Chapter I: Elementary Optics (2)

This experiment illustrates a case of the _obliteration_ of structure by obstructing the passage of the diffraction spectra to the eye-piece. The next experiment shows how the appearance of fine structure may be _created_ by manipulating the spectra.

When a diaphragm such as that shown in Fig. 29 is placed at the back of the objective, so as to cut off each alternate one of the upper row of spectra in Fig. 27_a_, that row will obviously become identical with the lower one, and if the theory holds good, we should find the image of the upper lines identical with that of the lower. On replacing the eye-piece, we see that it is so, the upper set of lines are doubled in number, a new line appearing in the centre of the space between each of the old (upper) ones, and upper and lower set having become to all appearance identical, as seen in Fig. 29_a_.

In the same way, if we stop off all but the outer spectra, as in Fig. 30, the lines are apparently again doubled, as seen in Fig. 30_a_.

A case of apparent creation of structure, similar in principle to the foregoing, though more striking, is afforded by a network of squares, as in Fig. 31, having sides _parallel_ to this page, which gives the spectra shown in Fig. 31_a_, consisting of vertical rows for the horizontal lines and horizontal rows for the vertical ones. But it is readily seen that two diagonal rows of spectra exist at right angles to the diagonals of the squares, just as would arise from sets of lines in the direction of the diagonals, so that if the theory holds good we ought to find, on obstructing all the other spectra and allowing only the diagonal ones to pass to the eye-piece, that the vertical and horizontal lines have disappeared and are replaced by two new sets of lines at _right angles to the diagonals_.

On inserting the diaphragm, Fig. 32, and replacing the eye-piece, we find in the place of the old network the one shown in Fig. 32_a_, the squares being, however, smaller in the proportion of 1 : √2, as they should be in accordance with the theory propounded.

An object such as _Pleurosigma angulatum_, which gives six diffraction spectra arranged as in Fig. 33, should, according to this theory, show markings in a hexagonal arrangement. For there will be one set of lines at right angles to _b_, _a_, _e_, another set at right angles to _c_, _a_, _f_, and a third at right angles to _g_, _a_, _d_. These three sets of lines will obviously produce the appearance shown in Fig. 33_a_.

A great variety of appearances may be produced with the same arrangement of spectra. Any two adjacent spectra with the central beam (as _b_, _c_, _a_) will form equilateral triangles and give hexagonal markings. Or by stopping off all but _g_, _c_, _e_ (or _b_, _d_, _f_), we again have the spectra in the form of equilateral triangles; but as they are now further apart, the sides of the triangles in the two cases being as √3 : 1, the hexagons will be smaller and three times as numerous. Their sides will also be arranged at a different angle to those of the first set. The hexagons may be entirely obliterated by admitting only the spectra _g_, _c_, or _g_, _f_, or _b_, _f_, etc., when new lines will appear at right angles, or obliquely inclined, to the median line. By varying the combinations of the spectra, therefore, different figures of varying size and positions are produced, all of which cannot, of course, represent the true structure. Not only, however, may the appearance of particular structure be obliterated or created, but it may even be _predicted_ before being seen under the microscope. If the position and relative intensity of the spectra in any particular case are given, the character of the resultant image, in some instances, may be worked out by mathematical calculations. A remarkable instance of such a prediction is to be found in the case recorded by Mr. Stephenson, where a mathematical student who had never seen a diatom, worked out the purely mathematical result of the interference of the six spectra _b-g_ of Fig. 33 (identical with _P. angulatum_), giving the drawing copied in Fig. 34. The special feature was the small markings between the hexagons, which had not, before this time, been noticed on _P. angulatum_. On more closely scrutinizing a valve, stopping out the central beam and allowing the six spectra only to pass, the small markings were found actually to exist, though they were so faint they had previously escaped observation until the result of the mathematical deduction had shown that they _ought_ to be seen.

These experiments seem to show that diffraction plays a very essential part in the formation of microscopical images, since dissimilar structures give identical images when the differences of their diffractive effect is removed, and conversely similar structures may give dissimilar images when their diffractive images are made dissimilar. Whilst a purely dioptric image answers point for point to the object on the stage, and enables a safe inference to be drawn as to the actual nature of that object, the visible indications of minute structure in a microscopical image are not always or necessarily conformable to the real nature of the object examined, so that nothing more can safely be inferred from the image as presented to the eye, than the presence in the object of such structural peculiarities as will produce the particular diffraction phenomena on which these images depend.

