Optical aberration

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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,

2353:

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.

2429:

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 (

2584:

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

3929:). 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 739:. 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 2462: 292: 417: 660:, 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 496:

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.

2500:

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,

4512:). 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 805:

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.

2392:; 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. 4658:

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

484:, 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. 66: 109: 728: 822: 720: 220: 5034: 814: 2316:, 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 ( 25: 2377:, 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 2345: 762: 156: 2504:, 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. 643:

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

635: 505: 4692:), 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. 252:. 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. 4508:). 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 ( 688:

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

1414: 2341:; 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. 4657:

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

2433:

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.

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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

2384:

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

512:

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

259:

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

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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

2590:

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

783:

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

642:

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

591:

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

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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

495:

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

3916:

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

2572:

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

2499:

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

2428:

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

2356:

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

710:

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

247:

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

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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,

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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

4254:

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,

2507:

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

2423:

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;

774:

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

612:

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

3856:

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

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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

2352:

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

804:

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

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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

2336:

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

4669:

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

687:

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

3513: 2508:

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

4688:. 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 ( 2521:

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.

3917:

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

4383:, 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 769:

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

407:

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.

2434:

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

2525:

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:

543:

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

651:

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

568:

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 (

2385:

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.

664:

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.

395:

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.

2603:); 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 1155: 3949:), 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, 2536:; 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 2512:

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*,

801:), 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. 3989:

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

2546:

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

1063: 2488:, 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 956: 788:

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

2797: 85:

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

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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

596:

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

4025:

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 (

2006: 1927: 2416:

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

797:

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

2083: 559:

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

2615:

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

521:). 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 1848: 1780: 2320:). 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 1647: 1589: 579:

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

3254: 4655: 3700: 3590: 1712: 533:

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

583:

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.

3773: 2388:

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

4146: 2472:

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

2530:

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.

2198: 2595:

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 (

4730:

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.

4580: 2244: 2452:

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.

1464: 4246: 3070: 3010: 2143: 4456: 3829:

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

2568:), 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 1409:{\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}}} 4348: 4301: 1531: 4502: 4186: 3040: 2980: 330:. 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 4846: 4381: 315:). 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 2357:

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'

1147: 295:

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

2573:

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

1104: 4820: 3173: 3143: 3827: 3800: 3247: 3220: 2950: 2923: 2896: 2869: 2163: 2111: 4054: 3619: 2711: 2662: 1490: 352:. Because of dispersion, different wavelengths of light come to focus at different points. Chromatic aberration does not appear when monochromatic light is used. 4022: 3113: 2819: 4080: 3987: 3967: 3947: 3914: 3882: 3847: 3193: 3090: 2839: 2682: 2633: 2716: 4464:

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

2448:(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 671:

was probably the discoverer of astigmation; the position of the astigmatic image lines was determined by Thomas Young; and the theory was developed by

2556:

In optical systems composed of lenses, the position, magnitude and errors of the image depend upon the refractive indices of the glass employed (see

2261:, a wavefront may be perfectly represented by a sufficiently large number of higher-order Zernike polynomials. However, wavefronts with very steep 3857:

determined by Chester More Hall in 1728, Klingenstierna in 1754 and by Dollond in 1757, who constructed the celebrated achromatic telescopes. (See

967: 5181: 866: 5049: 5054: 592:

will be reproduced, but attended by aberrations comparable in magnitude to ON. These aberrations are avoided if, according to Abbe, the

76: 1938: 1859: 609:

to characterize a superior achromatism, and, subsequently, by many writers to denote freedom from spherical aberration as well.

430:

Chromatic aberration occurs when different wavelengths are not focussed to the same point. Types of chromatic aberration are:

5206: 4504:, is, according to the experiments of Sir G. G. Stokes (Proc. Roy. Soc., 1878), the most suitable for visual instruments ( 2017: 4824: 4056:) be greater than that of the same lens, this being made of the more dispersive of the two glasses employed, it is termed 38: 2461: 5241: 1795: 1727: 2591:

image point of one color, another colour produces a disk of confusion; this is similar to the confusion caused by two

2441:

The condition for freedom from coma in the third order is also of importance for telescope objectives; it is known as

4804: 4771: 206: 52: 3508:{\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}} 2438:

All these rules are valid, inasmuch as the thicknesses and distances of the lenses are not to be taken into account.

601:

then becomes sin u'1/h1=sin u'2/h2. A system fulfilling this condition and free from spherical aberration is called

420:

Comparison of an ideal image of a ring (1) and ones with only axial (2) and only transverse (3) chromatic aberration

1604: 1546: 4706: 616: 307:, light from any given point on an object would pass through the lens and come together at a single point in the 4585: 3626: 3520: 1662: 3705: 2540:: this is corrected more with regard to aperture; objectives for groups more with regard to the field of view. 853:, thus individual aberration contributions to an overall wavefront may be isolated and quantified separately. 537:

of radius O'1R, and in a parallel plane at O'2 another one of radius O'2R2; between these two is situated the

4085: 622: 379: 2294: 2168: 5458: 5445:

website, Michael W. Davidson, Mortimer Abramowitz, Olympus America Inc., and The Florida State University

4526: 2203: 2400:

derived the aberrations of the third order; and in later times the method was pursued by Clerk Maxwell (

2378: 1422: 4191: 2116: 2481: 4678:

satisfying the sine condition for several colors, which therefore, according to his definition, were

4387:

and depend upon the aperture and focal length in the same manner as the primary chromatic errors do.

4350:, then for a third color, c, the focal length is different; that is, if c lies between a and b, then 2519:

corrected for the angle of aperture u* (the height of incidence h*) or the angle of field of view w*.

2389: 606: 121: 4523:

Should there be in two lenses in contact the same focal lengths for three colours a, b, and c, i.e.

255:

An image-forming optical system with aberration will produce an image which is not sharp. Makers of

4307: 4258: 2577:

when it shows the same kind of chromatic error as a thin positive lens, otherwise it is said to be

2420:

1905, 18, p. 941) founded his theory of aberrations on the differential geometry of surfaces.

1503: 784:

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

5233:

Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light

5438: 4467: 4151: 2286: 705: 130: 44: 4353: 416: 291: 4701: 2274: 1652: 1120: 857: 4763: 4756: 2564:, above). Since the index of refraction varies with the color or wavelength of the light (see 2308:

to evaluate the point image of an aberrated optical system taking into account the effects of

617:

Aberration of lateral object points (points beyond the axis) with narrow pencils — astigmatism

3849:

be not zero (equation 2)), and the dispersive powers must also be different (according to 4).

3045: 2985: 2443: 1495:

The first few Zernike polynomials, multiplied by their respective fitting coefficients, are:

1083: 166: 5039:

One or more of the preceding sentences incorporates text from a publication now in the

4788: 3148: 3118: 3015: 2955: 849:

that individually represent different types of aberrations. These Zernike coefficients are

829:

Circular wavefront profiles associated with aberrations may be mathematically modeled using

586: 499: 173: 126: 5232: 5134: 4796: 4751: 3886: 3805: 3778: 3225: 3198: 2928: 2901: 2874: 2847: 2148: 2096: 1594: 850: 473: 425: 369: 337: 296: 268: 224: 4028: 3595: 2687: 2638: 8: 2607:(This explains the gigantic focal lengths in vogue before the discovery of achromatism.) 2565: 2477: 2313: 2301: 1469: 792: 488: 341: 331: 256: 5138: 4004: 3095: 2801: 2369:, of the point O at some distance from the axis could be constructed. Writing Dξ'=ξ'-ξ' 5175: 4916: 4065: 3972: 3952: 3932: 3899: 3867: 3832: 3178: 3075: 2824: 2667: 2618: 2321: 2312:. The perfect point image in the presence of diffraction had already been described by 830: 736: 364: 264: 5237: 5212: 5202: 5118: 4908: 4885:. Reference may also be made to the treatise of Czapski-Eppenstein, pp. 155–161. 4800: 4767: 4414: 2266: 672: 5352: 817:

Image plane of a flat-top beam under the effect of the first 21 Zernike polynomials.

5168:

Die bilderzeugung in optischen Instrumenten vom Standpunkte der geometrischen Optik

5142: 4711: 2493: 2465: 2270: 2250: 740: 530: 389: 345: 276: 5380:, p. 373; K. Schwarzschild, Göttingen. Akad. Abhandl., 1905, 4, Nos. 2 and 3 5163: 4789: 4395: 2543: 2533: 2404:

1874–1875; (see also the treatises of R. S. Heath and L. A. Herman), M. Thiesen (

2282: 400: 384: 249: 181: 693: 682: 453: 177: 744: 5452: 5216: 5146: 5045: 5040: 4912: 2557: 2489: 2305: 834: 825:

Effect of Zernike aberrations in Log scale. The intensity minima are visible.

