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The Resolving Power Calculator applies the Rayleigh criterion to determine the minimum angular separation at which a circular aperture can distinguish two point sources. Using the formula θ = 1.22λ/D, it computes the diffraction-limited angular resolution, the corresponding spatial resolution at a given distance, and can solve for the required aperture diameter or operating wavelength.
Every optical instrument — telescopes, microscopes, cameras, the human eye — has a fundamental resolution limit set by diffraction at its aperture. Lord Rayleigh established that two point sources are just resolvable when the central maximum of one Airy disk falls on the first minimum of the other. This criterion provides the theoretical best-case resolution; atmospheric turbulence, lens aberrations, and detector pixel size often degrade performance further in practice.
When light from a distant point source passes through a circular aperture of diameter D, it does not form a perfect point image but instead produces an Airy diffraction pattern — a bright central disk surrounded by concentric rings. The angular radius of the first dark ring is:
$$\theta = 1.22\frac{\lambda}{D}$$
This is the Rayleigh criterion: two point sources separated by angle θ are just barely resolvable. Sources closer together appear as a single blurred spot.
The factor 1.22 comes from the first zero of the Bessel function J₁ that describes diffraction by a circular aperture (the exact value is 1.21966...).
The spatial resolution at distance d is the linear separation corresponding to the angular resolution:
$$\Delta x = \theta \cdot d = 1.22\frac{\lambda d}{D}$$
Key implications:
For a telescope with focal length f, the Airy disk radius on the detector is r = 1.22λf/D = 1.22λN, where N = f/D is the f-number. This sets the minimum useful pixel size for the detector.
The angular resolution in arcseconds tells you the smallest angle your instrument can resolve. For context, the full Moon subtends about 1800 arcseconds, and good astronomical seeing limits ground-based telescopes to about 1 arcsecond regardless of aperture. The spatial resolution converts this angle to a physical separation at your specified distance — useful for estimating whether a telescope can resolve binary stars or a microscope can separate cellular structures.
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A 100 mm aperture telescope resolves 1.39 arcseconds at 550 nm — sufficient to split many binary stars. At the Moon's distance (384,400 km), this corresponds to about 2.6 km resolution, meaning individual craters several km across are barely resolvable.
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The dark-adapted human pupil (D ≈ 6 mm) has a diffraction limit of about 23 arcseconds. At a reading distance of 6 m, this corresponds to about 0.67 mm — roughly the size of a period on a standard eye chart, matching the 20/20 vision standard.
The Rayleigh criterion states that two point sources are just resolvable when the central maximum of one Airy pattern coincides with the first minimum of the other. The angular separation at this limit is θ = 1.22λ/D. It is a practical (not absolute) definition — sources slightly closer than this limit can sometimes be distinguished with advanced image processing.
The factor 1.22 (more precisely, 1.21966) is the first zero of the Bessel function J₁(x)/x divided by π. It arises from the mathematical solution to diffraction by a circular aperture. A rectangular slit would give a factor of 1.0 instead.
Yes, but for microscopes the resolution limit is usually expressed as the Abbe diffraction limit: d = 0.61λ/NA, where NA is the numerical aperture. This is equivalent to the Rayleigh criterion adapted for imaging close objects rather than distant stars.
Space telescopes operate above the atmosphere, which causes turbulent blurring (seeing). A ground-based telescope rarely achieves better than ~1 arcsecond resolution regardless of aperture size, while a space telescope achieves its full diffraction-limited resolution. Adaptive optics can partially correct atmospheric effects.
In fluorescence microscopy, super-resolution techniques (STED, PALM, STORM) bypass the diffraction limit by exploiting molecular switching, achieving resolutions below 50 nm. In astronomy, interferometry combines multiple telescopes to achieve the resolution of a much larger effective aperture.
Resolution improves (θ decreases) with shorter wavelength. X-ray telescopes can theoretically resolve much finer detail than optical telescopes with the same aperture. Conversely, radio telescopes (λ ~ cm to m) need very large dishes or interferometric arrays to achieve arcsecond-level resolution.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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