Physical significance of numerical aperture

Physical significance of NA

Numerical aperture NA = sin θ₀ = √(n₁² − n₂²) describes how much light a fiber accepts and guides.

1. Light-gathering power: NA fixes the acceptance cone with half-angle θ₀. Larger NA means a wider cone, so more light from a source (LED, laser) can be launched, giving higher coupling efficiency.
2. Acceptance angle: θ₀ = sin⁻¹(NA). For NA = 0.2, θ₀ ≈ 11.5°.
3. Number of modes: the V-number is V = (2πa/λ) NA, and the number of modes in a multimode step-index fiber is about V²/2. Higher NA means more modes.
4. Dispersion and bandwidth: more modes lead to more intermodal dispersion, so a very high NA reduces the information capacity. A very low NA, on the other hand, makes the fiber more sensitive to bending loss, because the guiding is weak.
5. NA cannot exceed 1 for air.

Typical values: single-mode fibers 0.1 to 0.14; multimode fibers 0.2 to 0.5.

Dependence on the refractive indices

NA = √(n₁² − n₂²) = n₁ √(2Δ),   Δ = (n₁ − n₂)/n₁

• NA increases when n₁ (core) increases or n₂ (cladding) decreases; it depends only on the difference of the squares n₁² − n₂².
• If n₁ = n₂, then NA = 0 and no light is guided.
• It is independent of the core diameter.
• Example: n₁ = 1.48, n₂ = 1.46 gives NA = √(2.1904 − 2.1316) = 0.242, so θ₀ ≈ 14°.