689: 447: 404: 374: 304: 240: 841:

over a circle of unit radius. A complex, aberrated wavefront profile may be

219: 4684: 2412:

1895, 21, p. 410), and particularly successfully by K. Schwarzschild (

2324:, and in characterizing optical systems with respect to their aberrations. 1058:{\displaystyle Z_{n}^{-m}(\rho ,\phi )=R_{n}^{m}(\rho )\,\sin(m\,\phi ),\!} 842: 668: 477: 280: 5196: 5058:. Vol. 1 (11th ed.). Cambridge University Press. pp. 54–61. 2332:

The preceding review of the several errors of reproduction belongs to the

775:

diminished; in practice, this generally occurs. This ray, named by Abbe a

108: 4762:(2nd ed.). Philadelphia: Harcourt Brace College Publishers. p. 3892: 2537: 2309: 2289:

definition in the wavefront. In this case, other fitting methods such as

1785: 951:{\displaystyle Z_{n}^{m}(\rho ,\phi )=R_{n}^{m}(\rho )\,\cos(m\,\phi )\!} 846: 628: 587:

Aberration of elements, i.e. smallest objects at right angles to the axis

500:

Aberration of axial points (spherical aberration in the restricted sense)

481: 4920: 4896: 1852:"45° Astigmatism", a cylindrical shape oriented at ±45° from the X axis 727: 605:(Greek a-, privative, plann, a wandering). This word was first used by 2517:

attend smaller values of w. The practical optician names such systems:

2278: 1536: 838: 492: 349: 272: 4394:, the abscissae are focal lengths, and the ordinates wavelengths. The 2792:{\displaystyle {\dfrac {df}{f}}={\dfrac {dn}{(n-1)}}={\dfrac {1}{n}},} 821: 692:

in the narrower sense only; other errors of coma have been treated by

3858: 2262: 5122: 719: 2476:

such pair of planes (this was proven with increasing generality by

2290: 1717: 1117:, and ρ is the normalized radial distance. The radial polynomials 813: 554: 735:

Even if the image is sharp, it may be distorted compared to ideal

1107: 1077: 860:

Zernike polynomials. The even Zernike polynomials are defined as

4188:(e.g. if the lenses be made of the same glass), this reduces to 2898:

be the powers corresponding to the lenses of refractive indices

5258:

Bericht uber die Ergebnisse einiger dioptrischer Untersuchungen

4847:"Aberration: Understanding optical imperfections in telescopes" 2317: 2258: 1114: 232: 675:. A bibliography by P. Culmann is given in Moritz von Rohr's 2281:, are not well modeled by Zernike polynomials, which tend to 1110: 683:

Aberration of lateral object points with broad pencils — coma

244: 4455: 2344: 1931:"X-Coma", comatic image flaring in the horizontal direction 761: 2254: 2664:, and the powers, or reciprocals of the focal lengths, be 2010:"Y-Coma", comatic image flaring in the vertical direction 2001:{\displaystyle a_{7}\times (3\rho ^{2}-2)\rho \sin(\phi )} 1922:{\displaystyle a_{6}\times (3\rho ^{2}-2)\rho \cos(\phi )} 3896:. For the construction of an achromatic collective lens ( 634: 504: 2515:

zones of astigmatism, curvature of field and distortion,

2484:

in 1926, see summary in Walther, A., J. Opt. Soc. Am. A

279:

discuss the general features of reflected and refracted

75:

is largely based on an article in the out-of-copyright

5123:"Allgemeine Theorie der monochromat. Aberrationen, etc" 2561: 2456: 517:); and conversely with a dispersive surface or lenses ( 2395:

The aberrations can also be expressed by means of the

1400: 1384: 699: 5288:

Theorie und Geschichte des photographischen Objectivs

4672:

Theorie und Geschichte des photographischen Objectivs

4588: 4529: 4470: 4392:

Theorie und Geschichte des photographischen Objectivs

4356: 4310: 4261: 4194: 4154: 4088: 4068: 4031: 4007: 3975: 3955: 3935: 3902: 3870: 3835: 3808: 3781: 3708: 3629: 3598: 3523: 3257: 3228: 3201: 3181: 3151: 3121: 3098: 3078: 3048: 3018: 2988: 2958: 2931: 2904: 2877: 2850: 2827: 2804: 2775: 2741: 2721: 2719: 2690: 2670: 2641: 2621: 2468:

assist in the elimination of atmospheric distortion.

2206: 2171: 2151: 2119: 2099: 2078:{\displaystyle a_{8}\times (6\rho ^{4}-6\rho ^{2}+1)} 2020: 1941: 1862: 1798: 1730: 1665: 1607: 1549: 1506: 1472: 1425: 1158: 1123: 1086: 970: 869: 696:

and Moritz von Rohr, and later by Allvar Gullstrand.

441: 4707:

