1902 Encyclopedia > Microscope > Objectives. Numerical Aperture.

(Part 6)


It has been seen that one of the principal points in the construction of microscopic objectives to which the attention of their makers has been constantly directed has been the enlargement of their "aperture,"—this term being understood to mean, not their absolute opening as expressed by linear measure, but their capacity for receiving and bringing to a remote conjugate focus the rays diverging from the several points of a near object. The aperture of an objective has been usually estimated by its "angle of aperture,"—that is, by the degree of divergence of the most extreme rays proceeding from the axial point of the object to the niargin of the objective (fig. 15) which take part in the formation of the image. It is pointed out, however, by Professor Abbe that, in the case of single lenses used as objectives, their apertures are really proportional, not to their respective angles of aperture, but to the ratio between the actual diameter or clear opening of each to its focal distance, a ratio which is simply expressed by the sine of its semiangle. And in the case of combinations of lenses it can be demonstrated mathematically that their respective apertures are determinable—other conditions by the same—by the ratio of the, diameters of their back lenses, so far as these are really utilized, to their respective focal lengths—this ratio being expressed, as before, by the sine of the semiangle of aperture (sin u).

The difference between these two modes of comparison can be readily made obvious by reference to the theoretical maximum of 180º, which is attained by openinjout the boundaries of the angle abc (fig. 15) until they come into the same straight line, the sine of the semiangle (90º) then becoming unity. For, while an objective having an angle of 60º would count by comparison of angles as having only one-third of the theoretical maximum, its real aperture would be half that maximum since the sine of its semiangle (30º)=1/2. And, as the sines of angles beyond 60º increase very slowly, an objective of 120º angle will have as much as 87 per cent. of the theoretical maximum of aperture, although its angle is only two-thirds, or 66·6 per cent., of 180º. It hence becomes obvious that little is really gained in real aperture by the opening-out of the angle of microscopic objectives to its greatest practicable limit (which may be taken as 170º), while such extension—even if unattended with any loss either of definition or of colour-correction—necessarily involves a great reduction alike in the working distance and in the focal depth or penetration of the combination, as will be presently explained.

Numerical Aperture.—It has now been demonstrated by Professor Abbe that, independently of the advantages already specified as derivable from the application of the immersion system to objectives of short focus and wide aperture, the real aperture of an immersion objective is considerably greater than that of a dry or air objective of the same angle—the comparative apertures of objectives working through different media being in the compound ratio of two factors, viz., the sines of their respective semiangles of aperture and the refractive indices of the "immersion" fluids. It is the product of these (n sin u) that gives what is termed by Professor Abbe the "numerical aperture,"—which serves, therefore, as the only true standard of comparison, not only between dry or air and water or oil immersion lenses, but also between immersion lenses adapted to work respectively with water, oil, or any other interposed fluid. That the angle of aperture expressed by the same number of degrees must correspond with very different working apertures in dry, water immersion, and oil or homogeneous immersion objectives becomes evident when we consider what happens when divergent pencils of rays pass from one medium into another of higher refractive index. For such divergent pencils, proceeding from air into water or oil, will be closed together or compressed; so that the rays which, when an object is mounted in air, spread out over the whole hemisphere then form comparatively narrow pencils, and can thus be utilized by an immersion objective of smaller aperture than is required in a dry objective to admit the most diverging rays of air-pencils. It follows, therefore, that a given angle in a water or oil immersion objective represents a much larger aperture than does the same angle in an air-objective; and thus it comes to pass that by opening out the angle of immersion objectives they may be made to receive and utilize rays of much greater divergence than can possibly enter dry objectives of even maximum aperture.

The following table, abridged from that given by Professor Abbe for every 0·02 of numerical aperture from 0·50 up to the maximum of 1·52, brings this contrast into clear view:—

== TABLE ==

Thus, taking as a standard of comparison a dry objective of the maximum theoretical angle of 180º, whose numerical aperture is the sine of 90º, or 1·00, we find this standard equalled by a water-immersion objective whose angle of aperture is no more than 97 1/2, and by an oil or homogeneous immersion objective of only 82º—the numerical apertures of these, obtained by multiplying the sines of their respective semiangles by the refractive index of water or of oil, being 1·00 in each case. Each, therefore, will have as great a power of receiving and utilizing divergent rays as any dry objective can even theoretically possess.

