4681:
aberration of the axis point, which is still present to disturb the image, after par-axial rays of different colors are united by an appropriate combination of glasses. If a collective system be corrected for the axis point for a definite wavelength, then, on account of the greater dispersion in the negative components — the flint glasses, — overcorrection will arise for the shorter wavelengths (this being the error of the negative components), and under-correction for the longer wavelengths (the error of crown glass lenses preponderating in the red). This error was treated by Jean le Rond d'Alembert, and, in special detail, by C. F. Gauss. It increases rapidly with the aperture, and is more important with medium apertures than the secondary spectrum of par-axial rays; consequently, spherical aberration must be eliminated for two colors, and if this be impossible, then it must be eliminated for those particular wavelengths which are most effectual for the instrument in question (a graphical representation of this error is given in M. von Rohr,
2364:
image ray may be defined by the points (ξ', η'), and (x', y'), in the planes I' and II'. The origins of these four plane coordinate systems may be collinear with the axis of the optical system; and the corresponding axes may be parallel. Each of the four coordinates ξ', η', x', y' are functions of ξ, η, x, y; and if it be assumed that the field of view and the aperture be infinitely small, then ξ, η, x, y are of the same order of infinitesimals; consequently by expanding ξ', η', x', y' in ascending powers of ξ, η, x, y, series are obtained in which it is only necessary to consider the lowest powers. It is readily seen that if the optical system be symmetrical, the origins of the coordinate systems collinear with the optical axis and the corresponding axes parallel, then by changing the signs of ξ, η, x, y, the values ξ', η', x', y' must likewise change their sign, but retain their arithmetical values; this means that the series are restricted to odd powers of the unmarked variables.
2440:
of view as seen from the object and is expressed as an angular measurement. Higher order aberrations in telescope design can be mostly neglected. For microscopes it cannot be neglected. For a single lens of very small thickness and given power, the aberration depends upon the ratio of the radii r:r', and is a minimum (but never zero) for a certain value of this ratio; it varies inversely with the refractive index (the power of the lens remaining constant). The total aberration of two or more very thin lenses in contact, being the sum of the individual aberrations, can be zero. This is also possible if the lenses have the same algebraic sign. Of thin positive lenses with n=1.5, four are necessary to correct spherical aberration of the third order. These systems, however, are not of great practical importance. In most cases, two thin lenses are combined, one of which has just so strong a positive aberration (
2595:
constants of reproduction. These constants are determined by the data of the system (radii, thicknesses, distances, indices, etc., of the lenses); therefore their dependence on the refractive index, and consequently on the color, are calculable. The refractive indices for different wavelengths must be known for each kind of glass made use of. In this manner the conditions are maintained that any one constant of reproduction is equal for two different colors, i.e. this constant is achromatized. For example, it is possible, with one thick lens in air, to achromatize the position of a focal plane of the magnitude of the focal length. If all three constants of reproduction be achromatized, then the
Gaussian image for all distances of objects is the same for the two colors, and the system is said to be in
3940:). In practice, however, it is often more useful to avoid the second condition by making the lenses have contact, i.e. equal radii. According to P. Rudolph (Eder's Jahrb. f. Photog., 1891, 5, p. 225; 1893, 7, p. 221), cemented objectives of thin lenses permit the elimination of spherical aberration on the axis, if, as above, the collective lens has a smaller refractive index; on the other hand, they permit the elimination of astigmatism and curvature of the field, if the collective lens has a greater refractive index (this follows from the Petzval equation; see L. Seidel, Astr. Nachr., 1856, p. 289). Should the cemented system be positive, then the more powerful lens must be positive; and, according to (4), to the greater power belongs the weaker dispersive power (greater
750:. 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
2473:
671:, 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 O' and O" are separately formed (in neighboring planes ellipses are formed), and in a plane between O' and O" a circle of least confusion. The interval O'O", termed the astigmatic difference, increases, in general, with the angle W made by the principal ray OP with the axis of the system, i.e. with the field of view. Two
303:
507:
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.
2511:
dealt with by means of the approximation theory; in most cases, however, the analytical difficulties were too great for older calculation methods but may be ameliorated by application of modern computer systems. Solutions, however, have been obtained in special cases. At the present time constructors almost always employ the inverse method: they compose a system from certain, often quite personal experiences, and test, by the trigonometrical calculation of the paths of several rays, whether the system gives the desired reproduction (examples are given in A. Gleichen,
428:
4523:). For ordinary photography, however, there is this disadvantage: the image on the focusing-screen and the correct adjustment of the photographic sensitive plate are not in register; in astronomical photography this difference is constant, but in other kinds it depends on the distance of the objects. On this account the lines D and G' are united for ordinary photographic objectives; the optical as well as the actinic image is chromatically inferior, but both lie in the same place; and consequently the best correction lies in F (this is known as the
816:
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 a' cos 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.
2403:; in 1840, J. Petzval constructed his portrait objective, from similar calculations which have never been published. The theory was elaborated by S. Finterswalder, who also published a posthumous paper of Seidel containing a short view of his work; a simpler form was given by A. Kerber. A. Konig and M. von Rohr have represented Kerber's method, and have deduced the Seidel formulae from geometrical considerations based on the Abbe method, and have interpreted the analytical results geometrically.
4669:
the difficulty by constructing fluid lenses between glass walls. Fraunhofer prepared glasses which reduced the secondary spectrum; but permanent success was only assured on the introduction of the Jena glasses by E. Abbe and O. Schott. In using glasses not having proportional dispersion, the deviation of a third colour can be eliminated by two lenses, if an interval be allowed between them; or by three lenses in contact, which may not all consist of the old glasses. In uniting three colors an
495:, 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.
77:
120:
2327:, 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 (
739:
2388:, then Dξ' and Dη' are the aberrations belonging to ξ, η and x, y, and are functions of these magnitudes which, when expanded in series, contain only odd powers, for the same reasons as given above. On account of the aberrations of all rays which pass through O, a patch of light, depending in size on the lowest powers of ξ, η, x, y which the aberrations contain, will be formed in the plane I'. These degrees, named by
833:
731:
231:
5045:
2515:, Leipzig and Berlin, 1902). The radii, thicknesses and distances are continually altered until the errors of the image become sufficiently small. By this method only certain errors of reproduction are investigated, especially individual members, or all, of those named above. The analytical approximation theory is often employed provisionally, since its accuracy does not generally suffice.
654:
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 (Gr. a-, privative, stigmia, a point). Naming the central ray passing through the entrance pupil the
825:
36:
4703:), so that it is eliminated in the image of the whole microscope. The best telescope objectives, and photographic objectives intended for three-color work, are also apochromatic, even if they do not possess quite the same quality of correction as microscope objectives do. The chromatic differences of other errors of reproduction seldom have practical importance.
263:. In an imaging system, it occurs when light from one point of an object does not converge into (or does not diverge from) a single point after transmission through the system. Aberrations occur because the simple paraxial theory is not a completely accurate model of the effect of an optical system on light, rather than due to flaws in the optical elements.
2356:
773:
167:
4519:). In a similar manner, for systems used in photography, the vertex of the color curve must be placed in the position of the maximum sensibility of the plates; this is generally supposed to be at G'; and to accomplish this the F and violet mercury lines are united. This artifice is specially adopted in objectives for astronomical photography (
699:
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—formerly the only one considered—is
646:
516:
1425:
2352:; and with each order of infinite smallness, i.e. with each degree of approximation to reality (to finite objects and apertures), a certain number of aberrations is associated. This connection is only supplied by theories which treat aberrations generally and analytically by means of indefinite series.
4668:
must be equal for the two kinds of glass employed. This follows by considering equation (4) for the two pairs of colors ac and bc. Until recently no glasses were known with a proportional degree of absorption; but R. Blair (Trans. Edin. Soc., 1791, 3, p. 3), P. Barlow, and F. S. Archer overcame
2444:
vide supra) as the other a negative; the first must be a positive lens and the second a negative lens; the powers, however: may differ, so that the desired effect of the lens is maintained. It is generally an advantage to secure a great refractive effect by several weaker than by one high-power lens.
2410:
of the system and its differential coefficients, instead of by the radii, &c., of the lenses; these formulae are not immediately applicable, but give, however, the relation between the number of aberrations and the order. Sir
William Rowan Hamilton (British Assoc. Report, 1833, p. 360) thus
2395:
are consequently only odd powers; the condition for the formation of an image of the mth order is that in the series for Dξ' and Dη' the coefficients of the powers of the 3rd, 5th...(m-2)th degrees must vanish. The images of the Gauss theory being of the third order, the next problem is to obtain an
523:
Let S (fig. 1) be any optical system, rays proceeding from an axis point O under an angle u1 will unite in the axis point O'1; and those under an angle u2 in the axis point O'2. If there is refraction at a collective spherical surface, or through a thin positive lens, O'2 will lie in front of O'1 so
270:
need to correct optical systems to compensate for aberration. Aberrations are particularly impactful in telescopes, where they can significantly degrade the quality of observed celestial objects. Understanding and correcting these optical imperfections are crucial for astronomers to achieve clear and
4035:
vanish, a certain value can be assigned to it which will produce, by the addition of the two lenses, any desired chromatic deviation, e.g. sufficient to eliminate one present in other parts of the system. If the lenses I. and II. be cemented and have the same refractive index for one color, then its
2601:
In practice it is more advantageous (after Abbe) to determine the chromatic aberration (for instance, that of the distance of intersection) for a fixed position of the object, and express it by a sum in which each component conlins the amount due to each refracting surface. In a plane containing the
794:
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
653:
A point O (fig. 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
602:
If rays issuing from O (fig. 1) are concurrent, it does not follow that points in a portion of a plane perpendicular at 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 N
578:
is the image formed by the component S2, 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 O is
506:
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
3927:
positive) it follows, by means of equation (4), that a collective lens I. of crown glass and a dispersive lens II. of flint glass must be chosen; the latter, although the weaker, corrects the other chromatically by its greater dispersive power. For an achromatic dispersive lens the converse must be
2583:
of the distances of intersection, of magnifications, and of monochromatic aberrations. If mixed light be employed (e.g. white light) all these images are formed and they cause a confusion, named chromatic aberration; for instance, instead of a white margin on a dark background, there is perceived a
2510:
Practical methods solve this problem with an accuracy which mostly suffices for the special purpose of each species of instrument. The problem of finding a system which reproduces a given object upon a given plane with given magnification (insofar as aberrations must be taken into account) could be
2439:
Aberration of the third order of axis points is dealt with in all text-books on optics. It is very important in telescope design. In telescopes aperture is usually taken as the linear diameter of the objective. It is not the same as microscope aperture which is based on the entrance pupil or field
2367:
The nature of the reproduction consists in the rays proceeding from a point O being united in another point O'; in general, this will not be the case, for ξ', η' vary if ξ, η be constant, but x, y variable. It may be assumed that the planes I' and II' are drawn where the images of the planes I and
721:
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
258:
to be spread out over some region of space rather than focused to a point. Aberrations cause the image formed by a lens to be blurred or distorted, with the nature of the distortion depending on the type of aberration. Aberration can be defined as a departure of the performance of an optical system
4474:
The focal lengths are made equal for the lines C and F. In the neighborhood of 550 nm the tangent to the curve is parallel to the axis of wavelengths; and the focal length varies least over a fairly large range of color, therefore in this neighborhood the color union is at its best. Moreover,
4314:
varies within the spectrum. This fact was first ascertained by J. Fraunhofer, who defined the colors by means of the dark lines in the solar spectrum; and showed that the ratio of the dispersion of two glasses varied about 20% from the red to the violet (the variation for glass and water is about
4265:
If a constant of reproduction, for instance the focal length, be made equal for two colors, then it is not the same for other colors, if two different glasses are employed. For example, the condition for achromatism (4) for two thin lenses in contact is fulfilled in only one part of the spectrum,
2518:
In order to render spherical aberration and the deviation from the sine condition small throughout the whole aperture, there is given to a ray with a finite angle of aperture u* (width infinitely distant objects: with a finite height of incidence h*) the same distance of intersection, and the same
2434:
The aberrations of the third order are: (1) aberration of the axis point; (2) aberration of points whose distance from the axis is very small, less than of the third order — the deviation from the sine condition and coma here fall together in one class; (3) astigmatism; (4) curvature of the field;
785:
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
623:
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
3867:
Newton failed to perceive the existence of media of different dispersive powers required by achromatism; consequently he constructed large reflectors instead of refractors. James
Gregory and Leonhard Euler arrived at the correct view from a false conception of the achromatism of the eye; this was
2594:
If, in the first place, monochromatic aberrations be neglected — in other words, the
Gaussian theory be accepted — then every reproduction is determined by the positions of the focal planes, and the magnitude of the focal lengths, or if the focal lengths, as ordinarily happens, be equal, by three
2363:
A ray proceeding from an object point O (fig. 5) can be defined by the coordinates (ξ, η). Of this point O in an object plane I, at right angles to the axis, and two other coordinates (x, y), the point in which the ray intersects the entrance pupil, i.e. the plane II. Similarly the corresponding
815:
The constancy of 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. 4). It requires the
4688:
The condition for the reproduction of a surface element in the place of a sharply reproduced point — the constant of the sine relationship must also be fulfilled with large apertures for several colors. E. Abbe succeeded in computing microscope objectives free from error of the axis point and
2347:
in which definite aberrations are discussed separately; it is well suited to practical needs, for in the construction of an optical instrument certain errors are sought to be eliminated, the selection of which is justified by experience. In the mathematical sense, however, this selection is
4680:
The
Gaussian theory is only an approximation; monochromatic or spherical aberrations still occur, which will be different for different colors; and should they be compensated for one color, the image of another color would prove disturbing. The most important is the chromatic difference of
698:
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
3524:
2519:
sine ratio as to one neighboring the axis (u* or h* may not be much smaller than the largest aperture U or H to be used in the system). The rays with an angle of aperture smaller than u* would not have the same distance of intersection and the same sine ratio; these deviations are called
4699:. While, however, the magnification of the individual zones is the same, it is not the same for red as for blue; and there is a chromatic difference of magnification. This is produced in the same amount, but in the opposite sense, by the oculars, which Abbe used with these objectives (
2532:
Spherical aberration and changes of the sine ratios are often represented graphically as functions of the aperture, in the same way as the deviations of two astigmatic image surfaces of the image plane of the axis point are represented as functions of the angles of the field of view.