Optical telescope § The five Seidel aberrations

2327: 3249:with the color. Then the following relations hold: 1651:"Y-Tilt", the deviation of the overall beam in the 1593:"X-Tilt", the deviation of the overall beam in the 845:with Zernike polynomials to yield a set of fitting 472:The introduction of simple auxiliary terms, due to 5406: 4755: 4750: 4649: 4574: 4496: 4375: 4342: 4295: 4240: 4180: 4140: 4074: 4048: 4016: 3981: 3961: 3941: 3908: 3876: 3841: 3821: 3794: 3767: 3694: 3613: 3584: 3507: 3241: 3214: 3187: 3167: 3137: 3107: 3084: 3064: 3034: 3004: 2974: 2944: 2917: 2890: 2863: 2833: 2813: 2791: 2705: 2676: 2656: 2627: 2238: 2192: 2157: 2137: 2105: 2077: 2000: 1921: 1842: 1774: 1706: 1641: 1583: 1525: 1484: 1458: 1408: 1141: 1098: 1057: 950: 263:Aberration can be analyzed with the techniques of 5390: 5113: 5111: 1843:{\displaystyle a_{5}\times \rho ^{2}\sin(2\phi )} 1775:{\displaystyle a_{4}\times \rho ^{2}\cos(2\phi )} 1230: 1229: 1228: 1186: 1149:have no azimuthal dependence, and are defined as 1054: 947: 5450: 5194: 4795:. Cambridge: John Wiley & Sons Inc. p. 2551: 16:Deviation from perfect paraxial optical behavior 5409:Grundzuge der Theorie der optischen Instrumente 5393:Grundzüge der Theorie der optischen Instrumente 4304:50%). If, therefore, for two colors, a and b, 808: 572:); if entirely in front, it is the exit pupil ( 360:The most common monochromatic aberrations are: 5108: 4744: 437:Lateral (or "transverse") chromatic aberration 434:Axial (or "longitudinal") chromatic aberration 4881:, xxi. 325, by means of Sir W. R. Hamilton's 3994:and were employed by P. Rudolph in the first 2165:is the azimuthal angle around the pupil with 4062:For two thin lenses separated by a distance 3864:Glass with weaker dispersive power (greater 1720:wavefront resulting from being out of focus 1642:{\displaystyle a_{2}\times \rho \sin(\phi )} 1584:{\displaystyle a_{1}\times \rho \cos(\phi )} 355: 5341:Die Bilderzeugung in optischen Instrumenten 5201:(2nd ed.). San Diego: Academic Press. 5087:A Course of Lectures on Natural Philosophy. 4901:The Transactions of the Royal Irish Academy 677:Die Bilderzeugung in optischen Instrumenten 53:Learn how and when to remove these messages 5439:Microscope Objectives: Optical Aberrations 5180:: CS1 maint: location missing publisher ( 5158: 5156: 5117: 4650:{\displaystyle (n_{c}-n_{b})(n_{a}-n_{b})} 3695:{\displaystyle k_{1}/k_{2}=-dn_{2}/dn_{1}} 3585:{\displaystyle df=k_{1}dn_{1}+k_{2}dn_{2}} 2601:Monochromatic Aberration of the Axis Point 1707:{\displaystyle a_{3}\times (2\rho ^{2}-1)} 1361: 714: 239:is a property of optical systems, such as 227:. 2: A lens with less chromatic aberration 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: 2246:are the wavefront errors in wavelengths. 1376: 1319: 1280: 1253: 1044: 1031: 940: 927: 799:symmetrical or holosymmetrical objectives 513:long as the angle u2 is greater than u1 ( 207:Learn how and when to remove this message 5335: 5333: 5229: 5162: 5044: 4963: 4961: 4959: 4957: 4955: 4953: 4951: 4949: 4947: 4945: 4894: 4786: 4454: 3768:{\displaystyle f_{1}/f_{2}=-n_{1}/n_{2}} 2460: 2343: 837:in the 1930s, Zernike's polynomials are 820: 812: 760: 726: 718: 633: 503: 415: 411: 290: 218: 5353:"New Laser Improves VLT's Capabilities" 5153: 4888: 4582:, then the relative partial dispersion 4141:{\displaystyle D=v_{1}f_{1}+v_{2}f_{2}} 711:surface is concave towards the system. 5451: 3925:the latter being the best vide supra, 3890:; that with greater dispersive power, 750:Systems free of distortion are called 5421:A. Konig in M. v. Rohr's collection, 5330: 5280: 4942: 4933: 2193:{\displaystyle 0\leq \phi \leq 2\pi } 5230:Born, Max; Wolf, Emil (1999-10-13). 4662:is derived; there is yet a residual 4390:In fig. 6, taken from M. von Rohr's 2457:Practical elimination of aberrations 2297:may yield improved fitting results. 2113:is the normalized pupil radius with 754:(orthos, right, skopein to look) or 456:, rays of light proceeding from any 149: 102: 59: 18: 4821:"Comparison of Optical Aberrations" 4780: 4575:{\displaystyle f_{a}=f_{b}=f_{c}=f} 2339:infinitely small of a certain order 2239:{\displaystyle a_{0},\ldots ,a_{8}} 2087:"Third order spherical aberration" 961:and the odd Zernike polynomials as 700:Curvature of the field of the image 452:In a perfect optical system in the 322:Aberrations fall into two classes: 13: 4938:(in German). Göttingen: Dieterich. 4398:used are shown in adjacent table. 2841:the dispersive power of the glass. 2480:in 1858, by Bruns in 1895, and by 2408:1890, 35, p. 804), H. Bruns ( 2382:the numerical orders of the image, 1459:{\displaystyle R_{n}^{m}(\rho )=0} 581:perpendicular height of incidence, 442:Theory of monochromatic aberration 399:In addition to these aberrations, 14: 5470: 5432: 4241:{\displaystyle D=(f_{1}+f_{2})/2} 4082:the condition for achromatism is 2328:Analytic treatment of aberrations 2138:{\displaystyle 0\leq \rho \leq 1} 34:This article has multiple issues. 5032: 4666:but it can always be neglected. 2844:Two thin lenses in contact: let 2502:Lehrbuch der geometrischen Optik 627:For Astigmatism of the eye, see 613:light reaching the image plane. 154: 107: 64: 23: 5415: 5399: 5383: 5370: 5345: 5318: 5306: 5293: 5267: 5250: 5223: 5188: 5091: 5079: 4823:. Edmund Optics. Archived from 4724: 2318:Extended Nijboer-Zernike theory 2269:structure, such as produced by 2200:, and the fitting coefficients 1382: 42:or discuss these issues on the 5236:. Cambridge University Press. 5062: 4934:Gauss, Carl Friedrich (1841). 4927: 4863: 4839: 4813: 4644: 4618: 4615: 4589: 4227: 4201: 3492: 3473: 3457: 3438: 3432: 3384: 3381: 3366: 3360: 3312: 3309: 3290: 2821:is called the dispersion, and 2764: 2752: 2072: 2034: 1995: 1989: 1977: 1955: 1916: 1910: 1898: 1876: 1837: 1828: 1769: 1760: 1701: 1679: 1636: 1630: 1578: 1572: 1447: 1441: 1352: 1335: 1323: 1320: 1313: 1296: 1284: 1281: 1266: 1254: 1244: 1234: 1215: 1203: 1180: 1174: 1048: 1038: 1028: 1022: 1001: 989: 944: 934: 924: 918: 897: 885: 731:Fig. 3b: Pincushion distortion 1: 4737: 4660:achromatism of a higher order 4343:{\displaystyle f_{a}=f_{b}=f} 4296:{\displaystyle dn_{2}/dn_{1}} 2575:chromatically under-corrected 2552:Chromatic or color aberration 1526:{\displaystyle a_{0}\times 1} 779:(not to be confused with the 623:Astigmatism (optical systems) 180:in tone and meet Knowledge's 82:, which was produced in 1911. 5391:Czapski; Eppenstein (1903). 