But, as the actual angle of either a water or an oil immersion objective can be opened out to the same extent as that of an air or dry objective, it follows that the aperture of the former can be augmented far beyond even the theoretical maximum of the latter. Thus the numerical aperture of a water-ininiersion lens of the maximum angle of 180º is 1·33, or one-third greater than that of an air-lens of the same angle; and this aperture would be given by an oil-immersion objective of only 122º. Again, the numerical aperture of an oil-immersion objective having the theoretical maximum angle of 180º would be 1·52, or more than onehalf greater than that of an air-lens of the same angle. And the numerical apertures corresponding to angles of 170º, which have been actually attained in both cases, fall very little short of the proportions just given.

So, again, an oil-immersion objective whose angle of aperture is only 60º has as high a numerical aperture (0·76) as a water immersion objective of 69 1/2, or as a dry objective of 99º; and a dry objective of 140º has no greater a numerical aperture (0·94) than a water-immersion of 90º or an oil-immersion of 76 1/2.

This important doctrine may be best made practically intelligible by a comparison of the relative diameters of the back lenses of dry with those of water and oil immersion objectives of the same powey, from an "air-angle" of 60º to an "oil-angle" of 180º,—these diameters expressing, in each case, the opening between the extreme pencil-forminsg rays at their issue from the posterior surface of the combination, to meet in its conjugate focus for the formation of the image, the relation of which opening in each case to the focal length of the combination is the real measure of its aperture (fig. 16). Thus the dry objective of 60º angle (5 in fig. 16) has its air-angle represented by sin u=1/2=0·50 numerical aperture. The dry objective of 97º (4) has its air-angle represented by sin u ==3/4 =0·75 numerical aperture. And the dry objective having the (theoretical) angle of 180º (3) has its air-angle represented by sin u =1·00 numerical aperture,—this corresponding to 96º water-angle and 82º oil-angle. But the water-immersion lens having the (theoretical) angle of 180º (2) has its water-angle represente by n sin u = 1·33 numerical aperture. And the oil-immersion lens having the (theoretical) angle of 180º (1) has its oil-angle represented by n sin u = 1·52 "numerical aperture." [Footnote 268-1] These theoretical apertures for water and oil immersion lenses having been found as nearly attainable in practice as the theoretical maximum for dry objectives, such lenses can utilize rays from objects mounted in balsam or other dense media, which are entirely lost for the image (since the do not exist physically) when the same object is in air or is observed through a filia of air. And this loss cannot be compensated by an increase of illumination; because the rays whichare lost are different rays physically from those obtained by any illumination, however intense, through all aeriform medium.

It is by increasing the number of diffraction-spectra that the additional rays thus received by objectives of great numerical aperture impart to them an increased resolving power for lined and dotted objects,—the truth of the image formed by the recombination of these spectra being (as already shown) essentially dependent oil the number of them that the objective may be capable of receiving.

But whilst the resolving power of microscopic objectives increases in the ratio oi their respective numerical apertures, and whilst their illuminating power (dependent upon the quantity of light that passes through them) increases with the square of the numerical aperture, the case is reversed with another most important quality,—that of penetration or focal depth; for this diminishes as the numerical aperture increases, until nothing but what is precisely in the focal plane can be even discerned with objectives possessed of the highest resolving power. Thus, the penetrating power of an objective of 60º air-angle being expressed as 2·000, an extension of that angle to 76 1/2º reduces it to 1·613, all extension to 89º reduces it to 1·429, and an extension to 99º reduces it to 1·316; further extension to 118 1/2º reduces it to 1·163, while an objective whose air-ailgle is 140º has a penetrating powerof only 1·064. So, again, the oil-immersion objective which has the numerical aperture of 1·00 corresponding to the theoretical air-angle of 180º has a penetrating power of 1·000; this is brought down to ·752 when its angle is so increased as to make its numerical aperture 1·33, equalling the theoretical maximum of a water-immersion objective, and is ·658 at the theoretical maximum (1·52) of an oil-objective-

