Airy Disk Diameter Calculations

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What the Airy disk is

A lens with a circular aperture does not focus a point of light to a perfect point. Diffraction spreads the light into a pattern: a bright central disk, called the Airy disk, surrounded by fainter rings. The first dark ring marks the edge of the central disk. Its diameter sets the smallest spot the lens can form, if no other errors are present.

The Airy disk size depends on two things only: the wavelength of the light and the f-number of the lens. The f-number is the focal length divided by the diameter of the entrance pupil, often written N.

The formula

The diameter of the Airy disk, measured to the first dark ring, is 2.44 multiplied by the wavelength multiplied by the f-number: d = 2.44 λN. The radius to the first dark ring is 1.22 λN. Both results are in the same length units as the wavelength.

For visible light, a common reference wavelength is 550 nanometres, which is 0.55 micrometres. Convert to the unit you need before calculating, because mixing nanometres and micrometres is one of the most common errors.

A worked example

Take a wavelength of 550 nanometres and an f-number of 8. The diameter is 2.44 × 0.55 µm × 8, which is about 10.7 µm. At an f-number of 2.8, the same wavelength gives about 3.8 µm. The lens stopped down to f/8 forms a larger minimum spot than the same lens at f/2.8.

The result matters when it is compared with the pixel spacing of a sensor. If the Airy disk is much smaller than a pixel, diffraction is unlikely to be the limiting factor. If it is several pixels across, the image will be softer than the sensor could record.

Assumptions to state

The formula assumes a diffraction-limited lens, a circular unobstructed aperture, and a single wavelength or a well-defined effective wavelength. Real lenses have aberrations, which add blur on top of diffraction. Reflecting telescopes and some lenses have a central obstruction, which changes the pattern. In those cases the formula describes the ideal limit, not the measured image.

Common mistakes

Using the focal length instead of the aperture diameter when calculating the f-number. Using a different unit for the wavelength from the unit used for the result. Treating the Airy disk as the sharpness of the image: the disk is the smallest spot possible, and real images are usually larger.

Using the calculator

The calculator on the home page takes an f-number and a wavelength and returns the Airy disk diameter. Use it to check your arithmetic and to see how the result scales with aperture. It is not a substitute for measuring a lens.

Working through a real choice of aperture

Suppose you are choosing between a lens stopped to f/5.6 and one stopped to f/11 for a landscape at 550 nanometres. At f/5.6 the Airy disk is about 7.5 µm across, and at f/11 it is about 14.8 µm. If your sensor has 4 µm pixels, the f/5.6 disk spans roughly two pixels, while the f/11 disk spans about four. The smaller aperture gives more depth of field, but it also spreads each point of light over more pixels, which softens fine detail.

The comparison does not tell you which aperture to choose. It tells you what each choice costs in diffraction terms, so the decision can be made with the depth of field needed for the scene in view.

Relating the result to pixels and sampling

A useful rule of thumb in imaging is to compare the Airy disk with the pixel pitch. When the disk is around one pixel or less, diffraction does little to limit the captured detail. When it covers several pixels, the image is limited by diffraction, and extra pixels add little. The Nyquist limit of the sensor, described in the sensor article, sets the finest detail the pixels can record, and the Airy disk sets the finest detail the lens can deliver. The image is limited by whichever is coarser.

Measurement and verification

The formula gives an ideal value. To check a real lens, photograph a point-like source or a fine test target at a range of apertures and measure the width of the image. A measured spot larger than the Airy disk indicates aberrations or focus error. A measured spot close to the Airy disk indicates a lens that is close to diffraction-limited at that aperture.

Deriving the factor 1.22 and 2.44

The number 1.22 comes from the first zero of the Bessel function of the first kind of order one, which describes the intensity pattern of a circular aperture. The first zero occurs at a radial distance of 3.83 in normalised units, and converting that into an angle or a length gives the factor 1.22 for the radius. Doubling it gives 2.44 for the full diameter. Readers do not need the Bessel function to use the formula, but knowing where the factor comes from helps when a result looks wrong.

The derivation assumes a uniformly illuminated circular aperture and a distant point source, which is the standard setting for the Airy pattern. Textbooks such as Hecht and Born and Wolf give the full derivation.

Effect of a central obstruction

Many telescopes and some compact lenses have a central obstruction, such as a secondary mirror. The obstruction changes the diffraction pattern. The central peak becomes slightly narrower, but the energy moves into the rings, which increases their brightness. For visual and imaging purposes, the change matters most when the obstruction is large relative to the aperture. The simple formula describes an unobstructed aperture, so it is an approximation for obstructed systems.

Working through a change of wavelength

Compare two colours through the same f/8 lens. At 450 nanometres (blue), the Airy diameter is about 8.8 micrometres. At 650 nanometres (red), it is about 12.7 micrometres. The difference is large enough to matter at the pixel level, which is one reason that colour performance and diffraction must be considered together in a camera system.

Common uses in practice

Designers use the Airy diameter to set a target for lens performance, to decide whether a sensor can benefit from more pixels, and to predict the smallest spot a point source can produce. Field testers use it to judge whether a measured spot is close to the ideal. Amateur astronomers use the equivalent angular formula to judge whether a telescope can resolve a planned target.

Checking a calculation step by step

Work through the formula in a fixed order: convert the wavelength to the length unit you want, multiply by the f-number, then multiply by 2.44. Keeping the units visible at each step prevents the most common error, which is mixing nanometres with micrometres. Compare the result with a rough estimate, such as 2.4 times the wavelength in micrometres, times the f-number. A large disagreement signals a units error.

Common scenarios and typical values

For a fast lens at f/2.8 and green light, the Airy diameter is near 3.8 micrometres, which is smaller than most current pixels. At f/8 it is near 10.7 micrometres, which spans several pixels on a small sensor. Using these benchmarks lets you judge a lens setting quickly, without repeating the arithmetic each time.

Limits of the single number

The Airy diameter describes the ideal spot for one wavelength and one aperture. A polychromatic image, an off-axis point and a real lens each change the result. Use the figure as a benchmark for the best case, and test a real lens to see how far it falls short.

Going further

The result is most useful when it is compared with what a lens actually delivers. Testing a real lens at the same aperture shows how far the measured spot falls short of the Airy diameter, and that gap is the place to look for aberrations or focus error.

Summary

The key point of this article is the one stated at its start. Check the formula, use the stated assumptions, and confirm any result with a test before relying on it in practice.

Sources

Eugene Hecht, Optics, Pearson (any current edition).
Max Born and Emil Wolf, Principles of Optics, Cambridge University Press (any current edition).
ISO 12233: Photography, Electronic still picture imaging, Resolution and spatial frequency responses.

Next step

Return to the pillar guide or the Diffraction summary to read the related articles.

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General information only. This page explains optical principles and does not replace lens design work, lens testing or advice from a qualified optical engineer.