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The '''Strehl ratio''' is a measure of the quality of optical image formation, originally proposed<ref>Strehl, K. 1895, ''Aplanatische und fehlerhafte Abbildung im Fernrohr'', '''Zeitschrift für Instrumentenkunde''' 15 (Oct.), 362-370.</ref><ref>Strehl, K. 1902, ''Über Luftschlieren und Zonenfehler'', '''Zeitschrift für Instrumentenkunde''', 22 (July), 213-217.</ref> by [[Karl Strehl]] (1864-1940) after whom the term is named. Used variously in situations where optical resolution is compromised due to lens [[Optical aberration|aberration]]s or due to imaging through the [[Astronomical seeing|turbulent atmosphere]], the Strehl ratio has a value between 0 and 1, with a "perfect" (unaberrated) optical system attaining the value of unity.

==Mathematical definition==
The Strehl ratio <math>S</math> is frequently defined<ref name="Saeak1">{{Citation| last = Sacek| first = Vladimir| title = 6.5. Strehl ratio| work = Notes on amateur telescope optics| date = July 14, 2006| url = http://www.telescope-optics.net/Strehl.htm| accessdate = March 2, 2011}}</ref> as the ratio of the peak image [[Irradiance|intensity]] from a [[point source]] compared to the maximum attainable intensity using an ideal optical system limited only by [[diffraction]] over the system's [[aperture]]. It is also often expressed in terms not of the peak intensity but the intensity at the image center (intersection of the optical axis with the focal plane) due to an on-axis source; in most important cases these definitions result in a very similar figure (or identical figure, when the point of peak intensity must be exactly at the center due to symmetry). Using the latter definition, the Strehl ratio <math>S</math> can be computed in terms of <math>\delta(x,y)</math>, the offset of the [[wavefront]] due to an on-axis point source, compared to that produced by an ideal focusing system over the [[aperture]] A(x,y). Using [[Fraunhofer diffraction]] theory, one computes the wave amplitude using the [[Fourier transform]] of the aberrated pupil function evaluated at 0,0 (center of the image plane) where the phase factors of the [[Fraunhofer diffraction (mathematics)#Cartesian co-ordinates|Fourier transform formula]] are reduced to unity. Since the Strehl ratio refers to intensity, it is found from the squared [[Absolute value#Complex numbers|magnitude]] of that amplitude:
:<math>S = |\langle e^{i\phi} \rangle|^2
= |\langle e^{i2\pi\delta/\lambda} \rangle|^2</math>
where ''i'' is the [[imaginary unit]], <math>\phi =2\pi\delta/\lambda</math> is the [[phase (waves)|phase]] error over the aperture at wavelength λ, and the average of the complex quantity inside the brackets is taken over the aperture A(x,y).

The Strehl ratio can be estimated using only the [[statistic]]s of the phase deviation
<math>\phi</math>, according to the formula of Mahajan<ref>{{Citation | last = Mahajan | first = Virendra | title = Strehl ratio for primary aberrations in terms of their aberration variance | journal = J. Opt. Soc. Am. | volume = 73 | issue = 6 | pages = 860–861 | year = 1983 | url = http://www.opticsinfobase.org/josa/abstract.cfm?uri=josa-73-6-860 | doi = 10.1364/JOSA.73.000860 }}</ref><ref>http://www.wolframalpha.com/entities/calculators/Strehl_ratio_formula/av/uo/vo/ Strehl ratio formula</ref>
:<math>S \approx {e^{-\sigma^2}}</math>
where sigma (σ) is the [[root mean square deviation]] over the aperture of the wavefront phase:
<math>\sigma^2 = \langle (\phi - \bar\phi)^2 \rangle</math>.