Hence it is clear that, as some of the qualities to be sought in microscopic objectives are absolutely incompatible, a preference is to be accorded to objectives of greatest resolving power but very little penetration, or to those of moderate resolving power and great penetration, according to the uses to which they are to be applied; and some general principles will now be laid down in regard to this matter, based alike on science and experience.

In the first place, a marked distinction is to be drawn between those objectives of low or moderate power which are to be worked, dioptrically and those of high power which are to be worked diffractively. The objects on which the former are to be for the most part used are either minute transparent bodies having solid forms which the observer should be able to take in as wholes (as in the case of Polycystina, the larger diatoms, Infusoria, &c.); or transparent sections, dissections, or injections, whose parts lie in. different planes, the general relations of which he desires to study, while reserving their details for more special scrutiny; or opaque objects, whose structure can only be apprehended from the examination of their surfaces, when the inequalities of those surfaces are seen in their relations to each other. In all these cases. it is desirable that microscopic vision should resemble ordinary vision as much as possible. If the eye were so constructed as to enable us to discern only those parts of an object that lie precisely in the plane to whichwe focus it, our visual conceptions of the forms and relations of these parts, and consequently of the object as a whole, would in general be very inadequate, and often erroneous. It is because, while focussing our eye successively oil the several planes of the object, we can see the relation of each to what is nearer and more remote that we can readily acquire a visual conception of its shape as a whole, and that unmistakable perception of solid form which is given by the combination of the two dissimilar perspectives of' near objects in binocular vision (p. 273) could not possibly be formed if our vision were strictly limited to the exact plane for which our eyes are focussed.

Hence it is obvious that, in the case of objectives of low and moderate, amplification, focal depth or penetration is a quality for the want of which no other excellence can compensate,—the opening-out of their apertures being only advantageous in so far as it does not seriously interfere with their penetrating power. It is, no doubt, quite possible to construct a 1 inch objective with an aperture so large that, when the requisite amplification has been gained by deep eye-piecing, it shall resolve the lined "tests" ordinarily used for a 1/4, or to construct an objective of 4/10 inch focus which shall in like manner do the ordinary work of a 1/8. But, as such objectives are thereby spoiled for their own proper work, the loss to the microscopist is but poorly compensated by his ability to resolve with them, under such deep eye-pieces as cannot be habitually used without serious risk to the eye-sight, the lined and dotted tests which can be much better shown under objectives of shorter focus and wider aperture, with eye-pieces of low amplification. For, whilst deep eye-pieces cannot be habitually employed for continuous observation, without putting a strain upon the eyes resembling that which results from the constant use of a magnifying glass, even the very highest objectives may be used continuously for long periods in combination with shallow eye-pieces, with scarcely any fatigue, and therefore (it is probable) without sensible injury. [Footnote 269-1]

In estimating the goodness of a microscopic objective, five distinct qualities have to be separately considered:—(1) its working distance, or the actual interval between its front lens and the object on which it is focussed; (2) its penetrating power, or focal depth; (3) the flatness of its field; (4) its definition, or power of diving a distinct image of all well-marked features of an object, and especially of their boundary lines; and (5) its resolving power, by which it separates closely approximated lines, dots, or striae.

