Airy Disk

Where the pattern comes from

An ideal point source far from a uniformly illuminated circular aperture of diameter D produces, in the far field, the Fraunhofer diffraction pattern of the aperture function. That function is a disk of radius D/2, and its 2D Fourier transform — derived by Airy in 1835 — is:

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

J₁ is the first-order Bessel function of the first kind. The function 2·J₁(x)/x is the radial analogue of sinc(x) for circular apertures. Plotting I(θ) gives:

• Bright central disk from θ = 0 out to the first zero — contains ~84% of total power.
• First bright ring with peak ~1.75% of central intensity (7% of total power).
• Subsequent rings dropping rapidly: 0.42%, 0.16%, 0.078%... of central intensity.

The Rayleigh resolution criterion

The first dark ring of an Airy pattern occurs where J₁(x) = 0 for the first nonzero x, which is x ≈ 3.8317. Solving for θ:

sin θ_min = 3.8317 / π · λ / D ≈ 1.2197 · λ / D
        ≈ 1.22 · λ / D     (the Rayleigh criterion)

Rayleigh's resolution rule says two point sources are just resolved when the central peak of one's Airy pattern falls on the first dark ring of the other. At that separation, the combined intensity dips by ~26% between the peaks — perceptible but marginal.

Criterion	Separation (λ/D)	Contrast in dip	Use
Rayleigh	1.22	~26% dip	Standard textbook
Dawes	~1.02 (0.97·λ/D-ish, empirical)	Visual binary star limit (Dawes 1867)	Amateur astronomy
Sparrow	~0.95	No dip — peaks merge	Aggressive limit
Abbe (microscopy)	d = 0.61 λ / NA	≈ Rayleigh in NA form	Microscope spec

Worked example — Hubble Space Telescope

Hubble has primary mirror D = 2.4 m. At green-yellow visual peak λ = 550 nm:

θ_min = 1.22 · 550 × 10⁻⁹ / 2.4
      = 2.795 × 10⁻⁷ rad
      = 2.795 × 10⁻⁷ × (180/π) × 3600   arcsec
      ≈ 0.058 arcsec

For comparison: the Moon subtends ~0.5° (1800 arcsec) — so 0.058 arcsec is about 1/30000 of the Moon's diameter. At lunar distance (384,000 km), that's a resolution of ~110 m on the Moon's surface — finer than most lunar landers. Ground-based telescopes hit ~1 arcsec from atmospheric seeing; adaptive optics or space telescopes recover the diffraction limit.

Costed claims — resolution across instruments

Instrument	D	λ	θ = 1.22 λ/D
Human eye (pupil)	5 mm	550 nm	27.7 arcsec (1/2 arcmin)
iPhone 15 main camera	~5 mm aperture (f/1.78, 24 mm)	550 nm	27.7 arcsec ≈ pixel pitch at sensor
Hubble Space Telescope	2.4 m	550 nm	0.058 arcsec
James Webb Space Telescope	6.5 m	2 µm (NIR)	0.078 arcsec
VLT (Paranal)	8.2 m	550 nm	0.017 arcsec (with adaptive optics)
Event Horizon Telescope (VLBI)	~Earth diameter	1.3 mm	20 µarcsec
Confocal microscope (NA = 1.4)	oil immersion	500 nm	d ≈ 218 nm (Abbe form)

Real apertures aren't Airy

Ideal pattern requires (a) circular aperture, (b) uniform illumination, (c) far field, (d) no aberrations. Real telescopes deviate:

• Central obstruction (secondary mirror). A Cassegrain telescope has an annular aperture. Ring intensities rise; the central disk shrinks slightly but the pattern has more energy in sidelobes.
• Spider vanes. The struts that hold the secondary in place create diffraction spikes — the four-pointed stars you see in Hubble images come from Hubble's spider.
• Apodization. Designing an aperture with edge tapering (e.g., a Gaussian or cosine profile) suppresses ring intensity at the cost of a slightly wider central peak. Used in coronagraphs for exoplanet imaging.
• Aberrations. Defocus, spherical aberration, coma, astigmatism all distort the pattern. Hubble's 1990 launch flaw was a spherical aberration that smeared the central disk — fixed by COSTAR in 1993.

JavaScript — Airy pattern and resolution

// Bessel J₁ via series (good for small x, decent for x < 20)
function besselJ1(x) {
  if (Math.abs(x) < 1e-9) return x / 2;
  // Series: J₁(x) = sum_{k=0}^∞ (-1)^k / (k! (k+1)!) · (x/2)^{2k+1}
  let term = x / 2;
  let sum = term;
  for (let k = 1; k < 60; k++) {
    term *= -(x / 2) * (x / 2) / (k * (k + 1));
    sum += term;
    if (Math.abs(term) < 1e-14) break;
  }
  return sum;
}

// Airy intensity at angle θ (radians)
function airyIntensity(theta, D, lambda, I_0 = 1) {
  const x = Math.PI * D * Math.sin(theta) / lambda;
  if (Math.abs(x) < 1e-9) return I_0;
  const j1 = besselJ1(x);
  return I_0 * (2 * j1 / x) ** 2;
}