==The Airy disk==
{{main|Airy disk}}
{{multiple image
| image1 = Airy-pattern.svg
| alt1 = Airy pattern
| caption1 = Computer-generated image of the [[Airy disk]]
| image2 = Airy Pattern.svg
| alt2 = Airy function
| caption2 = Graph of the Airy intensity function vs. normalized radius
}}
Due to [[diffraction]], even a focusing system which is perfect according to [[geometrical optics]] will have a limited spatial [[Image resolution#Spatial resolution|resolution]]. In the usual case of a uniform circular aperture, the [[point spread function]] (PSF) which describes the image formed from an object with no spatial extent (a "point source"), is given by the [[Airy disk]] as illustrated here. For a circular aperture, the peak [[Irradiance|intensity]] found at the center of the Airy disk defines the point source image intensity required for a Strehl ratio of unity. An imperfect optical system using the same physical aperture will generally produce a broader PSF in which the peak intensity is reduced according to the factor given by the Strehl ratio. An optical system with only minor imperfections in this sense may be referred to as "diffraction limited" as its PSF closely resembles the Airy disk; a Strehl ratio of greater than .8 is frequently cited as a criterion for the use of that designation.

Note that for a given aperture the size of the Airy disk grows linearly with the wavelength <math>\lambda</math>, and consequently the peak intensity falls according to <math>\lambda^{-2}</math> so that the reference point for unity Strehl ratio is changed. Typically, as wavelength is increased, an imperfect optical system will have a broader PSF with a decreased peak intensity. However the peak intensity of the reference Airy disk would have decreased even more at that longer wavelength, resulting in a ''better'' Strehl ratio at longer wavelengths (typically) even though the actual image resolution is poorer.

==Usage==
The ratio is commonly used to assess the quality of [[astronomical seeing]] in the presence of [[atmospheric turbulence]] and assess the performance of any [[adaptive optics|adaptive optical]] correction system. It is also used for the selection of short exposure images in the [[lucky imaging]]
method.

In industry, the Strehl ratio has become a popular way to summarize the performance of an optical design because it gives the performance of a real system, of finite cost and complexity, relative to a theoretically perfect system, which would be infinitely expensive and complex to build and would still have a finite point spread function. It provides a simple method to decide whether a system with a Strehl ratio of, for example, 0.95 is good enough, or whether twice as much should be spent to try to get a Strehl ratio of perhaps 0.97 or 0.98.

==Limitations==
Characterizing the form of the point-spread function by a single number, as the Strehl Ratio does, will be meaningful and sensible only if the point-spread function is little distorted from its ideal (aberration-free) form, which will be true for a well-corrected system that operates close to the diffraction limit. That includes most telescopes and [[microscope]]s, but excludes most photographic systems, for example. The Strehl ratio has been linked via the work of Marechal to an aberration tolerancing theory which is very useful to designers of well-corrected optical systems, allowing a meaningful link between the aberrations of [[geometrical optics]] and the diffraction theory of physical optics. A significant shortcoming of the Strehl ratio as a method of image assessment is that, although it is relatively easy to calculate for an optical design prescription on paper, it is normally difficult to measure for a real optical system, not least because the theoretical maximum peak intensity is not readily available.

==See also==
*[[Circle of confusion]]
*[[Fraunhofer diffraction]]
*[[Fraunhofer diffraction (mathematics)]]
*[[Huygens–Fresnel principle]]
*[[Optical transfer function]]

==References==
<references/>

==External links==
*[http://www.rfroyce.com/standards.htm Discussion page] R.F. Royce' explanation of Strehl ratio in lay terms
*[http://www2.keck.hawaii.edu/optics/aochar/Strehl_meter2.htm Strehl meter] W.M. Keck Observatory Strehl calculator page
*[http://scienceworld.wolfram.com/physics/StrehlRatio.html Definition page] Eric Weisstein's World of Physics
*[http://www.telescope-optics.net/Strehl.htm Strehl ratio] Telescope Optics Net practical explanation of Strehl ratio for amateur telescope makers

[[Category:Astronomical imaging]]
[[Category:Optics]]

Root group - Revision history

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