Editing Wavelength (section)

== Interference and diffraction ==
=== Double-slit interference ===
{{Main|Interference (wave propagation)}}

[[File:Interferometer path differences.JPG|thumb|Pattern of light intensity on a screen for light passing through two slits. The labels on the right refer to the difference of the path lengths from the two slits, which are idealized here as point sources.]]

When sinusoidal waveforms add, they may reinforce each other (constructive interference) or cancel each other (destructive interference) depending upon their relative phase. This phenomenon is used in the [[Interferometry|interferometer]]. A simple example is an experiment due to [[Thomas Young (scientist)|Young]] where light is passed through [[Double-slit experiment|two slits]].<ref name=Sluder>
{{cite book
 |title=Digital microscopy
 |author1=Greenfield Sluder  |author2=David E. Wolf
 |name-list-style=amp |url=https://archive.org/details/digitalmicroscop00gree
 |url-access=registration
 |page=[https://archive.org/details/digitalmicroscop00gree/page/15 15]
 |chapter=IV. Young's Experiment: Two-Slit Interference
 |isbn=978-0-12-374025-0
 |edition=3rd
 |year=2007
 |publisher=Academic Press
 }}</ref>
As shown in the figure, light is passed through two slits and shines on a screen. The path of the light to a position on the screen is different for the two slits, and depends upon the angle θ the path makes with the screen. If we suppose the screen is far enough from the slits (that is, ''s'' is large compared to the slit separation ''d'') then the paths are nearly parallel, and the path difference is simply {{nowrap|''d'' sin ''θ''}}. Accordingly, the condition for constructive interference is:<ref name=Halliday>
{{cite book
 |title=Fundamentals of Physics
 |chapter-url=https://books.google.com/books?id=RVCE4EUjDCgC&pg=PT965
 |page=965
 |chapter=§35-4 Young's interference experiment
 |last1=Halliday|last2=Resnick|last3=Walker
 |isbn=978-81-265-1442-7
 |year=2008
 |edition=Extended 8th
 |publisher=Wiley-India
 }}</ref>
<math display="block"> d \sin \theta = m \lambda \ , </math>
where ''m'' is an integer, and for destructive interference is:
<math display="block"> d \sin \theta = (m + 1/2 )\lambda \ . </math>

Thus, if the wavelength of the light is known, the slit separation can be determined from the interference pattern or ''fringes'', and ''vice versa''.

For multiple slits, the pattern is<ref name=Harris>
{{cite book |author=Kordt Griepenkerl|title=Handbook of physics |chapter-url=https://books.google.com/books?id=c60mCxGRMR8C&pg=PA307 |pages=307 ''ff'' |editor1=John W Harris |editor2=Walter Benenson |editor3=Horst Stöcker |editor4=Holger Lutz |chapter= §9.8.2 Diffraction by a grating |isbn=0-387-95269-1 |year=2002 |publisher=Springer}}
</ref>
<math display="block">I_q = I_1 \sin^2 \left( \frac {q\pi g \sin \alpha} {\lambda} \right)  /  \sin^2  \left( \frac{ \pi g \sin \alpha}{\lambda}\right) \ , </math>
where ''q'' is the number of slits, and ''g'' is the grating constant. The first factor, ''I''<sub>1</sub>, is the single-slit result, which modulates the more rapidly varying second factor that depends upon the number of slits and their spacing. In the figure ''I''<sub>1</sub> has been set to unity, a very rough approximation.

The effect of interference is to ''redistribute'' the light, so the energy contained in the light is not altered, just where it shows up.<ref name= Murphy>
{{cite book
 |title=Fundamentals of light microscopy and electronic imaging
 |url=https://books.google.com/books?id=UFgdjxTULJMC&pg=PA64
 |page=64
 |author= Douglas B. Murphy
 |isbn=0-471-23429-X
 |year=2002
 |publisher=Wiley/IEEE
 }}</ref>

=== Single-slit diffraction ===
{{Main|Diffraction|Diffraction formalism}}
[[File:Double-slit diffraction pattern.png|thumb|200px|Diffraction pattern of a double slit has a single-slit [[Envelope (waves)|envelope]].]]

The notion of path difference and constructive or destructive interference used above for the double-slit experiment applies as well to the display of a single slit of light intercepted on a screen. The main result of this interference is to spread out the light from the narrow slit into a broader image on the screen. This distribution of wave energy is called [[diffraction]].

Two types of diffraction are distinguished, depending upon the separation between the source and the screen: [[Fraunhofer diffraction]] or far-field diffraction at large separations and [[Fresnel diffraction]] or near-field diffraction at close separations.

In the analysis of the single slit, the non-zero width of the slit is taken into account, and each point in the aperture is taken as the source of one contribution to the beam of light (''Huygens' wavelets''). On the screen, the light arriving from each position within the slit has a different path length, albeit possibly a very small difference. Consequently, interference occurs.

In the Fraunhofer diffraction pattern sufficiently far from a single slit, within a [[small-angle approximation]], the intensity spread ''S'' is related to position ''x'' via a squared [[sinc function]]:<ref>
{{cite book
 | title = Optical scattering: measurement and analysis
 | edition = 2nd
 | author = John C. Stover
 | publisher = SPIE Press
 | year = 1995
 | isbn = 978-0-8194-1934-7
 | page = 64
 | url = https://books.google.com/books?id=ot0tjJL72uUC&q=single-slit+diffraction+sinc-function&pg=PA65
 }}</ref>
<math display="block">S(u) = \mathrm{sinc}^2(u) = \left( \frac {\sin \pi u}{\pi u} \right) ^2 \ ; </math>
with
<math display="block">u = \frac {x L}{\lambda R} \ , </math>
where ''L'' is the slit width, ''R'' is the distance of the pattern (on the screen) from the slit, and λ is the wavelength of light used. The function ''S'' has zeros where ''u'' is a non-zero integer, where are at ''x'' values at a separation proportion to wavelength.

=== 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.