How this calculation works
Even a perfect circular aperture cannot focus a point into a geometrical point. Diffraction produces an Airy pattern whose first dark ring defines a useful size convention. This page reports that first-zero radius and diameter. The same angular scale is used by the Rayleigh criterion for two point sources. It is a conventional diffraction comparison, not a universal statement that any two real objects at that separation will be visible.
First zero angle ≈ 1.219669891 λ/D; focal radius ≈ fθA worked example
At 550 nm, a 100 mm aperture and 1000 mm focal length give a first-zero radius of about 0.00670818 mm. The corresponding Airy diameter is 0.0134164 mm. The angular scale is about 1.3837 arcseconds. Doubling aperture at unchanged focal length halves the angular scale and focal-plane spot dimensions in this ideal uniformly illuminated model.
Good to know
The pupil is unobstructed, circular and uniformly illuminated. Central obstructions and apodization alter the pattern, so their results cannot be inferred by merely subtracting obstructed area. This page does not calculate a measured modulation transfer function or guarantee image sharpness. Read the first-zero size as a specifically defined model output rather than a complete lens specification.
Reference: OpenStax — circular aperture diffraction (opens in a new tab) ↗
What this calculation answers
Even a perfect circular aperture cannot focus a point into a geometrical point. Diffraction produces an Airy pattern whose first dark ring defines a useful size convention. This page reports that first-zero radius and diameter.
The same angular scale is used by the Rayleigh criterion for two point sources. It is a conventional diffraction comparison, not a universal statement that any two real objects at that separation will be visible.
The Telescope eyepiece magnification field and exit pupil calculator relates eyepiece focal length to angular magnification and exit pupil; increasing that magnification alone does not change this aperture diffraction scale.
Use the right input conventions
Enter clear aperture diameter, not radius, and use wavelength in nanometres. Focal length and aperture use millimetres. The wavelength entry assumes propagation in air or an equivalent wavelength already appropriate to the model.
The implemented expression is paraxial. Settings that produce a first-zero angle above 0.1 radians are rejected because the small-angle focal-plane mapping would no longer match the stated approximation.
Interpret the result
At fixed aperture, changing focal length changes the physical spot size but does not change the angular Rayleigh scale. At fixed focal ratio, the focal-plane Airy diameter follows wavelength. Those are distinct ways of comparing instruments.
A camera pixel size alone does not establish resolution. Sampling, optical aberrations, obstruction, atmospheric seeing and source contrast all interact with the diffraction pattern and are outside this single-aperture calculation.
Work through the example
At 550 nm, a 100 mm aperture and 1000 mm focal length give a first-zero radius of about 0.00670818 mm. The corresponding Airy diameter is 0.0134164 mm.
The angular scale is about 1.3837 arcseconds. Doubling aperture at unchanged focal length halves the angular scale and focal-plane spot dimensions in this ideal uniformly illuminated model.
Where the model stops
The pupil is unobstructed, circular and uniformly illuminated. Central obstructions and apodization alter the pattern, so their results cannot be inferred by merely subtracting obstructed area.
This page does not calculate a measured modulation transfer function or guarantee image sharpness. Read the first-zero size as a specifically defined model output rather than a complete lens specification.
A few questions, answered
Does a longer focal length improve angular resolution?
Not at fixed aperture in this model; it enlarges the diffraction spot in the focal plane.
What changes in the comparison?
Double clear aperture at fixed focal length and wavelength.