3928:
adopted. This is, at the present day, the ordinary type, e.g., of telescope objective; the values of the four radii must satisfy the equations (2) and (4). Two other conditions may also be postulated: one is always the elimination of the aberration on the axis; the second either the
4394:, and vice versa; these algebraic results follow from the fact that towards the red the dispersion of the positive crown glass preponderates, towards the violet that of the negative flint. These chromatic errors of systems, which are achromatic for two colors, are called the
780:
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
418:
are effects which shift the position of the focal point. Piston and tilt are not true optical aberrations, since when an otherwise perfect wavefront is altered by piston and tilt, it will still form a perfect, aberration-free image, only shifted to a different position.
2445:
By one, and likewise by several, and even by an infinite number of thin lenses in contact, no more than two axis points can be reproduced without aberration of the third order. Freedom from aberration for two axis points, one of which is infinitely distant, is known as
2536:
The final form of a practical system consequently rests on compromise; enlargement of the aperture results in a diminution of the available field of view, and vice versa. But the larger aperture will give the larger resolution. The following may be regarded as typical:
554:
The largest opening of the pencils, which take part in the reproduction of O, i.e., the angle 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
662:
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
579:
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 (
2396:
image of 5th order, or to make the coefficients of the powers of 3rd degree zero. This necessitates the satisfying of five equations; in other words, there are five alterations of the 3rd order, the vanishing of which produces an image of the 5th order.
675:
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.
406:
Although defocus is technically the lowest-order of the optical aberrations, it is usually not considered as a lens aberration, since it can be corrected by moving the lens (or the image plane) to bring the image plane to the optical focus of the lens.
2614:); and since this disk becomes the less harmful with an increasing image of a given object, or with increasing focal length, it follows that the deterioration of the image is proportional to the ratio of the aperture to the focal length, i.e. the
1166:
3960:), that is to say, crown glass; consequently the crown glass must have the greater refractive index for astigmatic and plane images. In all earlier kinds of glass, however, the dispersive power increased with the refractive index; that is,
2547:; necessary corrections are — for astigmatism, curvature of field and distortion; errors of the aperture only slightly regarded; examples — photographic widest angle objectives and oculars. Between these extreme examples stands the
2523:
and the constructor endeavors to reduce these to a minimum. The same holds for the errors depending upon the angle of the field of view, w: astigmatism, curvature of field and distortion are eliminated for a definite value, w*,
812:), 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 tan w' / tan w = 1.
4000:
increased; but some of the Jena glasses by E. Abbe and O. Schott were crown glasses of high refractive index, and achromatic systems from such crown glasses, with flint glasses of lower refractive index, are called the
2557:
have small fields of view and aberrations on axis are very important. Therefore, zones will be kept as small as possible and design should emphasize simplicity. Because of this these lenses are the best for analytical
1074:
2499:, 415–422 (1989)). For a single pair of planes (e.g. for a single focus setting of an objective), however, the problem can in principle be solved perfectly. Examples of such a theoretically perfect system include the
967:
799:
or magnification of the image. For N to be constant for all values of w, a' tan w'/a tan w must also be constant. If the ratio a'/a be sufficiently constant, as is often the case, the above relation reduces to the
2808:
96:
It should be brought up to date to reflect subsequent history or scholarship (including the references, if any). When you have completed the review, replace this notice with a simple note on this article's talk
2348:
arbitrary; the reproduction of a finite object with a finite aperture entails, in all probability, an infinite number of aberrations. This number is only finite if the object and aperture are assumed to be
607:
sin u'1/sin u1=sin u'2/sin u2, holds for all rays reproducing the point O. If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence; the
4036:
effect for that one color is that of a lens of one piece; by such decomposition of a lens it can be made chromatic or achromatic at will, without altering its spherical effect. If its chromatic effect (
2017:
1938:
2427:
1905, 4, No. 1), who thus discovered the aberrations of the 5th order (of which there are nine), and possibly the shortest proof of the practical (Seidel) formulae. A. Gullstrand (vide supra, and
808:
i.e. tan w'/ tan w= a constant. This simple relation (see Camb. Phil. Trans., 1830, 3, p. 1) is fulfilled in all systems which are symmetrical with respect to their diaphragm (briefly named
2094:
570:
for both the hole and the limiting margin of the lens. The component S1 of the system, situated between the aperture stop and the object O, projects an image of the diaphragm, termed by Abbe the
2626:
In a very thin lens, in air, only one constant of reproduction is to be observed, since the focal length and the distance of the focal point are equal. If the refractive index for one color be
532:). The caustic, in the first case, resembles the sign > (greater than); in the second < (less than). If the angle u1 is very small, O'1 is the Gaussian image; and O'1 O'2 is termed the
1859:
1791:
2331:). 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
1658:
1600:
590:
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
3265:
4666:
3711:
3601:
1723:
544:
with aperture u2. If the pencil with the angle u2 is that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at O'1 there is a circular
594:
i.e. their distance from the axis. This distance replaces the angle u in the preceding considerations; and the aperture, i.e., the radius of the entrance pupil, is its maximum value.
3784:
2399:
The expression for these coefficients in terms of the constants of the optical system, i.e. the radii, thicknesses, refractive indices and distances between the lenses, was solved by
4157:
2483:
The classical imaging problem is to reproduce perfectly a finite plane (the object) onto another plane (the image) through a finite aperture. It is impossible to do so perfectly for
2541:
Largest aperture; necessary corrections are — for the axis point, and sine condition; errors of the field of view are almost disregarded; example — high-power microscope objectives.
2209:
2606:
in spherical aberration. For infinitely distant objects the radius Of the chromatic disk of confusion is proportional to the linear aperture, and independent of the focal length (
4741:
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.
4591:
2255:
2463:
S1/r(n'−n) = 0, where r is the radius of a refracting surface, n and n' the refractive indices of the neighboring media, and S the sign of summation for all refracting surfaces.
1475:
4257:
3081:
3021:
2154:
3840:
must have different algebraic signs, or the system must be composed of a collective and a dispersive lens. Consequently the powers of the two must be different (in order that
2579:), it follows that a system of lenses (uncorrected) projects images of different colors in somewhat different places and sizes and with different aberrations; i.e. there are
1420:{\displaystyle R_{n}^{m}(\rho )=\!\sum _{k=0}^{(n-m)/2}\!\!\!{\frac {(-1)^{k}\,(n-k)!}{k!\,((n+m)/2-k)!\,((n-m)/2-k)!}}\;\rho ^{n-2\,k}\quad {\mbox{if }}n-m{\mbox{ is even}}}
4359:
4312:
1542:
4513:
4197:
3051:
2991:
341:. Monochromatic aberrations are caused by the geometry of the lens or mirror and occur both when light is reflected and when it is refracted. They appear even when using
4857:
4392:
326:). Real lenses do not focus light exactly to a single point, however, even when they are perfectly made. These deviations from the idealized lens performance are called
2368:
II are formed by rays near the axis by the ordinary
Gaussian rules; and by an extension of these rules, not, however, corresponding to reality, the Gauss image point O'
1158:
306:
Reflection from a spherical mirror. Incident rays (red) away from the center of the mirror produce reflected rays (green) that miss the focal point, F. This is due to
2584:
colored margin, or narrow spectrum. The absence of this error is termed achromatism, and an optical system so corrected is termed achromatic. A system is said to be
1115:
4831:
3184:
3154:
3838:
3811:
3258:
3231:
2961:
2934:
2907:
2880:
2174:
2122:
4065:
3630:
2722:
2673:
5192:
1501:
363:. Because of dispersion, different wavelengths of light come to focus at different points. Chromatic aberration does not appear when monochromatic light is used.
4033:
3124:
2830:
4091:
3998:
3978:
3958:
3925:
3893:
3858:
3204:
3101:
2850:
2693:
2644:
2727:
5065:
4475:
this region of the spectrum is that which appears brightest to the human eye, and consequently this curve of the secondary on spectrum, obtained by making
87:
2459:(4) After eliminating the aberration On the axis, coma and astigmatism, the relation for the flatness of the field in the third order is expressed by the
682:
was probably the discoverer of astigmation; the position of the astigmatic image lines was determined by Thomas Young; and the theory was developed by
2472:
2567:
In optical systems composed of lenses, the position, magnitude and errors of the image depend upon the refractive indices of the glass employed (see
2272:, a wavefront may be perfectly represented by a sufficiently large number of higher-order Zernike polynomials. However, wavefronts with very steep
3868:
determined by
Chester More Hall in 1728, Klingenstierna in 1754 and by Dollond in 1757, who constructed the celebrated achromatic telescopes. (See
978:
877:
5217:
5060:
603:
will be reproduced, but attended by aberrations comparable in magnitude to ON. These aberrations are avoided if, according to Abbe, the
49:
5252:
4815:
4782:
217:
63:
4467:
1949:
1870:
620:
to characterize a superior achromatism, and, subsequently, by many writers to denote freedom from spherical aberration as well.
