagation constant $β$ ($n_{eff}=β/k_0$). Both views agree when many modes.

2. Ray Propagation (what exam expects)

• Step-index MMF: ray angle $θ$ to axis, TIR at core-cladding requires $θ ≤ θ_{max}=cos^{-1}(n_2/n_1)=sin^{-1}(NA/n_1)$? Actually complementary. Axial ray shortest, extreme ($θ_{max}$) longest path:

⇒ delay difference

per km (modal dispersion). This is why step MMF pulses spread.

• Graded-index: $n(r)$ parabolic ⇒ ray equation $d/ds(n dr/ds)=∇n$ gives sinusoidal trajectory with period

focusing, so all modes have nearly equal group velocity ⇒ low dispersion.

• Single-mode: only one ray (fundamental mode) ⇒ no modal dispersion.

3. Wave Modes

For step fibre, solving Helmholtz in cylindrical coordinates with Bessel $J_l(κ r)$ in core, $K_l(γ r)$ in cladding, continuity gives eigenvalue equation whose solutions $β_{lm}$ are modes. Count ≈ $V²/4$ for large $V$; cutoff $V_c=2.405$ for LP11.

• Mode field diameter slightly larger than core for SMF.
• Effective index $n_2 < n_{eff} < n_1$.

4. Polarization and Evanescent Field

Total internal reflection leaves evanescent field in cladding decaying as $e^{-γ r}$ over ~$λ$; cladding must be thick enough (≥ few µm) to prevent loss to coating.

5. Worked Intuition

Step-index fibre length 1 km, n1=1.48, Δ=0.01: Δτ≈ n1 L Δ /c ≈1.48·1000·0.01/3e8≈49 ns (step) vs GRIN Δτ≈ n1 L Δ²/8c≈0.06 ns (real).

6. Exam Points

• Describe ray propagation via TIR (step) vs sinusoidal (graded) — 5 marks, very frequent.
• Mention mode view and $V²/4$ estimate — 3-5 marks, frequent.
• Explain modal dispersion due to path length difference — 5 marks, very frequent.
• Evanescent field in cladding — 3 marks.