The Airy Disk: Why No Lens Ever Images a Perfect Point

A perfect lens still can't focus to a point

Grind a lens with zero aberrations, polish it flawlessly, and point it at a single distant star. The image you get is still not a point of light. It is a bright central disc surrounded by a faint ring, then a fainter ring, then another — a pattern that exists purely because light is a wave and the lens has a finite aperture. This is the Airy pattern, named for George Biddell Airy, who derived it in 1835, and its central bright disc is the Airy disk. No amount of optical perfection removes it; it is a hard limit set by diffraction, not a manufacturing defect.

Diffraction through a circular aperture

Any wave passing through an opening spreads out at its edges — this is diffraction, and it applies to water waves in a harbour mouth exactly as it applies to light in a lens aperture. For a circular aperture of diameter D illuminated by a plane wave of wavelength λ, working out the Fraunhofer (far-field) diffraction integral gives an intensity pattern built from the first-order Bessel function J₁, not the simple sinc function you get from a single slit:

I(θ) = I₀ · [ 2·J₁(x) / x ]²        where   x = (π·D/λ)·sin(θ)

J₁(x) = first-order Bessel function of the first kind
θ     = angle from the optical axis

The first dark ring and the size of the disc

The intensity first drops to exactly zero at the first zero of J₁(x)/x, which occurs at x = 3.8317. Substituting back gives the angular radius of the Airy disk — the radius, from the centre, of the first dark ring that bounds it:

sin(θ₁) ≈ θ₁ = 1.22 · λ / D        (for small angles, θ₁ in radians)

The central disc contains about 84% of the total diffracted energy; the first bright ring around it holds about 7%, and each successive ring holds progressively less. Notice the two levers in that formula: the angular size of the blur shrinks as the aperture D grows, and grows as the wavelength λ grows. Both facts have huge practical consequences.

The Rayleigh criterion

Because every point source produces its own Airy pattern, two point sources close together in angle produce two overlapping patterns, and at some separation the eye or sensor can no longer tell them apart. Lord Rayleigh proposed a practical, if somewhat arbitrary, dividing line in 1879: two sources are just resolved when the peak of one Airy pattern sits exactly on the first dark ring of the other.

θ_Rayleigh = 1.22 · λ / D      (minimum resolvable angular separation)

At exactly this separation there is a shallow dip of roughly 26% between the two combined peaks — visible but not dramatic, which is why the Rayleigh criterion is a convention rather than a hard physical wall. A related, stricter standard is the Sparrow criterion, where the dip between the two peaks disappears entirely; a looser one is the Dawes limit, an empirically tuned rule of thumb historically popular among visual astronomers for splitting double stars.

Why this sets real limits

The 1.22·λ/D relationship is not a footnote — it is the reason astronomical telescopes are built as large as engineering and funding allow (Hubble's 2.4 m mirror at visible wavelengths resolves roughly 0.05 arcseconds, close to its diffraction limit), the reason camera lenses lose sharpness when stopped down to very small apertures even though depth of field improves (diffraction blur grows as the aperture shrinks), and the reason optical microscopy cannot resolve structures much finer than about half the wavelength of light used, which is exactly why electron microscopy (with its far shorter de Broglie wavelength) and clever tricks like structured illumination and STED microscopy exist to beat the classical diffraction limit.

Computing the pattern

Rendering the Airy pattern numerically only requires a fast, accurate evaluation of J₁, which is usually done with a rational polynomial or asymptotic approximation rather than a series expansion, for speed:

function airyIntensity(theta, wavelength, aperture) {
  const x = (Math.PI * aperture / wavelength) * Math.sin(theta);
  if (Math.abs(x) < 1e-6) return 1;           // limit at x -> 0 is 1
  const j1 = besselJ1(x);                       // rational/asymptotic approx
  const term = 2 * j1 / x;
  return term * term;
}

Frequently asked questions

Try it live

Everything above runs in your browser — open Airy Disk and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.