Editing Optical aberration (section)

==Theory of monochromatic aberration==
{{See also|Lens (optics)}}
In a perfect optical system in the [[optics#Classical optics|classical theory of optics]],<ref>Thiesen, M. (1890) ''Berlin. Akad. Sitzber.''; and (1892) xxxv. 799; ''Berlin. Phys. Ges. Verh.''; Bruns, H. (1895) ''Leipzig. Math.  Phys. Ber.'', xxi. 325, by means of Sir W. R. Hamilton's ''characteristic function''. Reference may also be made to the treatise of Czapski-Eppenstein, pp.&nbsp;155–161.</ref><ref name="Hamilton">{{Cite journal |last=Hamilton |first=W. R. |date=1828 |title=Theory of Systems of Rays |url=https://www.jstor.org/stable/30078906 |journal=The Transactions of the Royal Irish Academy |volume=15 |pages=69–174 |jstor=30078906 |issn=0790-8113}}</ref> rays of light proceeding from any ''object point'' unite in an ''image point''; and therefore the ''object space'' is reproduced in an ''image space.'' The introduction of simple auxiliary terms, due to [[Carl Friedrich Gauss|Gauss]],<ref>{{Cite book |last=Gauss |first=Carl Friedrich |title=Dioptrische Untersuchungen |publisher=Dieterich |year=1841 |location=Göttingen |language=de}}</ref><ref name=EB1911/> named the [[focal length]]s and [[focal plane]]s, permits the determination of the image of any object for any system. The Gaussian theory, however, is only true so long as the angles made by all rays with the optical axis (the symmetrical axis of the system) are infinitely small, i.e., with infinitesimal objects, images and lenses; in practice these conditions may not be realized, and the images projected by uncorrected systems are, in general, ill-defined and often blurred if the aperture or field of view exceeds certain limits.<ref name=EB1911/>

The investigations of [[James Clerk Maxwell]]<ref>Maxwell, James Clerk (1856) ''Phil.Mag.,'' and (1858) ''Quart. Journ. Math.''.</ref> and [[Ernst Abbe]]<ref group="note">The investigations of Ernst Abbe on geometrical optics, originally published only in his university lectures, were first compiled by S. Czapski in 1893. See full reference below.</ref> showed that the properties of these reproductions, i.e., the relative position and magnitude of the images, are not special properties of optical systems, but necessary consequences of the supposition (per Abbe) of the reproduction of all points of a space in image points, and are independent of the manner in which the reproduction is effected. These authors showed, however, that no optical system can justify these suppositions, since they are contradictory to the fundamental laws of reflection and refraction.  Consequently, the Gaussian theory only supplies a convenient method of approximating reality; realistic optical systems fall short of this unattainable ideal. Currently, all that can be accomplished is the projection of a single plane onto another plane; but even in this, aberrations always occurs and it may be unlikely that these will ever be entirely corrected.<ref name=EB1911/>

===Aberration of axial points (spherical aberration in the restricted sense)===
[[File:ABERR1.svg|right|frame|'''Figure 1''']]

