Angular Resolution Calculator

Angular resolution describes the smallest angular separation at which an optical system can distinguish two point sources. A telescope may show two nearby stars as separate points only when their diffraction patterns are sufficiently distinct. A microscope faces the same physical limitation when separating tiny features. For an ideal circular aperture, the Rayleigh criterion estimates this limit as 1.22 times the wavelength divided by the aperture diameter. Wavelength and diameter must use the same base unit, and the resulting angle is in radians. Diffraction occurs because light behaves as a wave. Even a perfect lens does not focus a point source into an infinitely small point; it produces an Airy pattern with a bright central disk and surrounding rings. Under the Rayleigh convention, two equally bright sources are just resolved when the center of one Airy disk falls on the first minimum of the other. The factor 1.22 comes from the first zero of the diffraction pattern for a circular opening. Other aperture shapes and resolution definitions can use different constants. A larger aperture improves resolution because it narrows the diffraction pattern. A shorter wavelength also improves resolution, which is why blue light provides a smaller theoretical angle than red light for the same instrument. The calculator converts the selected wavelength and aperture units to meters, evaluates the Rayleigh expression, and reports radians, degrees, arcminutes, and arcseconds. Arcseconds are especially convenient in astronomy, while radians are often preferred in optical engineering calculations. The result is an ideal diffraction limit, not a guarantee of practical image quality. Atmospheric turbulence often dominates ground-based telescope resolution unless adaptive optics or short-exposure methods are used. Lens aberrations, focus error, detector pixel size, alignment, contrast, source brightness, and mechanical vibration can also reduce usable detail. In microscopy, numerical aperture and refractive index may be more directly useful than a simple clear diameter. Use this calculation to compare ideal systems, choose a plausible aperture, or check optical scale, then include real operating conditions when specifying equipment. Extremely small angles also require consistent significant figures; input precision should reflect how accurately wavelength and clear aperture are actually known.