Numerical Aperture

Equivalent forms

NA = n_0 \sin\theta_{max}

\theta_{max} = \arcsin(NA/n)

A single sine sets both how much light an instrument grabs and how fine the detail it can resolve.

Symbol	Name	SI unit	Dimension	Range
$n$	Medium index Refractive index of the medium in the acceptance cone	dimensionless	1	1 – 1.6
$theta_max$	Half-acceptance angle Half-angle of the maximal cone of rays accepted or emitted	rad	1	0 – 1.39
$NA$	Numerical apertureoutput Dimensionless light-gathering parameter	dimensionless	1	0 – 1.6

Unit systems

Symbol	SI	CGS	Imperial
$n$	dimensionless	dimensionless	dimensionless
$theta_max$	rad	rad	deg
$NA$	dimensionless	dimensionless	dimensionless

Where it holds

Valid in the paraxial-to-moderate-angle regime for a well-corrected system. For optical fibers,

NA = \sqrt(n

_core^2 - n_clad^2); the n*sin(theta_max) form applies at the entrance face in the launch medium.

Discovery

Ernst Abbe · 1873

While developing microscope optics for Carl Zeiss, Abbe introduced numerical aperture to quantify a lens's light-gathering and resolving ability, showing that resolution improves directly with NA.

Try this

How does a hair-thin glass fiber decide which rays of light it will accept and which it rejects?

Light enters a fiber from air through a cone with a half-acceptance angle of 30 degrees, with the input medium index 1.0. What is the numerical aperture?

Research status: stable

Real-world applications

Microscope objectives are rated by NA, which sets their resolution limit.
Optical-fiber NA determines how easily light couples in and how much modal dispersion occurs.
Photolithography uses high-NA immersion lenses to print ever-smaller chip features.
Laser focusing optics trade NA for spot size and depth of focus.

Common misconceptions

NA can exceed 1 in air — in $air (n = 1)$ it is capped at 1; only immersion media push it higher.
Higher magnification automatically means higher resolution — resolution is set by NA, not magnification.
NA only matters for microscopes — it governs fiber coupling, photolithography, and laser focusing alike.

Experimental verification

Measuring the far-field divergence cone emerging from a single-mode fiber, or the maximum tilt at which a microscope objective still forms a sharp image, reproduces

NA = n*\sin (\theta _\max )

to within a few percent.

Derivation

By the conservation of etendue, the largest meridional ray a system can accept makes a half-angle theta_max with the axis.

Projecting onto the optical invariant gives the light-gathering measure

NA = n*\sin (\theta _\max )

, where n is the index of the medium the cone sits in.

For a step-index fiber, applying Snell's law at the face plus total internal reflection at the core-cladding boundary yields

NA = \sqrt(n

_core^2 - n_clad^2).

Limiting cases

\theta_{max} \to 0⟶ NA \to 0A pencil-thin cone gathers almost no light and resolves almost nothing.

\theta_{max} \to 90^\circ⟶ NA \to nA full hemisphere of acceptance saturates NA at the medium index — why oil-immersion lenses

(n \sim 1.5)

reach

NA \sim 1.4

.

What if…

What if you immerse the front lens in

oil (n = 1.52)

?

For the same physical cone, NA scales up by the index, reaching $\sim 1.4$ and sharpening resolution — the principle behind oil-immersion microscopy.

What if you halve the acceptance angle?

NA drops roughly proportionally (for small angles), the system gathers about a quarter as much light, and the resolvable detail coarsens.

1

NA of a fiber entrance in air

Given ·

n:: 1
theta max:: 0.5236

Find · NA

Steps

Apply $NA = n*\sin (\theta _\max )$ .
$\sin (30\,\mathrm{deg}) = 0.5$ .
$NA = 1.0*0.5 = 0.5$ .

Result ·

NA = n*\sin (\theta _\max ) = 1.0*\sin (30\,\mathrm{deg}) = 1.0*0.5 = 0.5

.

Rayleigh Criterion→Critical Angle (Total Internal Reflection)→Snell's Law→