Rayleigh Criterion Calculator
Calculate rayleigh criterion with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
Formula
theta = 1.22 x lambda / D
Where theta is the minimum angular resolution in radians, lambda is the wavelength of light, and D is the diameter of the aperture. The factor 1.22 comes from the first zero of the Bessel function describing diffraction through a circular aperture.
Worked Examples
Example 1: Telescope Angular Resolution
Problem: A telescope has a 200mm aperture. What is its angular resolution at 550nm (green light)?
Solution: Angular resolution = 1.22 x wavelength / aperture diameter\n= 1.22 x 550e-9 m / 0.200 m\n= 3.355e-6 radians\nConvert to arcseconds: 3.355e-6 x (180/pi) x 3600 = 0.692 arcseconds\nRayleigh limit shorthand: 140 / 200 = 0.700 arcseconds (close agreement)
Result: Angular Resolution: 0.692 arcseconds (can resolve features separated by this angle)
Example 2: Minimum Resolvable Distance at Range
Problem: A camera with a 50mm lens (50mm aperture) observes objects 1 km away at 550nm. What is the smallest feature it can resolve?
Solution: Angular resolution = 1.22 x 550e-9 / 0.050 = 1.342e-5 radians\nMinimum separation = angular resolution x distance\n= 1.342e-5 x 1000 m = 0.01342 m = 13.42 mm\nAiry disk radius = 1.22 x 550e-9 x (200/50) = 2.684 micrometers
Result: Minimum Resolvable Distance: 13.42 mm at 1 km range
Frequently Asked Questions
What is the Rayleigh criterion and why does it matter in optics?
The Rayleigh criterion defines the minimum angular separation at which two point sources of light can be distinguished as separate objects through an optical system. It was established by Lord Rayleigh in 1879 and states that two sources are just resolvable when the central maximum of one diffraction pattern falls on the first minimum of the other. This criterion is fundamental in determining the resolving power of telescopes, microscopes, cameras, and even the human eye. Without this physical limit, we could theoretically build infinitely powerful optical instruments, but diffraction imposes a hard boundary on resolution that depends on wavelength and aperture size.
What role does wavelength play in the Rayleigh criterion calculation?
Wavelength is directly proportional to the angular resolution limit, so shorter wavelengths provide better resolving power. Blue light at 450nm gives about 22 percent better resolution than red light at 650nm through the same aperture. This is why electron microscopes, which use electron beams with extremely short de Broglie wavelengths, can resolve features far smaller than optical microscopes. In astronomy, observing at shorter wavelengths (such as ultraviolet or X-ray) can reveal finer details than visible light observations with the same aperture. Radio telescopes require enormous dish diameters specifically because radio wavelengths are millions of times longer than visible light.
How do I apply the Rayleigh criterion to telescope selection and comparison?
To compare telescopes using the Rayleigh criterion, calculate the angular resolution for each by dividing 1.22 times the observing wavelength by the aperture diameter. A practical shorthand for visible light (550nm) is the Dawes limit, which approximates to 116 divided by the aperture in millimeters, giving arcseconds. For example, a 150mm telescope resolves about 0.77 arcseconds, while a 250mm telescope resolves about 0.46 arcseconds. This tells you whether a telescope can split close double stars or resolve fine planetary detail. However, atmospheric seeing typically limits ground-based resolution to about 1-2 arcseconds regardless of aperture size.
What is the difference between the Rayleigh criterion and the Dawes limit?
The Rayleigh criterion and Dawes limit are two different standards for defining optical resolution. The Rayleigh criterion is based on diffraction theory and places the central maximum of one source at the first minimum of the other, producing a roughly 26 percent intensity dip between the two peaks. The Dawes limit is empirically derived from actual observations of double stars and represents a slightly tighter separation where a trained observer can still detect two sources. The Dawes limit is approximately 116/D arcseconds (where D is in millimeters) compared to the Rayleigh limit of about 140/D arcseconds. This means the Dawes limit allows resolution at about 83 percent of the Rayleigh separation.
Can the Rayleigh criterion be overcome with modern techniques?
Several techniques can achieve resolution beyond the classical Rayleigh limit. Super-resolution microscopy methods like STED, PALM, and STORM in biological imaging can resolve features 10-20 times smaller than the diffraction limit by exploiting fluorescence switching. Interferometry combines signals from multiple separated telescopes to synthesize an effective aperture equal to their baseline separation. The Event Horizon Telescope used this principle with radio dishes spanning the globe to image a black hole. Computational methods like deconvolution and structured illumination also push past the Rayleigh limit. However, these techniques have their own limitations including noise sensitivity and specialized sample requirements.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.