ABCD the earth's orbit, and s the true position of a star.
When the earth is at A, in consequence of aberration, the star
always parallel to AC, i.e. the ecliptic, and since it is equal
to the ratio of the velocity
of light to the velocity of the earth, it is necessarily constant. This constant length subtends an angle of about 4o" at the earth; the " constant of aberration " is half this angle. The generally accepted value is 2o•445", due to Struve; the last two figures are uncertain, and all that can be definitely affirmed is that the value lies between 2o•43" and 20.48". The minor axis, on the other hand, is not constant, but, as we have already seen, depends on the latitude, being the product of
the major axis into the sine of the latitude.
Assured that his explanation was true, Bradley corrected his observations for aberration, but he found that there still remained a residuum which was evidently not a parallax, for it did not exhibit an annual cycle. He reverted to his early idea of a nutation of the earth's axis, and was rewarded by the discovery that the earth did possess such an oscillation (see
ASTRONOMY). Bradley recognized the fact that the t.,c as experimental determination of the aberration constant
gave the ratio of the velocities of light and of the ca.) earth; hence, if the velocity of the earth be known, the velocity of light is determined. In recent years LaQ' much attention has been given to the nature of the FIG. 4• propagation of light from the heavenly bodies to the earth, the argument generally being centred about the relative effect of the motion of the aether on the velocity of light. This subject is discussed in the articles AETHER and LIGHT.
II. ABERRATION IN OPTICAL SYSTEMS
Aberration in optical systems, i.e. in lenses or mirrors or a series of them, may be defined as the nonconcurrence of rays from the points of an object after transmission through the system; it happens generally that an image formed by such a system is irregular, and consequently the correction of optical systems for aberration is of fundamental importance to the instrumentmaker. Reference should be made to the articles REFLEXION, REFRACTION, and CAUSTIC for the general characters of reflected and refracted rays (the article LENS considers in detail the properties of this instrument, and should also be consulted); in this article will be discussed the nature, varieties and modes of aberrations mainly from the practical point of view, i.e. that of the opticalinstrument maker.
Aberrations may be divided in two classes: chromatic (Gr. xpwµa, colour) aberrations, caused by the composite nature of
B
A
is displaced to a point a, its displacement sa being parallel to the earth's motion at A; when the earth is at B, the star appears at b; and so on
throughout an orbital revolution of the earth. Every star, therefore, describes an apparent orbit, which, if the line joining the sun and the star be perpendicular to the plane ABCD, will be exactly similar to that of the 'earth, i.e. almost a circle. As the star decreases in latitude, this circle will be viewed more and more obliquely, becoming a flatter 1
and flatter ellipse until, with A zero latitude, it degenerates into a straight line (fig. 4).
The major axis of any such aberrational ellipse is
Lat. 6e
the light generally applied (e.g. white light), which is dispersed by refraction, and monochromatic (Gr. µovos, one) aberrations produced without dispersion. Consequently the monochromatic class includes the aberrations at reflecting surfaces of any coloured light, and at refracting surfaces of monochromatic or light of single wave length.
(a) Monochromatic Aberration.
The elementary theory of optical systems leads to the theorem: Rays of light proceeding from any " object point " unite in an " image point "; and therefore an " object space " is reproduced in an " image space." The introduction of simple auxiliary terms, due to C. F. Gauss (Dioptrische Untersuchungen, Gottingen, 1841), named the focal lengths and focal planes, permits the determination of the image of any object for any system (see LENS). 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 are not realized, and the images projected by uncorrected systems are, in general, ill defined and often completely blurred, if the aperture or field of view exceeds certain limits. The investigations of James Clerk Maxwell (Phil.Mag., 1856; Quart. Journ. Math., 1858, and Ernst Abbe') 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 (in Abbe) of the reproduction of all points of a space in image points (Maxwell assumes a less general hypothesis), and are independent of the manner in which the reproduction is effected. These authors proved, however, that no optical system can justify these suppositions, since they are contradictory to the fundamental laws of reflexion and refraction. Consequently the Gaussian theory only supplies a convenient method of approximating to reality; and no constructor would attempt to realize this unattainable ideal. All that at present can be attempted is, to reproduce a single plane in another plane; but even this has not been altogether satisfactorily accomplished, aberrations always occur, and it is improbable that these will ever be entirely corrected.
