Numerical Aperture Calculator
Calculate the numerical aperture (NA) of an optical fiber from the refractive indices of its core and cladding. NA determines light-gathering ability and the maximum acceptance angle of the fiber.
About this calculator
Numerical aperture (NA) quantifies how much light an optical fiber or lens can collect. For a step-index optical fiber, it is derived from the condition for total internal reflection at the core-cladding interface: NA = √(n_core² − n_cladding²), where n_core is the refractive index of the fiber core and n_cladding is the refractive index of the surrounding cladding material. A higher NA means the fiber accepts light over a wider cone of angles. The acceptance half-angle θ_max is related by sin(θ_max) = NA. NA also governs resolution in microscopy: higher NA objectives resolve finer detail because they capture more of the diffracted light from the specimen. Typical single-mode fibers have NA ≈ 0.1–0.2, while multimode fibers range up to 0.5.
How to use
Suppose a fiber has a core refractive index of 1.50 and a cladding refractive index of 1.46. Enter coreIndex = 1.50 and claddingIndex = 1.46. The calculator computes: NA = √(1.50² − 1.46²) = √(2.25 − 2.1316) = √(0.1184) ≈ 0.344. This means the fiber can accept light entering at angles up to arcsin(0.344) ≈ 20.1° from the fiber axis — the acceptance cone. A larger index difference between core and cladding would yield a higher NA and a wider acceptance angle.
Frequently asked questions
What does numerical aperture tell you about an optical fiber's performance?
Numerical aperture describes two key performance traits of an optical fiber: its light-gathering efficiency and its bandwidth-distance product. A higher NA means the fiber collects light from a wider cone, making it easier to couple light sources into the fiber. However, higher NA multimode fibers also suffer from greater modal dispersion — different light paths travel different lengths and arrive at slightly different times — which limits data transmission speed over long distances. Single-mode fibers use a very small NA and core diameter to allow only one propagation mode, dramatically increasing bandwidth.
How is numerical aperture different for a microscope objective compared to an optical fiber?
For a microscope objective, NA = n · sin(θ), where n is the refractive index of the medium between the lens and the sample (air, water, or oil) and θ is the half-angle of the maximum light cone the objective can accept. For an optical fiber, NA = √(n_core² − n_cladding²), derived from the geometry of total internal reflection. Both definitions capture the same physical idea — the range of angles over which the system collects or emits light — but arise from different optical configurations. Oil-immersion objectives use high-n immersion oils to push NA above 1.0, something impossible in air.
Why must the core refractive index always be greater than the cladding index in an optical fiber?
Total internal reflection — the mechanism that keeps light trapped inside the fiber core — can only occur when light travels from a denser (higher n) to a rarer (lower n) medium. If the cladding index were equal to or greater than the core index, light striking the core-cladding boundary would refract out rather than reflect back, and the fiber would leak light instead of guiding it. This is also why the formula NA = √(n_core² − n_cladding²) would yield an imaginary number if n_cladding ≥ n_core, signaling that the configuration cannot support guided propagation.