Airy Disk
When light passes through a circular aperture, it does not form a perfect point—it creates a diffraction pattern of concentric rings called an Airy pattern. This sets the fundamental resolution limit for every telescope, microscope, and camera.
Adjust the Aperture to see how a larger opening produces a tighter Airy disk. Switch to Two Sources and adjust Separation to see the Rayleigh criterion in action. The Profile Line shows the 1D intensity cross-section.
The Airy Pattern
When a plane wave passes through a circular aperture of diameter D, the
resulting far-field intensity pattern is described by the Airy function:
I(θ) = I₀ [2J₁(x)/x]², where
x = πD sin(θ)/λ and J₁ is the first-order
Bessel function of the first kind. The central bright disk contains about 84% of the
total light, surrounded by progressively fainter rings.
The first dark ring
The first zero of J₁(x) occurs at x ≈ 3.8317, giving the angular radius
of the first dark ring as θ ≈ 1.22 λ/D. This is the
famous factor of 1.22 that appears throughout optics. A larger aperture (bigger D) or
shorter wavelength (smaller λ) produces a smaller Airy disk and finer resolution.
The Rayleigh Criterion
Two point sources are considered just resolved when the central maximum of one
Airy pattern falls on the first minimum of the other. This occurs at an angular separation
of θ_R = 1.22 λ/D—the Rayleigh criterion.
Below this angle, the two patterns merge into one and the sources are unresolvable.
In practice, the Rayleigh limit is somewhat conservative. The Sparrow criterion (when the combined intensity profile has no dip between the peaks) gives a slightly tighter limit. But the Rayleigh criterion remains the standard benchmark for optical resolution.
Why Bigger Telescopes See Finer Detail
The diffraction limit θ = 1.22 λ/D means that the minimum
resolvable angle is inversely proportional to the aperture diameter. A 10-meter telescope
resolves angles 100 times smaller than a 10-centimeter telescope at the same wavelength.
This is why astronomers build ever-larger mirrors—not primarily to collect more light
(though that helps), but to achieve finer angular resolution.
Radio telescopes, working at wavelengths millions of times longer than visible light, need proportionally larger apertures. The Event Horizon Telescope that imaged the black hole in M87 achieved Earth-sized resolution by linking radio dishes across the globe in a technique called very long baseline interferometry.