This, and related general questions, have been treated—besides the abovementioned authors—by M. Thiesen (Berlin. Akad. Sitzber., 1890, XXXV. 799; Berlin.Phys.Ges.Verh., 1892) and H. Bruns (Leipzig. Math. Phys. Ber., 1895, xxi. 325) by means of Sir W. R. Hamilton s "characteristic function" (Irish Acad. Trans., "Theory of Systems of Rays," 1828, et seq.). Reference may also be made to the treatise of CzapskiEppenstein, pp. 155161.
A review of the simplest cases of aberration will now be given. (r) Aberration of axial points (Spherical aberration in the restricted sense). If S (figs) be apy optical system, rays proceeding from an axis point 0 under an angle u, will unite in the axis point O',; and those under an angle u2 in the axis point O'2. If there be refraction at a collective spherical surface, or through a thin positive lens, O'2 will lie in front of O'I so long as the angle u2 is greater than u, (" under correction "); and conversely with a dispersive surface or lenses (" over correction "). The caustic, in the first case, resembles the sign > (greater than); in the second < (less than). If the angle uI be very small, O', is the Gaussian image; and O', O'2 is termed the " longitudinal aberration," and O',R the " lateral aberration " of the pencils with aperture u2. If the pencil with the angle u2 be 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 " disk of confusion" of radius O',R, and in a parallel plane at O'2 another one of radius 0'2R2; between these two is situated the "disk of least confusion."
The largest opening of the pencils, which take part in the reproduction of 0, 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 " stop" or "diaphragm"; Abbe used the term " aperture stop " for both the hole and the limiting margin of the
' The investigations of E. Abbe on geometrical optics, originally published only in his university lectures, were first compiled by S. Czapski in 1893: See below, AUTHORITIES.lens. The component SI of the system, situated between the aperture stop and the object 0, projects an image of the diaphragm, termed by Abbe the " entrance pupil "; the " exit pupil " is the image formed by the component S2, which is placed behind the aperture stop. All rays which issue from 0 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 0 is the angle u subtended by the entrance pupil at this point, the magnitude of the aberration will be determined by the position and diameter of the entrance pupil. If the system be entirely behind the aperture stop, then this is itself the entrance pupil (" front stop ") ; if entirely in front, it is the exit pupil (" back stop ").
If the object point be infinitely distant, all rays received by the first member of the system are parallel, and their intersections, after traversing the system, vary according to their " perpendicular height of incidence," i.e. their distance from the axis. This distance replaces the angle u in the preceding considerations; and the aperture, i.e. the radius of the entrance pupil, is its maximum value.
(2) Aberration of elements, i.e. smallest objects at right angles to the axis.—If rays issuing from 0 (fig. 5) be concurrent, it does not follow
that points in a
portion of a plane perpendicular at N O to the axis will be also con o current, even if the part of the plane be very small. With
a considerable
aperture, the
neighbouring
point N will be reproduced, but attended by aberrations comparable in magnitude to ON. These aberrations are avoided if, according to Abbe, the " sine condition," sin u'I/sin uI = sin u'2/sin U2, holds for all rays reproducing the point O. If the object point 0 be infinitely distant, uI and u2 are to be replaced by hi and h2, the perpendicular heights of incidence; the " sine condition " then becomes sin u',/hi=sin u'2/h2. A system fulfilling this condition and free from spherical aberration is called
aplanatic " (Greek a, privative, ~rXavi7, a wandering). This word was first used by Robert Blair (d. 1828), professor of practical astronomy at Edinburgh University, to characterize a superior achromatism, and, subsequently, by many writers to denote freedom from spherical aberration. Both the aberration of axis points, and the deviation from the sine condition, rapidly increase in most (uncorrected) systems with the aperture.