Further investigations and experiments led Abbe to discard so much of his theoretical conclusions relating to superimposed images having a distinct character as well as a different origin, and as to their capability of being separated and examined apart from each other. In a later paper he writes: “I no longer maintain in principle the distinction between the absorption image or direct dioptrical image and the diffraction image, nor do I hold that the microscopical image of an object consists of two superimposed images of different origin or a different mode of production. Thus it appears that both the absorption image and the diffraction image he held to be equally of diffraction origin; but while a lens of small aperture would give the former with facility, it would be powerless to reveal the latter, because of its limited capacity to gather in the strongly-deflected rays due to the excessively minute bodies the microscopical objective has to deal with.”[11]

Abbe’s theory of vision has been questioned by mathematicians, and since his death Lord Rayleigh went more deeply into the question of “the theory of the formation of optical images,” with special reference to the microscope and telescope. He has shown that two lines cannot be fairly resolved unless their components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the aperture; also, that the measure of resolution is only possible with a square aperture, or one bounded by straight lines, parallel to the lines resolved.

Lord Rayleigh’s Theory of the Formation of Optical Images, with Special Reference to the Microscope.[12]

Of the two methods adopted, that of Helmholtz’s consists in tracing the image representative of a mathematical point in the object, the point being regarded as self-luminous; that of Abbe’s the typical object was not, as we have seen, a luminous _point_, but a _grating_ illuminated by plane waves of light. In the latter method, Lord Rayleigh argues that the complete representation of the object requires the co-operation of all the spectra which are focussed in the principal focal plane of the objective; when only a few are present the representation is imperfect, and wholly fails when there is only one. He then proceeds to show, by the aid of diagrams and mathematical formula, how the resolving power can be adduced.

On further criticism of the Abbe spectrum theory, he observes “that although the image ultimately formed may be considered to be due to the spectra focussed to a given point, the degree of conformity of the image to the object is another question. The consideration of the case of a very fine grating, which might afford no lateral spectra at all, shows the incorrectness of the usually accepted idea that if all the spectra are utilised the image will still be incomplete, so that the theory (originally promulgated by Abbe) requires a good deal of supplementing; while it is inapplicable when the incident light is not parallel, and when the object is, for example, a double point and not a grating. Even in the case of a grating, the spectrum theory is inapplicable, if the grating is self-luminous; for in this case no spectra can be formed since the radiations from the different elements of the grating have no permanent phase-relations.” For these reasons Lord Rayleigh advises that the question should be reconsidered from the older point of view, according to which the typical object is a point and not a grating. Such treatment will show that the theory of resolving power is essentially the same for all instruments. The peculiarities of the microscope, arising from the divergence-angles not being limited to be small, and from the different character of the illumination, are theoretically only differences of detail. These investigations can be extended to gratings, and the results so obtained confirm for the most part the conclusions of the spectrum theory.

Furthermore, that the function of the condenser in microscopic practice in throwing upon the object the image of the lamp-flame is to cause the object to behave, at any rate in some degree, as if it were self-luminous, and thus to obviate the sharply-marked interference bands which arise when permanent and definite phase-relations are permitted to exist between the radiations which issue from various points of the object. This is capable of mathematical proof; and in the case where the illumination is such that each point of the row or of the grating radiates independently, the limit to resolution is seen to depend only on the width of the aperture, and thus to be the same for all forms of aperture as for those of the rectangular. That Abbe’s theory of microscopic vision is fairly open to the criticisms passed on it by Lord Rayleigh must be taken for granted.

Definition of Aperture; Principles of Microscopic Vision.

It must be well within the last half-century that the achromatic objective-glass for the microscope was brought to perfection and its value became generally recognised. Prior to the discovery of the achromatic principle in the construction of lenses it was assumed that the formation of the microscopic image took place (as we have already seen) on ordinary dioptric principles. As the image is formed in the camera or telescope, so it was said to be in the microscope. This belief existed, it will be remembered, at a time when dry objectives only were in favour and the use of the term _angle of aperture_ was misunderstood, when it was supposed that the different media with diffraction-indices were used; and the angle of the radiant pencil was believed not only to admit of a comparison of two apertures in the same medium, but likewise to admit of a standard of comparison when the media were entirely different in their refractive qualities.