Let {{mvar|S}} ('''Figure 1''') be any optical system, rays proceeding from an axis point {{mvar|O}} under an angle {{math|''u''{{sub|1}}}} will unite in the axis point {{math|''{{prime|O}}''{{sub|1}}}}; and those under an angle {{math|''u''{{sub|2}}}} in the axis point {{math|''{{prime|O}}''{{sub|2}}}}.  If there is refraction at a collective spherical surface, or through a thin positive lens, {{math|''{{prime|O}}''{{sub|2}}}} will lie in front of {{math|''{{prime|O}}''{{sub|1}}}} so long as the angle {{math|''u''{{sub|2}}}} is greater than {{math|''u''{{sub|1}}}} (''under correction''); and conversely with a dispersive surface or lenses (''over correction'').  The caustic, in the first case, resembles the sign '>' (greater than); in the second '<' (less than).  If the angle {{math|''u''{{sub|1}}}} is very small, {{math|''{{prime|O}}''{{sub|1}}}} is the Gaussian image; and {{math|''{{prime|O}}''{{sub|1}} ''{{prime|O}}''{{sub|2}}}} is termed the ''longitudinal aberration'', and {{math|''{{prime|O}}''{{sub|1}}''R''}} the ''lateral aberration'' of the [[Pencil (physics)|pencils]] with aperture {{math|''u''{{sub|2}}}}. If the pencil with the angle {{math|''u''{{sub|2}}}} is that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at {{math|''{{prime|O}}''{{sub|1}}}} there is a circular ''disk of confusion'' of radius {{math|''{{prime|O}}''{{sub|1}}''R''}}, and in a parallel plane at {{math|''{{prime|O}}''{{sub|2}}}} another one of radius {{math|''{{prime|O}}''{{sub|2}}''R''{{sub|2}}}}; between these two is situated the ''disk of least confusion''.<ref name=EB1911/>

The largest opening of the pencils, which take part in the reproduction of {{mvar|O}}, i.e., the angle {{mvar|u}}, is generally determined by the margin of one of the lenses or by a hole in a thin plate placed between, before, or behind the lenses of the system.  This hole is termed the ''stop'' or ''diaphragm''; Abbe used the term ''[[aperture]] stop'' for both the hole and the limiting margin of the lens.  The component {{math|''S''{{sub|1}}}} of the system, situated between the aperture stop and the object {{mvar|O}}, projects an image of the diaphragm, termed by Abbe the ''entrance pupil''; the ''exit pupil'' is the image formed by the component {{math|''S''{{sub|2}}}}, which is placed behind the aperture stop.  All rays which issue from O and pass through the aperture stop also pass through the entrance and exit pupils, since these are images of the aperture stop.  Since the maximum aperture of the pencils issuing from {{mvar|O}} is the angle u subtended by the entrance pupil at this point, the magnitude of the aberration will be determined by the position and diameter of the entrance pupil.  If the system be entirely behind the aperture stop, then this is itself the entrance pupil (''front stop''); if entirely in front, it is the exit pupil (''back stop'').<ref name=EB1911/>

If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their ''perpendicular height of incidence,'' i.e. their distance from the axis.  This distance replaces the angle {{mvar|u}} in the preceding considerations; and the aperture, i.e., the radius of the entrance pupil, is its maximum value.<ref name=EB1911/>

====Aberration of elements, i.e. smallest objects at right angles to the axis====
If rays issuing from {{mvar|O}} ('''Figure 1''') are concurrent, it does not follow that points in a portion of a plane perpendicular at {{mvar|O}} to the axis will be also concurrent, even if the part of the plane be very small.  As the diameter of the lens increases (i.e., with increasing aperture), the neighboring point {{mvar|N}} will be reproduced, but attended by aberrations comparable in magnitude to {{mvar|ON}}. These aberrations are avoided if, according to Abbe, the ''sine condition'', {{math|1= sin ''{{prime|u}}''{{sub|1}}/sin ''u''{{sub|1}} = sin ''{{prime|u}}''{{sub|2}}/sin ''u''{{sub|2}}}}, holds for all rays reproducing the point {{mvar|O}}. If the object point {{mvar|O}} is infinitely distant, {{math|''u''{{sub|1}}}} and {{math|''u''{{sub|2}}}} are to be replaced by {{math|''h''{{sub|1}}}} and {{math|''h''{{sub|2}}}}, the perpendicular heights of incidence; the ''sine condition'' then becomes {{math|1= sin ''{{prime|u}}''{{sub|1}}/''h''{{sub|1}} = sin ''{{prime|u}}''{{sub|2}}/''h''{{sub|2}}}}. A system fulfilling this condition and free from spherical aberration is called ''aplanatic'' (Greek {{Transliteration|grc|a-}}, privative; {{Transliteration|grc|plann}}, a wandering).  This word was first used by [[Robert Blair (astronomer)|Robert Blair]] to characterize a superior achromatism, and, subsequently, by many writers to denote freedom from spherical aberration as well.<ref name=EB1911/>

Since the aberration increases with the distance of the ray from the center of the lens, the aberration increases as the lens diameter increases (or, correspondingly, with the diameter of the aperture), and hence can be minimized by reducing the aperture, at the cost of also reducing the amount of light reaching the image plane.