5068:Maxwell, James Clerk (1856) 4680:aplanatic for several colors 3092:denote the total power, and 2414:Göttingen. Akad. Abhandl., 2295:singular value decomposition 1788:shape along the X or Y axis 809:Zernike model of aberrations 344:, the variation of a lens's 7: 5407:Czapski-Eppenstein (1903). 5290:, Berlin, 1899, p. 248 4897:"Theory of Systems of Rays" 4695: 4518:freedom from chemical focus 4497:{\displaystyle f_{C}=f_{F}} 4181:{\displaystyle v_{1}=v_{2}} 3998:(photographic objectives). 2334:Abbe theory of aberrations, 286: 120:to comply with Knowledge's 10: 5475: 5376:A. Konig in M. von Rohr's 5097:Gullstrand, Allvar (1890) 4936:Dioptrische Untersuchungen 4879:Leipzig. Math. Phys. Ber. 4851:www.jameswebbdiscovery.com 4754:; Wheeler, Gerald (1992). 4376:{\displaystyle f_{c}<f} 2410:Leipzig. Math. Phys. Ber., 723:Fig. 3a: Barrel distortion 703: 626: 620: 454:classical theory of optics 445: 423: 223:1: Imaging by a lens with 5195:Schroeder, D. J. (2000). 4787:Guenther, Robert (1990). 4682:; such systems he termed 1142:{\displaystyle R_{n}^{m}} 662:astigmatic image surfaces 356:Monochromatic aberrations 311:(or, more generally, the 5313:München. Akad. Sitzber., 5301:Munchen. Acad. Abhandl. 5147:10.1002/andp.19053231504 5099:Skand. Arch. f. Physiol. 4895:Hamilton, W. R. (1828). 4875:Berlin. Phys. Ges. Verh. 4873:; and (1892) xxxv. 799; 4717: 4510:pure actinic achromatism 3927:Monochromatic Aberration 2562:Monochromatic aberration 2406:Berlin. Akad. Sitzber., 2402:Proc. London Math. Soc., 539:disk of least confusion. 523:longitudinal aberration, 248:from the predictions of 133:may contain suggestions. 118:may need to be rewritten 5327:, Leipzig, 1895-6-7-8-9 5303:, 1891, 17, p. 519 5264:1857, vols. xxiv. xxvi. 5055:Encyclopædia Britannica 4883:characteristic function 3065:{\displaystyle r''_{2}} 3005:{\displaystyle r''_{1}} 2397:characteristic function 1535:"Piston", equal to the 1099:{\displaystyle n\geq m} 715:Distortion of the image 706:Petzval field curvature 654:first principal section 260:accurate observations. 78:Encyclopædia Britannica 5389:Formulae are given in 5085:Young, Thomas (1807), 4871:Berlin. Akad. Sitzber. 4702:Aberrations of the eye 4651: 4576: 4498: 4460: 4377: 4344: 4297: 4250:condition for oculars. 4242: 4182: 4142: 4076: 4050: 4018: 3983: 3963: 3943: 3910: 3878: 3843: 3823: 3796: 3769: 3696: 3615: 3586: 3509: 3243: 3216: 3189: 3169: 3168:{\displaystyle dn_{2}} 3139: 3138:{\displaystyle dn_{1}} 3109: 3086: 3066: 3036: 3035:{\displaystyle r'_{2}} 3006: 2976: 2975:{\displaystyle r'_{1}} 2946: 2919: 2892: 2865: 2835: 2815: 2793: 2707: 2678: 2658: 2629: 2469: 2349: 2279:aerodynamic flowfields 2275:atmospheric turbulence 2240: 2194: 2159: 2139: 2107: 2079: 2002: 1923: 1844: 1776: 1708: 1643: 1585: 1527: 1486: 1460: 1410: 1227: 1143: 1100: 1059: 952: 826: 818: 766: 732: 724: 639: 509: 487:The investigations of 421: 300: 228: 5443:Molecular Expressions 5325:Beiträge zur Dioptrik 5315:1898, 28, p. 395 5262:Akad. Sitzber., Wien, 4758:Physics: A World View 4652: 4577: 4499: 4458: 4378: 4345: 4298: 4243: 4183: 4143: 4077: 4051: 4019: 3984: 3964: 3944: 3923:Fraunhofer Condition, 3911: 3879: 3844: 3824: 3822:{\displaystyle f_{2}} 3797: 3795:{\displaystyle f_{1}} 3770: 3697: 3616: 3587: 3510: 3244: 3242:{\displaystyle n_{2}} 3217: 3215:{\displaystyle n_{1}} 3190: 3170: 3140: 3110: 3087: 3067: 3037: 3007: 2977: 2947: 2945:{\displaystyle n_{2}} 2920: 2918:{\displaystyle n_{1}} 2893: 2891:{\displaystyle f_{2}} 2866: 2864:{\displaystyle f_{1}} 2836: 2816: 2794: 2708: 2679: 2659: 2630: 2570:chromatic differences 2464: 2436:Herschel's condition. 2361:, with coordinates ξ' 2347: 2241: 2195: 2160: 2158:{\displaystyle \phi } 2140: 2108: 2106:{\displaystyle \rho } 2080: 2003: 1924: 1845: 1777: 1709: 1644: 1586: 1528: 1487: 1461: 1411: 1187: 1144: 1101: 1060: 953: 824: 816: 764: 730: 722: 637: 552:; Abbe used the term 507: 419: 412:Chromatic aberrations 338:Chromatic aberrations 294: 222: 5260:, Buda Pesth, 1843; 4690:compensating oculars 4586: 4527: 4506:optical achromatism, 4468: 4354: 4308: 4259: 4192: 4152: 4086: 4066: 4049:{\displaystyle df/f} 4029: 4005: 3973: 3953: 3933: 3900: 3868: 3833: 3806: 3779: 3706: 3627: 3614:{\displaystyle df=0} 3596: 3521: 3255: 3226: 3199: 3179: 3149: 3119: 3096: 3076: 3046: 3016: 2986: 2956: 2929: 2902: 2875: 2848: 2825: 2802: 2717: 2706:{\displaystyle f+df} 2688: 2668: 2657:{\displaystyle n+dn} 2639: 2619: 2204: 2169: 2149: 2117: 2097: 2018: 1939: 1860: 1796: 1784:"0° Astigmatism", a 1728: 1663: 1605: 1547: 1504: 1470: 1423: 1156: 1121: 1084: 968: 867: 851:linearly independent 468:is reproduced in an 464:; and therefore the 426:Chromatic aberration 370:Spherical aberration 297:spherical aberration 225:chromatic aberration 174:improve this article 5277:, 1856, p. 289 5198:Astronomical optics 5139:1905AnP...323..941G 5133:(18). Upsala: 941. 5074:Quart. Journ. Math. 4877:; Bruns, H. (1895) 4869:Thiesen, M. (1890) 4827:on December 6, 2011 4385:secondary spectrum, 3592:. For achromatism 3431: 3407: 3359: 3335: 3061: 3031: 3001: 2971: 2586:stable achromatism. 2304:were introduced by 1485:{\displaystyle n-m} 1440: 1173: 1138: 1021: 988: 917: 884: 831:Zernike polynomials 489:James Clerk Maxwell 332:monochromatic light 267:. The articles on 257:optical instruments 5459:Geometrical optics 5299:S. Finterswalder, 5127:Annalen der Physik 5119:Gullstrand, Allvar 4752:Kirkpatrick, Larry 4664:tertiary spectrum, 4647: 4572: 4514:actinic correction 4494: 4461: 4373: 4340: 4293: 4238: 4178: 4138: 4072: 4046: 4017:{\displaystyle df} 4014: 4001:Instead of making 3979: 3959: 3939: 3906: 3874: 3839: 3819: 3792: 3765: 3692: 3621:, hence, from (3), 3611: 3582: 3505: 3419: 3395: 3347: 3323: 3239: 3212: 3185: 3165: 3135: 3108:{\displaystyle df} 3105: 3082: 3072:respectively; let 3062: 3049: 3032: 3019: 3002: 2989: 2972: 2959: 2942: 2915: 2888: 2861: 2831: 2814:{\displaystyle dn} 2811: 2789: 2784: 2769: 2735: 2703: 2674: 2654: 2635:, and for another 2625: 2605:relative aperture. 2470: 2350: 2322:numerical aperture 2302:circle polynomials 2236: 2190: 2155: 2135: 2103: 2075: 1998: 1919: 1840: 1772: 1704: 1639: 1581: 1523: 1482: 1456: 1426: 1406: 1404: 1388: 1159: 1139: 1124: 1096: 1055: 1007: 971: 948: 903: 870: 827: 819: 767: 758:(straight lines). 737:pinhole projection 733: 725: 658:meridional section 645:axis of the pencil 640: 527:lateral aberration 510: 422: 334:, hence the name. 301: 265:geometrical optics 229: 5423:Die Bilderzeugung 5378:Die Bilderzeugung 5208:978-0-08-049951-2 5105:, 53, pp. 2, 185. 