It was during my tenure of office as secretary of the Royal Microscopical Society (1867 to 1873), that the aperture question, and also that of _numerical aperture_, came under discussion, both being met by the majority of the Fellows of the Society and practical opticians by a _non-possumus_.

Opticians alleged, that is, before the value of aperture became fully recognised (1860), that the achromatic objective had reached a stage of perfection, beyond which it was not possible to go; indeed, not only opticians, but physicists of high standing, as Professor Helmholtz, who made many important contributions to the theory of the microscope, and who, after duly weighing all the known physical laws on which the formation of images can be explained, emphatically stated that in his opinion “the limit of possible improvement of the microscope as an instrument of discovery had been very nearly reached.” A quarter of a century ago I ventured to throw a doubt upon so questionable a statement. I determined, if possible, to submit the aperture question to an exhaustive examination. My views were accordingly submitted to two of the highest authorities in this country--Sir George Airy, the then Astronomer Royal, and Sir George Stokes, Professor of Physics at Cambridge University--both of whom agreed with me that the possible increase of aperture would be attended with great advantage to the objective, and open the way to an extension of power resolution in the microscope.[13] The discussion afterwards took a warm turn, as will be seen on reference to “The Monthly Microscopical Journals” of 1874, 1875 and 1876.

The confusion into which the aperture question at this period had lapsed was no doubt due to the fact that its opponents had not yet grasped the true meaning of the term _aperture_. It was believed to be synonymous with “angular aperture,” much in use at the time. It will, however, appear quite unaccountable that even the older opticians should have confounded the latter with the former; and so entirely disregarded the fact that the angles of the pencil of light admitted by the objective cannot serve as a measure of its _aperture_, and that high refractive media can greatly reduce the value length of waves of light.

When the medium in which the objective works is the same as air, it is not that a comparison can be made by the angles of the radiant pencils only, but by their sines. For example, if two dry objectives admit pencils of 60° and 180°, their real apertures are not as 1 : 3, but as 1 : 2 only. Aperture in fact is computed by mathematicians by tracing the rays from the back focus through the system of lenses to the front focus, the front focus being the point at which the whole cone of rays converge as free as may be from aberration. If the front focus be in air, no pencil greater than 82°, “double the angle of total reflection,” can _emerge_ from the plane front of the lens; and, obviously, if no greater cone can emerge to a focus one way, neither can any greater cone enter the body of the lens from the radiant. This angle, then, of 82°, must be regarded as the limit for dry lenses or objectives.

This limit, it will be seen on more careful examination, is very nearly the maximum angle that can be computed for a lens to have a front focus in air. This can be proved by the consideration of the angle of the image of rays, as they are radiated from the object itself in balsam: for although this angle of image rays viewed as nascent from a self-luminous object capable of scattering rays in all directions, may be 180° in the substance of the balsam and cover-glass, of the 180° only 82° of the central portion will emerge into air--all rays beyond this limit are internally reflected at the cover-glass. This cone, then, of 82° becomes 180° in air, and a large part must necessarily be lost by reflection at the first incidence on the plane front of the lens. But with a formula permitting the use of a water medium between the front lens and the cover-glass, the aperture of the image rays may reach 126°--double the critical angle from glass to water; and with an oil medium, the aperture will be found to be limited only by the form of the front lens that can be constructed by the optician.

To sum up, then, the effect of the immersion system, greatly assists in the correction of aberration, gives increased magnification and angular aperture, increase of working distance between the objective and object, and renders admissible the use of the thicker glass-cover.

The aperture question would in all probability have remained unsolved many years longer (ten or twelve years elapsed after I brought the question under discussion before opticians gave way), but for the fortunate circumstance that the eminent mathematical and practical optician, Professor Abbe, of Jena, was about to visit London. This came off in the early part of the seventies, when the late Mr. John Mayall and myself had the good fortune to interview him. The subject discussed was naturally the increase of aperture and the theory of microscopical vision. He readily at our request undertook to re-investigate the question in all its bearings on the microscope. It is almost unnecessary to add that the conclusions he came to, and the results obtained, have proved of inestimable value to the microscopist and practical optician, and it may well seem necessary to explain somewhat at greater length the conclusions the learned Professor came to, and by the adoption of which the microscope has been placed on a more scientific basis than it had before attained to. Several papers were published _in extenso_ in the “Journal of the Royal Microscopical Society,” and I am greatly indebted to Mr. Frank Crisp, LL.D., for an excellent _resumé_ of Abbe’s Monograph.[14]

The essential step in the consideration of aperture is, as I have said, to understand clearly what is meant by the term. It will at once be recognised that its definition must necessarily refer to its primary meaning of _opening_, and must, in the case of an optical instrument, define its capacity for receiving rays from the object, and transmitting them to the image received at the eye-piece.

