# 1911 Encyclopædia Britannica/Objective

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​OBJECTIVE, or Object Glass, the lens of any optical system which first receives the light from the object viewed; in a compound system the rays subsequently traverse the eye-piece. The theoretical investigations upon which the construction of an optical system having specified properties is based, are treated in the article Aberration, and, from another standpoint, in the article Diffraction. Here we deal with the methods by which the theoretical deductions are employed by the practical optician. It should be noted that the mathematical calculations provide data which are really only approximations, and consequently it is often found that a system constructed on such data requires modification before it fulfils the practical requirements. For example, take the case of a photographic objective. Calculations of the paths of two extreme rays in the meridional section of an oblique pencil of large aperture may prove that the rays intersect on a plane containing the axial focus, but similar calculations of many other rays would be necessary before the mean point of intersection could be settled with sufficient exactness. Suppose, however, that the optician has accurately realized the results of the mathematician, he can then determine the divergence of the practical from the theoretical properties by measuring the positions and conformation of the most distinct or mean foci, and, if sufficiently acquainted with the theory of the construction, he can modify one or more curvatures or thicknesses and so attain to a closer agreement with the ideal. Theory and practice co-operate in the realization of an original system. The order is not always the same, but generally the mathematician, by notoriously laborious calculations, supplies data which are at first closely followed by the constructor and afterwards modified in accordance with experimental observations.
In addition to the problem of constructing an original system, the optician has to deal with the reproduction of a realized system in different sizes. Two questions then arise: (1) To what degree of accuracy the radii of curvature can, or should, be repeated, and (2) to what degree of uniformity the surfaces can, or should be figured. With regard to the first point there is no great difficulty in working the requisite iron or brass tools of any curvature to within an error of 1/10th% of the radius; male and female templets being used for very deep curves, and the spherometer for tools of longer radii (by appropriate grinding together, the radii are alterable at will within narrow, but sufficient, limits). The accuracy attained in the grinding, however, is open to very perceptible modification by the subsequent polishing and figuring processes. This is particularly undesirable in the case of deep curves and large apertures. A variation in a radius of curvature may occasion a little spherical aberration at the axial focus, but if the amount be small it may be neutralized by imparting to the lens a parabolic form or its opposite. Such an artifice is frequently adopted in correcting large telescope objectives.
With optical systems which transmit large pencils with considerable obliquity (such as wide angle photographic objectives) the curves are very deep, and a departure from the true radius which would be tolerated in a telescope cannot be permitted here. Such lenses are usually tested by means of a master curve worked in glass. The master curve is fitted to the experimental lens, and an inspection of the interference fringes shows the quality of the fit—whether it be perfect, or too shallow or too deep. The Workman then modifies his polisher or stroke in order to correct the divergence. Flat surfaces are tested similarly. This test by contact has been strongly advocated and has been regarded as sufficient to detect all irregularities of any moment. This claim, however, is not justified, for the test is not sensitive to errors sufficient in amount to render a telescope objective almost valueless; but such errors are easily discernible by other optical devices. In general, accuracy in the radii of curvature is of primary importance and trueness of figuring is of secondary importance in photographic objectives, while the reverse holds with telescopic objectives; in wide angle microscopic objectives these two conditions are of equal moment. Eye pieces do not require the same degree of accuracy either in the curvature or the figuring.
A rough idea of the exactitude to which the figuring of the finest telescope objectives must be carried out is readily deduced. If two slips of paper, bearing printed letters 1/20 of an in. high be placed in almost exact alignment, one 31·2 in. from the eye and the other 39 in., and viewed in moderate daylight with the eye having a pupillary aperture of 1/8 of an in., one set of the letters will be legible while the other is not. In this case the difference of convergence or refracting power exercised by the eye in transferring its focus from one slip to the other is 1/156 or one quarter diopter. If an image on the retina is 1/4 diopter out of focus, then each point of the object is represented by a circle of confusion 0·0004 in. or 2′ 45″ in angular measure in diameter, the focal length of the eye being assumed to be 0·5 in. and the pupillary aperture 1/8 of an in. If the effective aperture of the pupil or the aperture of a pencil traversing the pupil be 1/𝑛th of this standard, the size of the disk of confusion will be the same (viz. 0·0004 in.) if the retinal image be n quarter diopters out of focus. In general, for a constant size of the circle of confusion or, in other words, the same amount of visual blurring, the apertures of the pencils traversing the pupil and the focussing errors (expressed in quarter diopters) vary inversely.
If a portion of a figured surface of a telescope objective differs in curvature from the major portion of the lens so as to form a circle of confusion on the retina of a diameter not less than 2′ 45″, it is clear that the lens is faulty, the image formed by the perfect portion being sharp and well defined, and that formed by the imperfect portion blurred to the extent above determined, and to a greater extent if we allow for the effect of diffraction in the formation of the image. For example, a protuberance 1 in. in diameter at the centre of an object glass of 12 in. aperture refracting to a separate focus would theoretically form a spurious disk of about 5 seconds diameter, which would subtend a diameter of 50 minutes at the retina under a power of 600.
