The fastest f-number for the human eye is about 2.1,^[8] corresponding to a diffraction-limited point spread function with approximately 1 μm diameter. However, at this f-number, spherical aberration limits visual acuity, while a 3 mm pupil diameter (f/5.7) approximates the resolution achieved by the human eye.^[9] The maximum density of cones in the human fovea is approximately 170,000 per square millimeter,^[10] which implies that the cone spacing in the human eye is about 2.5 μm, approximately the diameter of the point spread function at f/5.

Focused laser beam

A circular laser beam with uniform intensity across the circle (a flat-top beam) focused by a lens will form an Airy disk pattern at the focus. The size of the Airy disk determines the laser intensity at the focus.

Aiming sight

Some weapon aiming sights (e.g. FN FNC) require the user to align a peep sight (rear, nearby sight, i.e. which will be out of focus) with a tip (which should be focused and overlaid on the target) at the end of the barrel. When looking through the peep sight, the user will notice an Airy disk that will help center the sight over the pin.^{A circular laser beam with uniform intensity across the circle (a flat-top beam) focused by a lens will form an Airy disk pattern at the focus. The size of the Airy disk determines the laser intensity at the focus.}

Aiming sight

Some weapon aim

Some weapon aiming sights (e.g. FN FNC) require the user to align a peep sight (rear, nearby sight, i.e. which will be out of focus) with a tip (which should be focused and overlaid on the target) at the end of the barrel. When looking through the peep sight, the user will notice an Airy disk that will help center the sight over the pin.^[11]

Conditions for observation {\displaystyle \lambda } of the light. The last two conditions can be formally written as ${\displaystyle R>a^{2}/\lambda }$ .

In practice, the conditions for uniform illumination can be met by placing the source of the illumination far from the aperture. If the conditions for far field are not met (for example if the aperture is large), the far-field Airy diffraction pattern can also be obtained on a screen much closer to the aperture by using a lens right after the aperture (or the lens itself can form the aperture). The Airy pattern will then be formed at the focus of the lens rather than at infinity.

Hence, the focal spot of a uniform circular laser beam (a flattop beam) focused by a lens will also be an Airy pattern.

In a camera or imaging system an object far away gets imaged onto the film or detector plane by the objective lens, and the far field diffraction pattern is observed at the detector. The resulting image is a convolution of the ideal image with the Airy diffraction pattern due to diffraction from the iris aperture or due to the finite size of the lens. This leads to the finite resolution of a lens system described above.

The intensity of the Airy pattern follows the Fraunhofer diffraction pattern of a circular aperture, given by the squared modulus of the Fourier transform of the circular aperture:

${\displaystyle I(\theta )=I_{0}\left[{\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right]^{2}=I_{0}\left[{\frac {2J_{1}(x)}{x}}\right]^{2}}$

where ${\displaystyle I_{0}}$${\displaystyle I_{0}}$ is the maximum intensity of the pattern at the Airy disc center, ${\displaystyle J_{1}}$ is the Bessel function of the first kind of order one, ${\displaystyle k={2\pi }/{\lambda }}$ is the wavenumber, ${\displaystyle a}$ is the radius of the aperture, and ${\displaystyle \theta }$ is the angle of observation, i.e. the angle between the axis of the circular aperture and the line between aperture center and observation point. ${\displaystyle x=ka\sin \theta ={\frac {2\pi a}{\lambda }}{\frac {q}{R}}}$, where q is the radial distance from the observation point to the optical axis and R is its distance to the aperture. Note that the Airy disk as given by the above expression is only valid for large R, where Fraunhofer diffraction applies; calculation of the shadow in the near-field must rather be handled using Fresnel diffraction.

However the exact Airy pattern does appear at a finite distance if a lens is placed at the aperture. Then the Airy pattern will be perfectly focussed at the distance given by the lens's focal length (assuming collimated light incident on the aperture) given by the above equations.