1. The "working distance" of an objective has no fixed relation to its focal length,—the latter being estimated by its equality in power with a single lens of given radius of curvature (such as 1/4 inch, 1/4 inch, 1/12inch, &c.), while the former varies with the mode in which the combination is constructed and with the aperture given to it. For low and moderate powers, ranging up to 4/10 inch focus, good working distance is especially important, alike because it is closely related to penetrating power, and also because it facilitates the use of side-illumination for opaque objects. And in such objectives of high power as are to be used, not for the resolution of lined or dotted tests, but for the observation of living and moving objects of extreme minuteness, good working distance is no less important, on account of its relation to focal depth. In the case of those objectives, on the other hand, in which resolving power is made the first consideration, it is only needful that the working distance shall be such as to permit the interposition of a thin glass cover; and this, although necessarily diminished with the widening of the aperture, can be always obtained by the adoption of the immersion system.

2. The "penetrating power" or "focal depth" of an objective may be defined as consisting in the vertical range through which the parts of an object not precisely in the focal plane may be seen with sufficient distinctness to enable their relations with what lies exactly in that plane to be clearly traced out,—just as would be done by ordinary vision if the object were itself enlarged to the dimensions of its microscopic image. The close relation between this quality and the preceding becomes obvious when it is considered is that the longer the working distance of an objective the less will the distinctness of the image it forms be affected by any given alteration (say the 1/10000 of an inch) in its focal adjustment. Consequently, of two objectives having the same magnifying power but different working distances, that one will have the most focal depth whose working distance is the greater. On the other hand, as the penetrating power of an objective is reduced in direct accordance with the increase of its numerical aperture, it must be sacrificed wherever the highest resolving power is to be attained. Hence, as already remarked, this attribute will be very differently valued by different observers, according to the work on which they are respectively engaged. For the general purposes of biological research, not only with low or moderate (the reasons already stated), but also with high powers, a considerable amount of focal depth is essential. It is impossible, for example, to follow satisfactorily the movements of an Amoeba, or to study the "cyclosis" in the cell of a Vallisneria, or to trace the distribution of a nerve-thread, with an objective in which focal depth is so completely sacrificed to aperture that nothing can be discerned save what is precisely in the focal plane, since, instead of passing gradationally from one focal plane to another, as the observer can do with an objective of good penetration, he can only get a succession of "dissolving views," with an interval of "chaos" between each pair. When, on the other hand, it is desired to scrutinize with the greatest precision such minute details as are presented in one and the same focal plane (as, for example, those of the thinnest possible film of tissue spread out between a glass slide and its covering glass), the microscopist will prefer an objective in which focal depth is subordinated to aperture, for the sake of fhe resolving power which he can thus command. And it will often happen in biological research that it is advantageous thus to bring objectives of the latter class to bear upon objects which could not have been detected in the first instance save by objectives of much inferior resolving power but greater focal depth.

3. The "flatness of the field" afforded by the objective is a condition of great importance to the advantageous use of the microscope, since the extent of the area clearly seen at one time practically depends upon it. Many objectives are so constructed that, even when the object is perfectly flat, the foci of the central and peripheral parts of the field are so different that, when the adjustment is made for one, the other is more or less indistinct. Hence, when the central part of the area is in focus, no more information is gained respecting the peripheral than if the latter bad been altogether stopped out. With a really good objective, not only should the image be distinct over the whole field at once, but the marginal portion should be as free from colour as the central. As imperfection in this respect is often masked by the contraction of the aperture of the, diaphragm in the eye-piece, the relative merits of two objectives, as regards flatness of field should always be tested under an eye-piece giving a large aperture.

4. The "defining power" of an objective, which depends upon the completeness of its corrections for spherical and for chromatic aberration, and upon the accurate centring of its component lenses, is an attribute essential to its satisfactory performance, whatever may be its other qualities,—its importance in scientific research being such that no superiority in resolving power can compensate for the want of it; and, though it is possible to obtain perfect correction for spherical aberration up to the highest practicable limit of angle, yet the difficulty of securing it increases rapidly with the augmentation of aperture, the want of it being made perceptible, especially when deep eye-pieces are put on, by the blurring of clearly-marked lines or edges, and by general "fog." Perfect colour-correction, on the other hand, is not possible for dry lenses of the widest angle, on account of the irrationality of the secondary spectrum; but this may be neutralized by the use of the immersion system. As already stated, what has to be aimed at in the construction of microscopic objectives is not absolute colour-correction, but a slight degree of over-correction, which, by compensating the chromatic dispersion of the Huygenian eye-piece, shall produce an image free from false colour. As this can be secured far more easily in the construction of objectives of moderate than in those of very wide aperture, the cost of the former is proportionally small,—an additional reason for the preference to be given to them on other grounds, in regard to all save very special kinds of microscopic work.