Regarding 2′ 45″ as the maximum diameter of a geometric circle of confusion permissible in a telescopic object glass, we proceed to determine the heights of the protuberance or depression which causes it. If 𝑓 be the equivalent focal length of the eye-piece and F that of the objective (the back focal length in the case of the microscope), then the linear error at the focus of the eye-piece is 1/156𝑓$^2$, or, expressed as a variation of 1/F, 1/156(𝑓/F)$^2$, (＝∆1/F). If a lens has one side plane and is worked to a mathematically sharp edge, its thickness 𝑡 at the centre is (approximately) A$^2$/8𝑟, where A is the whole aperture and 𝑟 the radius; and if 𝑔 be the equivalent focal length and μ the refractive index, we may write 𝑟＝𝑔(μ−1) and obtain
| 𝑡＝A$^2$/8𝑔(μ−1) | (1). |
​It is clear that for lenses in which the focal length is large compared with the aperture, the thickness 𝑡 is independent of the shape of the lens so long as the focal length and aperture remain constant. Consequently a protuberance may be regarded as a thin meniscus lens with mathematically sharp edges accurately fitted to a perfectly regular spherical surface. Substituting for 1/𝑔 the 1/156 (𝑓/F)$^2$ obtained above it follows that
| 𝑡＝A$^2$/8(μ−1) 1/156 (𝑓/F)$^2$ | (2). |
The effective aperture of the eye has been supposed to be 1/8 in.; calling this P, it is then obvious that (since F/𝑓 is the magnifying power) P(F/𝑓) is the theoretical aperture of objective requisite to supply the 1/8 in. eye-pencil. Substituting P(F/𝑓) for A in equation (2) we obtain
| 𝑡＝P$^2$/8(μ−1)×156 | (3). |
This relation gives the thickness of a meniscus protuberance fitted to an objective (assumed to have an unlimited aperture) which fills the 1/8 in. pupil and occasions the maximum blurring permissible. If μ be 1·5, 𝑡 is equal to 1/39,936 in.
If the thickness 𝑡 correspond to the aperture A, then for another aperture 𝑎 to produce the same blurring we must have ∆′ (1/F)＝ ∆(1/F)A/𝑎, i.e. the focal length of the protuberance, and therefore the thickness 𝑡 must vary as A. Consider a telescope of 12 in. aperture, focal length of objective (F)＝180 in., focal length of eyepiece (𝑓) 0·3 in. and magnifying power (F/𝑓)=600. The aperture theoretically requisite to transmit the pupillary pencil 1/8 in. aperture is 1/8·600＝75 in. If the permissible protuberance cover the entire aperture of 75 in. its thickness would be 1/39,936 in. as above, but if restricted to a diameter of 1 in., then the maximum allowable thickness would be 1/75×1/39,936 in.＝say 1/3,000,000 in. Since the latter protuberance is assumed to fill only 1/75 of the aperture of the pupil of the eye, it produces an error in focussing equivalent to 75 quarter diopters or 75/156. If we take the power of the eyepiece to be 1/·3 in. and subtract from it 75/156, we obtain 1/·35, so that AF is −·05 in.
Either the knife-edge test, or the more usual method of testing figuring by examining the out-of-focus disks formed on the retina when the eye-piece is inside and outside its correct focus, would certainly show the effect of this protuberance as a bright central spot when inside focus, and a dark central patch when outside; a practised eye can detect one-half the above error, and a quarter when the power is 1200 instead of 600. It may be noticed that, under the same circumstances, the error permissible in a reflecting telescope is only one quarter of that admitted in the refractor. In the case of a microscope objective of 10 in. back-focal-length used with a 1 in. eye-piece, the aperture required to transmit the pupillary pencil of 1/8 in. aperture is 11/4 in. Regarding the supposititious protuberance or depression as 1/20 in. in diameter, its thickness or depth must not exceed 1/39,936×0·05/1·25, or say 1/1,000,000 in. Therefore the accuracy of figuring required in the best microscopes does not fall far short of that required in telescopes.
The best optical workmanship, as applied to large reflecting surfaces, aims at reducing local protuberances or depressions to within the limiting height or depth of one twelve-millionth part of their diameter (A) and the optical methods which detect these errors are exceedingly delicate. The finest spherometer detects errors down to about three-millionths of an inch, below which it is valueless. The same applies to the study of the interference fringes formed when a master curve is fitted. It will not show up such fine errors. The figuring of spherical surfaces 12 in. or more in diameter by abrasion with a polisher so that no part of the surface is elevated or depressed above the average level by more than the above defined amounts is commonly practised, but much technical knowledge is necessary for success. It is a sine qua non that the material of the polisher should be as plastic and inelastic as is consistent with a moderate degree of hardness. The best material for large work is Stockholm pitch from which the greater part of the turpentine has been removed by evaporation, and the abrasive used is the finest rouge and water. For small work certain waxes, more or less mixed with rouge or putty powder, are used. Water is used as the lubricant. During delicate figuring temperature changes must be carefully avoided, otherwise buckling and consequent bad figuring of the lens or a variation in the hardness of the polisher may supervene. The motion of the polisher must therefore be leisurely. Moreover, any surface must be allowed to attain a uniform temperature before testing. When, as often happens, an elevation or depression on a large lens apparently refuses to be dislodged by straightforward polishing, recourse is had to local retouching. The faulty parts are localized by optical tests and then rubbed down by small polishers of an inch or more in diameter. In this way a central protuberance 1 in. in diameter and 1/2,000,000 of an in. high standing on the centre of a large objective may be removed by a polisher less than an inch in diameter worked at 200 half inch strokes per minute and at a pressure of 6 ozs. in about a minute. Great care is required, for if the process be carried too far, the whole surface must be re-figured. Local retouching serves to remove those conspicuous zones of aberration to which certain photographic lenses of large relative aperture are necessarily liable. An annular channel is polished out at a mean distance equal to 7/10 of the semi-aperture from the centre of the lens, and this is carefully shaded off towards the centre and also towards the edge; this corrects the zone of rays which focus at a point short of the focus of the centre and edge rays. This correction is particularly necessary in the case of certain lenses designed for stellar photography. (H. D. T.)