// Rayleigh resolution
function rayleighAngle(D, lambda) {
  return 1.22 * lambda / D;  // radians
}

const lambda = 550e-9;

// Hubble at 550 nm
console.log(`Hubble: ${(rayleighAngle(2.4, lambda) * 180/Math.PI * 3600).toFixed(3)} arcsec`);
// 0.058 arcsec

// JWST at 2 µm
console.log(`JWST: ${(rayleighAngle(6.5, 2e-6) * 180/Math.PI * 3600).toFixed(3)} arcsec`);
// 0.078 arcsec

// Find first zero of J₁ numerically
function findFirstZero() {
  for (let x = 2; x < 5; x += 0.001) {
    if (besselJ1(x) * besselJ1(x + 0.001) < 0) return x;
  }
  return null;
}
console.log(`First J₁ zero: ${findFirstZero().toFixed(4)}`);  // ≈ 3.8317

// Two-source resolvability (combined intensity from two Airy patterns)
function combinedTwoSources(theta, separation, D, lambda) {
  // Two sources at ±separation/2
  const I1 = airyIntensity(theta - separation/2, D, lambda);
  const I2 = airyIntensity(theta + separation/2, D, lambda);
  return I1 + I2;  // incoherent: intensities add
}

// At Rayleigh separation, evaluate dip between peaks
const sep = rayleighAngle(0.1, lambda);  // 100 mm aperture, 550 nm
const peak = combinedTwoSources(sep/2, sep, 0.1, lambda);    // at one of the peaks
const middle = combinedTwoSources(0, sep, 0.1, lambda);      // halfway between
console.log(`Rayleigh dip: ${((peak - middle) / peak * 100).toFixed(1)}%`);
// ~26% — that's the standard Rayleigh dip

// Encircled energy
function encircledEnergy(theta_rad, D, lambda) {
  // Numerical integration of 2πθ·I(θ) up to theta_rad
  const N = 1000;
  let sum = 0;
  for (let i = 0; i < N; i++) {
    const t = (i + 0.5) / N * theta_rad;
    sum += 2 * Math.PI * t * airyIntensity(t, D, lambda);
  }
  return sum * theta_rad / N;
}

// 84% of energy is inside the first dark ring — check it
const totalIn5Rays = encircledEnergy(5 * rayleighAngle(0.1, lambda), 0.1, lambda);
const inAiryDisk = encircledEnergy(rayleighAngle(0.1, lambda), 0.1, lambda);
console.log(`Fraction in Airy disk: ${(inAiryDisk / totalIn5Rays * 100).toFixed(1)}%`);
// ~84%

Where the Airy disk sets the limit

• Astronomy. Every imaging telescope has 1.22 λ/D as its in-principle resolution. Ground-based: atmospherically limited unless using adaptive optics. Space: diffraction limited.
• Photography. Camera lens at f/22 in visible light has Airy disk ~10 µm — comparable to the pixel pitch on high-res sensors. Stopping down past the diffraction limit makes images softer.
• Microscopy. Abbe limit d = 0.61 λ/NA is the Airy criterion in NA terms. Visible light optical microscopy: ~200 nm. Electron microscopy uses λ ~ 5 pm to image atoms.
• Coronagraphs. To image planets near a host star, you must suppress the host's Airy rings far below the planet flux ratio. Apodized apertures and Lyot stops shape the pattern.
• Laser focusing. A Gaussian laser beam through a lens focuses to ~λ·f/D — analogous to the Airy disk. Sets the spot size in laser microscopy and laser machining.
• Optical tweezers / single-molecule imaging. The Airy disk diameter sets the diffraction-limited localization precision before super-resolution methods kick in.

Common mistakes

• Using sinc instead of 2J₁/x. sinc is the 1D (slit) pattern; the circular aperture gives 2J₁(x)/x. The first zeros differ: π vs ~3.832 — that's why circular has 1.22 instead of 1 in the formula.
• Forgetting the "1.22". A common shortcut writes "λ/D" as the resolution. That's the 1D (slit) limit. Circular apertures need the 1.22 factor — about 22% coarser resolution.
• Treating Rayleigh as a hard limit. Modern techniques (super-resolution microscopy, sparsity-based deconvolution, ML-based image reconstruction) routinely beat 1.22 λ/D when SNR is high enough.
• Using small-angle sin θ ≈ θ for very wide-angle apertures. For very high NA microscopes, you need the full sin θ form; for telescopes, θ is tiny and the approximation is fine.
• Ignoring central obstructions. Cassegrain telescopes have annular apertures, which change the relative ring brightness. The pattern is still close to Airy, but sidelobes can be 2× higher.
• Confusing physical pixel size with diffraction. A sensor can be limited by either: small pixels with bad optics give a blurry image; large pixels with great optics undersample the Airy disk and lose resolution. Match Nyquist (pixel pitch ~ Airy radius / 2).