===Aberration of lateral object points (points beyond the axis) with narrow pencils — astigmatism===
{{Main|Astigmatism (optical systems)}}
{{For |Astigmatism of the eye|Astigmatism}}

[[File:ABERR2.svg|right|frame|'''Figure 2''']]

A point {{mvar|O}} ('''Figure 2''') at a finite distance from the axis (or with an infinitely distant object, a point which subtends a finite angle at the system) is, in general, even then not sharply reproduced if the pencil of rays issuing from it and traversing the system is made infinitely narrow by reducing the aperture stop; such a pencil consists of the rays which can pass from the object point through the now infinitely small entrance pupil.  It is seen (ignoring exceptional cases) that the pencil does not meet the refracting or reflecting surface at right angles; therefore it is astigmatic (Greek {{Transliteration|grc|a-}}, privative; {{Transliteration|grc|stigmia}}, a point).  Naming the central ray passing through the entrance pupil the ''axis of the pencil'' or ''principal ray'', it can be said: the rays of the pencil intersect, not in one point, but in two focal lines, which can be assumed to be at right angles to the principal ray; of these, one lies in the plane containing the principal ray and the axis of the system, i.e. in the ''first principal section'' or ''meridional section'', and the other at right angles to it, i.e. in the second principal section or sagittal section.  We receive, therefore, in no single intercepting plane behind the system, as, for example, a focusing screen, an image of the object point; on the other hand, in each of two planes lines {{mvar|{{prime|O}}}} and {{mvar|{{pprime|O}}}} are separately formed (in neighboring planes ellipses are formed), and in a plane between {{mvar|{{prime|O}}}} and {{mvar|{{pprime|O}}}} a circle of least confusion.  The interval {{mvar|{{prime|O}}{{pprime|O}}}}, termed the astigmatic difference, increases, in general, with the angle {{mvar|W}} made by the principal ray {{mvar|OP}} with the axis of the system, i.e. with the field of view.  Two ''astigmatic image surfaces'' correspond to one object plane; and these are in contact at the axis point; on the one lie the focal lines of the first kind, on the other those of the second.  Systems in which the two astigmatic surfaces coincide are termed anastigmatic or stigmatic.<ref name=EB1911/>

[[Isaac Newton|Sir Isaac Newton]] was probably the discoverer of astigmation; the position of the astigmatic image lines was determined by Thomas Young;<ref>Young, Thomas (1807), ''A Course of Lectures on Natural Philosophy.''</ref> and the theory was developed by [[Allvar Gullstrand]].<ref>Gullstrand, Allvar (1890) ''Skand. Arch. f. Physiol.''; and (1901) ''Arch. f.  Ophth.'', 53, pp. 2, 185.</ref><ref name="gullstrand1900"/><ref name=EB1911/>  A bibliography by P. Culmann is given in Moritz von Rohr's ''Die Bilderzeugung in optischen Instrumenten''.<ref name=vonRohr>{{cite book |first=Moritz |last=von Rohr |author-link=Moritz von Rohr |title=Die bilderzeugung in optischen Instrumenten vom Standpunkte der geometrischen Optik |location=Berlin |date=1904}}</ref><ref name=EB1911/>

