Angular Resolution Calculator — Rayleigh & Abbe

Formula

How this is calculated

Every optical instrument is limited by diffraction — light bends around the aperture edge and creates a blurred Airy disk instead of a perfect point image. The Rayleigh criterion sets the minimum resolvable angle between two point sources as θ = 1.22 λ / D (in radians), where λ is the wavelength and D the aperture diameter in the same units. Two stars are "just resolved" when the central maximum of one Airy disk falls on the first minimum of the other. At 550 nm, a 200 mm aperture resolves down to about 0.69 arcseconds.

Dawes' limit is an empirical rule for optical telescopes: θ″ = 115.8 / D_mm arcseconds. It was derived from observations of close binary stars and is slightly tighter than the Rayleigh criterion because stars are points and human vision is sensitive to partial resolution. For practical visual astronomy at typical wavelengths (about 550 nm) both criteria give similar results.

For optical microscopes, Ernst Abbe derived d = λ / (2·NA), where NA = n·sin(α) is the numerical aperture (n is the refractive index of the immersion medium, α the half-angle of the objective cone). Oil-immersion objectives (NA up to 1.4) push visible-light resolution to about 200 nm. Real systems add aberrations, detector pixellation, and signal-to-noise limits that reduce practical resolution further.