Math 2.0

Optics, electromagnetic theory Wave optics and diffraction Object

Airy disk

Builds on Fraunhofer diffractionBessel function of the first kind

Description

Let a plane monochromatic wave of wavelength λ\lambda and wavenumber k=2π/λk = 2\pi/\lambda fall normally on a circular aperture of radius aa. Let J1J_1 be the Bessel function of the first kind of order 11, and let I0I_0 be the intensity on the axis. In the Fraunhofer regime, the intensity at angle θ\theta from the axis is I(θ)=I0(2J1(x)x)2,x=kasin⁡θ.I(\theta) = I_0 \left( \frac{2 J_1(x)}{x} \right)^2, \qquad x = k a \sin\theta. The pattern is a bright central disk surrounded by fainter rings. The Airy disk is the central bright disk, bounded by the first dark ring. This ring lies at the first positive zero x≈3.8317x \approx 3.8317 of J1J_1, that is, at sin⁡θ≈1.22 λD,D=2a.\sin\theta \approx 1.22 \, \frac{\lambda}{D}, \qquad D = 2a.

Remarks

The formula comes from the Fourier transform of the indicator function of a disk: in polar coordinates its angular integral gives J0J_0, and ∫0xsJ0(s) ds=xJ1(x)\int_0^x s J_0(s) \, ds = x J_1(x). Since 2J1(x)/x→12 J_1(x)/x \to 1 as x→0x \to 0, the expression is normalised so that I(0)=I0I(0) = I_0. About 84%84\% of the light lies inside the Airy disk. The size of the Airy disk sets the resolving power of a telescope or microscope with a circular aperture. In the Rayleigh criterion, two point sources are just resolved when the centre of one Airy pattern falls on the first dark ring of the other.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

Show the path as a graph
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 approximationconceptRefractive 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 diskobject
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. Refractive index concept
  7. Wave equation concept
  8. Physical optics definition
  9. Fresnel number definition
  10. Kirchhoff's diffraction formula statement
  11. Rayleigh–Sommerfeld diffraction statement
  12. Fresnel diffraction definition
  13. Fraunhofer diffraction definition
  14. Airy disk object

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Airy_disk
About this page
TagDB435TBQJT
URIm2://DB435TBQJT
Kindobject
WritingW0 stub
MSC78A45, 33C10
UsesFraunhofer diffraction, Bessel function of the first kind
Same as
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by