4453: 4452: 4075:{\displaystyle D} 3982:{\displaystyle n} 3962:{\displaystyle v} 3942:{\displaystyle v} 3909:{\displaystyle f} 3877:{\displaystyle v} 3842:{\displaystyle f} 3188:{\displaystyle f} 3085:{\displaystyle f} 2834:{\displaystyle n} 2783: 2768: 2734: 2677:{\displaystyle f} 2628:{\displaystyle n} 2544:Long focus lenses 2466:Laser guide stars 2450:Petzval equation, 2431:under-correction, 2267:spatial frequency 2091: 2090: 1539:of the wavefront 1403: 1387: 1359: 772:center of gravity 673:Allvar Gullstrand 535:disk of confusion 217: 216: 209: 199: 198: 182:quality standards 148: 147: 122:quality standards 101: 100: 57: 5466: 5426: 5419: 5413: 5412: 5403: 5397: 5396: 5387: 5381: 5374: 5368: 5367: 5365: 5363: 5357:ESO Announcement 5349: 5343: 5337: 5328: 5322: 5316: 5310: 5304: 5297: 5291: 5284: 5278: 5271: 5265: 5254: 5248: 5247: 5227: 5221: 5220: 5192: 5186: 5185: 5179: 5171: 5164:von Rohr, Moritz 5160: 5151: 5150: 5115: 5106: 5103:Arch. f. Ophth. 5095: 5089: 5083: 5077: 5066: 5060: 5059: 5038: 5036: 5035: 5029: 4940: 4939: 4931: 4925: 4924: 4892: 4886: 4867: 4861: 4860: 4858: 4857: 4843: 4837: 4836: 4834: 4832: 4817: 4811: 4810: 4794: 4784: 4778: 4777: 4761: 4748: 4731: 4728: 4712:Wavefront coding 4656: 4654: 4653: 4648: 4643: 4642: 4630: 4629: 4614: 4613: 4601: 4600: 4581: 4579: 4578: 4573: 4565: 4564: 4552: 4551: 4539: 4538: 4503: 4501: 4500: 4495: 4493: 4492: 4480: 4479: 4401: 4400: 4396:Fraunhofer lines 4382: 4380: 4379: 4374: 4366: 4365: 4349: 4347: 4346: 4341: 4333: 4332: 4320: 4319: 4302: 4300: 4299: 4294: 4292: 4291: 4279: 4274: 4273: 4247: 4245: 4244: 4239: 4234: 4226: 4225: 4213: 4212: 4187: 4185: 4184: 4179: 4177: 4176: 4164: 4163: 4147: 4145: 4144: 4139: 4137: 4136: 4127: 4126: 4114: 4113: 4104: 4103: 4081: 4079: 4078: 4073: 4058:hyper-chromatic. 4055: 4053: 4052: 4047: 4042: 4023: 4021: 4020: 4015: 3988: 3986: 3985: 3980: 3968: 3966: 3965: 3960: 3948: 3946: 3945: 3940: 3915: 3913: 3912: 3907: 3883: 3881: 3880: 3875: 3848: 3846: 3845: 3840: 3828: 3826: 3825: 3820: 3818: 3817: 3801: 3799: 3798: 3793: 3791: 3790: 3774: 3772: 3771: 3766: 3764: 3763: 3754: 3749: 3748: 3733: 3732: 3723: 3718: 3717: 3701: 3699: 3698: 3693: 3691: 3690: 3678: 3673: 3672: 3654: 3653: 3644: 3639: 3638: 3620: 3618: 3617: 3612: 3591: 3589: 3588: 3583: 3581: 3580: 3568: 3567: 3555: 3554: 3542: 3541: 3514: 3512: 3511: 3506: 3504: 3503: 3485: 3484: 3469: 3468: 3450: 3449: 3427: 3418: 3403: 3394: 3355: 3346: 3331: 3322: 3302: 3301: 3286: 3285: 3273: 3272: 3248: 3246: 3245: 3240: 3238: 3237: 3221: 3219: 3218: 3213: 3211: 3210: 3194: 3192: 3191: 3186: 3174: 3172: 3171: 3166: 3164: 3163: 3144: 3142: 3141: 3136: 3134: 3133: 3114: 3112: 3111: 3106: 3091: 3089: 3088: 3083: 3071: 3069: 3068: 3063: 3057: 3041: 3039: 3038: 3033: 3027: 3011: 3009: 3008: 3003: 2997: 2981: 2979: 2978: 2973: 2967: 2951: 2949: 2948: 2943: 2941: 2940: 2924: 2922: 2921: 2916: 2914: 2913: 2897: 2895: 2894: 2889: 2887: 2886: 2870: 2868: 2867: 2862: 2860: 2859: 2840: 2838: 2837: 2832: 2820: 2818: 2817: 2812: 2798: 2796: 2795: 2790: 2785: 2776: 2770: 2767: 2750: 2742: 2736: 2730: 2722: 2712: 2710: 2709: 2704: 2683: 2681: 2680: 2675: 2663: 2661: 2660: 2655: 2634: 2632: 2631: 2626: 2494:Maxwell fish-eye 2424:(5) distortion. 2253:synthesis using 2245: 2243: 2242: 2237: 2235: 2234: 2216: 2215: 2199: 2197: 2196: 2191: 2164: 2162: 2161: 2156: 2144: 2142: 2141: 2136: 2112: 2110: 2109: 2104: 2084: 2082: 2081: 2076: 2065: 2064: 2049: 2048: 2030: 2029: 2007: 2005: 2004: 1999: 1970: 1969: 1951: 1950: 1928: 1926: 1925: 1920: 1891: 1890: 1872: 1871: 1849: 1847: 1846: 1841: 1821: 1820: 1808: 1807: 1781: 1779: 1778: 1773: 1753: 1752: 1740: 1739: 1713: 1711: 1710: 1705: 1694: 1693: 1675: 1674: 1648: 1646: 1645: 1640: 1617: 1616: 1590: 1588: 1587: 1582: 1559: 1558: 1532: 1530: 1529: 1524: 1516: 1515: 1498: 1497: 1491: 1489: 1488: 1483: 1465: 1463: 1462: 1457: 1439: 1434: 1415: 1413: 1412: 1407: 1405: 1401: 1389: 1385: 1381: 1380: 1360: 1358: 1342: 1303: 1272: 1252: 1251: 1232: 1226: 1222: 1201: 1172: 1167: 1148: 1146: 1145: 1140: 1137: 1132: 1105: 1103: 1102: 1097: 1076:are nonnegative 1064: 1062: 1061: 1056: 1020: 1015: 987: 979: 957: 955: 954: 949: 916: 911: 883: 878: 833:. Developed by 743:, and there are 741:image distortion 669:Sir Isaac Newton 515:under correction 390:Image distortion 346:refractive index 212: 205: 194: 191: 185: 158: 157: 150: 143: 140: 134: 111: 103: 96: 93: 87: 80:Eleventh Edition 68: 67: 60: 49: 27: 26: 19: 5474: 5473: 5469: 5468: 5467: 5465: 5464: 5463: 5449: 5448: 5435: 5430: 5429: 5420: 5416: 5404: 5400: 5388: 5384: 5375: 5371: 5361: 5359: 5351: 5350: 5346: 5338: 5331: 5323: 5319: 5311: 5307: 5298: 5294: 5285: 5281: 5272: 5268: 5255: 5251: 5244: 5228: 5224: 5209: 5193: 5189: 5173: 5172: 5161: 5154: 5116: 5109: 5096: 5092: 5084: 5080: 5067: 5063: 5048:, ed. (1911). " 5033: 5031: 5030: 4943: 4932: 4928: 4893: 4889: 4868: 4864: 4855: 4853: 4845: 4844: 4840: 4830: 4828: 4819: 4818: 4814: 4807: 4785: 4781: 4774: 4749: 4745: 4740: 4735: 4734: 4729: 4725: 4720: 4698: 4638: 4634: 4625: 4621: 4609: 4605: 4596: 4592: 4587: 4584: 4583: 4560: 4556: 4547: 4543: 4534: 4530: 4528: 4525: 4524: 4488: 4484: 4475: 4471: 4469: 4466: 4465: 4361: 4357: 4355: 4352: 4351: 4328: 4324: 4315: 4311: 4309: 4306: 4305: 4287: 4283: 4275: 4269: 4265: 4260: 4257: 4256: 4248:, known as the 4230: 4221: 4217: 4208: 4204: 4193: 4190: 4189: 4172: 4168: 4159: 4155: 4153: 4150: 4149: 4132: 4128: 4122: 4118: 4109: 4105: 4099: 4095: 4087: 4084: 4083: 4067: 4064: 4063: 4038: 4030: 4027: 4026: 4006: 4003: 4002: 3974: 3971: 3970: 3954: 3951: 3950: 3934: 3931: 3930: 3901: 3898: 3897: 3869: 3866: 3865: 3854: 3834: 3831: 3830: 3813: 3809: 3807: 3804: 3803: 3786: 3782: 3780: 3777: 3776: 3759: 3755: 3750: 3744: 3740: 3728: 3724: 3719: 3713: 3709: 3707: 3704: 3703: 3686: 3682: 3674: 3668: 3664: 3649: 3645: 3640: 3634: 3630: 3628: 3625: 3624: 3597: 3594: 3593: 3576: 3572: 3563: 3559: 3550: 3546: 3537: 3533: 3522: 3519: 3518: 3499: 3495: 3480: 3476: 3464: 3460: 3445: 3441: 3423: 3414: 3399: 3390: 3351: 3342: 3327: 3318: 3297: 3293: 3281: 3277: 3268: 3264: 3256: 3253: 3252: 3233: 3229: 3227: 3224: 3223: 3206: 3202: 3200: 3197: 3196: 3180: 3177: 3176: 3175:the changes of 3159: 3155: 3150: 3147: 3146: 3129: 3125: 3120: 3117: 3116: 3097: 3094: 3093: 3077: 3074: 3073: 3053: 3047: 3044: 3043: 3023: 3017: 3014: 3013: 2993: 2987: 2984: 2983: 2963: 2957: 2954: 2953: 2936: 2932: 2930: 2927: 2926: 2909: 2905: 2903: 2900: 2899: 2882: 2878: 2876: 2873: 2872: 2855: 2851: 2849: 2846: 2845: 2826: 2823: 2822: 2803: 2800: 2799: 2774: 2751: 2743: 2740: 2723: 2720: 2718: 2715: 2714: 2689: 2686: 2685: 2669: 2666: 2665: 2640: 2637: 2636: 2620: 2617: 2616: 2554: 2534:Wide angle lens 2459: 2418:Ann. d. Phys., 