5. "Resolving power," being that by which very minute and closely approximated markings—whether lines, striae, dots, or apertures—can be separately discerned, is a function which is only of primary importance in objectives whose amplifying power specially fits them for the study of objects of this class. It appears from the mathematical researches of Professor Abbe that the maximum resolving power (with a theoretical angle of 180º) would be capable of separating 146,528 lines to the inch; but he considers the limit of visual resolution depending on the power of the eye to be about 1/118000 of an inch; and this limit seems to have been nearly reached. To make such a separation distinctly perceptible, an amplification of at least 3000 linear would be requisite; and this can only be obtained either by the use of an objective of very high power (such as 1/25 inch focus) in combination with a low or medium eye-piece or by putting a very deep eye-piece upon an objective of lower power (such as a 1/8 inch),—the former method, for the reasons already given, being decidedly preferable. For the resolution of less closely approximated markings objectives of 1/20, 1/16, 1/12, and 1/8 inch answer very well; and the resolving power which they require may be obtained without any excessive widening of the aperture. For the loss of resolving power consequent upon the contraction of the angle of a water-immersion objective to 128 1/3º is only one-tenth of the theoretical maximum 128,212; while a reduction to 105 2/3º only lowers the number of separable lines to 102,184 to the inch,—thus diminishing the resolving power by little more than one-fifth, while the working distance and focal depth of the combination are greatly increased, and perfect definition is more certainly attainable. The inch is (according to the writer’s experience, which is confirmed by the theoretical deductions of Professor Abbe) the lowest objective in which resolving power should be made the primary qualification—the 1/6, 1/5, _, and 4/10 inch being specially suited to kinds of biological work in which this is far less important than focal depth and dioptric precision. This view is strengthened by the very important consideration that the resolving power given by wide aperture cannot be utilized, except by a method of illumination that causes light to pass through the object at an obliquity corresponding to that at which the most divergent rays enter the objective. Now, although in the case of objects whose markings are only superficial such obliquity may not be productive of false appearances (though even this is scarcely con-ceivable), it must have that effect when the object is thick enough to have an internal structure; and the experience of all biological observers who have carried out the most delicate and difficult investigations is in accord, not only as to the advantage of direct illumination, but as to the deceptiveness of the appearances given by oblique, and the consequent danger of error in any inferences drawn from the latter. Thus, for example, the admirable researches of Strassburger, Fleming, Klein, and others upon the changes which take, place in cell-nuclei during their subdivision can only be followed and verified (as the writer can personally testify) by examination of these objects under axial illumination, with objectives of an angle so moderate as to possess focal depth enough to follow the wonderful differentiation of component parts brought out by staining processes through their whole thickness.

The most perfect objectives for the ordinary purposes of scientific research, therefore, will be obviously those which combine exact definition and flatness of field with the widest aperture that can be given without an inconvenient reduction of working distance and loss of the degree of focal depth suitable to the work on which they are respectively to be employed. These last attributes are especially needed in the study of living and moving, objects; and, in the case of these, dry objectives are decidedly preferable to immersion, since the shifting of the slide which is requisite to enable the move-ment of the object to be followed is very apt to produce disarrange-ment of the interposed drop. And, owing to the solvent power which the essential oils employed for homogeneous immersion have for the ordinary cements and varnishes, such care is necessary in the use of objectives constructed to work with them as can only be given when the observer desires to make a very minute and critical examination of a securely-mounted object.