441:
Chromatic aberration occurs when different wavelengths are not focussed to the same point. Types of chromatic aberration are:
4515:, is, according to the experiments of Sir G. G. Stokes (Proc. Roy. Soc., 1878), the most suitable for visual instruments (
2028:
4835:
4067:) be greater than that of the same lens, this being made of the more dispersive of the two glasses employed, it is termed
633:
390:
2305:
1806:
1738:
132:
5469:
2602:
image point of one color, another colour produces a disk of confusion; this is similar to the confusion caused by two
2452:
The condition for freedom from coma in the third order is also of importance for telescope objectives; it is known as
427:
302:
2389:
3519:{\displaystyle f=f_{1}-f_{2}=(n_{1}-1)(1/r'_{1}-1/r''_{1})+(n2-1)(1/r'_{2}-1/r''_{2})=(n_{1}-1)k_{1}+(n_{2}-1)k_{2}}
2449:
All these rules are valid, inasmuch as the thicknesses and distances of the lenses are not to be taken into account.
612:
then becomes sin u'1/h1=sin u'2/h2. A system fulfilling this condition and free from spherical aberration is called
431:
Comparison of an ideal image of a ring (1) and ones with only axial (2) and only transverse (3) chromatic aberration
2492:
1615:
1557:
2400:
617:
318:, light from any given point on an object would pass through the lens and come together at a single point in the
4596:
3637:
3531:
1673:
3716:
2551:: this is corrected more with regard to aperture; objectives for groups more with regard to the field of view.
864:, thus individual aberration contributions to an overall wavefront may be isolated and quantified separately.
548:
of radius O'1R, and in a parallel plane at O'2 another one of radius O'2R2; between these two is situated the
177:
4096:
2297:
716:
141:
55:
2179:
4712:
2285:
868:
5456:
website, Michael W. Davidson, Mortimer
Abramowitz, Olympus America Inc., and The Florida State University
4537:
2214:
2454:
2411:
derived the aberrations of the third order; and in later times the method was pursued by Clerk Maxwell (
1433:
4202:
2127:
5243:
5145:
4762:
4689:
satisfying the sine condition for several colors, which therefore, according to his definition, were
4398:
and depend upon the aperture and focal length in the same manner as the primary chromatic errors do.
4361:, then for a third color, c, the focal length is different; that is, if c lies between a and b, then
3897:
2530:
corrected for the angle of aperture u* (the height of incidence h*) or the angle of field of view w*.
861:
484:
436:
380:
348:
307:
279:
235:
4534:
Should there be in two lenses in contact the same focal lengths for three colours a, b, and c, i.e.
266:
An image-forming optical system with aberration will produce an image which is not sharp. Makers of
4318:
4269:
2588:
when it shows the same kind of chromatic error as a thin positive lens, otherwise it is said to be
2576:
2488:
2431:
1905, 18, p. 941) founded his theory of aberrations on the differential geometry of surfaces.
2324:
2312:
1514:
803:
795:
of the image field. Referring to fig. 4, we have O'Q'/OQ = a' tan w'/a tan w = 1/N, where N is the
499:
352:
342:
267:
5244:
Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light
5449:
5186:
4927:
4478:
4162:
2332:
841:
747:
375:
275:
4364:
192:
1131:
5248:
5223:
5213:
5129:
4919:
4811:
4778:
4774:
4767:
4717:
4425:
2575:, above). Since the index of refraction varies with the color or wavelength of the light (see
2319:
to evaluate the point image of an aberrated optical system taking into account the effects of
2277:
683:
628:
Aberration of lateral object points (points beyond the axis) with narrow pencils — astigmatism
3860:
be not zero (equation 2)), and the dispersive powers must also be different (according to 4).
3056:
2996:
1506:
The first few Zernike polynomials, multiplied by their respective fitting coefficients, are:
1094:
188:
5153:
5050:
One or more of the preceding sentences incorporates text from a publication now in the
4799:
4722:
3159:
3129:
3026:
2966:
2504:
2476:
2281:
2261:
860:
that individually represent different types of aberrations. These Zernike coefficients are
840:
Circular wavefront profiles associated with aberrations may be mathematically modeled using
751:
541:
400:
356:
287:
4807:
3816:
3789:
3236:
3209:
2939:
2912:
2885:
2858:
2159:
2107:
230:
5174:
4406:
4039:
3606:
2698:
2649:
2554:
2544:
2293:
411:
395:
260:
2618:(This explains the gigantic focal lengths in vogue before the discovery of achromatism.)
1480:
704:
5149:
4015:
3106:
2812:
2380:, of the point O at some distance from the axis could be constructed. Writing Dξ'=ξ'-ξ'
119:
4076:
3983:
3963:
3943:
3910:
3878:
3843:
3189:
3086:
2835:
2678:
2629:
2323:. The perfect point image in the presence of diffraction had already been described by
1663:
738:
5463:
5056:
5051:
4896:. Reference may also be made to the treatise of Czapski-Eppenstein, pp. 155–161.
2568:
2500:
2316:
845:
700:
458:
415:
385:
315:
251:
5363:
828:
Image plane of a flat-top beam under the effect of the first 21 Zernike polynomials.
5179:
Die bilderzeugung in optischen Instrumenten vom Standpunkte der geometrischen Optik
4695:
1605:
853:
832:
679:
488:
291:
5391:, p. 373; K. Schwarzschild, Göttingen. Akad. Abhandl., 1905, 4, Nos. 2 and 3
4800:
2415:
1874–1875; (see also the treatises of R. S. Heath and L. A. Herman), M. Thiesen (
17:
3903:
2548:
2320:
1796:
857:
730:
639:
492:
824:
2289:
1547:
849:
755:
503:
360:
283:
5227:
5157:
4923:
836:
Effect of Zernike aberrations in Log scale. The intensity minima are visible.
3869:
2273:
852:
over a circle of unit radius. A complex, aberrated wavefront profile may be
2423:
1895, 21, p. 410), and particularly successfully by K. Schwarzschild (
2335:, and in characterizing optical systems with respect to their aberrations.
1069:{\displaystyle Z_{n}^{-m}(\rho ,\phi )=R_{n}^{m}(\rho )\,\sin(m\,\phi ),\!}
5207:
5069:. Vol. 1 (11th ed.). Cambridge University Press. pp. 54–61.
2343:
The preceding review of the several errors of reproduction belongs to the
786:
diminished; in practice, this generally occurs. This ray, named by Abbe a
4773:(2nd ed.). Philadelphia: Harcourt Brace College Publishers. p.
2301:
2300:
definition in the wavefront. In this case, other fitting methods such as
1728:
962:{\displaystyle Z_{n}^{m}(\rho ,\phi )=R_{n}^{m}(\rho )\,\cos(m\,\phi )\!}
598:
Aberration of elements, i.e. smallest objects at right angles to the axis
565:
511:
Aberration of axial points (spherical aberration in the restricted sense)
4931:
4907:
1863:"45° Astigmatism", a cylindrical shape oriented at ±45° from the X axis
616:(Greek a-, privative, plann, a wandering). This word was first used by
2528:
attend smaller values of w. The practical optician names such systems:
1118:
1088:
4466:
4405:, the abscissae are focal lengths, and the ordinates wavelengths. The
2803:{\displaystyle {\dfrac {df}{f}}={\dfrac {dn}{(n-1)}}={\dfrac {1}{n}},}
2355:
772:
703:
in the narrower sense only; other errors of coma have been treated by
2269:
1125:
464:
243:
5133:
2487:
such pair of planes (this was proven with increasing generality by
1128:, and ρ is the normalized radial distance. The radial polynomials
4465:
2471:
2354:
1121:
771:
746:
Even if the image is sharp, it may be distorted compared to ideal
737:
729:
645:
644:
515:
514:
426:
301:
255:
229:
871:
Zernike polynomials. The even Zernike polynomials are defined as
4199:(e.g. if the lenses be made of the same glass), this reduces to
2909:
be the powers corresponding to the lenses of refractive indices
2265:
5269:
Bericht uber die Ergebnisse einiger dioptrischer Untersuchungen
4858:"Aberration: Understanding optical imperfections in telescopes"
2328:
686:. A bibliography by P. Culmann is given in Moritz von Rohr's
160:
113:
70:
29:
2292:, are not well modeled by Zernike polynomials, which tend to
694:
Aberration of lateral object points with broad pencils — coma
1942:"X-Coma", comatic image flaring in the horizontal direction
2675:, and the powers, or reciprocals of the focal lengths, be
2021:"Y-Coma", comatic image flaring in the vertical direction
2012:{\displaystyle a_{7}\times (3\rho ^{2}-2)\rho \sin(\phi )}
1933:{\displaystyle a_{6}\times (3\rho ^{2}-2)\rho \cos(\phi )}
3907:. For the construction of an achromatic collective lens (
2526:
zones of astigmatism, curvature of field and distortion,
2495:
in 1926, see summary in Walther, A., J. Opt. Soc. Am. A
290:
discuss the general features of reflected and refracted
86:
is largely based on an article in the out-of-copyright
5134:"Allgemeine Theorie der monochromat. Aberrationen, etc"
2572:
528:); and conversely with a dispersive surface or lenses (
184:
137:
2406:
The aberrations can also be expressed by means of the
1411:
1395:
5299:
Theorie und Geschichte des photographischen Objectivs
4683:
Theorie und Geschichte des photographischen Objectivs
4599:
4540:
4481:
4403:
Theorie und Geschichte des photographischen Objectivs
4367:
4321:
4272:
4205:
4165:
4099:
4079:
4042:
4018:
3986:
3966:
3946:
3913:
3881:
3846:
3819:
3792:
3719:
3640:
3609:
3534:
3268:
3239:
3212:
3192:
3162:
3132:
3109:
3089:
3059:
3029:
2999:
2969:
2942:
2915:
2888:
2861:
2838:
2815:
2786:
2752:
2732:
2730:
2701:
2681:
2652:
2632:
2479:
assist in the elimination of atmospheric distortion.
2217:
2182:
2162:
2130:
2110:
2089:{\displaystyle a_{8}\times (6\rho ^{4}-6\rho ^{2}+1)}
2031:
1952:
1873:
1809:
1741:
1676:
1618:
1560:
1517:
1483:
1436:
1169:
1134:
1097:
981:
880:
707:
and Moritz von Rohr, and later by Allvar Gullstrand.
