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| [[Image:Ernst Abbe memorial.JPG|thumb|right|Memorial to [[Ernst Karl Abbe]], who approximated the diffraction limit of a microscope as <math>d=\frac{\lambda}{2n\sin{\theta}}</math>, where ''d'' is the resolvable feature size, ''λ'' is the wavelength of light, ''n'' is the index of refraction of the medium being imaged in, and ''θ'' (depicted as ''α'' in the inscription) is the half-angle subtended by the optical objective lens.]]
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| [[Image:Diffraction limit diameter vs angular resolution.svg|thumb|Log-log plot of aperture diameter vs angular resolution at the diffraction limit for various light wavelengths compared with various astronomical instruments. For example, the blue star shows that the Hubble Space Telescope is almost diffraction-limited in the visible spectrum at 0.1 arcsecs, whereas the red circle shows that the human eye should have a resolving power of 20 arcsecs in theory, though normally only 60 arcsecs.]]
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| The resolution of an optical imaging system{{spaced ndash}} a [[microscope]], [[telescope]], or [[camera]]{{spaced ndash}} can be limited by factors such as imperfections in the lenses or misalignment. However, there is a fundamental maximum to the resolution of any optical system which is due to [[diffraction]]. An optical system with the ability to produce images with [[angular resolution]] as good as the instrument's theoretical limit is said to be '''diffraction limited'''.<ref>{{cite book
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| | first = Max | last = Born
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| | coauthors = Emil Wolf
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| | title = Principles of Optics | publisher = Cambridge University Press | year = 1997 | isbn = 0-521-63921-2 }}</ref>
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| The resolution of a given instrument is proportional to the size of its [[objective lens|objective]], and inversely proportional to the [[wavelength]] of the light being observed. For telescopes with circular apertures, the size of the smallest feature in an image that is diffraction limited is the size of the [[Airy disc]]. As one decreases the size of the [[aperture]] in a [[lens (optics)|lens]] diffraction increases. At small apertures, such as [[F-stop|f/22]], most modern lenses are limited only by diffraction.
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| In [[astronomy]], a '''diffraction-limited''' observation is one that is limited only by the optical power of the instrument used. However, most observations from Earth are [[Astronomical seeing|seeing]]-limited due to [[Earth's atmosphere|atmospheric]] effects. Optical telescopes on the [[Earth]] work at a much lower resolution than the diffraction limit because of the distortion introduced by the passage of light through several kilometres of [[atmospheric turbulence|turbulent atmosphere]]. Some advanced observatories have recently started using [[adaptive optics]] technology, resulting in greater image resolution for faint targets, but it is still difficult to reach the diffraction limit using adaptive optics.
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| [[Radiotelescope]]s are frequently diffraction-limited, because the wavelengths they use (from millimeters to meters) are so long that the atmospheric distortion is negligible. Space-based telescopes (such as [[Hubble Space Telescope|Hubble]], or a number of non-optical telescopes) always work at their diffraction limit, if their design is free of [[optical aberration]].
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| ==The Abbe diffraction limit for a microscope==
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| The observation of sub-wavelength structures with microscopes is difficult because of the '''Abbe diffraction limit'''. [[Ernst Abbe]] found in 1873 that light with wavelength λ, traveling in a medium with refractive index n and converging to a spot with angle <math>\theta</math> will make a spot with diameter
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| :<math>d=\frac{ \lambda}{2 (n \sin \theta)}</math><ref>{{cite book|last=Lipson, Lipson and Tannhauser|title=Optical Physics|year=1998|publisher=Cambridge|location=United Kingdom|isbn=978-0-521-43047-0|pages=340}}</ref> | |
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| The denominator <math> n\sin \theta </math> is called the [[numerical aperture]] (NA) and can reach about 1.4 in modern optics, hence the Abbe limit is d=λ/2.8. Considering green light around 500 nm and a NA of 1, the Abbe limit is roughly d=λ/2=250 nm which is large compared to most nanostructures or biological cells which have sizes on the order of 1μm and internal organelles which are much smaller. To increase the resolution, shorter wavelengths can be used such as UV and X-ray microscopes. These techniques offer better resolution but are expensive, suffer from lack of contrast in biological samples and may damage the sample.
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| ==Obtaining higher resolution==
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| There are techniques for producing images that appear to have higher resolution than allowed by simple use of diffraction-limited optics.<ref name="U2">{{cite journal
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| | url=http://www.opfocus.org/index.php?topic=story&v=4&s=1
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| | author=Niek van Hulst
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| | title=Many photons get more out of diffraction
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| | journal=[[Optics & Photonics Focus]]
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| | volume=4
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| | issue=1
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| | year=2009
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| }}</ref> Although these techniques improve some aspect of resolution, they generally come at an enormous increase in cost and complexity. Usually the technique is only appropriate for a small subset of imaging problems, with several general approaches outlined below.