The following table expresses the magnifying powers of objectives constructed on the English scale of inches and parts of an inch, with the 10 inch body and the A and B eye-pieces usually supplied by English makers, and also specifies the angle of aperture which, in the writer’s judgment, is most suitable for each. He has the satisfac-tion of finding that his opinions on this latter point, which are based on long experience in the microscopic study of a wider range of animal and vegetable objects than has fallen within the purview of most of his contemporaries, are in accordance with the conclu-sions, drawn by Professor Abbe from his profound investigations into the theory of microscopic vision, [Footnote 270-1] which have been carried into practical accomplishment in the excellent productions of Mr Zeiss.

== TABLE ==

For ordinary biological work, the 1/8, 1/10, and 1/12 objectives, with angles of from 100º to 120º, will be found to answer extremely well if constructed on the water-immersion system.

Each of these powers should be tested upon objects most suited to determine its capacity for the particular kind of work on which it is to be employed; and, in such testing, the application of deeper eye-pieces than can be habitually employed with advantage will often serve to bring out marked differences between two objectives which seem to work almost equally well under those ordinarily used,—defects in definition or colour-correction, and want of light, which might otherwise have escaped notice, being thus made apparent. No single object is of such general utility for these purposes as a large well-marked Podura scale for the eye which has been trained to the use of a particular specimen of' it will soon learn to recognize by its means the qualities of any objective between 1 inch and 1/6 inch focus; and it may be safely asserted that the objective which most clearly and sharply exhibits its characteristic markings is the best for the ordinary work of the histologist.

For the special attribute of resolving power, on the other band, tests of an entirely different order are required; and these are fur-nished, as already stated, either by the more "difficult" diatoms or by the highest numbers of Nobert’s ruled test-plate. The diatom-valve at present most in use as a test for resolving power is the Amphipleura pellucida, the lines on which were long supposed to be more closely approximated than those of Nobert’s nineteenth band, being affirmed by Air Sollitt to range from 120 to 130 in 1/1000 of an inch. But the admirable photographs of this valve obtained by Colonel Dr Woodward have confirmed the conclusion long previously expressed by the writer, that this esti-mate was far too high, being, based on the "spurious lineation" produced by diffraction, and show that the striae on the largest valves do not exceed 91, while those on the smallest are never more numerous than 100, in 1/1000 of an inch. The same admirable manipulator has also obtained excellent photographs of another very difficult test-diatom, Surirella gemma, from which it appears- that its transverse striae count longitudinally at the rate of 72,000 to the inch, whilst the beaded appearances into which these may be resolved count transversely at the rate of 84,000 to the inch. Thus it appears that the complete resolution of these "vexatious" diatoms does not require by any means the maximum of aperture, but is probably dependent at least as much on the perfection of the corrections and the effectiveness of the illumination.

It must be understood that there is no intention in these remarks to undervalue the efforts which have been perseveringly made by the ablest constructors of microscopic objectives in the direction of enlargement of aperture. For these efforts, besides increasing the resolving power of the instrument, have done the great service of producing a vast improvement in the quality of those objectives of moderate aperture which are most valuable to the scientific biolo-gist; and the microscopist who wishes his armamentum to be com-plete will provide himself with objectives of those different qualities, as well as different powers, which shall best suit his particular requirements. [Footnote 270-2]


268-1 The dotted circles in the interior of 1 and 2, of the same diameter as 3, show the excess in the diameters of the back lenses of the water and oil objectives over that of the dry at their respective theoretical limits.

269-1 Hence, for work of this kind, the shallower eye-pieces and longer tubes of English microscopes are to be preferred to the deeper eye-pieces and shorter tubes of the ordinary Continental model, the shallowest, eye-pieces of the latter being usually equal in power to the ordinary B eye-pieces of the former.

270-1 See his paper on "The Relation of Aperture and Power in the Microscope," in Jour. Roy. Micros. Soc., 1882. pp. 300, 460.

270-2 See the remarks of Mr Dallinger,—whose experience in the application of the highest power to tile study of the minutest living objects is probably greater than that of any living observer,—in Jour. Roy. Micros. Soc., December 1882, p. 853.

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