4718:
Optical telescope § The five Seidel aberrations
3260:with the color. Then the following relations hold:
1662:"Y-Tilt", the deviation of the overall beam in the
1604:"X-Tilt", the deviation of the overall beam in the
856:with Zernike polynomials to yield a set of fitting
483:The introduction of simple auxiliary terms, due to
4766:
4660:
4585:
4507:
4386:
4353:
4306:
4251:
4191:
4151:
4085:
4059:
4027:
3992:
3972:
3952:
3919:
3887:
3852:
3832:
3805:
3778:
3705:
3624:
3595:
3518:
3252:
3225:
3198:
3178:
3148:
3118:
3095:
3075:
3045:
3015:
2985:
2955:
2928:
2901:
2874:
2844:
2824:
2802:
2716:
2687:
2667:
2638:
2249:
2203:
2168:
2148:
2116:
2088:
2011:
1932:
1853:
1785:
1717:
1652:
1594:
1536:
1495:
1469:
1419:
1152:
1109:
1068:
961:
274:Aberration can be analyzed with the techniques of
5124:
5122:
1854:{\displaystyle a_{5}\times \rho ^{2}\sin(2\phi )}
1786:{\displaystyle a_{4}\times \rho ^{2}\cos(2\phi )}
1241:
1240:
1239:
1197:
1160:have no azimuthal dependence, and are defined as
1065:
958:
4806:. Cambridge: John Wiley & Sons Inc. p.
27:Deviation from perfect paraxial optical behavior
5420:Grundzuge der Theorie der optischen Instrumente
5404:Grundzüge der Theorie der optischen Instrumente
4315:50%). If, therefore, for two colors, a and b,
583:); if entirely in front, it is the exit pupil (
371:The most common monochromatic aberrations are:
448:Lateral (or "transverse") chromatic aberration
445:Axial (or "longitudinal") chromatic aberration
4892:, xxi. 325, by means of Sir W. R. Hamilton's
4005:and were employed by P. Rudolph in the first
2176:is the azimuthal angle around the pupil with
8:
4073:For two thin lenses separated by a distance
3875:Glass with weaker dispersive power (greater
1731:wavefront resulting from being out of focus
1653:{\displaystyle a_{2}\times \rho \sin(\phi )}
1595:{\displaystyle a_{1}\times \rho \cos(\phi )}
5352:Die Bilderzeugung in optischen Instrumenten
5212:(2nd ed.). San Diego: Academic Press.
5098:A Course of Lectures on Natural Philosophy.
4912:The Transactions of the Royal Irish Academy
688:Die Bilderzeugung in optischen Instrumenten
64:Learn how and when to remove these messages
5450:Microscope Objectives: Optical Aberrations
5191:: CS1 maint: location missing publisher (
5169:
5167:
4661:{\displaystyle (n_{c}-n_{b})(n_{a}-n_{b})}
3706:{\displaystyle k_{1}/k_{2}=-dn_{2}/dn_{1}}
3596:{\displaystyle df=k_{1}dn_{1}+k_{2}dn_{2}}
2612:Monochromatic Aberration of the Axis Point
1718:{\displaystyle a_{3}\times (2\rho ^{2}-1)}
1372:
250:is a property of optical systems, such as
238:. 2: A lens with less chromatic aberration
5038:
5036:
5034:
5032:
5030:
5028:
5026:
5024:
5022:
5020:
5018:
5016:
5014:
5012:
5010:
5008:
5006:
5004:
5002:
5000:
4998:
4996:
4994:
4992:
4990:
4988:
4986:
4984:
4982:
4980:
4978:
4976:
4649:
4636:
4620:
4607:
4598:
4571:
4558:
4545:
4539:
4499:
4486:
4480:
4372:
4366:
4339:
4326:
4320:
4298:
4286:
4280:
4271:
4241:
4232:
4219:
4204:
4183:
4170:
4164:
4143:
4133:
4120:
4110:
4098:
4078:
4049:
4041:
4017:
3985:
3965:
3945:
3912:
3880:
3845:
3824:
3818:
3797:
3791:
3770:
3761:
3755:
3739:
3730:
3724:
3718:
3697:
3685:
3679:
3660:
3651:
3645:
3639:
3608:
3587:
3574:
3561:
3548:
3533:
3510:
3491:
3475:
3456:
3434:
3425:
3410:
3401:
3362:
3353:
3338:
3329:
3308:
3292:
3279:
3267:
3244:
3238:
3217:
3211:
3191:
3170:
3161:
3140:
3131:
3108:
3088:
3064:
3058:
3034:
3028:
3004:
2998:
2974:
2968:
2947:
2941:
2920:
2914:
2893:
2887:
2866:
2860:
2837:
2814:
2785:
2751:
2731:
2729:
2700:
2680:
2651:
2631:
2257:are the wavefront errors in wavelengths.
2241:
2222:
2216:
2181:
2161:
2129:
2109:
2071:
2055:
2036:
2030:
1976:
1957:
1951:
1897:
1878:
1872:
1827:
1814:
1808:
1759:
1746:
1740:
1700:
1681:
1675:
1623:
1617:
1565:
1559:
1522:
1516:
1482:
1446:
1441:
1435:
1410:
1394:
1387:
1377:
1349:
1330:
1310:
1291:
1264:
1258:
1242:
1229:
1213:
1202:
1179:
1174:
1168:
1144:
1139:
1133:
1096:
1055:
1042:
1027:
1022:
991:
986:
980:
951:
938:
923:
918:
890:
885:
879:
810:symmetrical or holosymmetrical objectives
524:long as the angle u2 is greater than u1 (
218:Learn how and when to remove this message
5346:
5344:
4974:
4972:
4970:
4968:
4966:
4964:
4962:
4960:
4958:
4956:
4411:
3779:{\displaystyle f_{1}/f_{2}=-n_{1}/n_{2}}
848:in the 1930s, Zernike's polynomials are
831:
823:
5364:"New Laser Improves VLT's Capabilities"
4754:
4734:
4593:, then the relative partial dispersion
4152:{\displaystyle D=v_{1}f_{1}+v_{2}f_{2}}
722:surface is concave towards the system.
5184:
3936:the latter being the best vide supra,
3901:; that with greater dispersive power,
761:Systems free of distortion are called
5432:A. Konig in M. v. Rohr's collection,
2204:{\displaystyle 0\leq \phi \leq 2\pi }
7:
5241:Born, Max; Wolf, Emil (1999-10-13).
4673:is derived; there is yet a residual
4401:In fig. 6, taken from M. von Rohr's
2468:Practical elimination of aberrations
2308:may yield improved fitting results.
2124:is the normalized pupil radius with
765:(orthos, right, skopein to look) or
467:, rays of light proceeding from any
4832:"Comparison of Optical Aberrations"
4586:{\displaystyle f_{a}=f_{b}=f_{c}=f}
2350:infinitely small of a certain order
2250:{\displaystyle a_{0},\ldots ,a_{8}}
2098:"Third order spherical aberration"
972:and the odd Zernike polynomials as
711:Curvature of the field of the image
463:In a perfect optical system in the
333:Aberrations fall into two classes:
191:in tone and meet Knowledge (XXG)'s
4949:(in German). Göttingen: Dieterich.
4409:used are shown in adjacent table.
2852:the dispersive power of the glass.
2491:in 1858, by Bruns in 1895, and by
2419:1890, 35, p. 804), H. Bruns (
2393:the numerical orders of the image,
1470:{\displaystyle R_{n}^{m}(\rho )=0}
592:perpendicular height of incidence,
453:Theory of monochromatic aberration
410:In addition to these aberrations,
25:
4252:{\displaystyle D=(f_{1}+f_{2})/2}
4093:the condition for achromatism is
2339:Analytic treatment of aberrations
2149:{\displaystyle 0\leq \rho \leq 1}
131:to comply with Knowledge (XXG)'s
45:This article has multiple issues.
5043:
4677:but it can always be neglected.
2855:Two thin lenses in contact: let
2513:Lehrbuch der geometrischen Optik
638:For Astigmatism of the eye, see
624:light reaching the image plane.
165:
118:
75:
34:
4834:. Edmund Optics. Archived from
2329:Extended Nijboer-Zernike theory
2280:structure, such as produced by
2211:, and the fitting coefficients
1393:
53:or discuss these issues on the
5247:. Cambridge University Press.
4945:Gauss, Carl Friedrich (1841).
4655:
4629:
4626:
4600:
4238:
4212:
3503:
3484:
3468:
3449:
3443:
3395:
3392:
3377:
3371:
3323:
3320:
3301:
2832:is called the dispersion, and
2775:
2763:
2083:
2045:
2006:
2000:
1988:
1966:
1927:
1921:
1909:
1887:
1848:
1839:
1780:
1771:
1712:
1690:
1647:
1641:
1589:
1583:
1458:
1452:
1363:
1346:
1334:
1331:
1324:
1307:
1295:
1292:
1277:
1265:
1255:
1245:
1226:
1214:
1191:
1185:
1059:
1049:
1039:
1033:
1012:
1000:
955:
945:
935:
929:
908:
896:
742:Fig. 3b: Pincushion distortion
1:
4671:achromatism of a higher order
4354:{\displaystyle f_{a}=f_{b}=f}
4307:{\displaystyle dn_{2}/dn_{1}}
2586:chromatically under-corrected
2563:Chromatic or color aberration
1537:{\displaystyle a_{0}\times 1}
790:(not to be confused with the
634:Astigmatism (optical systems)
93:, which was produced in 1911.
5402:Czapski; Eppenstein (1903).
5079:Maxwell, James Clerk (1856)
4691:aplanatic for several colors
3103:denote the total power, and
2425:Göttingen. Akad. Abhandl.,
2306:singular value decomposition
1799:shape along the X or Y axis
820:Zernike model of aberrations
355:, the variation of a lens's
5418:Czapski-Eppenstein (1903).
5301:, Berlin, 1899, p. 248
4908:"Theory of Systems of Rays"
4529:freedom from chemical focus
4508:{\displaystyle f_{C}=f_{F}}
4192:{\displaystyle v_{1}=v_{2}}
4009:(photographic objectives).
2345:Abbe theory of aberrations,
5486:
5387:A. Konig in M. von Rohr's
5108:Gullstrand, Allvar (1890)
4947:Dioptrische Untersuchungen
4890:Leipzig. Math. Phys. Ber.
4862:www.jameswebbdiscovery.com
4765:; Wheeler, Gerald (1992).
4387:{\displaystyle f_{c}<f}
2421:Leipzig. Math. Phys. Ber.,
734:Fig. 3a: Barrel distortion
714:
637:
631:
465:classical theory of optics
456:
434:
234:1: Imaging by a lens with
5206:Schroeder, D. J. (2000).
4798:Guenther, Robert (1990).
4693:; such systems he termed
1153:{\displaystyle R_{n}^{m}}
673:astigmatic image surfaces
367:Monochromatic aberrations
322:(or, more generally, the
5324:München. Akad. Sitzber.,
5312:Munchen. Acad. Abhandl.
5158:10.1002/andp.19053231504
5110:Skand. Arch. f. Physiol.
4906:Hamilton, W. R. (1828).
4886:Berlin. Phys. Ges. Verh.
4884:; and (1892) xxxv. 799;
4521:pure actinic achromatism
3938:Monochromatic Aberration
2573:Monochromatic aberration
2417:Berlin. Akad. Sitzber.,
2413:Proc. London Math. Soc.,
550:disk of least confusion.
534:longitudinal aberration,
259:from the predictions of
144:may contain suggestions.
129:may need to be rewritten
5338:, Leipzig, 1895-6-7-8-9
5314:, 1891, 17, p. 519
5275:1857, vols. xxiv. xxvi.
5066:Encyclopædia Britannica
4894:characteristic function
3076:{\displaystyle r''_{2}}
3016:{\displaystyle r''_{1}}
2408:characteristic function
1546:"Piston", equal to the
1110:{\displaystyle n\geq m}
726:Distortion of the image
717:Petzval field curvature
665:first principal section
271:accurate observations.