(3) Aberration of lateral object points (points beyond the axis) with narrow pencils. Astigmatism.—A point 0 (fig. 6) at a finite distance from the axis (or with an infinitely distant object, a point which subtends a finite angle at the system) is, in general, even then not sharply reproduced, if the pencil of rays issuing from it and traversing the system is made infinitely narrow by reducing the aperture stop; such a pencil consists of the rays which can pass from the object point through the now infinitely small entrance pupil. It is seen (ignoring exceptional cases) that the pencil does not meet the refracting or reflecting surface at right angles; therefore it is astigmatic (Gr. a, privative, rrtyµa, a point). Naming the central ray passing through the entrance pupil the " axis of the pencil " or " principal ray," we can say: the rays of the pencil intersect, not in one point, but in two focal lines, which we can assume to be at right angles to the principal ray; of these, one lies in the plane containing the principal ray and
the axis of the system, i.e. in the " first principal section " or " meridional section," and the other at right angles to it, i.e. in the second principal section or sagittal section. We receive, therefore, in no single intercepting plane behind the system, as, for example, a focussing screen, an image of the object point; on the other hand, in each of two planes lines 0' and 0" are separately formed (in neighbouring planes ellipses are formed), and in a plane between 0' and 0" 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 " astigmatic image surfaces " correspond to one object plane; and these are in contact at the axis point; on the one lie the focal lines of the first kind, on the other those of the second. Systems in which the two astigmatic surfaces coincide are termed anastigmatic or stigmatic.
Sir Isaac Newton was probably the discoverer of astigmation; the position of the astigmatic image lines was determined by Thomas Young (A Course of Lectures on Natural Philosophy, 1807); and the theory has been recently developed by A. Gullstrand (Skand. Arch. f. physiol., 1890, 2, p. 269 ; Allgemeine Theorie der monochromat. Aberrationen, etc., Upsala, 1900; Arch. f. Ophth., 1901, 53, pp. 2, 185). A bibliography by P. Culmann is given in M. von Rohr's Die Bilderzeugung in optischen Instrumenten (Berlin, 1904).
(4) Aberration of lateral object points with broad pencils. Coma. —By opening the stop wider, similar deviations arise for lateral points as have been already discussed for axial points; but in this case they are much more complicated. The course of the rays in the meridional section is no longer symmetrical to the principal ray of the pencil; and on an intercepting plane there appears, instead of a luminous point, a patch of light, not symmetrical about a point, and often exhibiting a resemblance to a comet having its tail directed towards or away from the axis. From this appearance it takes its name. The unsymmetrical form of the meridional pencil—formerly the only one considered—is coma in the narrower sense only; other errors of coma have been treated by A. KSnig and M. von Rohr (op. cit.), and more recently by A. Gullstrand (op. cit.; Ann. d. Phys., 1905, 18, p. 941).
(5) Curvature of the field of the image.—If the above errors be eliminated, the two astigmatic surfaces united, and a sharp image obtained with a wide aperture—there remains the necessity to correct the curvature of the image surface, especially when the image is to be received upon a plane surface, e.g. in photography. In most cases the surface is concave towards the system.
(6) Distortion of the image.—If now the image be sufficiently sharp, inasmuch as the rays proceeding from every object point meet in an image point of satisfactory exactitude, it may happen that the image is distorted, i.e. not sufficiently like the object. This error consists in the different parts of the object being reproduced with different magnifications; for instance, the inner parts may differ in greater magnification than the outer (" barrelshaped distortion "), or conversely (" cushionshaped distortion") (see fig. 7). Systems free of this aberration are called " orthoscopic " (dpOor, right,
o'KOIrELY, to look).
This aberration is
quite distinct from
that of the sharpness
of reproduction ; in
unsharp reproduction,
the question of dis
tortion 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 " centre of gravity " of the patch may be regarded
as the image point, this being the point where the plane receiv
ing the image, e.g. a focussing screen, intersects the ray passing
through the middle of the stop. This assumption is justified if
a poor image on the focussing screen remains stationary when
the aperture is diminished; in practice, this generally occurs.
