Airy Disk Diffraction Radius

The engineering

A perfect lens focusing light through a circular aperture cannot make a point — diffraction spreads it into a bright central Airy disk ringed by faint fringes. The radius to the first dark ring sets the diffraction limit: the 1.22 factor comes from the first zero of the J₁ Bessel function that describes the circular-aperture pattern. Enter a focal length to get the physical spot size at the focal plane, otherwise you get the far-field angular resolution.

The Rayleigh criterion says two point sources are just resolved when one sits on the other's first null — so this same θ is your resolution limit. Handy check: for visible light the spot diameter in microns is roughly 1.35 × the f-number, so an f/8 lens can't beat about 11 µm regardless of how good the glass is. Real systems only approach this when aberrations are below the diffraction budget.

Where this math comes from

George Biddell Airy, then Lucasian Professor at Cambridge and later Astronomer Royal, worked out the diffraction pattern of a circular aperture in 1835 while studying the theoretical limits of telescope images. He solved the integral over the aperture and found the intensity followed a Bessel-function form, with the now-famous central bright disk bounded by the first zero at 1.22 λ/D.

Lord Rayleigh turned Airy's pattern into a practical resolving-power rule in 1879, and Max Born and Emil Wolf later canonized the full treatment in Principles of Optics — the standard derivation every optical engineer still cites for the 1.22 coefficient.

1. 1835George Biddell AiryDerives the circular-aperture diffraction pattern and its central disk.
2. 1879Lord RayleighStates the resolution criterion using the first Airy null.
3. 1959Born & WolfPrinciples of Optics codifies the modern derivation of the 1.22 λ/D limit.

See the full timeline of the math behind every calculator →

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