= Airy disk
{c}
{title2=$\theta_1\simeq1.22\lambda/D$}
{wiki}
= Airy disc
{c}
{synonym}
The <Airy disk> is the central bright region, up to the first dark ring, in the scalar <diffraction> pattern of a uniformly illuminated, unobscured circular <aperture>. The whole pattern includes the weaker surrounding rings; its radius is distinct from its <full width at half maximum>.
For a circular <optical pupil> of <diameter> $D$ under normal, uniform monochromatic <plane wave> illumination, <Fraunhofer diffraction> gives
$$
\frac{I(\theta)}{I_0}=P(u)=\left[\frac{2J_1(u)}{u}\right]^2,\qquad u=\frac{\pi D\sin\theta}{\lambda},\qquad P(0)=1.
$$
Here $I_0$ is the on-axis <light intensity>, $\theta$ is the <angle> from the <optical axis>, $\lambda$ is the <wavelength> in the propagation medium, and $J_1$ is the order-one <Bessel function of the first kind>. The <paraxial approximation> replaces $\sin\theta$ by $\theta$, measured in <radians>. The first positive zero $j_{1,1}=3.831705970\ldots$ gives
$$
\sin\theta_1=\frac{j_{1,1}}{\pi}\frac{\lambda}{D},\qquad \theta_1\simeq1.219669891\frac{\lambda}{D}\simeq1.22\frac{\lambda}{D}.
$$
This is the angular radius of the <Airy disk>; the angular diameter is twice as large. The <Airy disk derivation> explains the origin of the <Bessel function of the first kind> and of the numerical coefficient.<Oscillations and Waves by Richard Fitzpatrick/Two-Dimensional Fourier Optics>[equations 10.78–10.85 and Table 10.1: circular-aperture response and first zero]{ref}
\Image[/airy-disk-radial-intensity.png]
{title=Ideal circular-aperture intensity: central disk, dark rings and radial profile}
The image and both radial curves show the same calculated <light intensity>, divided by its peak. The image's <logarithmic scale> makes the weak rings visible; its colour is a display choice. The linear curve shows how small the ring peaks are compared with the centre. A dark ring marks a zero of <wave amplitude>, produced by cancellation across the <optical pupil>; it is not the edge of all transmitted <light>. The first <Airy disk> contains about $83.78\%$ of the total <radiant flux>, as calculated in <Airy disk normalization and encircled energy>.
For an original design example, take an ideal <optical lens> in air with <focal length> $f=100\,\mathrm{mm}$, circular <optical pupil> <diameter> $D=5.00\,\mathrm{mm}$ and <wavelength> $\lambda=550\,\mathrm{nm}$. In the <focal plane>, the radial coordinate $r\simeq f\theta$ gives
$$
\theta_1\simeq1.34164\times10^{-4}\,\mathrm{rad},\qquad r_1\simeq1.219669891\frac{\lambda f}{D}=13.4164\,\mathrm{\mu m}.
$$
The first-zero diameter is $26.8327\,\mathrm{\mu m}$. Solving $P(u_{1/2})=1/2$ gives $u_{1/2}=1.616339948\ldots$, so the diameter measured by <full width at half maximum> is only
$$
w_{1/2}=\frac{2u_{1/2}}{\pi}\frac{\lambda f}{D}\simeq1.028993970\frac{\lambda f}{D}=11.3189\,\mathrm{\mu m}.
$$
Thus a quoted “spot diameter” is incomplete without its definition. Doubling $D$ halves both widths at fixed <wavelength> and <focal length>; increasing <magnification> alone changes the image scale without improving object-space discrimination.
The <Airy disk> supplies a calculable reference for a <point spread function>, allowing <diffraction> to be distinguished from <defocus> and <optical aberrations>. Its beauty is that one circular boundary determines an entire sequence of bright and dark rings through a <Fourier transform>. The ideal formula assumes an unobscured pupil, uniform illumination, ideal <optical focus>, a single <wavelength>, and a scalar description with small output <angles>. A blocked centre, nonuniform pupil transmission or a significant pupil <wave phase> error changes the response. High <numerical aperture> may require <polarization of an electromagnetic wave> and a vector field description. The <circular-aperture diffraction laboratory experiment> tests the far-field pattern directly; the <Rayleigh criterion> uses its first zero to define a two-source convention.
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