89:Encyclopædia Britannica
5400:Formulae are given in
5096:Young, Thomas (1807),
4882:Berlin. Akad. Sitzber.
4713:Aberrations of the eye
4662:
4587:
4509:
4471:
4388:
4355:
4308:
4261:condition for oculars.
4253:
4193:
4153:
4087:
4061:
4029:
3994:
3974:
3954:
3921:
3889:
3854:
3834:
3807:
3780:
3707:
3626:
3597:
3520:
3254:
3227:
3200:
3180:
3179:{\displaystyle dn_{2}}
3150:
3149:{\displaystyle dn_{1}}
3120:
3097:
3077:
3047:
3046:{\displaystyle r'_{2}}
3017:
2987:
2986:{\displaystyle r'_{1}}
2957:
2930:
2903:
2876:
2846:
2826:
2804:
2718:
2689:
2669:
2640:
2480:
2360:
2290:aerodynamic flowfields
2286:atmospheric turbulence
2251:
2205:
2170:
2150:
2118:
2090:
2013:
1934:
1855:
1787:
1719:
1654:
1596:
1538:
1497:
1471:
1421:
1238:
1154:
1111:
1070:
963:
837:
829:
777:
743:
735:
650:
520:
498:The investigations of
432:
311:
239:
5454:Molecular Expressions
5336:Beiträge zur Dioptrik
5326:1898, 28, p. 395
5273:Akad. Sitzber., Wien,
4769:Physics: A World View
4663:
4588:
4510:
4469:
4389:
4356:
4309:
4254:
4194:
4154:
4088:
4062:
4030:
3995:
3975:
3955:
3934:Fraunhofer Condition,
3922:
3890:
3855:
3835:
3833:{\displaystyle f_{2}}
3808:
3806:{\displaystyle f_{1}}
3781:
3708:
3627:
3598:
3521:
3255:
3253:{\displaystyle n_{2}}
3228:
3226:{\displaystyle n_{1}}
3201:
3181:
3151:
3121:
3098:
3078:
3048:
3018:
2988:
2958:
2956:{\displaystyle n_{2}}
2931:
2929:{\displaystyle n_{1}}
2904:
2902:{\displaystyle f_{2}}
2877:
2875:{\displaystyle f_{1}}
2847:
2827:
2805:
2719:
2690:
2670:
2641:
2581:chromatic differences
2475:
2447:Herschel's condition.
2372:, with coordinates ξ'
2358:
2252:
2206:
2171:
2169:{\displaystyle \phi }
2151:
2119:
2117:{\displaystyle \rho }
2091:
2014:
1935:
1856:
1788:
1720:
1655:
1597:
1539:
1498:
1472:
1422:
1198:
1155:
1112:
1071:
964:
835:
827:
775:
741:
733:
648:
563:; Abbe used the term
518:
430:
423:Chromatic aberrations
349:Chromatic aberrations
305:
233:
5271:, Buda Pesth, 1843;
4701:compensating oculars
4597:
4538:
4517:optical achromatism,
4479:
4365:
4319:
4270:
4203:
4163:
4097:
4077:
4060:{\displaystyle df/f}
4040:
4016:
3984:
3964:
3944:
3911:
3879:
3844:
3817:
3790:
3717:
3638:
3625:{\displaystyle df=0}
3607:
3532:
3266:
3237:
3210:
3190:
3160:
3130:
3107:
3087:
3057:
3027:
2997:
2967:
2940:
2913:
2886:
2859:
2836:
2813:
2728:
2717:{\displaystyle f+df}
2699:
2679:
2668:{\displaystyle n+dn}
2650:
2630:
2215:
2180:
2160:
2128:
2108:
2029:
1950:
1871:
1807:
1795:"0° Astigmatism", a
1739:
1674:
1616:
1558:
1515:
1481:
1434:
1167:
1132:
1095:
979:
878:
862:linearly independent
479:is reproduced in an
475:; and therefore the
437:Chromatic aberration
381:Spherical aberration
308:spherical aberration
236:chromatic aberration
185:improve this article
5288:, 1856, p. 289
5209:Astronomical optics
5150:1905AnP...323..941G
5144:(18). Upsala: 941.
5085:Quart. Journ. Math.
4888:; Bruns, H. (1895)
4880:Thiesen, M. (1890)
4838:on December 6, 2011
4396:secondary spectrum,
3603:. For achromatism
3442:
3418:
3370:
3346:
3072:
3042:
3012:
2982:
2597:stable achromatism.
2315:were introduced by
1496:{\displaystyle n-m}
1451:
1184:
1149:
1032:
999:
928:
895:
842:Zernike polynomials
500:James Clerk Maxwell
343:monochromatic light
278:. The articles on
268:optical instruments
5470:Geometrical optics
5310:S. Finterswalder,
5138:Annalen der Physik
5130:Gullstrand, Allvar
4763:Kirkpatrick, Larry
4675:tertiary spectrum,
4658:
4583:
4525:actinic correction
4505:
4472:
4384:
4351:
4304:
4249:
4189:
4149:
4083:
4057:
4028:{\displaystyle df}
4025:
4012:Instead of making
3990:
3970:
3950:
3917:
3885:
3850:
3830:
3803:
3776:
3703:
3632:, hence, from (3),
3622:
3593:
3516:
3430:
3406:
3358:
3334:
3250:
3223:
3196:
3176:
3146:
3119:{\displaystyle df}
3116:
3093:
3083:respectively; let
3073:
3060:
3043:
3030:
3013:
3000:
2983:
2970:
2953:
2926:
2899:
2872:
2842:
2825:{\displaystyle dn}
2822:
2800:
2795:
2780:
2746:
2714:
2685:
2665:
2646:, and for another
2636:
2616:relative aperture.
2481:
2361:
2333:numerical aperture
2313:circle polynomials
2247:
2201:
2166:
2146:
2114:
2086:
2009:
1930:
1851:
1783:
1715:
1650:
1592:
1534:
1493:
1467:
1437:
1417:
1415:
1399:
1170:
1150:
1135:
1107:
1066:
1018:
982:
959:
914:
881:
838:
830:
778:
769:(straight lines).
748:pinhole projection
744:
736:
669:meridional section
656:axis of the pencil
651:
538:lateral aberration
521:
433:
345:, hence the name.
312:
276:geometrical optics
240:
5434:Die Bilderzeugung
5389:Die Bilderzeugung
5219:978-0-08-049951-2
5116:, 53, pp. 2, 185.
4464:
4463:
4086:{\displaystyle D}
3993:{\displaystyle n}
3973:{\displaystyle v}
3953:{\displaystyle v}
3920:{\displaystyle f}
3888:{\displaystyle v}
3853:{\displaystyle f}
3199:{\displaystyle f}
3096:{\displaystyle f}
2845:{\displaystyle n}
2794:
2779:
2745:
2688:{\displaystyle f}
2639:{\displaystyle n}
2555:Long focus lenses
2477:Laser guide stars
2461:Petzval equation,
2442:under-correction,
2278:spatial frequency
2102:
2101:
1550:of the wavefront
1414:
1398:
1370:
783:center of gravity
684:Allvar Gullstrand
546:disk of confusion
228:
227:
220:
210:
209:
193:quality standards
159:
158:
133:quality standards
112:
111:
68:
18:Seidel aberration
16:(Redirected from
5477:
5437:
5430:
5424:
5423:
5414:
5408:
5407:
5398:
5392:
5385:
5379:
5378:
5376:
5374:
5368:ESO Announcement
5360:
5354:
5348:
5339:
5333:
5327:
5321:
5315:
5308:
5302:
5295:
5289:
5282:
5276:
5265:
5259:
5258:
5238:
5232:
5231:
5203:
5197:
5196:
5190:
5182:
5175:von Rohr, Moritz
5171:
5162:
5161:
5126:
5117:
5114:Arch. f. Ophth.
5106:
5100:
5094:
5088:
5077:
5071:
5070:
5049:
5047:
5046:
5040:
4951:
4950:
4942:
4936:
4935:
4903:
4897:
4878:
4872:
4871:
4869:
4868:
4854:
4848:
4847:
4845:
4843:
4828:
4822:
4821:
4805:
4795:
4789:
4788:
4772:
4759:
4742:
4739:
4723:Wavefront coding
4667:
4665:
4664:
4659:
4654:
4653:
4641:
4640:
4625:
4624:
4612:
4611:
4592:
4590:
4589:
4584:
4576:
4575:
4563:
4562:
4550:
4549:
4514:
4512:
4511:
4506:
4504:
4503:
4491:
4490:
4412:
4407:Fraunhofer lines
4393:
4391:
4390:
4385:
4377:
4376:
4360:
4358:
4357:
4352:
4344:
4343:
4331:
4330:
4313:
4311:
4310:
4305:
4303:
4302:
4290:
4285:
4284:
4258:
4256:
4255:
4250:
4245:
4237:
4236:
4224:
4223:
4198:
4196:
4195:
4190:
4188:
4187:
4175:
4174:
4158:
4156:
4155:
4150:
4148:
4147:
4138:
4137:
4125:
4124:
4115:
4114:
4092:
4090:
4089:
4084:
4069:hyper-chromatic.
4066:
4064:
4063:
4058:
4053:
4034:
4032:
4031:
4026:
3999:
3997:
3996:
3991:
3979:
3977:
3976:
3971:
3959:
3957:
3956:
3951:
3926:
3924:
3923:
3918:
3894:
3892:
3891:
3886:
3859:
3857:
3856:
3851:
3839:
3837:
3836:
3831:
3829:
3828:
3812:
3810:
3809:
3804:
3802:
3801:
3785:
3783:
3782:
3777:
3775:
3774:
3765:
3760:
3759:
3744:
3743:
3734:
3729:
3728:
3712:
3710:
3709:
3704:
3702:
3701:
3689:
3684:
3683:
3665:
3664:
3655:
3650:
3649:
3631:
3629:
3628:
3623:
3602:
3600:
3599:
3594:
3592:
3591:
3579:
3578:
3566:
3565:
3553:
3552:
3525:
3523:
3522:
3517:
3515:
3514:
3496:
3495:
3480:
3479:
3461:
3460:
3438:
3429:
3414:
3405:
3366:
3357:
3342:
3333:
3313:
3312:
3297:
3296:
3284:
3283:
3259:
3257:
3256:
3251:
3249:
3248:
3232:
3230:
3229:
3224:
3222:
3221:
3205:
3203:
3202:
3197:
3185:
3183:
3182:
3177:
3175:
3174:
3155:
3153:
3152:
3147:
3145:
3144:
3125:
3123:
3122:
3117:
3102:
3100:
3099:
3094:
3082:
3080:
3079:
3074:
3068:
3052:
3050:
3049:
3044:
3038:
3022:
3020:
3019:
3014:
3008:
2992:
2990:
2989:
2984:
2978:
2962:
2960:
2959:
2954:
2952:
2951:
2935:
2933:
2932:
2927:
2925:
2924:
2908:
2906:
2905:
2900:
2898:
2897:
2881:
2879:
2878:
2873:
2871:
2870:
2851:
2849:
2848:
2843:
2831:
2829:
2828:
2823:
2809:
2807:
2806:
2801:
2796:
2787:
2781:
2778:
2761:
2753:
2747:
2741:
2733:
2723:
2721:
2720:
2715:
2694:
2692:
2691:
2686:
2674:
2672:
2671:
2666:
2645:
2643:
2642:
2637:
2505:Maxwell fish-eye
2435:(5) distortion.