===Aberration of lateral object points with broad pencils — coma===
By opening the stop wider, similar deviations arise for lateral points as have been already discussed for axial points; but in this case they are much more complicated.  The course of the rays in the meridional section is no longer symmetrical to the principal ray of the pencil; and on an intercepting plane there appears, instead of a luminous point, a patch of light, not symmetrical about a point, and often exhibiting a resemblance to a comet having its tail directed towards or away from the axis.  From this appearance it takes its name.  The unsymmetrical form of the meridional pencil{{snd}} formerly the only one considered{{snd}} is [[coma (optics)|coma]] in the narrower sense only; other errors of coma have been treated by [[Arthur König]] and Moritz von Rohr,<ref name=vonRohr/> and later by Allvar Gullstrand.<ref name="gullstrand1900">{{cite journal |first=Allvar |last=Gullstrand |author-link=Allvar Gullstrand |title=Allgemeine Theorie der monochromat. Aberrationen, etc. |location=Upsala |date=1900 |journal=Annalen der Physik |volume=1905 |issue=18 |page=941 | doi = 10.1002/andp.19053231504 |bibcode=1905AnP...323..941G |url=https://zenodo.org/record/2434371 }}</ref><ref name=EB1911/>

===Curvature of the field of the image===
{{Main|Petzval field curvature}}
If the above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture—there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e.g. in photography.  In most cases the surface is concave towards the system.<ref name=EB1911/>

===Distortion of the image===
[[File:Barrel distortion.svg|thumb|right|'''Figure 3a''': Barrel distortion]]
[[File:Pincushion distortion.svg|thumb|right|'''Figure 3b''': Pincushion distortion]]
{{main|Distortion (optics)}}
Even if the image is sharp, it may be distorted compared to ideal [[pinhole projection]].  In pinhole projection, the magnification of an object is inversely proportional to its distance to the camera along the optical axis so that a camera pointing directly at a flat surface reproduces that flat surface. Distortion can be thought of as stretching the image non-uniformly, or, equivalently, as a variation in magnification across the field. While "distortion" can include arbitrary deformation of an image, the most pronounced modes of distortion produced by conventional imaging optics is "barrel distortion", in which the center of the image is magnified more than the perimeter ('''Figure 3a'''). The reverse, in which the perimeter is magnified more than the center, is known as "pincushion distortion" ('''Figure 3b'''). This effect is called lens distortion or [[image distortion]], and there are algorithms to correct it.

Systems free of distortion are called ''orthoscopic'' ({{Transliteration|grc|orthos}}, right; {{Transliteration|grc|skopein}}, to look) or ''rectilinear'' (straight lines).

[[File:ABERR3rev.svg|left|frame|'''Figure 4''']]
This aberration is quite distinct from that of the sharpness of reproduction; in unsharp, reproduction, the question of distortion arises if only parts of the object can be recognized in the figure. If, in an unsharp image, a patch of light corresponds to an object point, the ''center of gravity'' of the patch may be regarded as the image point, this being the point where the plane receiving the image, e.g., a focusing screen, intersects the ray passing through the middle of the stop. This assumption is justified if a poor image on the focusing screen remains stationary when the aperture is diminished; in practice, this generally occurs. This ray, named by Abbe a ''principal ray'' (not to be confused with the ''principal rays'' of the Gaussian theory), passes through the center of the entrance pupil before the first refraction, and the center of the exit pupil after the last refraction.  From this it follows that correctness of drawing depends solely upon the principal rays; and is independent of the sharpness or curvature of the image field. Referring to '''Figure 4''', we have {{math|1= ''{{prime|O}}{{prime|Q}}''/''OQ'' = (''{{prime|a}}'' tan ''{{prime|w}}'')/(''a'' tan ''w'') = 1/''N''}}, where {{mvar|N}} is the ''scale'' or magnification of the image. For {{mvar|N}} to be constant for all values of {{mvar|w}}, {{math|(''{{prime|a}}'' tan ''{{prime|w}}'')/(''a'' tan ''w'')}} must also be constant. If the ratio {{math|''{{prime|a}}''/''a''}} be sufficiently constant, as is often the case, the above relation reduces to the ''condition of [[George Biddell Airy|Airy]],'' i.e. {{math|tan ''{{prime|w}}''/tan ''w''}} is a constant.<ref>{{cite journal |last= Airy |first= George Biddell |date= 1830 |title= On the Spherical Aberration of the Eye-Pieces of Telescopes |journal= Transactions of the Cambridge Philosophical Society |volume= 3 |pp= 1–58 |quote= It is evident that an object will be seen without distortion if its image, exactly similar to the object, be formed on a plane; and then the trigonometrical tangent of the angle, made with the axis of the lens by the axis of the pencil after refraction, will bear to the tangent of the angle before refraction a constant ratio: if the ratio be not constant, its difference from a constant ratio will indicate the degree of distortion. |url= https://archive.org/details/transactionsofca03camb/page/4/ |access-date= 2025-03-25}}</ref> This simple relation is fulfilled in all systems which are symmetrical with respect to their diaphragm (briefly named ''symmetrical or holosymmetrical objectives''), or which consist of two like, but different-sized, components, placed from the diaphragm in the ratio of their size, and presenting the same curvature to it (hemisymmetrical objectives); in these systems {{math|1=tan ''{{prime|w}}'' / tan ''w'' = 1}}.<ref name=EB1911/>

