Rayleigh's resolution limit
The minimum angular separation between two point sources that an optical system can resolve, limited by diffraction.
Rayleigh's limit: when two lights blur into one
Rayleigh's resolution limit defines the smallest angular separation at which two point sources of light can be distinguished as separate objects by an optical instrument. Below this threshold, diffraction causes the images to merge into a single blob. For a circular aperture, the limit occurs when the central bright disk of one diffraction pattern coincides with the first dark ring of the other, a criterion established by Lord Rayleigh in 1879. This is a hard physical boundary set by the wavelength of light and the aperture diameter, not by lens quality or magnification alone.
The mathematical form is simple: the minimum resolvable angle in radians equals approximately 1.22 times the wavelength divided by the aperture diameter. For visible light around 550 nanometers and a 10 mm aperture, this yields about 67 microradians, or roughly 14 arcseconds. Longer wavelengths (infrared, radio) have worse resolution for the same aperture; shorter wavelengths (ultraviolet, X-rays) achieve finer resolution. This is why radio telescopes must be enormously large to match the angular resolution of small optical instruments.
In practical imaging, the Rayleigh limit matters most where detail counts: astronomical observation, microscopy, and camera design. A telescope cannot be compensated by adding magnification if two stars fall below the resolution limit; they will always appear as one elongated smear. Microscopists working near the diffraction limit must choose objective apertures and illumination wavelengths carefully. The limit is independent of magnification but depends entirely on the diameter of the collecting optics.
Variants and competing criteria
Related standards exist for different optical systems and detection tasks. The Airy disk (the single bright spot at the center of the diffraction pattern) provides the geometric basis. The Sparrow criterion eliminates the dip between adjacent diffraction patterns, allowing slightly closer separation. The Fresnel number and diffraction regime shape what form the limit actually takes in near-field versus far-field imaging. These alternatives are used in specialized applications but Rayleigh remains the dominant reference in optical engineering and astronomy.
Practical systems rarely achieve the theoretical Rayleigh limit due to aberrations, atmospheric turbulence, detector noise, and imperfect apertures. Astronomers use adaptive optics, interferometry, or post-processing deconvolution to push past it. The limit itself is not a hard wall but marks the point where classical geometric optics breaks down and diffraction dominates; improving performance beyond this boundary requires either radically larger optics, fundamentally different physics, or computational correction.