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| ===Extending numerical aperture===
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| For a given [[numerical aperture]] (NA), the resolution of [[microscopy]] for flat objects under coherent illumination can be improved using [[interferometric microscopy]]. Using the partial images from a holographic recording of the distribution of the complex optical field, the large aperture image can be reconstructed numerically.<ref name="U1">{{cite journal
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| | url=http://www.opticsexpress.org/abstract.cfm?id=134719
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| | author=Y.Kuznetsova
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| | coauthors=A.Neumann, S.R.Brueck
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| | title=Imaging interferometric microscopy–approaching the linear systems limits of optical resolution
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| | journal=[[Optics Express]]
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| | volume=15
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| | issue=11
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| | pages=6651–6663
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| | year=2007
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| | doi=10.1364/OE.15.006651
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| |bibcode = 2007OExpr..15.6651K
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| | pmid=19546975}}</ref> Another technique, [[4Pi Microscope|4 PI Microscopy]] uses two opposing objectives to double the effective numerical aperture, effectively halving the diffraction limit.
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| Among sub-diffraction limited techniques, [[Microscopy#Structured_illumination|structured illumination]] holds the distinction of being one of the only methods that can work with simple reflectance without the need for special dyes or fluorescence and at very long working distances. In this method, multiple spatially modulated illumination patterns are used to double the effective numerical aperture. In principle, the technique can be used at any range and on any target provided that illumination can be controlled. Additionally, if exogenous contrast agents are used, the technique can also achieve more than a two-fold increase in resolution.
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| ===Near-field techniques===
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| The diffraction limit is only valid in the far field. Various [[Near and far field|near-field]] techniques that operate less than 1 wavelength of light away from the image plane can obtain substantially higher resolution. These techniques exploit the fact that the [[Evanescent wave|evanescent field]] contains information beyond the diffraction limit which can be used to construct very high resolution images, in principle beating the diffraction limit by a factor proportional to how far into the near field an imaging system extends. Techniques such as [[Total internal reflection fluorescence microscope|total internal reflectance microscopy]] and [[metamaterials]]-based [[superlens]] can image with resolution better than the diffraction limit by locating the [[objective lens]] extremely close (typically hundreds of nanometers) to the object. However, because these techniques cannot image beyond 1 wavelength, they cannot be used to image into objects thicker than 1 wavelength which limits their applicability. | |
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| ===Far-field techniques===
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| Far-field imaging techniques are most desirable for imaging objects that are large compared to the illumination wavelength but that contain fine structure. This includes nearly all biological applications in which cells span multiple wavelengths but contain structure down to molecular scales. In recent years several techniques have shown that sub-diffraction limited imaging is possible over macroscopic distances. These techniques usually exploit optical [[Nonlinear optics|nonlinearity]] in a material's reflected light to generate resolution beyond the diffraction limit.
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| Among these techniques, the [[Stimulated Emission Depletion Microscope|STED Microscope]] has been one of the most successful. In STED, multiple laser beams are used to first excite, and then quench [[fluorescent]] dyes. The nonlinear response to illumination caused by the quenching process in which adding more light causes the image to become less bright generates sub-diffraction limited information about the location of dye molecules, allowing resolution far beyond the diffraction limit provided high illumination intensities are used.
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| ==Other waves==
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| The same equations apply to other wave based sensors, such as radar and the human ear.
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| As opposed to light waves (i.e., photons), massive particles have a different relationship between their quantum mechanical wavelength and their energy. This relationship indicates that the effective [[De Broglie wavelength|"de Broglie" wavelength]] is inversely proportional to the momentum of the particle. For example, an electron at an energy of 10 keV has a wavelength of 0.01 nm, allowing the electron microscope ([[Scanning electron microscope|SEM]] or [[Transmission electron microscopy|TEM]]) to achieve high resolution images. Other massive particles such as helium, neon, and gallium ions have been used to produce images at resolutions beyond what can be attained with visible light. Such instruments provide nanometer scale imaging, analysis and fabrication capabilities at the expense of system complexity.
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| == See also ==
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| * [[Rayleigh criterion]]
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| ==References==
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| <references/>
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| ==External links==
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| * {{cite web
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| | first = Erwin | last = Puts
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| | work = Leica R-Lenses
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| |date=September 2003
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| | format = PDF | publisher = [[Leica Camera]]
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| | url = http://en.leica-camera.com/assets/file/download.php?filename=file_1864.pdf
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| | title = Chapter 3: 180 mm and 280 mm lenses
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| }} Describes the Leica APO-Telyt-R 280mm f/4, a diffraction-limited photographic lens.
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| [[Category:Diffraction]]
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| [[Category:Telescopes]]
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| [[Category:Microscopes]]
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