The zeros of ${\displaystyle J_{1}(x)}$ are at ${\displaystyle x=ka\sin \theta \approx 3.8317,7.0156,10.1735,13.3237,16.4706\dots }$. From this, it follows that the first dark ring in the diffraction pattern occurs where ${\displaystyle ka\sin <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}=3.8317\dots <!--\; \dots \; -->However\; the\; exact\; Airy\; pattern}$does appear at a finite distance if a lens is placed at the aperture. Then the Airy pattern will be perfectly focussed at the distance given by the lens's focal length (assuming collimated light incident on the aperture) given by the above equations.

The zeros of ${\displaystyle J_{1}(x)}$ are at ${\displaystyle x=ka\sin \theta \approx 3.8317,7.0156,10.1735,13.3237,16.4706\dots }$. From this, it follows that the first dark ring in the diffraction pattern occurs where ${\displaystyle ka\sin {\theta }=3.8317\dots }$, or

If a lens is used to focus the Airy pattern at a finite distance, then the radius ${\displaystyle q_{1}}$ of the first dark ring on the focal plane is solely given by the numerical aperture A (closely related to the f-number) by

${\displaystyle q_{1}=R\sin \theta _{1}\approx 1.22{R}{\frac {\lambda }{d}}=1.22{\frac {\lambda }{2A}}}$

where the numerical aperture A is equal to the aperture's radius d/2 divided by R', the distance from the center of the Airy pattern to the edge of the aperture. Viewing the aperture of radius d/2 and lens as a camera (see diagram above) projecting an image onto a focal pl

where the numerical aperture A is equal to the aperture's radius d/2 divided by R', the distance from the center of the Airy pattern to the edge of the aperture. Viewing the aperture of radius d/2 and lens as a camera (see diagram above) projecting an image onto a focal plane at distance f, the numerical aperture A is related to the commonly-cited f-number N= f/d (ratio of the focal length to the lens diameter) according to ${\displaystyle A={\frac {r}{R'}}={\frac {r}{\sqrt {f^{2}+r^{2}}}}={\frac {1}{\sqrt {4N^{2}+1}}}}$; for N>>1 it is simply approximated as ${\displaystyle A\approx {\frac {1}{2N}}}$. This shows that the best possible image resolution of a camera is limited by the numerical aperture (and thus f-number) of its lens due to diffraction.

The half maximum of the central Airy disk (where ${\displaystyle J_{1}(x)={x}/{2{\sqrt {2}}}}$) occurs at ${\displaysty$

The half maximum of the central Airy disk (where ${\displaystyle J_{1}(x)={x}/{2{\sqrt {2}}}}$) occurs at ${\displaystyle x=1.61633\dots }$; the 1/e² point (where ${\displaystyle J_{1}(x)={x}/{2e}}$) occurs at ${\displaystyle x=2.58383\dots }$, and the maximum of the first ring occurs at ${\displaystyle x=5.13562\dots }$.

The intensity ${\displaystyle I_{0}}$ at the center of the diffraction pattern is related to the total power ${\displaystyle P_{0}}$ incident on the aperture by^[12]

where ${\displaystyle \mathrm {E} }$ is the source strength per unit area at the aperture, A is the area of the aperture (${\displaystyle A=\pi a^{2}}$) and R is the distance from the aperture. At the focal plane of a lens, ${\displaystyle I_{0}=(P_{0}A)/(\lambda ^{2}f^{2})}$. The intensity at the maximum of the first ring is about 1.75% of the intensity at the center of the Airy disk.

The expression for ${\displaystyle I(\theta )}$ above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:

${\display$

The expression for ${\displaystyle I(\theta )}$ above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:

where ${\displaystyle J_{0}}$ and ${\displaystyle J_{1}}$ are Bessel functions. Hence the fractions of the total power contained within the first, second, and third dark rings (where ${\displaystyle J_{1}(ka\sin \theta )=0}$) are 83.8%, 91.0%, and 93.8% respectively.^[13]

