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Short note: Numerical aperture

Short answer

Numerical aperture is the light-gathering ability of a fibre, defined as NA = n0 sin θa = sqrt(n1² − n2²) ≈ n1√(2Δ). It depends only on core and cladding refractive indices, not fibre size; larger NA eases coupling but increases modal dispersion. For n1=1.48, n2=1.46, NA=0.2425 and acceptance angle θa=14.0°.

Write a short note on numerical aperture (NA).

Numerical aperture (NA) is a measure of the light-gathering (acceptance) ability of an optical fibre. It is defined as the sine of the acceptance angle multiplied by the refractive index of the medium from which light enters.

NA = n0 sin θa

Expression

From Snell's law at the input face, n0 sin θa = n1 cos φc, and from the TIR condition sin φc = n2/n1:

NA = √(n1² − n2²) ≈ n1 √(2Δ),   Δ = (n1 − n2)/n1
θa = sin⁻¹(NA)        (launch from air, n0 = 1)

          \  θa
   air     \       core n1, cladding n2
   ---------*===========================
           /   acceptance cone 2θa

Points to note

  • NA depends only on n1 and n2, not on the fibre size.
  • Large NA: easy coupling of light from LEDs, but more modes and more modal dispersion, so less bandwidth.
  • Typical values: 0.1–0.15 (single mode), 0.2–0.3 (multimode).
  • NA also enters the V-number V = (2πa/λ)·NA, which fixes the number of modes (single mode if V < 2.405).

Example: n1 = 1.48, n2 = 1.46: NA = √(2.1904 − 2.1316) = 0.2425, θa = 14.0°.

✓ Answer checked by Claude Opus 5.5 (high effort)

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