2376: 2372: 2368: 2364: 2360: 2330: 2283:low-pass filter 2230: 2226: 2211: 2207: 2205: 2202: 2201: 2170: 2167: 2166: 2150: 2147: 2146: 2118: 2115: 2114: 2098: 2095: 2094: 2060: 2056: 2044: 2040: 2025: 2021: 2019: 2016: 2015: 1965: 1961: 1946: 1942: 1940: 1937: 1936: 1886: 1882: 1867: 1863: 1861: 1858: 1857: 1816: 1812: 1803: 1799: 1797: 1794: 1793: 1748: 1744: 1735: 1731: 1729: 1726: 1725: 1689: 1685: 1670: 1666: 1664: 1661: 1660: 1612: 1608: 1606: 1603: 1602: 1554: 1550: 1548: 1545: 1544: 1511: 1507: 1505: 1502: 1501: 1471: 1468: 1467: 1435: 1430: 1424: 1421: 1420: 1399: 1383: 1366: 1362: 1338: 1299: 1273: 1247: 1243: 1233: 1231: 1218: 1202: 1191: 1168: 1163: 1157: 1154: 1153: 1133: 1128: 1122: 1119: 1118: 1085: 1082: 1081: 1016: 1011: 980: 975: 969: 966: 965: 912: 907: 879: 874: 868: 865: 864: 811: 747:to correct it. 717: 708: 702: 685: 632: 625: 619: 594:sine condition, 589: 519:over correction 502: 450: 444: 428: 414: 385:Field curvature 358: 289: 250:paraxial optics 213: 202: 201: 200: 195: 189: 186: 171: 159: 155: 144: 138: 135: 125: 112: 97: 91: 88: 84: 69: 65: 28: 24: 17: 12: 11: 5: 5472: 5462: 5461: 5447: 5446: 5434: 5433:External links 5431: 5428: 5427: 5414: 5411:. p. 170. 5398: 5395:. p. 166. 5382: 5369: 5344: 5329: 5317: 5305: 5292: 5279: 5266: 5249: 5243:978-0521642224 5242: 5222: 5207: 5187: 5152: 5107: 5090: 5078: 5061: 5046:Chisholm, Hugh 4941: 4926: 4887: 4862: 4838: 4812: 4805: 4779: 4772: 4742: 4741: 4739: 4736: 4733: 4732: 4722: 4721: 4719: 4716: 4715: 4714: 4709: 4704: 4697: 4694: 4646: 4641: 4637: 4633: 4628: 4624: 4620: 4617: 4612: 4608: 4604: 4599: 4595: 4591: 4571: 4568: 4563: 4559: 4555: 4550: 4546: 4542: 4537: 4533: 4491: 4487: 4483: 4478: 4474: 4451: 4450: 4449:405.1 nm 4447: 4444: 4441: 4438: 4435: 4432: 4428: 4427: 4424: 4421: 4418: 4411: 4408: 4405: 4372: 4369: 4364: 4360: 4339: 4336: 4331: 4327: 4323: 4318: 4314: 4290: 4286: 4282: 4278: 4272: 4268: 4264: 4237: 4233: 4229: 4224: 4220: 4216: 4211: 4207: 4203: 4200: 4197: 4175: 4171: 4167: 4162: 4158: 4135: 4131: 4125: 4121: 4117: 4112: 4108: 4102: 4098: 4094: 4091: 4071: 4045: 4041: 4037: 4034: 4013: 4010: 3992:new achromats, 3978: 3958: 3938: 3905: 3873: 3853: 3852: 3851: 3850: 3838: 3816: 3812: 3789: 3785: 3762: 3758: 3753: 3747: 3743: 3739: 3736: 3731: 3727: 3722: 3716: 3712: 3689: 3685: 3681: 3677: 3671: 3667: 3663: 3660: 3657: 3652: 3648: 3643: 3637: 3633: 3622: 3610: 3607: 3604: 3601: 3579: 3575: 3571: 3566: 3562: 3558: 3553: 3549: 3545: 3540: 3536: 3532: 3529: 3526: 3516: 3502: 3498: 3494: 3491: 3488: 3483: 3479: 3475: 3472: 3467: 3463: 3459: 3456: 3453: 3448: 3444: 3440: 3437: 3434: 3430: 3426: 3422: 3417: 3413: 3410: 3406: 3402: 3398: 3393: 3389: 3386: 3383: 3380: 3377: 3374: 3371: 3368: 3365: 3362: 3358: 3354: 3350: 3345: 3341: 3338: 3334: 3330: 3326: 3321: 3317: 3314: 3311: 3308: 3305: 3300: 3296: 3292: 3289: 3284: 3280: 3276: 3271: 3267: 3263: 3260: 3236: 3232: 3209: 3205: 3184: 3162: 3158: 3154: 3132: 3128: 3124: 3104: 3101: 3081: 3060: 3056: 3052: 3030: 3026: 3022: 3000: 2996: 2992: 2970: 2966: 2962: 2939: 2935: 2912: 2908: 2885: 2881: 2858: 2854: 2842: 2830: 2810: 2807: 2788: 2782: 2779: 2773: 2766: 2763: 2760: 2757: 2754: 2749: 2746: 2739: 2733: 2729: 2726: 2702: 2699: 2696: 2693: 2673: 2653: 2650: 2647: 2644: 2624: 2612: 2579:overcorrected. 2553: 2550: 2549: 2548: 2541: 2531: 2458: 2455: 2454: 2453: 2439: 2374: 2370: 2366: 2362: 2358: 2329: 2326: 2233: 2229: 2225: 2222: 2219: 2214: 2210: 2189: 2186: 2183: 2180: 2177: 2174: 2154: 2134: 2131: 2128: 2125: 2122: 2102: 2089: 2088: 2085: 2074: 2071: 2068: 2063: 2059: 2055: 2052: 2047: 2043: 2039: 2036: 2033: 2028: 2024: 2012: 2011: 2008: 1997: 1994: 1991: 1988: 1985: 1982: 1979: 1976: 1973: 1968: 1964: 1960: 1957: 1954: 1949: 1945: 1933: 1932: 1929: 1918: 1915: 1912: 1909: 1906: 1903: 1900: 1897: 1894: 1889: 1885: 1881: 1878: 1875: 1870: 1866: 1854: 1853: 1850: 1839: 1836: 1833: 1830: 1827: 1824: 1819: 1815: 1811: 1806: 1802: 1790: 1789: 1782: 1771: 1768: 1765: 1762: 1759: 1756: 1751: 1747: 1743: 1738: 1734: 1722: 1721: 1714: 1703: 1700: 1697: 1692: 1688: 1684: 1681: 1678: 1673: 1669: 1657: 1656: 1649: 1638: 1635: 1632: 1629: 1626: 1623: 1620: 1615: 1611: 1599: 1598: 1591: 1580: 1577: 1574: 1571: 1568: 1565: 1562: 1557: 1553: 1541: 1540: 1533: 1522: 1519: 1514: 1510: 1481: 1478: 1475: 1455: 1452: 1449: 1446: 1443: 1438: 1433: 1429: 1417: 1416: 1398: 1395: 1392: 1379: 1375: 1372: 1369: 1365: 1357: 1354: 1351: 1348: 1345: 1341: 1337: 1334: 1331: 1328: 1325: 1322: 1318: 1315: 1312: 1309: 1306: 1302: 1298: 1295: 1292: 1289: 1286: 1283: 1279: 1276: 1271: 1268: 1265: 1262: 1259: 1256: 1250: 1246: 1242: 1239: 1236: 1225: 1221: 1217: 1214: 1211: 1208: 1205: 1200: 1197: 1194: 1190: 1185: 1182: 1179: 1176: 1171: 1166: 1162: 1136: 1131: 1127: 1095: 1092: 1089: 1066: 1065: 1053: 1050: 1047: 1043: 1040: 1037: 1034: 1030: 1027: 1024: 1019: 1014: 1010: 1006: 1003: 1000: 997: 994: 991: 986: 983: 978: 974: 959: 958: 946: 943: 939: 936: 933: 930: 926: 923: 920: 915: 910: 906: 902: 899: 896: 893: 890: 887: 882: 877: 873: 810: 807: 781:principal rays 716: 713: 704:Main article: 701: 698: 684: 681: 649:principal ray, 621:Main article: 618: 615: 599:sine condition 588: 585: 561:entrance pupil 501: 498: 443: 440: 439: 438: 435: 424:Main article: 413: 410: 393: 392: 387: 382: 377: 372: 367: 357: 354: 340:are caused by 303:With an ideal 288: 285: 243:, that causes 215: 214: 197: 196: 162: 160: 153: 146: 145: 115: 113: 106: 99: 98: 72: 70: 63: 58: 32: 31: 29: 22: 15: 9: 6: 4: 3: 2: 5471: 5460: 5457: 5456: 5454: 5444: 5440: 5437: 5436: 5424: 5418: 5410: 5402: 5394: 5386: 5379: 5373: 5358: 5354: 5348: 5342: 5339:M. von Rohr, 5336: 5334: 5326: 5321: 5314: 5309: 5302: 5296: 5289: 5286:M. von Rohr, 5283: 5276: 5270: 5263: 5259: 5253: 5245: 5239: 5235: 5234: 5226: 5218: 5214: 5210: 5204: 5200: 5199: 5191: 5183: 5177: 5169: 5165: 5159: 5157: 5148: 5144: 5140: 5136: 5132: 5128: 5124: 5120: 5114: 5112: 5104: 5101:; and (1901) 5100: 5094: 5088: 5082: 5075: 5071: 5065: 5057: 5056: 5051: 5047: 5042: 5041:public domain 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: 4974: 4972: 4970: 4968: 4966: 4964: 4962: 4960: 4958: 4956: 4954: 4952: 4950: 4948: 4946: 4937: 4930: 4922: 4918: 4914: 4910: 4906: 4902: 4898: 4891: 4884: 4880: 4876: 4872: 4866: 4852: 4848: 4842: 4826: 4822: 4816: 4808: 4806:0-471-60538-7 4802: 4798: 4793: 4792: 4791:Modern Optics 4783: 4775: 4773:0-03-000602-3 4769: 4765: 4760: 4759: 4753: 4747: 4743: 4727: 4723: 4713: 4710: 4708: 4705: 4703: 4700: 4699: 4693: 4691: 4687: 4686: 4681: 4675: 4673: 4667: 