Angular Resolution
Calculate the angular resolution of any optical system using the Rayleigh criterion θ = 1.22 × λ / D. Free online physics calculator with instant radians, degrees, arcseconds results, and comparison charts for telescopes and cameras.
About This Calculator
The Angular Resolution Calculator uses the Rayleigh criterion to determine the diffraction-limited angular resolution of any optical system. Whether you are designing a telescope, choosing a camera lens, studying microscopy, or simply curious about the limits of human vision, this calculator provides instant results in radians, degrees, and arcseconds.
The fundamental formula, derived by Lord Rayleigh, is θ = 1.22 × λ / D, where θ is the angular resolution in radians, λ is the wavelength of light in meters, and D is the aperture diameter in meters. The factor 1.22 accounts for the first minimum of the Airy diffraction pattern produced by a circular aperture. A smaller θ means better resolution — the ability to distinguish finer details or separate closely spaced objects.
The calculator also compares your result against reference optical systems: the human eye (2 mm pupil by day, 8 mm at night), the Hubble Space Telescope (2.4 m mirror), and the James Webb Space Telescope (6.5 m mirror). The interactive chart shows how resolution varies with wavelength across the visible spectrum.
Key Applications
Angular resolution is critical in astronomy (telescope design), photography (lens selection), microscopy (specimen detail), remote sensing (satellite imagery), and defense (target recognition). Understanding the diffraction limit helps engineers and scientists optimize optical instruments for their specific use cases.
How to Use
Enter the wavelength of light (default 550 nm for green visible light) and the aperture diameter. Click Calculate to see the angular resolution in all three units and a comparison with well-known optical systems worldwide.
Frequently Asked Questions
What is the Rayleigh criterion for angular resolution?
The Rayleigh criterion states that two point sources are resolvable when the central maximum of one Airy pattern falls on the first minimum of the other. The formula is θ = 1.22 × λ / D, where θ is angular resolution in radians, λ is the wavelength of light, and D is the aperture diameter.
What is the angular resolution of the human eye?
The human eye has an angular resolution of approximately 0.02 degrees (about 70 arcseconds) during the day with a 2 mm pupil, and about 0.005 degrees (17 arcseconds) at night with an 8 mm dilated pupil, assuming 550 nm green light.
How does aperture diameter affect angular resolution?
Larger apertures produce better (smaller) angular resolution. The Rayleigh criterion θ = 1.22 × λ / D shows resolution is inversely proportional to aperture diameter. Doubling the aperture diameter halves the angular resolution value, allowing you to distinguish finer details.
What is the angular resolution of the Hubble Space Telescope?
The Hubble Space Telescope has a 2.4-meter primary mirror, giving it an angular resolution of about 0.05 arcseconds at 550 nm visible light. This is roughly 125 times better than the human eye, allowing Hubble to capture incredibly detailed images of distant galaxies and nebulae.
What wavelength should I use for angular resolution calculations?
For visible light optics, use 550 nm (green light, at the center of the visible spectrum). For infrared systems use 1000-10000 nm, for ultraviolet use 100-400 nm, and for radio telescopes use millimeter to meter wavelengths. The calculator accepts any wavelength in nanometers.
How do I convert angular resolution from radians to arcseconds?
To convert angular resolution from radians to degrees, multiply by 180/π. To convert to arcseconds, multiply the radians value by 180/π × 3600 = 206265. Alternatively, use the formula: arcseconds = (1.22 × λ / D) × 206265.
What is the angular resolution of the James Webb Space Telescope?
The James Webb Space Telescope (JWST) has a 6.5-meter segmented primary mirror, giving it an angular resolution of approximately 0.02 arcseconds at 550 nm. JWST observes primarily in infrared wavelengths, where resolution is about 0.1 arcseconds.
Can angular resolution be improved beyond the diffraction limit?
The Rayleigh criterion defines the diffraction limit for conventional optics. Techniques like adaptive optics, aperture synthesis (used in radio astronomy), and super-resolution microscopy can overcome this limit. In astronomy, interferometry combines multiple telescopes to achieve the resolution of a much larger aperture.