The constancy of {{math|''{{prime|a}}''/''a''}} necessary for this relation to hold was pointed out by R. H. Bow (Brit.  Journ.  Photog., 1861), and Thomas Sutton (Photographic Notes, 1862); it has been treated by O. Lummer and by M. von Rohr (Zeit. f.  Instrumentenk., 1897, 17, and 1898, 18, p.&nbsp;4). It requires the middle of the aperture stop to be reproduced in the centers of the entrance and exit pupils without spherical aberration. M. von Rohr showed that for systems fulfilling neither the Airy nor the Bow-Sutton condition, the ratio {{math|(''{{prime|a}}'' cos ''{{prime|w}}'')/(''a'' tan ''w'')}} will be constant for one distance of the object.  This combined condition is exactly fulfilled by holosymmetrical objectives reproducing with the scale 1, and by hemisymmetrical, if the scale of reproduction be equal to the ratio of the sizes of the two components.<ref name=EB1911/>

===Zernike model of aberrations===

[[File:ZernikeAiryImage.jpg|360px|thumb|Image plane of a flat-top beam under the effect of the first 21 Zernike polynomials]]
[[File:ZernikeLogAiryImage.jpg|360px|thumb|Effect of Zernike aberrations in Log scale. The intensity minima are visible.]]

Circular wavefront profiles associated with aberrations may be mathematically modeled using [[Zernike polynomial]]s.  Developed by [[Frits Zernike]] in the 1930s, Zernike's polynomials are [[orthogonal]] over a circle of unit radius. A complex, aberrated wavefront profile may be [[curve-fitted]] with Zernike polynomials to yield a set of fitting [[coefficient]]s that individually represent different types of aberrations.  These Zernike coefficients are [[linearly independent]], thus individual aberration contributions to an overall wavefront may be isolated and quantified separately.

There are [[even and odd functions|even and odd]] Zernike polynomials. The even Zernike polynomials are defined as

<math display="block">Z^{m}_n(\rho,\phi) = R^m_n(\rho)\,\cos(m\,\phi)</math>

and the odd Zernike polynomials as

<math display="block">Z^{-m}_n(\rho,\phi) = R^m_n(\rho)\,\sin(m\,\phi)</math>

where {{mvar|m}} and {{mvar|n}} are nonnegative [[integer]]s with {{math|''n'' ≥ ''m''}}, {{mvar|ϕ}} is the [[azimuth]]al [[angle]] in [[radian]]s, and {{mvar|ρ}} is the normalized radial distance.  The radial polynomials <math>R^m_n</math> have no azimuthal dependence, and are defined as

<math display="block">R^m_n(\rho) = 
\begin{cases}
\sum_{k=0}^{(n-m)/2} \!\!\! \frac{(-1)^k\,(n-k)!}{k!\,\left({n+m \over 2}-k\right)!\,\left({n-m \over 2}-k\right)!} \;\rho^{n-2\,k}, & \text{if } n-m \text{ is even} \\
0, & \text{if } n-m \text{ is odd.}
\end{cases}</math>