This ray, named by Abbe a " principal ray " (not to be confused
with the "principal rays" of the Gaussian theory), passes
through the centre of the entrance pupil before the first refraction,
and the centre of the exit pupil after the last refraction. From this it follows that correctness of drawing depends solely upon the principal rays; and is independent of the sharpness or curvature of the image field. Referring to fig. 8, we have O'Q'/OQ =a' tan w'/a tan w= 1/N, where N is the " scale " 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 " condition of Airy," i.e. tan w'J tan w=a constant. This simple relation (see Camb. Phil. Trans., 1830, 3, p. I) is fulfilled in all systems which are symmetrical with respect to their diaphragm (briefly named " symmetrical or holosymmetrical objectives "), or which consist of two like, but differentsized, 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. The constancy of a'/a necessary for this re
lation to hold was pointed out Q by R. H. Bow (Brit. Journ. Photog., 1861), and Thomas Sutton (Photographic Notes, 1862); it has been treated by 0. Lummer and by M. von Rohr (Zeit. f. Instrumentenk., 1897, 17, and 1898, 18, p. 4).
It requires the middle of the aperture stop to be reproduced in the centres of the entrance and exit pupils without spherical aberration. M. von Rohr showed that for systems fulfilling neither the Airy nor the BowSutton condition, the ratio a' tan 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.
Analytic Treatment of Aberrations.—The preceding review of the several errors of reproduction belongs to the " Abbe theory of aberrations," 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 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 " infinitely small of a certain order"; 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 connexion is only supplied by theories which treat aberrations generally and analytically by means of indefinite series.
A ray proceeding from an object point 0 (fig. 9) can be defined by the coordinates (E, n) of this point 0 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 image ray may be defined by the points (t' n'), 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
, rj , x', y' are functions of E, n, x, y; and if it be assumed that the field of view and the aperture be infinitely small, then t, n, x, y are of the same order of infinitesimals; consequently by expanding ', n', x', y' in ascending powers of E, i7, 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
Barrel shaped Cushion shaped Distorted image
Ogect
and the corresponding axes parallel, then by changing the signs of 07, x, y, the values 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.
The nature of the reproduction consists in the rays proceeding from a point 0 being united in another point 0'; in general, this will not be the case, for ', r7' vary if E, n 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 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' , with coordinates E'o, 17'o, of the point 0 at some distance from the axis could be constructed.
Writing 0E'='—'o and Or7'=p'—'o, then 0E' and 077' are the aberrations belonging to , tl 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 0, a patch of light, depending in size on the lowest powers of E, ri, x, y which the aberrations contain, will be formed in the plane I'. These degrees, named by J. Petzval (Bericht fiber die Ergebnisse einiger dioptrischer Untersuchungen,. Buda Pesth, 1843; Akad. Sitzber., Wien, 18J7, vols.xxiv.xxvi.) "the numerical orders of the image," are consequently only odd powers; the condition for the formation of an image of the mth order is that in the series for AE' and An' the coefficients of the powers of the 3rd, 5th . . . (m2)th degrees must vanish. The images of the Gauss theory being of the third order, the next problem is to obtain an 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.
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 L. Seidel (Asir. Nach., 1856, p. 289) ; in 1840, J. Petzval constructed his portrait objective, unexcelled even at the present day, from similar calculations, which have never been published (see M. von Rohr, Theorie and Geschichte des photographischen Objectless, Berlin, .1899, p. 248). The theory was elaborated by S. Finterswalder (Munchen. Akad. Abhandl., 1891, 17, p. 519), who also published a posthumous paper of Seidel containing a short view of his work (Munchen. Akad. Sitzber., 1898, 28, p. 395) ; a simpler form was given by A. Kerber (Beitrage zur Dioptrik, Leipzig, 1895—6—7—8—9). A. Konig and M. von Rohr (see M. von Rohr, Die Bilderzeugung in optischen Instrumenten, pp. 317323) 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 (pp. 212316).
