Diffraction-Limited Spot Diameter (Airy Disc)

George Biddell Airy worked out in 1835 what happens when light passes a round hole: not a point, but a bright central disc surrounded by faint rings, with the first dark ring at an angle of $1.22\lambda/D$. At the focus of a lens that becomes a disc of diameter $2.44\lambda f/D$, first null to first null, and it is the floor under every focusing and imaging system ever built. No lens focuses tighter than its own aperture permits, however perfect the glass.

The most useful reading of the formula is the one it does not look like: since $d = 2.44\lambda (f/D)$, the answer depends only on the RATIO $f/D$ — the f-number — and not on either quantity alone. Every f/8 lens in the world makes the same Airy disc at the same wavelength, whatever its focal length. That is why photographers talk about f-numbers, and why diffraction softening sets in at the same aperture setting on a phone camera and on a view camera.

Rayleigh's 1879 criterion is the same geometry put to work on resolution: two point sources are just resolved when the centre of one Airy disc falls on the first null of the other, at an angular separation of $1.22\lambda/D$. Everything about telescope design follows. Aperture is the only thing that buys resolution and magnification buys none, which is why amateur astronomers talk about inches of glass rather than about power. It is also why lithography chased ever shorter wavelengths — mercury lines at 436 nm, then deep ultraviolet at 193, now extreme ultraviolet at 13.5 — since the limit scales straight with $\lambda$.

Two honesty notes about the number itself. The aperture that counts is the part ACTUALLY FILLED with light, not the diameter engraved on the barrel: an oversized lens stopped down by an iris upstream has the iris's diffraction, and a beam covering only the middle third of a fine objective is diffracting off its own edge. And the central disc holds only 84% of the energy — 16% is out in the rings, which is why even a diffraction-limited spot scatters light where you did not want it, and why contrast in a microscope is a different problem from resolution.

And the disagreement with the Gaussian page, since a reader who computes both will find them different. On the same $f$, $D$ and $\lambda$, this formula gives 1.92 times what $4M^2\lambda f/(\pi D)$ gives. Both are correct. This one assumes uniform illumination of the pupil — what a telescope looking at a star, or a microscope under Köhler illumination, genuinely has — and quotes a first-null diameter. The other assumes a Gaussian tapering to 13.5% at the rim and quotes a $1/e^2$ diameter. Two illuminations, two width definitions, no contradiction. The commonly quoted "about 30% apart" figure comes from the case where a Gaussian is truncated at 1.5:1 so that it actually fills the aperture, which widens the Gaussian answer by half and brings the two within about 28%.

One last practical limit. Below roughly f/2, a simple spherical lens's aberration is far larger than its Airy disc, and this equation stops describing the limit you are at and starts describing the limit you are nowhere near. Getting to it needs an aspheric, a corrected multi-element objective, or in the extreme case adaptive correction.