Numerical Aperture

What numerical aperture measures

When a lens images a point of light, it can only collect the rays that fall within a certain cone. The numerical aperture quantifies how wide that cone is, combined with the medium it travels through:

NA = n · sin θ

Here n is the refractive index of the medium between the specimen and the front element of the lens (air, water, glycerin, or oil), and θ is the half-angle of the maximum cone of light the lens accepts — measured from the optical axis to the steepest ray that still makes it into the aperture. A wide acceptance cone means the lens samples light leaving the object over a broad range of angles.

Because sin θ can never exceed 1, a lens working in air is hard-capped at NA < 1 (real dry objectives top out near NA ≈ 0.95). The only way to break past 1 is to raise n: water immersion reaches NA ≈ 1.27, and oil immersion (n ≈ 1.515, index-matched to the coverslip glass) reaches NA ≈ 1.40. That single trick — filling the gap with a high-index liquid — is why the best light microscopes resolve roughly twice as fine as their dry counterparts.

Why NA controls resolution

Light is a wave, so it diffracts. A perfect point source does not image as a point — it images as a bright central blob surrounded by faint rings, the Airy disk. Two nearby points are only distinguishable when their Airy disks are separated enough. The Rayleigh criterion states that the smallest resolvable separation is:

d = 0.61 · λ / NA          (Rayleigh, incoherent)
d = λ / (2 · NA)           (Abbe diffraction limit)

The physical intuition: fine detail in an object radiates light at steep angles (high spatial frequencies). A larger NA captures those steep rays, while a small NA throws them away — so it cannot reconstruct the detail. Widening the cone shrinks the diffraction limit-set Airy disk and sharpens what you can see.

For green light (λ = 550 nm) an oil-immersion objective at NA = 1.4 resolves about 0.24 µm by Rayleigh, or 0.20 µm by Abbe — close to half the wavelength, the practical ceiling of conventional far-field microscopy. To beat that, you need either shorter wavelengths (UV, electron microscopy) or super-resolution tricks (STED, PALM/STORM) that sidestep the diffraction limit entirely.

Objective	NA	Resolution at 550 nm (0.61λ/NA)
4× dry scanning lens	0.10	≈ 3.4 µm
10× dry	0.25	≈ 1.3 µm
40× dry	0.65	≈ 0.52 µm
40× dry (high-NA)	0.95	≈ 0.35 µm
60× water immersion	1.20	≈ 0.28 µm
100× oil immersion	1.40	≈ 0.24 µm

Light-gathering and brightness

NA does double duty: besides resolution, it sets how much light the lens collects. The solid angle subtended by the cone scales with sin²θ, so image brightness in a microscope rises roughly with NA². Doubling the NA quadruples the light per unit area in the image — which is why high-NA objectives are prized for dim fluorescence work, where every photon counts.

The same cone also fixes the depth of field. A steep, wide cone converges sharply, so only a thin slice of the sample is in focus:

DOF ≈ n · λ / NA²

For the NA = 1.4 oil objective at 550 nm this gives a depth of field of only about 0.4 µm — thinner than many cells. Drop to a 10× NA = 0.25 lens and the DOF balloons to roughly 9 µm. This is the same trade-off photographers make: stopping a camera lens down to a small aperture (low NA) buys deep focus at the cost of light and ultimate sharpness.

NA, f-number, and the acceptance cone

Photographers rarely say "numerical aperture" — they say f-number (f/#). The two describe the same cone. For a lens focusing parallel light from infinity:

N = 1 / (2 · NA)        →        NA = 1 / (2 · N)

f-number (N)	Numerical aperture (NA)	Half-angle θ
f/1.0	0.50	30°
f/1.4	0.36	21°
f/2.8	0.18	10°
f/8	0.063	3.6°
f/16	0.031	1.8°

The fastest practical photographic lenses (f/1.0–f/1.2) sit near NA ≈ 0.4–0.5, far below microscope objectives, because they must image a wide field rather than a single near-axis point.

Numerical aperture of an optical fiber

The same concept describes how light enters a waveguide. For a step-index optical fiber with a core index n_core and cladding index n_clad, total internal reflection guides only those rays that strike the core–cladding boundary steeply enough. Working through the geometry gives the fiber's acceptance NA:

NA = √(n_core² − n_clad²)
θ_max = arcsin(NA / n_outside)        (n_outside ≈ 1 for air)

A typical multimode fiber with n_core = 1.475 and n_clad = 1.460 has NA = √(1.475² − 1.460²) ≈ 0.21, an acceptance half-angle of about 12°. Light launched within that cone is guided; steeper rays leak into the cladding and are lost. A higher-NA fiber couples more light from a broad source like an LED, but the extra path-length spread of high-angle modes increases modal dispersion, blurring fast pulses — so high-bandwidth links use low-NA single-mode fiber instead.

Worked numbers

• Oil vs. dry, same magnification. A 100× oil objective at NA 1.40 resolves 0.24 µm; a 100× dry objective is limited to NA ≈ 0.95 and 0.35 µm. The oil version sees about 1.5× finer — enough to separate adjacent bacteria.
• Half-angle from NA. NA 1.40 in oil (n = 1.515) means sin θ = 1.40/1.515 = 0.924, so θ ≈ 67.5° — the lens collects a cone almost a full hemisphere wide.
• Brightness gain. Going from NA 0.65 to NA 1.30 raises image brightness by (1.30/0.65)² = 4×, hugely valuable for weak fluorescent signals.
• Camera analog. An f/1.4 portrait lens (NA ≈ 0.36) gathers four times the light of an f/2.8 lens (NA ≈ 0.18) and renders a much shallower depth of field.

Common mistakes

• Forgetting the index n. NA = n·sin θ, not just sin θ. Quoting an oil objective's angle without including n = 1.515 understates its NA and its resolving power.
• Thinking NA can exceed 1 in air. Since sin θ ≤ 1 and air has n = 1, dry NA is capped below 1. Only a high-index immersion medium pushes past it.
• Confusing magnification with resolution. Empty magnification adds size without detail. Resolution is set by NA and λ; cranking magnification past ~1000×NA just enlarges blur.
• Mismatching immersion medium. Using a dry objective with oil, or oil meant for a different temperature/index, breaks the index match at the coverslip and slashes the effective NA.
• Ignoring the condenser NA. In transmitted-light microscopy the achievable resolution depends on both objective and condenser NA: d ≈ 1.22λ/(NA_obj + NA_cond). A closed-down condenser wastes the objective's NA.
• Assuming high NA is always better. High NA means a paper-thin depth of field and a short working distance. For thick or live specimens a lower-NA, longer-working-distance lens is often the right tool.