Numerical Aperture and Resolution

1. What numerical aperture measures

Point a lens (or a microscope objective, or the end face of an optical fiber) at a light source, and it can only collect light rays that fall within some maximum half-angle θ from its optical axis — rays arriving more obliquely simply miss the lens. Numerical aperture (NA) is a single number that captures how wide that acceptance cone is, combined with the refractive index of the medium the light is travelling through when it enters the lens.

NA governs two things that at first look unrelated: how much light a lens can gather (its "speed" or brightness), and — far more fundamentally — the smallest feature size it can ever resolve, no matter how good the glass or how careful the polishing.

2. The formula: NA = n·sinθ

where n is the refractive index of the medium between the lens and the object (1.0 for air, ~1.33 for water, ~1.51 for typical immersion oil), and θ is the half-angle of the widest cone of light the lens can accept or emit, measured from the optical axis.

Because sin θ can never exceed 1, a lens working in air can never have NA > 1 — the theoretical ceiling for a dry objective. High-end dry microscope objectives reach NA ≈ 0.95 (θ ≈ 72°), pushing right up against that ceiling. To go higher, you have to change the medium, not just widen the cone — which is exactly what immersion objectives do (see section 5).

3. Diffraction and the Airy disc

Even a mathematically perfect, aberration-free lens cannot form an infinitely sharp point image of a point source. Because the lens aperture is finite, it clips the incoming spherical wavefront, and diffraction spreads the image of a point into a central bright spot surrounded by faint concentric rings — the Airy pattern, first analysed by George Biddell Airy in 1835. The radius of the central disc (to its first dark ring) is:

A bigger NA gathers a wider cone of diffracted light and squeezes the Airy disc smaller — that's the entire physical reason a higher numerical aperture buys you finer resolution.

4. The Rayleigh criterion and Abbe's diffraction limit

Two point sources are considered just resolved under Lord Rayleigh's 1879 criterion when the centre of one source's Airy disc falls exactly on the first dark ring of the other's. This gives the minimum resolvable separation:

A closely related formula, derived by Ernst Abbe in 1873 from diffraction-grating theory rather than point-source imaging, describes the smallest periodic structure (e.g. a diffraction grating or specimen with fine stripes) a microscope can resolve when illuminated with a full cone of light from a matching condenser:

Both formulas say the same thing in spirit: resolution improves linearly with numerical aperture and linearly with shorter wavelength. Neither formula depends at all on magnification — a common misconception is that "zooming in more" reveals finer detail, but past the diffraction limit, extra magnification only enlarges the blur (so-called "empty magnification").

5. Beating air: immersion objectives

Since NA = n sin θ and sin θ can never exceed 1, the only way past NA = 1 is to raise n. Oil-immersion objectives fill the gap between the front lens and the specimen with oil matched to the refractive index of the glass coverslip (n ≈ 1.51 — deliberately chosen to equal typical crown glass). This does two things at once:

• It raises the achievable NA well past 1 (up to about 1.4–1.5 for the best objectives), directly shrinking the diffraction limit via the Abbe formula.
• It eliminates the refraction that would otherwise occur at the glass-air interface, so rays that would have been bent away and lost (or totally internally reflected, per our companion article on Snell's law and TIR) instead travel in a straight line into the objective, meaning oil immersion both gathers more light and gathers it from a wider real-world angle.

Water-immersion objectives (n ≈ 1.33) split the difference, useful for live-cell imaging where oil isn't practical, reaching NA up to roughly 1.2.

6. Numerical aperture trade-offs

• Depth of field shrinks as NA grows. A high-NA objective resolves exquisitely fine lateral detail but has an extremely thin in-focus slice (depth of field falls roughly as 1/NA²), which is why high-power oil objectives need very precise, vibration-free focusing.
• Working distance shrinks as NA grows. A wider acceptance cone geometrically forces the front lens element much closer to the specimen — 60× oil objectives often work at a fraction of a millimetre from the coverslip.
• In fiber optics, NA also sets the acceptance cone for light coupling into the fiber core and directly relates to the critical angle discussed in our Snell's law article: NA = √(n_core² − n_clad²). Higher NA fibers accept light more easily (good for cheap LED sources) but support more propagation modes, which increases modal dispersion and limits bandwidth — one reason long-haul telecom fiber deliberately uses low-NA, single-mode designs.