Basic principles (TIR, Acceptance angle and Numerical Aperture)

Basic principles (TIR, Acceptance angle and Numerical Aperture)

1. Total Internal Reflection (TIR) — Guiding Principle

If light travels from denser ($n_1$) to rarer ($n_2<n_1$), Snell's $n_1 sinθ_1 = n_2 sinθ_2$. For $θ_1 > θ_c = sin^{-1}(n_2/n_1)$, $sinθ_2>1$ impossible ⇒ no refraction, 100% reflection inside denser medium — TIR. Fibre guides by TIR at core-cladding.

2. Acceptance Angle ($θ_a$) and Cone

Light entering fibre from air ($n_0≈1$) at angle $θ_0$ to axis refracts into core at $θ_r$, then hits core-cladding at $φ =90°-θ_r$. For TIR need $φ ≥ θ_c$ ⇒

Derivation: $n_0 sinθ_a = n_1 sinθ_r$, and $sinθ_r = cosφ ≤ cosθ_c = √[1-(n_2/n_1)²]$ ⇒ above.

Rays within acceptance cone half-angle $θ_a$ are guided; outside leak as refracted.

For fibre in air $n_0=1$:

3. Numerical Aperture (NA)

NA = measure of light-gathering ability:

(For air, $NA =√[n_1²-n_2²]$). Also

where $Δ=(n_1-n_2)/n_1$.

• Large $Δ$ (large index step) ⇒ large $NA$ ⇒ larger acceptance cone and more modes, but more modal dispersion.

Typical: single-mode $NA≈0.12$ ($θ_a≈7°$), multimode $NA≈0.22$ ($≈12°$).

4. Inter-relation

$NA$ determines number of modes:

where $a$ = core radius. Single-mode when $V<2.405$ (cutoff of first higher mode).

5. Worked Examples

(a) n1=1.48, n2=1.46, air: NA=√(1.48²-1.46²)=√(0.0588)≈0.242, θ_a≈14° ✓
(b) Want to increase NA? Raise Δ by Ge-doping core.
(c) Fibre in water n0=1.33: effective NA reduces? Actually NA= n0 sinθ_a still √[n1²-n2²] independent of n0? Wait formula NA=n0 sinθ_a =√[n1²-n2²] so θ_a = sin^{-1}(NA/n0) ⇒ smaller θ in water.
(d) V-number for a=25 µm (core 50 µm), λ=1300 nm, NA=0.22 ⇒ V=2π·25e-6·0.22/1.3e-6≈26.6 ⇒ multimode.

6. Exam Points

• State TIR condition $θ_1>θ_c$ — 3 marks, frequent.
• Derive $NA=√[n_1²-n_2²]$ and $θ_a =sin^{-1}(NA/n_0)$ — 5-7 marks, very frequent (almost certain).
• Compute $θ_a, NA$ for given $n_1,n_2$ — 5 marks, very frequent.
• Mention acceptance cone diagram — 5 marks, frequent.