Editing Wavelength (section)

=== Diffraction-limited resolution ===
{{main|Angular resolution|Diffraction-limited system}}

Diffraction is the fundamental limitation on the [[Angular resolution|resolving power]] of optical instruments, such as [[telescope]]s (including [[radiotelescope]]s) and [[microscopes]].<ref name=Saxby>
{{cite book
 |author=Graham Saxby
 |chapter-url=https://books.google.com/books?id=e5mC5TXlBw8C&pg=PA57
 |page=57 |title=The science of imaging
 |chapter=Diffraction limitation
 |isbn=0-7503-0734-X
 |year=2002
 |publisher=CRC Press
 }}</ref>
For a circular aperture, the diffraction-limited image spot is known as an [[Airy disk]]; the distance ''x'' in the single-slit diffraction formula is replaced by radial distance ''r'' and the sine is replaced by 2''J''<sub>1</sub>, where ''J''<sub>1</sub> is a first order [[Bessel function]].<ref>
{{cite book
 | title = Introduction to Modern Optics
 | author = Grant R. Fowles
 | publisher = Courier Dover Publications
 | year = 1989
 | isbn = 978-0-486-65957-2
 | pages = 117–120
 | url = https://books.google.com/books?id=SL1n9TuJ5YMC&q=Airy-disk+Bessel+slit+diffraction+sin&pg=PA119
 }}</ref>

The resolvable ''spatial'' size of objects viewed through a microscope is limited according to the [[Angular resolution#The_Rayleigh_criterion|Rayleigh criterion]], the radius to the first null of the Airy disk, to a size proportional to the wavelength of the light used, and depending on the [[numerical aperture]]:<ref>
{{cite book
 | title = Handbook of biological confocal microscopy
 | edition = 2nd
 | author = James B. Pawley
 | publisher = Springer
 | year = 1995
 | isbn = 978-0-306-44826-3
 | page = 112
 | url = https://books.google.com/books?id=16Ft5k8RC-AC&pg=PA112
 }}</ref>
<math display="block">r_{Airy} = 1.22 \frac {\lambda}{2\,\mathrm{NA}} \ , </math>
where the numerical aperture is defined as <math>\mathrm{NA} = n \sin \theta\;</math> for θ being the half-angle of the cone of rays accepted by the [[microscope objective]].

The ''angular'' size of the central bright portion (radius to first null of the [[Airy disk]]) of the image diffracted by a circular aperture, a measure most commonly used for telescopes and cameras, is:<ref>
{{cite book
 | title = Reflecting Telescope Optics I: Basic Design Theory and Its Historical Development
 | author = Ray N. Wilson
 | publisher = Springer
 | year = 2004
 | isbn = 978-3-540-40106-3
 | page = 302
 | url = https://books.google.com/books?id=PuN7l2A2uzQC&q=telescope+diffraction-limited+resolution+sinc&pg=PA302
 }}</ref>
<math display="block">\delta = 1.22 \frac {\lambda}{D} \ , </math>
where ''λ'' is the wavelength of the waves that are focused for imaging, ''D'' the [[entrance pupil]] diameter of the imaging system, in the same units, and the angular resolution ''δ'' is in radians.

As with other diffraction patterns, the pattern scales in proportion to wavelength, so shorter wavelengths can lead to higher resolution.