Angular Resolution Calculator
Angular resolution is the smallest angle between two points that an optical system can still tell apart. Diffraction sets a fundamental limit: light bending around the edge of an aperture blurs every point into a small disc, and once two discs overlap too much they merge into one. This calculator uses the Rayleigh criterion, theta equals 1.22 times the wavelength divided by the aperture diameter, to work out that limit for a lens, camera, telescope or microscope with a circular aperture. You enter the wavelength of the light and the diameter of the aperture, and it returns the resolution in arcseconds, radians, degrees and microradians. A larger aperture gives a smaller angle and therefore finer detail, which is the main reason astronomers build ever bigger telescope mirrors, and shorter wavelengths resolve more than longer ones. The result is the theoretical diffraction limit, so it is the best an ideal instrument could achieve. Real optics fall short of it because of aberrations, imperfect focus, and for ground telescopes the blurring of the atmosphere. Use it to compare instruments, to check whether an aperture is large enough to separate a double star or resolve fine detail, and to understand why bigger optics see more.
This is the theoretical diffraction limit for a perfect circular aperture. Real instruments resolve a little less because of optical aberrations, focus errors and, for ground-based telescopes, atmospheric seeing.
How it works
The Rayleigh criterion states that two point sources are just resolvable when the peak of one diffraction disc lands on the first dark ring of the other. For a circular aperture that separation corresponds to an angle of theta equals 1.22 times the wavelength divided by the aperture diameter, measured in radians. The factor 1.22 comes from the first zero of the Airy diffraction pattern. To make the number easier to read, radians are converted to arcseconds by multiplying by 206,265, to degrees by multiplying by 180 over pi, and to microradians by multiplying by one million.
Worked example
Take green light with a wavelength of 550 nanometres and a 100 millimetre telescope aperture. Converting to metres, the wavelength is 5.5 times 10 to the minus 7 metres and the aperture is 0.1 metres. The Rayleigh criterion gives theta equal to 1.22 times 5.5 times 10 to the minus 7, divided by 0.1, which is 6.71 times 10 to the minus 6 radians. Multiplying by 206,265 converts radians to arcseconds, giving about 1.38 arcseconds. That is the finest detail this aperture can resolve in theory, so a double star closer together than about 1.38 arcseconds would blur into a single point.
Related calculators
- Wavelength Calculator: wavelength and frequency.
- Lens and Mirror Equation Calculator: focal length and image distance.
- De Broglie Wavelength Calculator: matter wavelength.
- Scientific Notation Calculator: work with powers of ten.