Rayleigh Criterion

Equivalent forms

\sin\theta_{min} \approx 1.22 \lambda / D

\theta_{min} D = 1.22 \lambda

A single ratio of wavelength to aperture sets the resolution limit of every telescope, microscope, and eye in the universe.

Symbol	Name	SI unit	Dimension	Range
$lambda$	Wavelength Wavelength of the light	m	L	4e-7 – 7e-7
$D$	Aperture diameter Diameter of the circular aperture	m	L	0.001 – 10
$theta_min$	Minimum resolvable angleoutput Smallest angular separation that can be resolved	rad	1	0 – 0.01

Unit systems

Symbol	SI	CGS	Imperial
$lambda$	m	cm	m
$D$	m	cm	m
$theta_min$	rad	rad	rad

Where it holds

Valid for circular apertures in the far field (Fraunhofer regime). Real instruments often perform worse due to aberrations, atmospheric seeing, or detector pixel size.

Discovery

Lord Rayleigh · 1879

Rayleigh formalized the resolution criterion while studying optical instruments. The 1.22 factor comes from the first zero of the Bessel function J_1, which describes circular aperture diffraction.

Try this

Why can't a telescope resolve two stars closer than a certain angle?

A telescope has aperture diameter 0.1 m operating at wavelength 550 nm. Find the minimum angular separation it can resolve.

Research status: stable

Real-world applications

Telescope design: Hubble's 2.4 m mirror resolves 0.05 arcsec at visible wavelengths.
Microscopy: limits maximum useful magnification before going to electron or super-resolution methods.
Radar resolution: shorter wavelengths or larger antennas give finer angular separation.
Human eye: pupil $\sim 3\,\mathrm{mm}$ gives $\sim 1$ arcmin resolution, matching cone spacing.

Common misconceptions

Bigger magnification means better resolution — only larger aperture helps below the diffraction limit.
The 1.22 factor is for slits — it is specific to circular apertures (slits give 1.0).
The eye is diffraction-limited — it is actually limited by cone spacing and aberrations in normal vision.

Experimental verification

Confirmed by direct observation of double stars at the resolution limit and by laboratory measurements of point-source diffraction patterns through circular pinholes.

Derivation

A circular aperture of diameter D produces an Airy pattern; the first zero of the diffracted intensity occurs at

\sin (\theta ) = 1.22*\lambda /D (1.22

being from the first root of the Bessel function J_1).

Two sources are resolvable when their angular separation equals this value — the Rayleigh criterion.

Limiting cases

D \to \infty

⟶

\theta _\min \to 0

Infinitely large aperture has no diffraction limit.

\lambda \to 0

⟶

\theta _\min \to 0

Shorter wavelengths give finer resolution — why electron microscopes outperform optical ones.

D = \lambda

⟶

\theta _\min \approx 1.22\,\mathrm{rad} \approx 70^{\circ}

Aperture comparable to wavelength means almost no directionality.

What if…

What if you use UV light

(\lambda = 200\,\mathrm{nm})

instead of visible?

Resolution improves by factor $\sim 2.75$ , reaching the same limit as a 2.75x larger visible-light aperture.

What if the aperture is rectangular instead of circular?

The factor changes from 1.22 to 1.0 (for the long dimension) and the diffraction pattern becomes asymmetric.

1

Telescope angular resolution

Given ·

lambda:: 5.5e-7
D:: 0.1

Find · theta_min

Steps

Apply Rayleigh criterion: $\theta _\min = 1.22 * \lambda / D$
Substitute: $\theta _\min = 1.22 * 5.5e-7 / 0.1$
Compute: $\theta _\min = 6.71e-6\,\mathrm{rad}$
Convert: $6.71e-6 * (180/pi) * 3600 \approx 1.38$ arcsec

Result ·

\theta _\min = 1.22 * 550e-9 / 0.1 = 6.71e-6\,\mathrm{rad} \approx 1.38

arcsec

Single Slit Diffraction→Young's Double Slit Interference→