The aberrations can also be expressed by means of the "characteristic function " 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 derived the aberrations of the third order; and in later times the method was pursued by Clerk Maxwell (Prot. London Math. Soc., 1874—1875; see also the treatises of R. S. Heath and L. A. Herman), M. Thiesen (Berlin. Akad. Sitzber., 1890, 35, p. 804), H. Bruns (Leipzig. Math. Phys. Bea, 1895, 21, p. 410), and particularly successfully by K. Schwartzschild (Gottingen. Akad. Abhandl., 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 Ann. d. Phys., 1905, 18, p. 941) founded his theory of aberrations on the differential geometry of surfaces.
The aberrations of the third order are: (1) aberration of the axis point; (2) aberration of points whose distance from, the
Aberra axis is very small, less than of the third order—the tions of deviation from the sine condition and coma here fall the third together in one class; (3) astigmatism; (4) curvature
order. of the field; (5) distortion.
(1) Aberration of the third order of axis points is dealt with in all textbooks on optics. It is important for telescope objectives, since their apertures are so small as to permit higher orders to 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 (thepower 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 (" undercorrection," 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 highpower lens. 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 " Herschel's condition. All these rules are valid; inasmuch as the thicknesses and distances of the lenses are not to be taken into account.
(2) The condition for freedom from coma in the third order is also of importance for telescope objectives; it is known as " Fraunhofer's condition." (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 " Petzval equation," r/r(n'—n)= o, where r is the radius of a refracting surface, n and n' the refractive indices of the neighbouring media, and E the sign of summation for all refracting surfaces.
Practical Elimination of Aberrations.—The existence of an optical system, which reproduces absolutely a finite plane on another with pencils of finite aperture, is doubtful; but practical systems 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 (in so far as aberrations must be taken into account) could be dealt with by means of the approximation theory; in most cases, however, the analytical difficulties are too great. Solutions, however, have been obtained in special cases (see A. Konig in M. von Rohr's Die Bilderzeugung, p. 373; K. Schwarzschild, Gottingen. Akad. Abhandl., 1905, 4, Nos. 2 and 3). 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, Lehrbuch der geometrischen Optik, 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.
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* (with infinitely distant objects: with a finite height of incidence h*) the same distance of intersection, and the same sine ratio as to one neighbouring the axis (u* or h* may not be much smaller
than the largest aperture U or H to be used in the system). The ,the
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 "zones," and the constructor endeavours 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*; " zones of astigmatism, curvature of field and distortion " attend smaller values of w. The practical optician names such systems: "corrected for the angle of aperture u* (the height of incidence h*), or the angle of field of view w*." 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.
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. The following may be regarded as typical: (I) Largest aperture; necessary corrections are—for the axis point, and sine condition; errors of the field of view are almost disregarded; example—highpower microscope objectives. (2) Largest field of view; 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 ordinary photographic objective: the portrait objective is corrected more with regard to aperture; objectives for groups more with regard to the field of view. (3) Telescope objectives have usually not very large apertures, and small fields of view; they should, however, possess zones as small as possible, and be built in the simplest manner. They are the best for analytical computation.
(b) Chromatic or Colour Aberration. '
In opti_al systems composed of lenses, the position, magnitude and errors of the image depend upon the refractive indices of the glass employed (see LENS, and above, " Monochromatic Aberration "). Since the index of refraction varies with the colour " or wave length of the light (see DISPERSION), it follows that a system of lenses (uncorrected) projects images of different colours .in somewhat different places and sizes and with different aberrations; i.e. there are " chromatic differences " 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 since they are all ultimately intercepted by a plane (the retina of the eye, a focussing screen of a camera, &c.), they cause a confusion, named chromatic aberration; for instance, instead of a white margin on a dark background, there is perceived a coloured 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 " chromatically undercorrected " when it shows the same kind of chromatic error as a thin positive lens, otherwise it is said to be " overcorrected."
