Math 2.0

Optics, electromagnetic theory Definition

Rayleigh criterion

Builds on Point spread functionAiry diskBessel function of the first kind

Definition

Let an aberration-free optical system with a circular aperture of diameter DD image two incoherent point sources of the same wavelength λ\lambda. Let θ\theta be the angle between the directions of the two sources. By the Rayleigh criterion, the two sources are just resolved when the following holds. The centre of the Airy disk of one source lies on the first dark ring of the Airy pattern of the other. This happens when sin⁡θ=j1,1π λD≈1.22 λD,\sin \theta = \frac{j_{1,1}}{\pi} \, \frac{\lambda}{D} \approx 1.22 \, \frac{\lambda}{D}, where j1,1≈3.8317j_{1,1} \approx 3.8317 is the first positive zero of the Bessel function J1J_1. The sources are resolved when θ\theta is at least this value.

Remarks

The constant comes from the point spread function of the circular aperture. At angle ϑ\vartheta from the centre it is proportional to (2J1(v)/v)2(2 J_1(v)/v)^2 with v=πDsin⁡(ϑ)/λv = \pi D \sin(\vartheta) / \lambda. Its first zero is at v=j1,1v = j_{1,1}, which gives the formula. For a slit of width aa the same rule gives sin⁡θ=λ/a\sin \theta = \lambda / a, since the first zero of (sin⁡u/u)2(\sin u / u)^2 is at u=πu = \pi.

Example: a telescope with D=0.1D = 0.1 m at λ=500\lambda = 500 nm resolves two stars when θ≥1.22×5×10−6\theta \ge 1.22 \times 5 \times 10^{-6} rad, that is about 6.1×10−66.1 \times 10^{-6} rad.

Non-example: two sources at half this angle are not resolved by the criterion, since the centre of each pattern lies inside the central disc of the other.

The criterion is a convention, not a theorem; other conventions, such as the Sparrow criterion, use a slightly different separation.

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Bessel function of the first kind, concept, no evidence levelBessel function of th…conceptFourier transform, concept, no evidence levelFourier transformconceptGreen's function, concept, no evidence levelGreen's functionconceptHelmholtz equation, concept, no evidence levelHelmholtz equationconceptParaxial approximation, concept, no evidence levelParaxial approximationconceptPoint spread function, concept, no evidence levelPoint spread functionconceptRefractive index, concept, no evidence levelRefractive indexconceptWave equation, concept, no evidence levelWave equationconceptPhysical optics, definition, no evidence levelPhysical opticsdefinitionFresnel number, definition, no evidence levelFresnel numberdefinitionKirchhoff's diffraction formula, statement, no evidence levelKirchhoff's diffracti…statementRayleigh–Sommerfeld diffraction, statement, no evidence levelRayleigh–Sommerfeld d…statementFresnel diffraction, definition, no evidence levelFresnel diffractiondefinitionFraunhofer diffraction, definition, no evidence levelFraunhofer diffractiondefinitionAiry disk, object, no evidence levelAiry diskobjectRayleigh criterion, definition, no evidence levelRayleigh criteriondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Bessel function of the first kind concept
  2. Fourier transform concept
  3. Green's function concept
  4. Helmholtz equation concept
  5. Paraxial approximation concept
  6. Point spread function concept
  7. Refractive index concept
  8. Wave equation concept
  9. Physical optics definition
  10. Fresnel number definition
  11. Kirchhoff's diffraction formula statement
  12. Rayleigh–Sommerfeld diffraction statement
  13. Fresnel diffraction definition
  14. Fraunhofer diffraction definition
  15. Airy disk object
  16. Rayleigh criterion definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Angular_resolution
About this page
TagY46NWBWB5C
URIm2://Y46NWBWB5C
Kinddefinition
WritingW0 stub
MSC78A10, 78A45
DefinesRayleigh criterion
UsesPoint spread function, Airy disk, Bessel function of the first kind
Changed3 November 2026

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