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Fiber Optic Numerical Aperture

Calculates the light-accepting cone angle of an optical fiber from the refractive indices of its core and cladding.

Written by the Vorzep Team

OpticsPhotonicsFiber Optics

Fiber Optic Numerical Aperture CalculatorNA = √(ncore² − ncladding²)

NA = √( ncore² − ncladding² )
NA = numerical aperture  ·  ncore = core refractive index  ·  ncladding = cladding refractive index
⟹ Solve⟷NA, ncore, ncladding
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Presets:
Numerical Aperture—
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NA: —ncore: —ncladding: —
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Numerical Aperture Gauge—
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NA = √(ncore² − ncladding²)  ·  NA determines the light‑gathering ability and acceptance angle of an optical fiber.

Variables

SymbolQuantityUnit
NANumerical aperture
n_coreRefractive index of fiber core
n_claddingRefractive index of fiber cladding

What it means

The numerical aperture of an optical fibre characterises the range of angles at which light can be launched and guided. It depends on the refractive indices of the core and cladding. A larger NA means more light can be coupled but may increase modal dispersion. This is used to design fibre optic systems and to select fibres for specific applications. Understanding NA is essential for telecommunication engineers and for designing fibre‑based sensors.

Worked example

Fiber Numerical Aperture – Two Detailed Examples

Real‑World
Scenario: A fiber optic cable has core refractive index n_core = 1.48 and cladding n_cladding = 1.46. The numerical aperture NA = sqrt(n_core² - n_cladding²) = sqrt(1.48² - 1.46²) = sqrt(2.1904 - 2.1316) = sqrt(0.0588) = 0.2425. This NA determines the acceptance angle of the fiber. The communications engineer uses this to couple light efficiently from a laser source into the fiber.
ParameterValue
n_core1.48
n_cladding1.46
1NA = sqrt(1.48² - 1.46²) = sqrt(0.0588) = 0.2425
Result 0.2425 ✓ Fiber NA
Scenario: A step‑index fiber with n_core = 1.5 and n_cladding = 1.47 has NA = sqrt(1.5² - 1.47²) = sqrt(2.25 - 2.1609) = sqrt(0.0891) = 0.2985. This higher NA allows the fiber to capture more light, which is beneficial for short‑distance applications requiring high optical power transmission. The system designer uses the NA to match the source optics.
ParameterValue
n_core1.5
n_cladding1.47
1NA = sqrt(1.5² - 1.47²) = sqrt(2.25 - 2.1609) = sqrt(0.0891) = 0.2985
Result 0.2985 ✓ Higher NA
Insight: The numerical aperture of a fiber determines its light‑gathering ability. It is derived from the refractive index difference between core and cladding. A higher NA allows more light to enter but may increase modal dispersion.

Common mistakes

  • Fiber optic numerical aperture: NA = √(n_core² − n_cladding²) – for a step‑index fiber.
  • n_core: Refractive index of the core – > n_cladding.
  • NA: Dimensionless – related to the maximum acceptance angle: NA = n₀·sin(θ_max), where n₀ is the external medium (usually air).
  • NA determines: The light‑gathering ability and the number of modes.
  • For graded‑index fibers: NA varies across the core – use the maximum value.

Applications

Fiber optic numerical aperture, NA = √(n_core² − n_cladding²), characterises the light‑gathering ability of an optical fibre. It determines the maximum acceptance angle for light entering the fibre. Engineers use NA to design fibre optic communication systems, to select appropriate coupling optics, and to assess the performance of fibre sensors. A higher NA allows more light collection but may increase dispersion. This formula is essential for designing fibre optic components and for understanding the propagation of light in fibres. By calculating NA, professionals can ensure efficient light coupling and transmission.

  • Design of fibre optic communication links
  • Selection of connectors, couplers, and splices
  • Fibre optic sensor design for strain, temperature, etc.
  • Coupling of light sources (lasers, LEDs) to fibres
  • Education on fibre optics and light guidance