In the case of the telescope-objective, its capacity for receiving and transmitting rays is necessarily measured by the expression of its absolute diameter or “opening.” No such absolute measure can be applied in the case of the microscope objective, the largest constructed lenses of which having by no means the largest apertures, being, in fact, the lower powers of the instrument, whose apertures are for the most part but small. The capacity of a microscope objective for receiving and transmitting rays is, however, as will be seen, estimated by its _relative_ opening, that is, its opening in relation to its focal length. When this relative opening has been ascertained, it may be regarded as synonymous with that denoted in the telescope by _absolute_ opening. That this is so will be better appreciated by the following consideration:--

In a single lens, the rays admitted within one meridional plane evidently increase as the diameter of the lens (all other circumstances remaining the same), and in the microscope we have, at the back of the lens, the same conditions to deal with as are in front in the case of the telescope; the larger or smaller number of emergent rays will therefore be measured by the clear diameter, and as no rays can emerge that have not first been admitted, this will give the measure of the admitted rays under similar circumstances.

If the lenses compared have different focal lengths but the same clear “openings,” they will transmit the same number of rays to equal areas of an image at a definite distance, because they would admit the same number if an object were substituted for the image; that is, if the lens were used as a telescope-objective. But as the focal lengths are different, the amplification of the images is different also, and equal areas of these images correspond to different areas of the object from which the rays are collected. Therefore, the higher power lens with the same opening as the lower power, will admit a _greater_ number of rays in all from the same object, because it admits the _same_ number as the latter from a _smaller_ portion of the object. Thus, if the focal lengths of two lenses are as 2 : 1, and the first amplifies N diameters, the second will amplify 2 N with the same distance of the image, so that the rays which are collected _to_ a given field of 1 mm. diameter of the image are admitted _from_ a field of 1/N mm. in the first case, and of 1/(2N) mm. in the second. As the “opening” of the objective is estimated by the diameter (and not by the area) the higher power lens admits _twice_ as many rays as the lower power, because it admits the same number from a field of half the diameter, and, in general, the admission of rays by the same opening, but different powers, must be in the inverse ratio of the focal lengths.

In the case of the single lens, therefore, its aperture is determined by the ratio between the clear opening and the focal length. The same considerations apply to the case of a compound objective, substituting, however, for the clear opening of the single lens the diameter of the pencil at its emergence from the objective, that is, the clear utilised diameter of the back lens. All equally holds good whether the medium in which the objective is placed is the same in the case of the two objectives or different, as an alteration of the medium makes no difference in the power.

Illustration: 180° Water Angle. (Numerical Aperture 1·33.)

Illustration: 180° Air Angle. 96° Water Angle. 82° Oil Angle. (Numerical Aperture 1·00.)

Illustration: 97° Air Angle. (Numerical Aperture ·75.)

Illustration: 60° Air Angle. (Numerical Aperture ·50.)

Fig. 35.--Relative diameters of the (utilized) back lenses of various dry and immersion objectives of the same power (1/4-in.) from an air angle of 60° to an oil angle of 180°.]

Thus we arrive at a general proposition for all kinds of objectives: 1st, when the power is the same, the admission of rays (or aperture) varies with the diameter of the pencil at its emergence; 2nd, when the powers are different, the same aperture requires different openings in the ratio of the focal lengths, or conversely with the same opening the aperture is in inverse ratio to the focal lengths. We see, therefore, that just as in the telescope the absolute diameter of the object-glass defines its _aperture_, so in the microscope _the ratio between the utilised diameter of the back lens and the focal length_ of the objective defines its aperture also, and this is clearly a definition of aperture in its primary and only legitimate meaning as “opening;” that is, the capacity of the objective for admitting rays from the object and transmitting them to the image.

