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The diffraction theory of point-spread functions was first studied by [[George Biddell Airy|Airy]] in the nineteenth century. He developed an expression for the point-spread function amplitude and intensity of a perfect instrument, free of aberrations (the so-called [[Airy disc]]). The theory of aberrated point-spread functions close to the optimum focal plane was studied by [[Frits Zernike|Zernike]] and Nijboer in the 1930–40s. A central role in their analysis is played by Zernike's [[Zernike polynomials|circle polynomials]] that allow an efficient representation of the aberrations of any optical system with rotational symmetry. Recent analytic results have made it possible to extend Nijboer and Zernike's approach for point-spread function evaluation to a large volume around the optimum focal point. This extended Nijboer-Zernike (ENZ) theory allows studying the imperfect imaging of three-dimensional objects in [[confocal microscopy]] or astronomy under non-ideal imaging conditions. The ENZ-theory has also been applied to the characterization of optical instruments with respect to their aberration by measuring the through-focus intensity distribution and solving an appropriate [[inverse problem]].
== Appplications ==
=== Microscopy ===
[[File:63x 1.4NA Confocal Point Spread Function 2+3D.png|thumb|An example of an experimentally derived point spread function from a confocal microscope using a 63x 1.4NA oil objective. It was generated using Huygens Professional deconvolution software. Shown are views in xz, xy, yz and a 3D representation.]]
In microscopy, experimental determination of PSF requires sub-resolution (point-like) radiating sources. [[Quantum dot]]s and [[fluorescent]] [[bead]]s are usually considered for this purpose.<ref>Light transmitted through minute holes in a thin layer of silver vacuum or chemically deposited on a slide or cover-slip have also been used, as they are bright and do not photo-bleach.
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Theoretical models as described above, on the other hand, allow the detailed calculation of the PSF for various imaging conditions. The most compact [[diffraction limited]] shape of the PSF is usually preferred. However, by using appropriate optical elements (e.g., a [[spatial light modulator]]) the shape of the PSF can be engineered towards different applications.
=== Astronomy ===
[[File:Hubble PSF with flawed optics.jpg|thumb|The point spread function of [[Hubble Space Telescope]]'s [[Wide Field and Planetary Camera|WFPC]] camera before corrections were applied to its optical system.]]
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For ground-based optical telescopes, atmospheric turbulence (known as [[astronomical seeing]]) dominates the contribution to the PSF. In high-resolution ground-based imaging, the PSF is often found to vary with position in the image (an effect called anisoplanatism). In ground-based [[adaptive optics]] systems the PSF is a combination of the aperture of the system with residual uncorrected atmospheric terms.<ref>{{Cite web|url=http://www.telescope-optics.net/diffraction_image.htm|title=POINT SPREAD FUNCTION (PSF)|website=www.telescope-optics.net|access-date=2017-12-30}}</ref>
=== Lithography ===
[[File:Airy_spot_overlap.png|thumb|left|300px|'''Overlapped PSF peaks.''' When the peaks are as close as ~ 1 wavelength/NA, they are effectively merged. The FWHM is ~ 0.6 wavelength/NA at this point.]]
The PSF is also a fundamental limit to the conventional focused imaging of a hole,<ref name=nat_res>[http://www.lithoguru.com/scientist/litho_tutor/TUTOR23%20(Fall%2098).pdf The Natural Resolution]</ref> with the minimum printed size being in the range of 0.6-0.7 wavelength/NA, with NA being the [[numerical aperture]] of the imaging system.<ref>[https://www.weizmann.ac.il/mcb/ZviKam/ALM/L2_Resolution.pdf Principles and Practice of Light Microscopy]</ref><ref>[http://ww.lithoguru.com/scientist/litho_papers/2000_103_Corner%20Rounding%20and%20Line-end%20Shortening%20in%20OL.pdf Corner Rounding and Line-end Shortening]</ref> For example, in the case of an [[extreme ultraviolet lithography|EUV]] system with wavelength of 13.5 nm and NA=0.33, the minimum individual hole size that can be imaged is in the range of 25-29 nm. A [[phase-shift mask]] has 180-degree phase edges which allow finer resolution.<ref name=nat_res/>
=== Ophthalmology ===
Point spread functions have recently become a useful diagnostic tool in clinical [[ophthalmology]]. Patients are measured with a [[wavefront]] sensor, and special software calculates the PSF for that patient's eye. This method allows a physician to simulate potential treatments on a patient, and estimate how those treatments would alter the patient's PSF. Additionally, once measured the PSF can be minimized using an adaptive optics system. This, in conjunction with a [[Charge-coupled device|CCD]] camera and an adaptive optics system, can be used to visualize anatomical structures not otherwise visible ''in vivo'', such as cone photoreceptors.<ref>{{Cite journal|last=Roorda|first=Austin|last2=Romero-Borja|first2=Fernando|last3=Iii|first3=William J. Donnelly|last4=Queener|first4=Hope|last5=Hebert|first5=Thomas J.|last6=Campbell|first6=Melanie C. W.|author6-link=Melanie Campbell|date=2002-05-06|title=Adaptive optics scanning laser ophthalmoscopy|journal=Optics Express|language=EN|volume=10|issue=9|pages=405–412|doi=10.1364/OE.10.000405|issn=1094-4087|bibcode=2002OExpr..10..405R|pmid=19436374}}</ref>
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