The first few Zernike polynomials, multiplied by their respective fitting coefficients, are:<ref>{{Cite book|last=Schroeder, D. J.|url=https://www.worldcat.org/oclc/162132153|title=Astronomical optics|date=2000|publisher=Academic Press|isbn=978-0-08-049951-2|edition=2nd|location=San Diego|oclc=162132153}}</ref>
{|
|-
|<math>a_0 \times 1 </math>|| "Piston", equal to the [[mean value]] of the wavefront
|-
|<math>a_1\times \rho \cos(\phi)</math> || "X-Tilt", the deviation of the overall beam in the [[Sagittal ray#Optical systems|sagittal]] direction
|-
|<math>a_2\times \rho \sin(\phi)</math> || "Y-Tilt", the deviation of the overall beam in the [[Tangential ray#Optical systems|tangential]] direction
|-
|<math>a_3\times (2\rho^2-1)</math> || "Defocus", a [[Parabola|parabolic]] wavefront resulting from being out of focus
|-
|<math>a_4\times \rho^2 \cos(2\phi)</math> || "0° Astigmatism", a [[cylindrical]] shape along the X or Y axis
|-
|<math>a_5\times \rho^2 \sin(2\phi)</math> || "45° Astigmatism", a cylindrical shape oriented at ±45° from the X axis
|-
|<math>a_6\times (3\rho^2-2)\rho \cos(\phi)</math> || "X-Coma", comatic image flaring in the horizontal direction
|-
|<math>a_7\times (3\rho^2-2)\rho \sin(\phi)</math> || "Y-Coma", comatic image flaring in the vertical direction
|-
|<math>a_8\times (6\rho^4-6\rho^2+1)</math> || "Third order spherical aberration"
|}

where {{mvar|ρ}} is the normalized pupil radius with {{math|0 ≤ ''ρ'' ≤ 1}}, {{mvar|ϕ}} is the azimuthal angle around the pupil with {{math|0 ≤ ''ϕ'' ≤ 2''π''}}, and the fitting coefficients {{math|''a''{{sub|0}}, ..., ''a''{{sub|8}}}}  are the wavefront errors in wavelengths.

As in [[Fourier analysis|Fourier]] synthesis using [[sine]]s and [[cosine]]s, a wavefront may be perfectly represented by a sufficiently large number of higher-order Zernike polynomials.  However, wavefronts with very steep [[gradients]] or very high [[spatial frequency]] structure, such as produced by [[Wave propagation|propagation]] through [[atmospheric turbulence]] or [[turbulence|aerodynamic flowfields]], are not well modeled by Zernike polynomials, which tend to [[low-pass filter]] fine [[Three-dimensional space|spatial]] definition in the wavefront. In this case, other fitting methods such as [[fractals]] or [[singular value decomposition]] may yield improved fitting results.

The [[Zernike polynomials|circle polynomials]] were introduced by [[Frits Zernike]] to evaluate the point image of an aberrated optical system taking into account the effects of [[diffraction]]. The perfect point image in the presence of diffraction had already been described by [[George Biddell Airy|Airy]], as early as 1835. It took almost hundred years to arrive at a comprehensive theory and modeling of the point image of aberrated systems (Zernike and Nijboer). The  analysis by Nijboer and Zernike describes the intensity distribution close to the optimum focal plane. An extended theory that allows the calculation of the point image amplitude and  intensity over a much larger volume in the focal region was recently developed ([http://www.nijboerzernike.nl Extended Nijboer-Zernike theory]). This Extended Nijboer-Zernike theory of point image or 'point-spread function' formation has found applications in general research on image formation, especially for systems with a high [[numerical aperture]], and in characterizing optical systems with respect to their aberrations.<ref>{{cite book |title=[[Principles of Optics|Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light]]|first1=Max |last1=Born |first2=Emil |last2=Wolf |isbn=978-0521642224|date=1999-10-13 |publisher=Cambridge University Press }}</ref>