Sparrow criterion
A resolution limit used to determine the moment when two point sources of equal intensity can be considered resolved.
Sparrow criterion: when two bright dots stop looking like one
The Sparrow criterion defines a resolution limit in optical systems by measuring the moment when the intensity profile between two equally bright point sources transitions from having a central maximum to showing a minimum in the middle. Unlike the Rayleigh criterion, which uses the first diffraction minimum as its benchmark, Sparrow's measure focuses on the inflection point where the slope of the combined intensity curve becomes zero. This occurs when the two point spread functions (PSFs) overlap such that their derivative at the center point equals zero.
In practical terms, Sparrow resolution is finer than Rayleigh resolution by approximately 5 to 10 percent, depending on the optical system. Where Rayleigh places the resolution limit at a point where most observers can just detect a dip between two sources, Sparrow identifies the threshold before any dip appears at all. This makes it a useful metric for applications where visual or instrumental detection of separation matters more than strict diffraction theory.
Application and measurement
Astronomers, microscopy specialists, and imaging engineers use Sparrow criterion to assess performance of telescopes, microscope objectives, and imaging sensors. It is particularly valuable in adaptive optics systems and wavefront sensing, where near-diffraction-limited performance must be verified experimentally rather than assumed. The criterion requires measurement or calculation of the second derivative of intensity across the focal plane, making it more computationally involved than Rayleigh analysis but more directly tied to human perception of contrast.
The criterion is named after Paul A. Sparrow, who formulated it in the 1950s as an alternative to the Rayleigh criterion. It occupies a middle ground in resolution metrics: stricter than the Abbe diffraction limit but less severe than the Rayleigh standard. In spectroscopy, interferometry, and lithography, where sub-wavelength precision is demanded, Sparrow's definition often aligns better with actual performance limits than theoretical diffraction maxima.