4665: 4661: 4639: 4635: 4631: 4626: 4622: 4610: 4606: 4602: 4597: 4593: 4569: 4566: 4561: 4557: 4553: 4548: 4544: 4540: 4535: 4531: 4521: 4519: 4515: 4511: 4507: 4489: 4485: 4481: 4476: 4472: 4457: 4448: 4445: 4442: 4439: 4436: 4433: 4430: 4429: 4425: 4422: 4419: 4416: 4412: 4409: 4406: 4403: 4402: 4399: 4397: 4393: 4388: 4386: 4370: 4367: 4362: 4358: 4337: 4334: 4329: 4325: 4321: 4316: 4312: 4288: 4284: 4280: 4276: 4270: 4266: 4262: 4252: 4251: 4235: 4231: 4222: 4218: 4214: 4209: 4205: 4198: 4195: 4173: 4169: 4165: 4160: 4156: 4133: 4129: 4123: 4119: 4115: 4110: 4106: 4100: 4096: 4092: 4089: 4069: 4060: 4059: 4043: 4039: 4035: 4032: 4011: 4008: 3999: 3997: 3993: 3976: 3969:decreased as 3956: 3936: 3928: 3924: 3920: 3903: 3895: 3894: 3889: 3888: 3871: 3862: 3860: 3836: 3814: 3810: 3787: 3783: 3775:. Therefore 3760: 3756: 3751: 3745: 3741: 3737: 3734: 3729: 3725: 3720: 3714: 3710: 3687: 3683: 3679: 3675: 3669: 3665: 3661: 3658: 3655: 3650: 3646: 3641: 3635: 3631: 3623: 3608: 3605: 3602: 3599: 3577: 3573: 3569: 3564: 3560: 3556: 3551: 3547: 3543: 3538: 3534: 3530: 3527: 3524: 3517: 3500: 3496: 3489: 3486: 3481: 3477: 3470: 3465: 3461: 3454: 3451: 3446: 3442: 3435: 3428: 3424: 3420: 3415: 3411: 3408: 3404: 3400: 3396: 3391: 3387: 3378: 3375: 3372: 3369: 3363: 3356: 3352: 3348: 3343: 3339: 3336: 3332: 3328: 3324: 3319: 3315: 3306: 3303: 3298: 3294: 3287: 3282: 3278: 3274: 3269: 3265: 3261: 3258: 3251: 3250: 3234: 3230: 3207: 3203: 3182: 3160: 3156: 3152: 3130: 3126: 3122: 3102: 3099: 3079: 3058: 3054: 3050: 3028: 3024: 3020: 2998: 2994: 2990: 2968: 2964: 2960: 2937: 2933: 2910: 2906: 2883: 2879: 2856: 2852: 2843: 2828: 2808: 2805: 2786: 2780: 2777: 2771: 2761: 2758: 2755: 2747: 2744: 2737: 2731: 2727: 2724: 2700: 2697: 2694: 2691: 2671: 2651: 2648: 2645: 2642: 2622: 2614: 2613: 2611: 2608: 2606: 2602: 2598: 2594: 2588: 2587: 2581: 2580: 2576: 2571: 2567: 2563: 2559: 2558:Lens (optics) 2545: 2542: 2539: 2535: 2532: 2529: 2528: 2527: 2523: 2520: 2516: 2511: 2505: 2503: 2497: 2495: 2491: 2490:Luneburg lens 2487: 2483: 2479: 2475: 2474:more than one 2467: 2463: 2451: 2447: 2445: 2440: 2437: 2432: 2427: 2426: 2425: 2421: 2419: 2415: 2411: 2407: 2403: 2398: 2393: 2391: 2386: 2383: 2380: 2373:and Dη'=η'-η' 2354: 2346: 2342: 2340: 2335: 2325: 2323: 2319: 2315: 2311: 2307: 2306:Frits Zernike 2303: 2298: 2296: 2292: 2288: 2284: 2280: 2276: 2272: 2268: 2265:or very high 2264: 2260: 2256: 2252: 2247: 2231: 2227: 2223: 2220: 2217: 2212: 2208: 2187: 2184: 2181: 2178: 2175: 2172: 2152: 2132: 2129: 2126: 2123: 2120: 2100: 2086: 2069: 2066: 2061: 2057: 2053: 2050: 2045: 2041: 2037: 2031: 2026: 2022: 2014: 2013: 2009: 1992: 1986: 1983: 1980: 1974: 1971: 1966: 1962: 1958: 1952: 1947: 1943: 1935: 1934: 1930: 1913: 1907: 1904: 1901: 1895: 1892: 1887: 1883: 1879: 1873: 1868: 1864: 1856: 1855: 1851: 1834: 1831: 1825: 1822: 1817: 1813: 1809: 1804: 1800: 1792: 1791: 1787: 1783: 1766: 1763: 1757: 1754: 1749: 1745: 1741: 1736: 1732: 1724: 1723: 1719: 1716:"Defocus", a 1715: 1698: 1695: 1690: 1686: 1682: 1676: 1671: 1667: 1659: 1658: 1654: 1650: 1633: 1627: 1624: 1621: 1618: 1613: 1609: 1601: 1600: 1596: 1592: 1575: 1569: 1566: 1563: 1560: 1555: 1551: 1543: 1542: 1538: 1534: 1520: 1517: 1512: 1508: 1500: 1499: 1496: 1493: 1479: 1476: 1473: 1453: 1450: 1444: 1436: 1431: 1427: 1402: is even 1396: 1393: 1390: 1377: 1373: 1370: 1367: 1363: 1355: 1349: 1346: 1343: 1339: 1332: 1329: 1326: 1316: 1310: 1307: 1304: 1300: 1293: 1290: 1287: 1277: 1274: 1269: 1263: 1260: 1257: 1248: 1240: 1237: 1223: 1219: 1212: 1209: 1206: 1198: 1195: 1192: 1188: 1183: 1177: 1169: 1164: 1160: 1152: 1151: 1150: 1134: 1129: 1125: 1116: 1112: 1109: 1093: 1090: 1087: 1079: 1075: 1071: 1051: 1045: 1041: 1035: 1032: 1025: 1017: 1012: 1008: 1004: 998: 995: 992: 984: 981: 976: 972: 964: 963: 962: 941: 937: 931: 928: 921: 913: 908: 904: 900: 894: 891: 888: 880: 875: 871: 863: 862: 861: 859: 854: 852: 848: 844: 840: 836: 835:Frits Zernike 832: 823: 815: 806: 802: 800: 796: 794: 791:condition of 787: 782: 778: 777:principal ray 773: 763: 759: 757: 753: 748: 746: 742: 738: 729: 721: 712: 707: 697: 695: 691: 680: 678: 674: 670: 666: 663: 659: 655: 650: 646: 636: 630: 624: 614: 610: 608: 604: 600: 595: 584: 582: 577: 575: 571: 566: 562: 558: 556: 551: 547: 541: 540: 536: 532: 528: 525:and O'1R the 524: 520: 516: 506: 497: 494: 490: 485: 483: 479: 478:focal lengths 475: 471: 467: 463: 459: 455: 449: 448:Lens (optics) 436: 433: 432: 431: 427: 418: 409: 406: 402: 397: 391: 388: 386: 383: 381: 378: 376: 373: 371: 368: 366: 363: 362: 361: 353: 351: 347: 343: 339: 335: 333: 329: 325: 324:monochromatic 320: 319:of the lens. 318: 314: 313:image surface 310: 306: 298: 293: 284: 282: 278: 274: 270: 266: 261: 258: 253: 251: 246: 242: 238: 234: 226: 221: 211: 208: 193: 183: 179: 175: 169: 168: 165:reads like a 163:This article 161: 152: 151: 142: 132: 128: 123: 119: 116:This article 114: 110: 105: 104: 95: 83: 81: 79: 73:This article 71: 62: 61: 56: 54: 47: 46: 41: 40: 35: 30: 21: 20: 5442: 5422: 5417: 5408: 5401: 5392: 5385: 5377: 5372: 5360:. Retrieved 5356: 5347: 5340: 5324: 5320: 5312: 5308: 5300: 5295: 5287: 5282: 5274: 5269: 5261: 5257: 5256:J. Petzval, 5252: 5231: 5225: 5197: 5190: 5167: 5130: 5126: 5102: 5098: 5093: 5086: 5081: 5073: 5069: 5064: 5053: 4935: 4929: 4904: 4900: 4890: 4882: 4878: 4874: 4870: 4865: 4854:. Retrieved 4850: 4841: 4829:. Retrieved 4825:the original 4815: 4790: 4782: 4757: 4746: 4726: 4689: 4685:apochromatic 4683: 4679: 4676: 4671: 4668: 4663: 4659: 4522: 4517: 4513: 4509: 4505: 4462: 4391: 4389: 4384: 4253: 4249: 4061: 4057: 4000: 3995: 3991: 3926: 3922: 3918: 3891: 3885: 3863: 3855: 2609: 2604: 2600: 2596: 2592: 2589: 2585: 2582: 2578: 2574: 2569: 2555: 2547:computation. 2524: 2518: 2514: 2509: 2506: 2501: 2498: 2485: 2482:Carathéodory 2473: 2471: 2449: 2444:Fraunhofer's 2442: 2435: 2430: 2422: 2417: 2413: 2409: 2405: 2401: 2396: 2394: 2387: 2381: 2355: 2351: 2338: 2333: 2331: 2299: 2248: 2092: 1494: 1418: 1106:, Φ is the 1073: 1069: 1067: 960: 858:even and odd 855: 847:coefficients 843:curve-fitted 828: 803: 798: 790: 785: 780: 776: 771: 768: 755: 751: 749: 734: 709: 694:Arthur König 686: 676: 667: 661: 657: 653: 648: 644: 641: 611: 607:Robert Blair 602: 598: 593: 590: 580: 578: 573: 569: 564: 560: 553: 549: 545: 542: 538: 534: 526: 522: 518: 514: 511: 486: 482:focal planes 476:, named the 470:image space. 