If, by way of illustration, we compare a series of dry and oil-immersion objectives, and commencing with small air angles, progress up to 180° air angle, and then take an oil-immersion of 82° and progress again to 180° oil angle, the ratio of opening to power progresses also, and attains its maximum, not in the case of the air angle of 180° (when it is exactly equivalent to the oil angle of only 82°), but is greatest at the oil angle of 180°. If we assume the objectives to have the same power throughout we get rid of one of the factors of the ratio, and we have only to compare the diameters of the emergent beams, and can represent their relations by diagrams.

Fig. 35 illustrates five cases of different apertures of 1/4-in. objectives, viz.: those of dry objectives of 60°, 97°, and 180° air angle, a water-immersion of 180° water angle, and an oil-immersion of 180° oil angle. The inner dotted circles in the two latter cases are of the same size as that corresponding to the 180° air angle.

A dry objective of the maximum air angle of 180° is only able to utilise a diameter of back lens equal to twice the focal length, while an immersion lens of even only 100° utilises a _larger_ diameter, _i.e._, it is able to transmit more rays from the object to the image than any dry objective is capable of transmitting. Whenever the angle of an immersion lens exceeds twice the critical angle for the immersion fluid, _i.e._, 96° for water or 82° for oil, its aperture is in excess of that of a dry objective of 180°.

This excess will be _seen_ if we take an oil-immersion objective of, say 122° balsam angle, illuminating it so that the whole field is filled with the incident rays, and use it first on an object not mounted in balsam, but dry. We then have a _dry objective_ of nearly 180° angular aperture, for, as will be seen by reference to Fig. 36, the cover-glass is virtually the first surface of the objective, as the front lens, the immersion fluid, and the cover-glass are all approximately of the same index, and form, therefore, a front lens of extra thickness. When the object is close to the cover-glass the pencil radiating from it will be very nearly 180°, and the emergent pencil (observed by removing the eye-piece) will be seen to utilise as much of the back lens of the objective as is equal to twice the focal length, that is, the _inner_ of the two circles at the head of Fig. 35.

If now balsam be run in beneath the cover-glass so that the angle of the pencil taken up by the objective is no longer 180°, but 122° only (that is, _smaller_), the diameter of the emergent pencil is _larger_ than it was before, when the angle of the pencil was 180° in air, and will be approximately represented by the _outer_ circle of Fig. 35. As the power remains the same in both cases, the larger diameter denotes the greater aperture of the immersion objective over a dry objective of even 180° angle, and the excess of aperture is made plainly visible.

Having settled the principle, it is still necessary, however, to find a proper _notation_ for comparing apertures. The astronomer can compare the apertures of his various objectives by simply expressing them in inches, but this is obviously not available to the microscopist, who has to deal with the ratio of two varying quantities.

In consequence of a discovery made by Professor Abbe in 1873, that a general relation existed between the pencil admitted into the front of the objective and that emerging from the back of the objective, he was able to show that the ratio of the semi-diameter of the emergent pencil to the focal length of the objective could be expressed by the formula _n_ Sin _u_, _i.e._, by the sine of half the angle of aperture (_u_) multiplied by the refractive index of the medium (_n_) in front of the objective (_n_ being 1·0 for air, 1·33 for water, and 1·52 for oil or balsam).

When, then, the values in any given cases of the expression _n_ Sin _u_ (which is known as the “numerical aperture”) has been ascertained, the objectives are instantly compared as regards their aperture, and, moreover, as 180° in air is equal to 1·0 (since _n_ = 1·0 and the sine of half 180° = 1·0) we see, with equal readiness, whether the aperture is smaller or larger than that corresponding to 180° in air. Thus, suppose we desire to compare the apertures of three objectives, one a dry objective, the second a water immersion, and the third an oil immersion; these would be compared on the angular aperture view as, say 74° air angle, 85° water angle, and 118° oil angle, so that a calculation must be worked out to arrive at the actual relation between them. Applying, however, the _numerical_[15] notation, which gives ·60 for the dry objective, ·90 for the water immersion, and 1·30 for the oil immersion, their relative apertures are immediately recognised, and it is seen, for instance, that the aperture of the water immersion is somewhat less than that of a dry objective of 180°, and that the aperture of the oil immersion exceeds that of the latter by 30%.

