Rayleigh Criterion Calculator
Enter wavelength and aperture diameter to compute the diffraction-limited angular resolution θ=1.22λ/D. λ=550 nm, D=3 mm (eye) → θ≈46 arcsec; D=2.4 m (Hubble) → θ≈0.057 arcsec.
Input Data
Results
At a glance:The Rayleigh criterion (Rayleigh criterion): the standard for the minimum resolvable angle of a circular-aperture optical instrument (telescope, microscope, camera, eye). When light passes a circular aperture, a point source is not imaged as a perfect point but as an Airy disk diffraction pattern — a central bright spot surrounded by rings. Rayleigh sets that two points are just resolved when the centre of one Airy disk lies on the first dark ring of the other. At that point the minimum resolvable angle is θ_min=1.22λ/D, with λ the wavelength, D the aperture diameter and 1.22 the first zero of the circular-aperture Bessel function. Two points closer than θ_min cannot be resolved (they merge into one blur). Linear resolution d=L·θ_min, L the object distance. Classic examples: human pupil D=3 mm, λ=550 nm → θ_min=1.22×550e-9/(3e-3)≈2.24e-4 rad≈46 arcsec; Hubble D=2.4 m, λ=550 nm → θ_min≈2.8e-7 rad≈0.057 arcsec. The Rayleigh criterion is the diffraction limit — even flawless optics cannot surpass it. The only ways to raise resolution: (1) enlarge D (bigger telescope); (2) lower λ (electron microscope uses a wavelength 10⁵× shorter than visible, reaching atomic scale); (3) use interferometry (e.g. the Event Horizon Telescope) to effectively enlarge D. Applications: telescope design, microscope oil-immersion objectives, camera aperture vs resolution trade-off, human visual-acuity limits, optical-disc storage density.
Formula
Angular resolution: θ_min = 1.22λ/D
Angular (arcsec): θ_arcsec = θ_min × 206265
Linear resolution: d = L·θ_min
$$\theta_{\min} = \frac{1.22\lambda}{D}$$How to Use
- Enter wavelength λ (nm; visible ~550).
- Enter aperture diameter D (mm; pupil 3, Hubble 2400).
- Enter distance L (m; 0 for angular only).
- The tool gives angular resolution (rad, arcsec) and linear resolution d.
Case Studies
Eye and Hubble diffraction limit
Eye pupil D=3 mm, λ=550 nm → θ_min≈2.24e-4 rad≈46 arcsec (~0.013°).
Hubble D=2.4 m, same λ → θ_min≈2.8e-7 rad≈0.057 arcsec — ~800× sharper than the eye.
Even perfect 3 mm optics cannot beat 46 arcsec; only a bigger aperture or shorter λ helps.
Electron microscopy and interferometry
Electron wavelength at 100 kV ≈ 0.0037 nm, ~10⁵× shorter than visible.
An electron microscope reaches atomic-scale resolution (sub-angstrom).
The EHT links radio dishes worldwide, effectively D=Earth-diameter, imaging the black-hole shadow.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.