Editing Diffraction (section)

=== Diffraction-limited imaging ===
{{Main|Diffraction-limited system}}

[[Image:zboo lucky image 1pc.png|frame|The Airy disk around each of the stars from the 2.56 m telescope aperture can be seen in this ''[[lucky imaging|lucky image]]'' of the [[binary star]] [[zeta Boötis]].]]

The ability of an imaging system to resolve detail is ultimately limited by [[diffraction-limited|diffraction]].  This is because a plane wave incident on a circular lens or mirror is diffracted as described above. The light is not focused to a point but forms an Airy disk having a central spot in the focal plane whose radius (as measured to the first null) is
<math display="block"> \Delta x = 1.22 \lambda N ,</math>
where <math>\lambda</math> is the wavelength of the light and <math>N</math> is the [[f-number]] (focal length <math>f</math> divided by aperture diameter <math>D</math>) of the imaging optics; this is strictly accurate for <math>N \gg 1</math> ([[paraxial]] case).  In object space, the corresponding [[angular resolution]] is
<math display="block">  \theta \approx  \sin \theta = 1.22 \frac{\lambda}{D},</math>
where <math>D</math> is the diameter of the [[entrance pupil]] of the imaging lens (e.g., of a telescope's main mirror).

Two point sources will each produce an Airy pattern – see the photo of a binary star.  As the point sources move closer together, the patterns will start to overlap, and ultimately they will merge to form a single pattern, in which case the two point sources cannot be resolved in the image.  The [[Angular resolution#The Rayleigh criterion|Rayleigh criterion]] specifies that two point sources are considered "resolved" if the separation of the two images is at least the radius of the Airy disk, i.e. if the first minimum of one coincides with the maximum of the other.

Thus, the larger the aperture of the lens compared to the wavelength, the finer the resolution of an imaging system. This is one reason astronomical telescopes require large objectives, and why [[Objective (optics)#Microscope|microscope objective]]s require a large [[numerical aperture]] (large aperture diameter compared to working distance) in order to obtain the highest possible resolution.