Editing Optical aberration (section)

===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.