2264:synthesis using
2256:
2254:
2253:
2248:
2246:
2245:
2227:
2226:
2210:
2208:
2207:
2202:
2175:
2173:
2172:
2167:
2155:
2153:
2152:
2147:
2123:
2121:
2120:
2115:
2095:
2093:
2092:
2087:
2076:
2075:
2060:
2059:
2041:
2040:
2018:
2016:
2015:
2010:
1981:
1980:
1962:
1961:
1939:
1937:
1936:
1931:
1902:
1901:
1883:
1882:
1860:
1858:
1857:
1852:
1832:
1831:
1819:
1818:
1792:
1790:
1789:
1784:
1764:
1763:
1751:
1750:
1724:
1722:
1721:
1716:
1705:
1704:
1686:
1685:
1659:
1657:
1656:
1651:
1628:
1627:
1601:
1599:
1598:
1593:
1570:
1569:
1543:
1541:
1540:
1535:
1527:
1526:
1509:
1508:
1502:
1500:
1499:
1494:
1476:
1474:
1473:
1468:
1450:
1445:
1426:
1424:
1423:
1418:
1416:
1412:
1400:
1396:
1392:
1391:
1371:
1369:
1353:
1314:
1283:
1263:
1262:
1243:
1237:
1233:
1212:
1183:
1178:
1159:
1157:
1156:
1151:
1148:
1143:
1116:
1114:
1113:
1108:
1087:are nonnegative
1075:
1073:
1072:
1067:
1031:
1026:
998:
990:
968:
966:
965:
960:
927:
922:
894:
889:
844:. Developed by
754:, and there are
752:image distortion
680:Sir Isaac Newton
526:under correction
401:Image distortion
357:refractive index
223:
216:
205:
202:
196:
169:
168:
161:
154:
151:
145:
122:
114:
107:
104:
98:
91:Eleventh Edition
79:
78:
71:
60:
38:
37:
30:
21:
5485:
5484:
5480:
5479:
5478:
5476:
5475:
5474:
5460:
5459:
5446:
5441:
5440:
5431:
5427:
5417:
5415:
5411:
5401:
5399:
5395:
5386:
5382:
5372:
5370:
5362:
5361:
5357:
5349:
5342:
5334:
5330:
5322:
5318:
5309:
5305:
5296:
5292:
5283:
5279:
5266:
5262:
5255:
5240:
5239:
5235:
5220:
5205:
5204:
5200:
5183:
5173:
5172:
5165:
5128:
5127:
5120:
5107:
5103:
5095:
5091:
5078:
5074:
5059:, ed. (1911). "
5055:
5044:
5042:
5041:
4954:
4944:
4943:
4939:
4905:
4904:
4900:
4879:
4875:
4866:
4864:
4856:
4855:
4851:
4841:
4839:
4830:
4829:
4825:
4818:
4797:
4796:
4792:
4785:
4761:
4760:
4756:
4751:
4746:
4745:
4740:
4736:
4731:
4709:
4645:
4632:
4616:
4603:
4595:
4594:
4567:
4554:
4541:
4536:
4535:
4495:
4482:
4477:
4476:
4368:
4363:
4362:
4335:
4322:
4317:
4316:
4294:
4276:
4268:
4267:
4259:, known as the
4228:
4215:
4201:
4200:
4179:
4166:
4161:
4160:
4139:
4129:
4116:
4106:
4095:
4094:
4075:
4074:
4038:
4037:
4014:
4013:
3982:
3981:
3962:
3961:
3942:
3941:
3909:
3908:
3877:
3876:
3865:
3842:
3841:
3820:
3815:
3814:
3793:
3788:
3787:
3766:
3751:
3735:
3720:
3715:
3714:
3693:
3675:
3656:
3641:
3636:
3635:
3605:
3604:
3583:
3570:
3557:
3544:
3530:
3529:
3506:
3487:
3471:
3452:
3304:
3288:
3275:
3264:
3263:
3240:
3235:
3234:
3213:
3208:
3207:
3188:
3187:
3186:the changes of
3166:
3158:
3157:
3136:
3128:
3127:
3105:
3104:
3085:
3084:
3055:
3054:
3025:
3024:
2995:
2994:
2965:
2964:
2943:
2938:
2937:
2916:
2911:
2910:
2889:
2884:
2883:
2862:
2857:
2856:
2834:
2833:
2811:
2810:
2762:
2754:
2734:
2726:
2725:
2697:
2696:
2677:
2676:
2648:
2647:
2628:
2627:
2565:
2545:Wide angle lens
2470:
2429:Ann. d. Phys.,
2387:
2383:
2379:
2375:
2371:
2341:
2294:low-pass filter
2237:
2218:
2213:
2212:
2178:
2177:
2158:
2157:
2126:
2125:
2106:
2105:
2067:
2051:
2032:
2027:
2026:
1972:
1953:
1948:
1947:
1893:
1874:
1869:
1868:
1823:
1810:
1805:
1804:
1755:
1742:
1737:
1736:
1696:
1677:
1672:
1671:
1619:
1614:
1613:
1561:
1556:
1555:
1518:
1513:
1512:
1479:
1478:
1432:
1431:
1373:
1284:
1254:
1244:
1165:
1164:
1130:
1129:
1093:
1092:
977:
976:
876:
875:
822:
758:to correct it.
728:
719:
713:
696:
643:
636:
630:
605:sine condition,
600:
530:over correction
513:
461:
455:
439:
425:
396:Field curvature
369:
300:
261:paraxial optics
224:
213:
212:
211:
206:
200:
197:
182:
170:
166:
155:
149:
146:
136:
123:
108:
102:
99:
95:
80:
76:
39:
35:
28:
23:
22:
15:
12:
11:
5:
5483:
5481:
5473:
5472:
5462:
5461:
5458:
5457:
5445:
5444:External links
5442:
5439:
5438:
5425:
5422:. p. 170.
5409:
5406:. p. 166.
5393:
5380:
5355:
5340:
5328:
5316:
5303:
5290:
5277:
5260:
5254:978-0521642224
5253:
5233:
5218:
5198:
5163:
5118:
5101:
5089:
5072:
5057:Chisholm, Hugh
4952:
4937:
4898:
4873:
4849:
4823:
4816:
4790:
4783:
4753:
4752:
4750:
4747:
4744:
4743:
4733:
4732:
4730:
4727:
4726:
4725:
4720:
4715:
4708:
4705:
4657:
4652:
4648:
4644:
4639:
4635:
4631:
4628:
4623:
4619:
4615:
4610:
4606:
4602:
4582:
4579:
4574:
4570:
4566:
4561:
4557:
4553:
4548:
4544:
4502:
4498:
4494:
4489:
4485:
4462:
4461:
4460:405.1 nm
4458:
4455:
4452:
4449:
4446:
4443:
4439:
4438:
4435:
4432:
4429:
4422:
4419:
4416:
4383:
4380:
4375:
4371:
4350:
4347:
4342:
4338:
4334:
4329:
4325:
4301:
4297:
4293:
4289:
4283:
4279:
4275:
4248:
4244:
4240:
4235:
4231:
4227:
4222:
4218:
4214:
4211:
4208:
4186:
4182:
4178:
4173:
4169:
4146:
4142:
4136:
4132:
4128:
4123:
4119:
4113:
4109:
4105:
4102:
4082:
4056:
4052:
4048:
4045:
4024:
4021:
4003:new achromats,
3989:
3969:
3949:
3916:
3884:
3864:
3863:
3862:
3861:
3849:
3827:
3823:
3800:
3796:
3773:
3769:
3764:
3758:
3754:
3750:
3747:
3742:
3738:
3733:
3727:
3723:
3700:
3696:
3692:
3688:
3682:
3678:
3674:
3671:
3668:
3663:
3659:
3654:
3648:
3644:
3633:
3621:
3618:
3615:
3612:
3590:
3586:
3582:
3577:
3573:
3569:
3564:
3560:
3556:
3551:
3547:
3543:
3540:
3537:
3527:
3513:
3509:
3505:
3502:
3499:
3494:
3490:
3486:
3483:
3478:
3474:
3470:
3467:
3464:
3459:
3455:
3451:
3448:
3445:
3441:
3437:
3433:
3428:
3424:
3421:
3417:
3413:
3409:
3404:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3369:
3365:
3361:
3356:
3352:
3349:
3345:
3341:
3337:
3332:
3328:
3325:
3322:
3319:
3316:
3311:
3307:
3303:
3300:
3295:
3291:
3287:
3282:
3278:
3274:
3271:
3247:
3243:
3220:
3216:
3195:
3173:
3169:
3165:
3143:
3139:
3135:
3115:
3112:
3092:
3071:
3067:
3063:
3041:
3037:
3033:
3011:
3007:
3003:
2981:
2977:
2973:
2950:
2946:
2923:
2919:
2896:
2892:
2869:
2865:
2853:
2841:
2821:
2818:
2799:
2793:
2790:
2784:
2777:
2774:
2771:
2768:
2765:
2760:
2757:
2750:
2744:
2740:
2737:
2713:
2710:
2707:
2704:
2684:
2664:
2661:
2658:
2655:
2635:
2623:
2590:overcorrected.