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 constants of reproduction. These constants are determined by the data of the system (radii, thicknesses; distances, indices, &c., of the lenses); therefore their dependence on the refractive index, and consequently on the colour, are calculable (the formulae are given in CzapskiEppenstein, Grundziige der Theorie der optischen Instrumente (1903, p. 166). The refractive indices for different wave lengths 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 colours, 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 he achromatized, then the Gaussian image for all distances of objects is the same for the two colours, and the system is said to be in " stable achromatism."
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 contains the amount due to each refracting surface (sec CzapskiEppenstein, op. cit.pa 170; A. Konig in M. v. Rohr's collection, Die Bilderzeugung, p. 340). In a plane containing the image point of one colour, another colour produces a disk of confusion; this is similar to the confusion caused by two "zones " in spherical aberration. For infinitely distant objects the radius of the chromatic disk ofconfusion is proportional to the linear aperture, and independent of the focal length (vide supra," Monochromatic Aberration of the Axis Point "); 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 " relative aperture." (This explains the gigantic focal lengths in vogue before the discovery of achromatism.)
Examples.(a) 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 colour be n, and for another n+dn, and the powers, or reciprocals of the focal lengths, be 4) and 0+4), then (i) d4)/4) =do/(n–r) =1/v; do is called the dispersion, and v the dispersive power of the glass.
(b) Two thin lenses in contact: let 4)1 and 4)2 be the powers corresponding to the lenses of refractive indices nl and ;12 and radii r'1i r'i, and r'2, r"2 respectively; let 4) denote the total power, and d4), dni, dn2 the changes of 4), n1, and n2 with the colour. Then the following relations hold:
( 2) 0=4)1+y2 .(nl  I)(I/r 1 I/r"1) + (n2 I) (1/r'2 –I/r"2) (o11)ki+(n2 – I)k2; and
(3) d4) = kidn1 + k2 dn2. For achromatism d4) = o, hence, from (3),
(4) ki/k2= dn2/dni, or cki/Os= –Pi/v2. Therefore 01 and 4)2 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 4) be not zero (equation 2)), and the dispersive powers must also be different
(See TELESCOPE.)
Glass with weaker dispersive power (greater v) is named " crown glass "; that with greater dispersive power, " flint glass." For the construction of an achromatic collective lens (4) 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 di persive lens the converse must be adopted. This is, at thdrpresent day, the ordinary type,
e.g., of telescope objective (fig. 1o); 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 "Herschel" or "Fraunhofer condition," the
latter being the best (vide supra, " Monochromatic Aberration "). In practice, however, it is often ,
more useful to avoid the second condition by FIG. 10. making the lenses have contact, i.e. equal
radii. According to P. Rudolph (Eder's Jahrb. f. Photog., 1$91, 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; Asir. 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 v), 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, v decreased as it increased; but some of the Jena'glasl s by E. Abbe and O. Schott were crown
(according to 4).
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 determined by Chester More Hall in 1728, Klingenstierna in 1754 and by Dollond in 1757, who constructed the celebrated achromatic telescopes.
glasses of high refractive index, and achromatic systems from such crown glasses, with flint glasses of lower refractive index, are called the " new achromats," and were employed by P. Rudolph in the first " anastigmats " (photographic objectives).
Instead of making d(t) 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 H. be cemented and have the same refractive index for one colour, then its effect for that one colour 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 (d4/O) be greater than that of the same lens, this being made of the more dispersive of the two glasses employed, it is termed " hyperchromatic."
For two thin lenses separated by a distance D the condition for achromatism is D=(vifi+v2f,) (vi+v2); if v1=v2 (e.g. if the lenses be made of the same glass), this reduces to D =I (fi+f2), known as the "condition for oculars."
If a constant of reproduction, for instance the focal length, be made equal for two colours, then it is not the same for other colours, 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, since do 2 /dn 1 varies within the spectrum. This fact was first ascertained by J. Fraunhofer,''revho defined the colours 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 50%). If, therefore, for two colours, a and b, fa= f b = f, then for a third colour, c, the focal length is different, viz. if c lie between a and b, then fc 

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