Approximation using a Gaussian profile I 0 ′ exp ⁡ ( − q 2 2 σ 2 )   , {\displaystyle I(q)\approx I'_{0}\exp \left({\frac {-q^{2}}{2\sigma ^{2}}}\right)\ ,}

where ${\displaystyle I'_{0}}$ is the irradiance at the center of the pattern, ${\displaystyle q}$ represents the radial distance from the center of the pattern, and $$

where ${\displaystyle I'_{0}}$ is the irradiance at the center of the pattern, ${\displaystyle q}$ represents the radial distance from the center of the pattern, and ${\displaystyle \sigma }$ is the Gaussian RMS width (in one dimension). If we equate the peak amplitude of the Airy pattern and Gaussian profile, that is, ${\displaystyle I'_{0}=I_{0}}$, and find the value of ${\displaystyle \sigma }$ giving the optimal approximation to the pattern, we obtain^[14]

${\displaystyle \sigma \approx 0.42\lambda N\ ,}$

where N is the f-number. If, on the other hand, we wish to enforce that the Gaussian profile has the same volume as does the Airy pattern, then this becomes

${\displaystyle \sigma \approx 0.45\lambda N\ .}$f-number. If, on the other hand, we wish to enforce that the Gaussian profile has the same volume as does the Airy pattern, then this becomes

${\displaystyle \sigma \approx 0.45\lambda N\ .}$

In optical aberration theory, it is common to describe an imaging system as diffraction-limited if the Airy disk radius is larger than the RMS spotsize determined from geometric ray tracing (see Optical lens design). The Gaussian profile approximation provides an alternative means of comparison: using the approximation above shows that the RMS width ${\displaystyle \sigma }$ of the Gaussian approximation to the Airy disk is about one-third the Airy disk radius, i.e. ${\displaystyle 0.42\lambda N}$ as opposed to ${\displaystyle 1.22\lambda N}$.

Obscured Airy pattern

Similar equations can also be derived for the obscured Airy diffraction pattern^[15][16] which is the diffraction pattern from an annular aperture or beam, i.e. a uniform circular aperture (beam) obscured by a circular block at the center. This situation is relevant to many common reflector telescope designs that incorporate a secondary mirror, including Newtonian telescopes and [15]^[16] which is the diffraction pattern from an annular aperture or beam, i.e. a uniform circular aperture (beam) obscured by a circular block at the center. This situation is relevant to many common reflector telescope designs that incorporate a secondary mirror, including Newtonian telescopes and Schmidt–Cassegrain telescopes.

$$

where ${\displaystyle \epsilon }$ is the annular aperture obscuration ratio, or the ratio of the diameter of the obscuring disk and the diameter of the aperture (beam). ${\displaystyle \left(0\leq \epsilon <1\right)}$, and x is defined as above: ${\displaystyle x=ka\sin(\theta )\approx {\frac {\pi R}{\lambda N}}}$ where ${\displaystyle R}$ is the radial distance in the focal plane from the optical axis, ${\displaystyle \lambda }$ is the wavelength and ${\displaystyle N}$ is the f-number of the system. The fractional encircled energy (the fraction of the total energy contained within a circle of radius ${\displaystyle R}$ centered at the optical axis in the focal plane) is then given by:

${\displaystyle \epsilon \rightarrow 0}$ the formulas reduce to the unobscured versions above.

The practical effect of having a central obstruction in a telescope is that the central disc becomes slightly smaller, and the first bright ring becomes brighter at the expense of the central disc. This becomes more problematic with short focal length telescopes which require larger secondary mirrors.^[17]

Comparison to Gaussian beam focus

A circular laser beam with uniform intensity profile, focused by a lens, will form an Airy pattern at the focal plane of the lens. The intensity at the center of the focus will be ${\displaystyle I_{0,Airy}=(P_{0}A)/(\lambda ^{2}f^{2})}$ where ${\displaystyle P_{0}}$ is the total power of the beam, ${\displaystyle A=\pi D^{2}/4}$ is the area of the beam (${\displaystyle D}$ is the beam diameter), ${\displaystyle I_{0,Airy}=(P_{0}A)/(\lambda ^{2}f^{2})}$