2564:
2561:
2560:
2559:
2552:
2542:
2469:
2466:
2465:
2464:
2450:
2385:
2381:
2377:
2373:
2369:
2340:
2337:
2244:
2240:
2236:
2233:
2230:
2225:
2221:
2200:
2197:
2194:
2191:
2188:
2185:
2165:
2145:
2142:
2139:
2136:
2133:
2113:
2100:
2099:
2096:
2085:
2082:
2079:
2074:
2070:
2066:
2063:
2058:
2054:
2050:
2047:
2044:
2039:
2035:
2023:
2022:
2019:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1979:
1975:
1971:
1968:
1965:
1960:
1956:
1944:
1943:
1940:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1900:
1896:
1892:
1889:
1886:
1881:
1877:
1865:
1864:
1861:
1850:
1847:
1844:
1841:
1838:
1835:
1830:
1826:
1822:
1817:
1813:
1801:
1800:
1793:
1782:
1779:
1776:
1773:
1770:
1767:
1762:
1758:
1754:
1749:
1745:
1733:
1732:
1725:
1714:
1711:
1708:
1703:
1699:
1695:
1692:
1689:
1684:
1680:
1668:
1667:
1660:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1626:
1622:
1610:
1609:
1602:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1568:
1564:
1552:
1551:
1544:
1533:
1530:
1525:
1521:
1492:
1489:
1486:
1466:
1463:
1460:
1457:
1454:
1449:
1444:
1440:
1428:
1427:
1409:
1406:
1403:
1390:
1386:
1383:
1380:
1376:
1368:
1365:
1362:
1359:
1356:
1352:
1348:
1345:
1342:
1339:
1336:
1333:
1329:
1326:
1323:
1320:
1317:
1313:
1309:
1306:
1303:
1300:
1297:
1294:
1290:
1287:
1282:
1279:
1276:
1273:
1270:
1267:
1261:
1257:
1253:
1250:
1247:
1236:
1232:
1228:
1225:
1222:
1219:
1216:
1211:
1208:
1205:
1201:
1196:
1193:
1190:
1187:
1182:
1177:
1173:
1147:
1142:
1138:
1106:
1103:
1100:
1077:
1076:
1064:
1061:
1058:
1054:
1051:
1048:
1045:
1041:
1038:
1035:
1030:
1025:
1021:
1017:
1014:
1011:
1008:
1005:
1002:
997:
994:
989:
985:
970:
969:
957:
954:
950:
947:
944:
941:
937:
934:
931:
926:
921:
917:
913:
910:
907:
904:
901:
898:
893:
888:
884:
821:
818:
792:principal rays
727:
724:
715:Main article:
712:
709:
695:
692:
660:principal ray,
632:Main article:
629:
626:
610:sine condition
599:
596:
572:entrance pupil
512:
509:
454:
451:
450:
449:
446:
435:Main article:
424:
421:
404:
403:
398:
393:
388:
383:
378:
368:
365:
351:are caused by
314:With an ideal
299:
296:
254:, that causes
226:
225:
208:
207:
173:
171:
164:
157:
156:
126:
124:
117:
110:
109:
83:
81:
74:
69:
43:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5482:
5471:
5468:
5467:
5465:
5455:
5451:
5448:
5447:
5443:
5435:
5429:
5426:
5421:
5413:
5410:
5405:
5397:
5394:
5390:
5384:
5381:
5369:
5365:
5359:
5356:
5353:
5350:M. von Rohr,
5347:
5345:
5341:
5337:
5332:
5329:
5325:
5320:
5317:
5313:
5307:
5304:
5300:
5297:M. von Rohr,
5294:
5291:
5287:
5281:
5278:
5274:
5270:
5264:
5261:
5256:
5250:
5246:
5245:
5237:
5234:
5229:
5225:
5221:
5215:
5211:
5210:
5202:
5199:
5194:
5188:
5180:
5176:
5170:
5168:
5164:
5159:
5155:
5151:
5147:
5143:
5139:
5135:
5131:
5125:
5123:
5119:
5115:
5112:; and (1901)
5111:
5105:
5102:
5099:
5093:
5090:
5086:
5082:
5076:
5073:
5068:
5067:
5062:
5058:
5053:
5052:public domain
5039:
5037:
5035:
5033:
5031:
5029:
5027:
5025:
5023:
5021:
5019:
5017:
5015:
5013:
5011:
5009:
5007:
5005:
5003:
5001:
4999:
4997:
4995:
4993:
4991:
4989:
4987:
4985:
4983:
4981:
4979:
4977:
4975:
4973:
4971:
4969:
4967:
4965:
4963:
4961:
4959:
4957:
4953:
4948:
4941:
4938:
4933:
4929:
4925:
4921:
4917:
4913:
4909:
4902:
4899:
4895:
4891:
4887:
4883:
4877:
4874:
4863:
4859:
4853:
4850:
4837:
4833:
4827:
4824:
4819:
4817:0-471-60538-7
4813:
4809:
4804:
4803:
4802:Modern Optics
4794:
4791:
4786:
4784:0-03-000602-3
4780:
4776:
4771:
4770:
4764:
4758:
4755:
4748:
4738:
4735:
4728:
4724:
4721:
4719:
4716:
4714:
4711:
4710:
4706:
4704:
4702:
4698:
4697:
4692:
4686:
4684:
4678:
4676:
4672:
4650:
4646:
4642:
4637:
4633:
4621:
4617:
4613:
4608:
4604:
4580:
4577:
4572:
4568:
4564:
4559:
4555:
4551:
4546:
4542:
4532:
4530:
4526:
4522:
4518:
4500:
4496:
4492:
4487:
4483:
4468:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4440:
4436:
4433:
4430:
4427:
4423:
4420:
4417:
4414:
4413:
4410:
4408:
4404:
4399:
4397:
4381:
4378:
4373:
4369:
4348:
4345:
4340:
4336:
4332:
4327:
4323:
4299:
4295:
4291:
4287:
4281:
4277:
4273:
4263:
4262:
4246:
4242:
4233:
4229:
4225:
4220:
4216:
4209:
4206:
4184:
4180:
4176:
4171:
4167:
4144:
4140:
4134:
4130:
4126:
4121:
4117:
4111:
4107:
4103:
4100:
4080:
4071:
4070:
4054:
4050:
4046:
4043:
4022:
4019:
4010:
4008:
4004:
3987:
3980:decreased as
3967:
3947:
3939:
3935:
3931:
3914:
3906:
3905:
3900:
3899:
3882:
3873:
3871:
3847:
3825:
3821:
3798:
3794:
3786:. Therefore
3771:
3767:
3762:
3756:
3752:
3748:
3745:
3740:
3736:
3731:
3725:
3721:
3698:
3694:
3690:
3686:
3680:
3676:
3672:
3669:
3666:
3661:
3657:
3652:
3646:
3642:
3634:
3619:
3616:
3613:
3610:
3588:
3584:
3580:
3575:
3571:
3567:
3562:
3558:
3554:
3549:
3545:
3541:
3538:
3535:
3528:
3511:
3507:
3500:
3497:
3492:
3488:
3481:
3476:
3472:
3465:
3462:
3457:
3453:
3446:
3439:
3435:
3431:
3426:
3422:
3419:
3415:
3411:
3407:
3402:
3398:
3389:
3386:
3383:
3380:
3374:
3367:
3363:
3359:
3354:
3350:
3347:
3343:
3339:
3335:
3330:
3326:
3317:
3314:
3309:
3305:
3298:
3293:
3289:
3285:
3280:
3276:
3272:
3269:
3262:
3261:
3245:
3241:
3218:
3214:
3193:
3171:
3167:
3163:
3141:
3137:
3133:
3113:
3110:
3090:
3069:
3065:
3061:
3039:
3035:
3031:
3009:
3005:
3001:
2979:
2975:
2971:
2948:
2944:
2921:
2917:
2894:
2890:
2867:
2863:
2854:
2839:
2819:
2816:
2797:
2791:
2788:
2782:
2772:
2769:
2766:
2758:
2755:
2748:
2742:
2738:
2735:
2711:
2708:
2705:
2702:
2682:
2662:
2659:
2656:
2653:
2633:
2625:
2624:
2622:
2619:
2617:
2613:
2609:
2605:
2599:
2598:
2592:
2591:
2587:
2582:
2578:
2574:
2570:
2569:Lens (optics)
2562:
2556:
2553:
2550:
2546:
2543:
2540:
2539:
2538:
2534:
2531:
2527:
2522:
2516:
2514:
2508:
2506:
2502:
2501:Luneburg lens
2498:
2494:
2490:
2486:
2485:more than one
2478:
2474:
2467:
2462:
2458:
2456:
2451:
2448:
2443:
2438:
2437:
2436:
2432:
2430:
2426:
2422:
2418:
2414:
2409:
2404:
2402:
2397:
2394:
2391:
2384:and Dη'=η'-η'
2365:
2357:
2353:
2351:
2346:
2338:
2336:
2334:
2330:
2326:
2322:
2318:
2317:Frits Zernike
2314:
2309:
2307:
2303:
2299:
2295:
2291:
2287:
2283:
2279:
2276:or very high
2275:
2271:
2267:
2263:
2258:
2242:
2238:
2234:
2231:
2228:
2223:
2219:
2198:
2195:
2192:
2189:
2186:
2183:
2163:
2143:
2140:
2137:
2134:
2131:
2111:
2097:
2080:
2077:
2072:
2068:
2064:
2061:
2056:
2052:
2048:
2042:
2037:
2033:
2025:
2024:
2020:
2003:
1997:
1994:
1991:
1985:
1982:
1977:
1973:
1969:
1963:
1958:
1954:
1946:
1945:
1941:
1924:
1918:
1915:
1912:
1906:
1903:
1898:
1894:
1890:
1884:
1879:
1875:
1867:
1866:
1862:
1845:
1842:
1836:
1833:
1828:
1824:
1820:
1815:
1811:
1803:
1802:
1798:
1794:
1777:
1774:
1768:
1765:
1760:
1756:
1752:
1747:
1743:
1735:
1734:
1730:
1727:"Defocus", a
1726:
1709:
1706:
1701:
1697:
1693:
1687:
1682:
1678:
1670:
1669:
1665:
1661:
1644:
1638:
1635:
1632:
1629:
1624:
1620:
1612:
1611:
1607:
1603:
1586:
1580:
1577:
1574:
1571:
1566:
1562:
1554:
1553:
1549:
1545:
1531:
1528:
1523:
1519:
1511:
1510:
1507:
1504:
1490:
1487:
1484:
1464:
1461:
1455:
1447:
1442:
1438:
1413: is even
1407:
1404:
1401:
1388:
1384:
1381:
1378:
1374:
1366:
1360:
1357:
1354:
1350:
1343:
1340:
1337:
1327:
1321:
1318:
1315:
1311:
1304:
1301:
1298:
1288:
1285:
1280:
1274:
1271:
1268:
1259:
1251:
1248:
1234:
1230:
1223:
1220:
1217:
1209:
1206:
1203:
1199:
1194:
1188:
1180:
1175:
1171:
1163:
1162:
1161:
1145:
1140:
1136:
1127:
1123:
1120:
1104:
1101:
1098:
1090:
1086:
1082:
1062:
1056:
1052:
1046:
1043:
1036:
1028:
1023:
1019:
1015:
1009:
1006:
1003:
995:
992:
987:
983:
975:
974:
973:
952:
948:
942:
939:
932:
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919:
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911:
905:
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899:
891:
886:
882:
874:
873:
872:
870:
865:
863:
859:
855:
851:
847:
846:Frits Zernike
843:
834:
826:
819:
817:
813:
811:
807:
805:
802:condition of
798:
793:
789:
788:principal ray
784:
774:
770:
768:
764:
759:
757:
753:
749:
740:
732:
725:
723:
718:
710:
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689:
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677:
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670:
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621:
619:
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588:
586:
582:
577:
573:
569:
567:
562:
558:
552:
551:
547:
543:
539:
536:and O'1R the
535:
531:
527:
517:
510:
508:
505:
501:
496:
494:
490:
489:focal lengths
486:
482:
478:
474:
470:
466:
460:
459:Lens (optics)
452:
447:
444:
443:
442:
438:
429:
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413:
408:
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399:
397:
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373:
372:
366:
364:
362:
358:
354:
350:
346:
344:
340:
336:
335:monochromatic
331:
330:of the lens.
329:
325:
324:image surface
321:
317:
309:
304:
297:
295:
293:
289:
285:
281:
277:
272:
269:
264:
262:
257:
253:
249:
245:
237:
232:
222:
219:
204:
194:
190:
186:
180:
179:
176:reads like a
174:This article
172:
163:
162:
153:
143:
139:
134:
130:
127:This article
125:
121:
116:
115:
106:
94:
92:
90:
84:This article
82:
73:
72:
67:
65:
58:
57:
52:
51:
46:
41:
32:
31:
19:
5453:
5433:
5428:
5419:
5412:
5403:
5396:
5388:
5383:
5371:. Retrieved
5367:
5358:
5351:
5335:
5331:
5323:
5319:
5311:
5306:
5298:
5293:
5285:
5280:
5272:
5268:
5267:J. Petzval,
5263:
5242:
5236:
5208:
5201:
5178:
5141:
5137:
5113:
5109:
5104:
5097:
5092:
5084:
5080:
5075:
5064:
4946:
4940:
4915:
4911:
4901:
4893:
4889:
4885:
4881:
4876:
4865:. Retrieved
4861:
4852:
4840:. Retrieved
4836:the original
4826:
4801:
4793:
4768:
4757:
4737:
4700:
4696:apochromatic
4694:
4690:
4687:
4682:
4679:
4674:
4670:
4533:
4528:
4524:
4520:
4516:
4473:
4402:
4400:
4395:
4264:
4260:
4072:
4068:
4011:
4006:
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3937:
3933:
3929:
3902:
3896:
3874:
3866:
2620:
2615:
2611:
2607:
2603:
2600:
2596:
2593:
2589:
2585:
2580:
2566:
2558:computation.