469: 466:object space 465: 461: 460:unite in an 458:object point 457: 451: 429: 398: 394: 359: 336: 327: 323: 321: 316: 312: 308: 302: 262: 254: 236: 230: 203: 190:October 2020 187: 164: 136: 127:You can help 117: 89: 77: 74: 50: 43: 37: 36:Please help 33: 5441:section of 5362:22 February 5275:Astr. Nach. 5273:L. Seidel, 5072:and (1858) 4426:Violet Hg. 3996:anastigmats 3893:flint glass 3887:crown glass 3884:) is named 2538:normal lens 2310:diffraction 2271:propagation 1786:cylindrical 756:rectilinear 752:orthoscopic 629:Astigmatism 462:image point 380:Astigmatism 317:aberrations 309:image plane 176:to make it 5070:Phil.Mag., 5050:Aberration 4907:: 69–174. 4856:2024-08-07 4738:References 2952:and radii 2610:Examples: 2597:vide supra 2566:dispersion 2446:condition. 2379:J. Petzval 1655:direction 1653:tangential 1597:direction 1537:mean value 856:There are 839:orthogonal 745:algorithms 570:front stop 565:exit pupil 493:Ernst Abbe 446:See also: 350:wavelength 342:dispersion 273:refraction 269:reflection 237:aberration 39:improve it 5217:162132153 5176:cite book 5170:. Berlin. 4913:0790-8113 4831:March 26, 4632:− 4603:− 3859:telescope 3738:− 3659:− 3487:− 3452:− 3409:− 3376:− 3337:− 3304:− 3275:− 2759:− 2390:L. Seidel 2263:gradients 2221:… 2188:π 2182:≤ 2179:ϕ 2176:≤ 2153:ϕ 2130:≤ 2127:ρ 2124:≤ 2101:ρ 2058:ρ 2051:− 2042:ρ 2032:× 1993:ϕ 1987:⁡ 1981:ρ 1972:− 1963:ρ 1953:× 1914:ϕ 1908:⁡ 1902:ρ 1893:− 1884:ρ 1874:× 1835:ϕ 1826:⁡ 1814:ρ 1810:× 1767:ϕ 1758:⁡ 1746:ρ 1742:× 1718:parabolic 1696:− 1687:ρ 1677:× 1634:ϕ 1628:⁡ 1622:ρ 1619:× 1576:ϕ 1570:⁡ 1564:ρ 1561:× 1518:× 1477:− 1445:ρ 1394:− 1371:− 1364:ρ 1347:− 1330:− 1308:− 1261:− 1238:− 1210:− 1189:∑ 1178:ρ 1108:azimuthal 1091:≥ 1046:ϕ 1036:⁡ 1026:ρ 999:ϕ 993:ρ 982:− 942:ϕ 932:⁡ 922:ρ 895:ϕ 889:ρ 603:aplanatic 574:back stop 550:diaphragm 328:chromatic 131:talk page 45:talk page 5453:Category 5425:, p. 340 5166:(1904). 5121:(1900). 4921:30078906 4696:See also 4459:Figure 6 3919:Herschel 3429:″ 3405:′ 3357:″ 3333:′ 3059:″ 3029:′ 2999:″ 2969:′ 2492:and the 2348:Figure 5 2291:fractals 2273:through 1595:sagittal 1492:is odd. 1386:if 1078:integers 765:Figure 4 638:Figure 2 555:aperture 508:Figure 1 287:Overview 277:caustics 167:textbook 139:May 2009 92:May 2016 5135:Bibcode 5043:: 2713:, then 2478:Maxwell 2287:spatial 2259:cosines 2251:Fourier 1115:radians 531:pencils 529:of the 365:Defocus 178:neutral 172:Please 5240: 5215: 5205: 5037: 4919: 4911: 4803: 4770: 4413:Green 4255:since 3222:, and 3012:, and 2510:zones, 2249:As in 2093:where 1068:where 563:; the 401:piston 241:lenses 233:optics 129:. The 4917:JSTOR 4718:Notes 4446:454.1 4443:486.2 4440:546.1 4437:589.3 4434:656.3 4431:767.7 4148:; if 3702:, or 3515:; and 2593:zones 2285:fine 2255:sines 1111:angle 1080:with 786:scale 474:Gauss 348:with 245:light 86:page. 5405:See 5364:2013 5238:ISBN 5213:OCLC 5203:ISBN 5182:link 5131:1905 4909:ISSN 4833:2012 4801:ISBN 4768:ISBN 4368:< 3802:and 2925:and 2871:and 2684:and 2560:and 2365:, η' 2314:Airy 2300:The 2257:and 1419:and 1072:and 793:Airy 690:coma 557:stop 546:stop 491:and 480:and 405:tilt 403:and 375:Coma 326:and 305:lens 281:rays 275:and 5143:doi 5052:". 4797:130 4764:410 4674:). 4520:). 4516:or 3921:or 3861:.) 2293:or 2277:or 1984:sin 1905:cos 1823:sin 1755:cos 1625:sin 1567:cos 1466:if 1113:in 1033:sin 929:cos 656:or 647:or 576:). 548:or 231:In 5455:: 5355:. 5332:^ 5211:. 5178:}} 5174:{{ 5155:^ 5141:. 5129:. 5125:. 5110:^ 4944:^ 4915:. 4905:15 4903:. 4899:. 4849:. 4799:. 4766:. 4423:G' 4415:Hg 4404:A' 3195:, 3145:, 3115:, 3042:, 2982:, 2599:, 2496:. 2145:, 679:. 283:. 271:, 235:, 48:. 5366:. 5246:. 5219:. 5184:) 5149:. 5145:: 5137:: 5076:. 4923:. 4859:. 4835:. 4809:. 4776:. 4645:) 4640:b 4636:n 4627:a 4623:n 4619:( 4616:) 4611:b 4607:n 4598:c 4594:n 4590:( 4570:f 4567:= 4562:c 4558:f 4554:= 4549:b 4545:f 4541:= 4536:a 4532:f 4490:F 4486:f 4482:= 4477:C 4473:f 4420:F 4417:. 4410:D 4407:C 4371:f 4363:c 4359:f 4338:f 4335:= 4330:b 4326:f 4322:= 4317:a 4313:f 4289:1 4285:n 4281:d 4277:/ 4271:2 4267:n 4263:d 4236:2 4232:/ 4228:) 4223:2 4219:f 4215:+ 4210:1 4206:f 4202:( 4199:= 4196:D 4174:2 4170:v 4166:= 4161:1 4157:v 4134:2 4130:f 4124:2 4120:v 4116:+ 4111:1 4107:f 4101:1 4097:v 4093:= 4090:D 4070:D 4044:f 4040:/ 4036:f 4033:d 4012:f 4009:d 3977:n 3957:v 3937:v 3904:f 3872:v 3837:f 3815:2 3811:f 3788:1 3784:f 3761:2 3757:n 3752:/ 3746:1 3742:n 3735:= 3730:2 3726:f 3721:/ 3715:1 3711:f 3688:1 3684:n 3680:d 3676:/ 3670:2 3666:n 3662:d 3656:= 3651:2 3647:k 3642:/ 3636:1 3632:k 3609:0 3606:= 3603:f 3600:d 3578:2 3574:n 3570:d 3565:2 3561:k 3557:+ 3552:1 3548:n 3544:d 3539:1 3535:k 3531:= 3528:f 3525:d 3501:2 3497:k 3493:) 3490:1 3482:2 3478:n 3474:( 3471:+ 3466:1 3462:k 3458:) 3455:1 3447:1 3443:n 3439:( 3436:= 3433:) 3425:2 3421:r 3416:/ 3412:1 3401:2 3397:r 3392:/ 3388:1 3385:( 3382:) 3379:1 3373:2 3370:n 3367:( 3364:+ 3361:) 3353:1 3349:r 3344:/ 3340:1 3329:1 3325:r 3320:/ 3316:1 3313:( 3310:) 3307:1 3299:1 3295:n 3291:( 3288:= 3283:2 3279:f 3270:1 3266:f 3262:= 3259:f 3235:2 3231:n 3208:1 3204:n 3183:f 3161:2 3157:n 3153:d 3131:1 3127:n 3123:d 3103:f 3100:d 3080:f 3055:2 3051:r 3025:2 3021:r 2995:1 2991:r 2965:1 2961:r 2938:2 2934:n 2911:1 2907:n 2884:2 2880:f 2857:1 2853:f 2829:n 2809:n 2806:d 2787:, 2781:n 2778:1 2772:= 2765:) 2762:1 2756:n 2753:( 2748:n 2745:d 2738:= 2732:f 2728:f 2725:d 2701:f 2698:d 2695:+ 2692:f 2672:f 2652:n 2649:d 2646:+ 2643:n 2623:n 2486:6 2375:0 2371:0 2367:0 2363:0 2359:0 2232:8 2228:a 2224:, 2218:, 2213:0 2209:a 2185:2 2173:0 2133:1 2121:0 2073:) 2070:1 2067:+ 2062:2 2054:6 2046:4 2038:6 2035:( 2027:8 2023:a 1996:) 1990:( 1978:) 1975:2 1967:2 1959:3 1956:( 1948:7 1944:a 1917:) 1911:( 1899:) 1896:2 1888:2 1880:3 1877:( 1869:6 1865:a 1838:) 1832:2 1829:( 1818:2 1805:5 1801:a 1770:) 1764:2 1761:( 1750:2 1737:4 1733:a 1702:) 1699:1 1691:2 1683:2 1680:( 1672:3 1668:a 1637:) 1631:( 1614:2 1610:a 1579:) 1573:( 1556:1 1552:a 1521:1 1513:0 1509:a 1480:m 1474:n 1454:0 1451:= 1448:) 1442:( 1437:m 1432:n 1428:R 1397:m 1391:n 1378:k 1374:2 1368:n 1356:! 1353:) 1350:k 1344:2 1340:/ 1336:) 1333:m 1327:n 1324:( 1321:( 1317:! 1314:) 1311:k 1305:2 1301:/ 1297:) 1294:m 1291:+ 1288:n 1285:( 1282:( 1278:! 1275:k 1270:! 1267:) 1264:k 1258:n 1255:( 1249:k 1245:) 1241:1 1235:( 1224:2 1220:/ 1216:) 1213:m 1207:n 1204:( 1199:0 1196:= 1193:k 1184:= 1181:) 1175:( 1170:m 1165:n 1161:R 1135:m 1130:n 1126:R 1094:m 1088:n 1074:n 1070:m 1052:, 1049:) 1042:m 1039:( 1029:) 1023:( 1018:m 1013:n 1009:R 1005:= 1002:) 996:, 990:( 985:m 977:n 973:Z 945:) 938:m 935:( 925:) 919:( 914:m 909:n 905:R 901:= 898:) 892:, 886:( 881:m 876:n 872:Z 795:, 631:. 299:. 210:) 204:( 192:) 188:( 184:. 170:. 141:) 137:( 124:. 94:) 90:( 55:) 51:(

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