2535:
2529:
2525:
2520:
2517:
2512:
2509:
2496:
2493:Carathéodory
2484:
2482:
2460:
2455:Fraunhofer's
2453:
2446:
2441:
2433:
2428:
2424:
2420:
2416:
2412:
2407:
2405:
2398:
2392:
2366:
2362:
2349:
2344:
2342:
2310:
2259:
2103:
1505:
1429:
1117:, Φ is the
1084:
1080:
1078:
971:
869:even and odd
866:
858:coefficients
854:curve-fitted
839:
814:
809:
801:
796:
791:
787:
782:
779:
766:
762:
760:
745:
720:
705:Arthur König
697:
687:
678:
672:
668:
664:
659:
655:
652:
622:
618:Robert Blair
613:
609:
604:
601:
591:
589:
584:
580:
575:
571:
564:
560:
556:
553:
549:
545:
537:
533:
529:
525:
522:
497:
493:focal planes
487:, named the
481:image space.
480:
477:object space
476:
472:
471:unite in an
469:object point
468:
462:
440:
409:
405:
370:
347:
338:
334:
332:
327:
323:
319:
313:
273:
265:
247:
241:
214:
201:October 2020
198:
175:
147:
138:You can help
128:
100:
88:
85:
61:
54:
48:
47:Please help
44:
5452:section of
5373:22 February
5286:Astr. Nach.
5284:L. Seidel,
5083:and (1858)
4437:Violet Hg.
4007:anastigmats
3904:flint glass
3898:crown glass
3895:) is named
2549:normal lens
2321:diffraction
2282:propagation
1797:cylindrical
767:rectilinear
763:orthoscopic
640:Astigmatism
473:image point
391:Astigmatism
328:aberrations
320:image plane
187:to make it
5081:Phil.Mag.,
5061:Aberration
4918:: 69–174.
4867:2024-08-07
4749:References
2963:and radii
2621:Examples:
2608:vide supra
2577:dispersion
2457:condition.
2390:J. Petzval
1666:direction
1664:tangential
1608:direction
1548:mean value
867:There are
850:orthogonal
756:algorithms
581:front stop
576:exit pupil
504:Ernst Abbe
457:See also:
361:wavelength
353:dispersion
284:refraction
280:reflection
248:aberration
50:improve it
5228:162132153
5187:cite book
5181:. Berlin.
4924:0790-8113
4842:March 26,
4643:−
4614:−
3870:telescope
3749:−
3670:−
3498:−
3463:−
3420:−
3387:−
3348:−
3315:−
3286:−
2770:−
2401:L. Seidel
2274:gradients
2232:…
2199:π
2193:≤
2190:ϕ
2187:≤
2164:ϕ
2141:≤
2138:ρ
2135:≤
2112:ρ
2069:ρ
2062:−
2053:ρ
2043:×
2004:ϕ
1998:
1992:ρ
1983:−
1974:ρ
1964:×
1925:ϕ
1919:
1913:ρ
1904:−
1895:ρ
1885:×
1846:ϕ
1837:
1825:ρ
1821:×
1778:ϕ
1769:
1757:ρ
1753:×
1729:parabolic
1707:−
1698:ρ
1688:×
1645:ϕ
1639:
1633:ρ
1630:×
1587:ϕ
1581:
1575:ρ
1572:×
1529:×
1488:−
1456:ρ
1405:−
1382:−
1375:ρ
1358:−
1341:−
1319:−
1272:−
1249:−
1221:−
1200:∑
1189:ρ
1119:azimuthal
1102:≥
1057:ϕ
1047:
1037:ρ
1010:ϕ
1004:ρ
993:−
953:ϕ
943:
933:ρ
906:ϕ
900:ρ
614:aplanatic
585:back stop
561:diaphragm
339:chromatic
142:talk page
56:talk page
5464:Category
5436:, p. 340
5177:(1904).
5132:(1900).
4932:30078906
4707:See also
4470:Figure 6
3930:Herschel
3440:″
3416:′
3368:″
3344:′
3070:″
3040:′
3010:″
2980:′
2503:and the
2359:Figure 5
2302:fractals
2284:through
1606:sagittal
1503:is odd.
1397:if
1089:integers
776:Figure 4
649:Figure 2
566:aperture
519:Figure 1
298:Overview
288:caustics
178:textbook
150:May 2009
103:May 2016
5146:Bibcode
5054::
2724:, then
2489:Maxwell
2298:spatial
2270:cosines
2262:Fourier
1126:radians
542:pencils
540:of the
376:Defocus
189:neutral
183:Please
5251:
5226:
5216:
5048:
4930:
4922:
4814:
4781:
4424:Green
4266:since
3233:, and
3023:, and
2521:zones,
2260:As in
2104:where
1079:where
574:; the
412:piston
252:lenses
244:optics
140:. The
4928:JSTOR
4729:Notes
4457:454.1
4454:486.2
4451:546.1
4448:589.3
4445:656.3
4442:767.7
4159:; if
3713:, or
3526:; and
2604:zones
2296:fine
2266:sines
1122:angle
1091:with
797:scale
485:Gauss
359:with
256:light
97:page.
5416:See
5375:2013
5249:ISBN
5224:OCLC
5214:ISBN
5193:link
5142:1905
4920:ISSN
4844:2012
4812:ISBN
4779:ISBN
4379:<
3813:and
2936:and
2882:and
2695:and
2571:and
2376:, η'
2325:Airy
2311:The
2268:and
1430:and
1083:and
804:Airy
701:coma
568:stop
557:stop
502:and
491:and
416:tilt
414:and
386:Coma
337:and
316:lens
292:rays
286:and
5154:doi
5063:".
4808:130
4775:410
4685:).
4531:).
4527:or
3932:or
3872:.)
2304:or
2288:or
1995:sin
1916:cos
1834:sin
1766:cos
1636:sin
1578:cos
1477:if
1124:in
1044:sin
940:cos
667:or
658:or
587:).
559:or
242:In
5466::
5366:.
5343:^
5222:.
5189:}}
5185:{{
5166:^
5152:.
5140:.
5136:.
5121:^
4955:^
4926:.
4916:15
4914:.
4910:.
4860:.
4810:.
4777:.
4434:G'
4426:Hg
4415:A'
3206:,
3156:,
3126:,
3053:,
2993:,
2610:,
2507:.
2156:,
690:.
294:.
282:,
246:,
59:.
5377:.
5257:.
5230:.
5195:)
5160:.
5156::
5148::
5087:.
4934:.
4870:.
4846:.
4820:.
4787:.
4656:)
4651:b
4647:n
4638:a
4634:n
4630:(
4627:)
4622:b
4618:n
4609:c
4605:n
4601:(
4581:f
4578:=
4573:c
4569:f
4565:=
4560:b
4556:f
4552:=
4547:a
4543:f
4501:F
4497:f
4493:=
4488:C
4484:f
4431:F
4428:.
4421:D
4418:C
4382:f
4374:c
4370:f
4349:f
4346:=
4341:b
4337:f
4333:=
4328:a
4324:f
4300:1
4296:n
4292:d
4288:/
4282:2
4278:n
4274:d
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4243:/
4239:)
4234:2
4230:f
4226:+
4221:1
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4213:(
4210:=
4207:D
4185:2
4181:v
4177:=
4172:1
4168:v
4145:2
4141:f
4135:2
4131:v
4127:+
4122:1
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4112:1
4108:v
4104:=
4101:D
4081:D
4055:f
4051:/
4047:f
4044:d
4023:f
4020:d
3988:n
3968:v
3948:v
3915:f
3883:v
3848:f
3826:2
3822:f
3799:1
3795:f
3772:2
3768:n
3763:/
3757:1
3753:n
3746:=
3741:2
3737:f
3732:/
3726:1
3722:f
3699:1
3695:n
3691:d
3687:/
3681:2
3677:n
3673:d
3667:=
3662:2
3658:k
3653:/
3647:1
3643:k
3620:0
3617:=
3614:f
3611:d
3589:2
3585:n
3581:d
3576:2
3572:k
3568:+
3563:1
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3555:d
3550:1
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3542:=
3539:f
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3508:k
3504:)
3501:1
3493:2
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3482:+
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3450:(
3447:=
3444:)
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3427:/
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3408:r
3403:/
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3396:(
3393:)
3390:1
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3381:n
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3375:+
3372:)
3364:1
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3355:/
3351:1
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3336:r
3331:/
3327:1
3324:(
3321:)
3318:1
3310:1
3306:n
3302:(
3299:=
3294:2
3290:f
3281:1
3277:f
3273:=
3270:f
3246:2
3242:n
3219:1
3215:n
3194:f
3172:2
3168:n
3164:d
3142:1
3138:n
3134:d
3114:f
3111:d
3091:f
3066:2
3062:r
3036:2
3032:r
3006:1
3002:r
2976:1
2972:r
2949:2
2945:n
2922:1
2918:n
2895:2
2891:f
2868:1
2864:f
2840:n
2820:n
2817:d
2798:,
2792:n
2789:1
2783:=
2776:)
2773:1
2767:n
2764:(
2759:n
2756:d
2749:=
2743:f
2739:f
2736:d
2712:f
2709:d
2706:+
2703:f
2683:f
2663:n
2660:d
2657:+
2654:n
2634:n
2497:6
2386:0
2382:0
2378:0
2374:0
2370:0
2243:8
2239:a
2235:,
2229:,
2224:0
2220:a
2196:2
2184:0
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2132:0
2084:)
2081:1
2078:+
2073:2
2065:6
2057:4
2049:6
2046:(
2038:8
2034:a
2007:)
2001:(
1989:)
1986:2
1978:2
1970:3
1967:(
1959:7
1955:a
1928:)
1922:(
1910:)
1907:2
1899:2
1891:3
1888:(
1880:6
1876:a
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1843:2
1840:(
1829:2
1816:5
1812:a
1781:)
1775:2
1772:(
1761:2
1748:4
1744:a
1713:)
1710:1
1702:2
1694:2
1691:(
1683:3
1679:a
1648:)
1642:(
1625:2
1621:a
1590:)
1584:(
1567:1
1563:a
1532:1
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1465:0
1462:=
1459:)
1453:(
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1439:R
1408:m
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1252:1
1246:(
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1231:/
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1215:(
1210:0
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1204:k
1195:=
1192:)
1186:(
1181:m
1176:n
1172:R
1146:m
1141:n
1137:R
1105:m
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1060:)
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1050:(
1040:)
1034:(
1029:m
1024:n
1020:R
1016:=
1013:)
1007:,
1001:(
996:m
988:n
984:Z
956:)
949:m
946:(
936:)
930:(
925:m
920:n
916:R
912:=
909:)
903:,
897:(
892:m
887:n
883:Z
806:,
642:.
310:.
221:)
215:(
203:)
199:(
195:.
181:.
152:)
148:(
135:.
105:)
101:(
66:)
62:(
20:)
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