The advantage of immersion, in comparison with dry objectives, becomes at once apparent. Instead of consisting merely in a diminution of the loss of light by reflection or increased working distance, it is seen that a wide-angled immersion objective has a larger aperture than a dry objective of maximum angle, so that for any of the purposes for which aperture is essential an immersion must necessarily be preferred to a dry objective.

That pencils of identical angular extension but in different media are different physically, will cease to appear in any way paradoxical if we recall the simple optical fact that rays, which in air are spread out over the whole hemisphere, are in a medium of higher refractive index such as oil _compressed_ into a cone of 82° round the perpendicular, _i.e._, twice the critical angle. A cone exceeding twice the critical angle of the medium will therefore embrace a _surplus_ of rays which do not exist even in the hemisphere when the object is in air.

The whole aperture question, notwithstanding the innumerable perplexities which heretofore surrounded it, is in reality completely solved by these two simple considerations: First, that “aperture” is to be applied in its ordinary meaning as representing the greater or less capacity of the objective for receiving and transmitting rays; and second, that when so applied the aperture of an objective is determined by the ratio between its opening and its focal length; the objective that utilises the larger back lens (or opening) relatively to its focal length having necessarily the larger aperture. It would hardly, therefore, serve any useful purpose if we were here to discuss the various erroneous ideas that gave rise to the contention that 180° in air must be the maximum aperture. Amongst these was the suggestion that the larger emergent beams of immersion objectives were due to the fact that the immersion fluid abolished the refractive action of the first plane surface which, in the case of air, prevented there being any pencil exceeding 82° within the glass. Also the very curious mistake which arose from the assumption that a hemisphere did not magnify an object at its centre because the rays passed through without refraction. A further erroneous view has, however, been so widespread that it seems to be desirable to devote a few lines to it, especially as it always appears at first sight to be both simple and conclusive.

If a dry objective is used upon an object in air, as in Fig. 37, the angle may approach 180°, but when the object is mounted in balsam, as in Fig. 37_a_, the angle at the object cannot exceed 82°, all rays outside that limit (shown by dotted lines) being reflected back at the cover-glass and not emerging into air. On using an immersion objective, however, the immersion fluid which replaces the air above the cover-glass allows the rays formerly reflected back to pass through to the objective, so that the angle at the object may again be nearly 180° as with the dry lens. The action of the immersion objective was, therefore, supposed to be simply that it repaired the loss in angle which was occasioned when the object was transferred from air to balsam, and merely restored the conditions existing in Fig. 37_a_ with the dry objective on a dry object.

As the result of this erroneous supposition, it followed that an immersion objective could have no advantage over a dry objective, except in the case of the latter being used upon a balsam-mounted object, its aperture then being (as was supposed) “cut down.” The error lies simply in overlooking the fact that the rays which are reflected back when the object is mounted in balsam (Fig. 37_a_) are not rays which are found when the object is in air (Fig. 37), but are _additional and different_ rays which do not exist in air, as they cannot be emitted in a substance of so low a refractive index.

Lastly, it should also be noted that it is numerical and not angular aperture which measures the quantity of light admitted to the objective by different pencils.

First take the case of the medium being the same. The popular notion of a pencil of light may be illustrated by Fig. 38, which assumes that there is equal intensity of emission in all directions, so that the quantity of light contained in any given pencils may be compared by simply comparing the contents of the solid cones. The Bouguer-Lambert law, however, shows that the quantity of light emitted by any bright point varies with the obliquity of the direction of emission, being _greater_ in a perpendicular than in an oblique direction. The rays are less intense in proportion as they are more inclined to the surface which emits them, so that a pencil is not correctly represented by Fig. 38, but by Fig. 38_a_, the density of the rays decreasing continuously from the vertical to the horizontal, and the squares of the sines of the semi-angles (_i.e._, of the numerical aperture) constituting the true measure of the quantity of light contained in any solid pencil.

If, again, the media are of different refractive indices, as air (1·0), water (1·33), and oil (1·52), the total amount of light emitted over the whole 180° from radiant points in these media under a given illumination is not the same, but is _greater_ in the case of the media of greater refractive indices in the ratio of the squares of those indices (_i.e._, as 1·0, 1·77 and 2·25). The quantity of light in pencils of different angle and in different media must therefore be compared by squaring the product of the sines and the refractive indices, _i.e._ (_n_ Sin _u_^2), for the square of the numerical aperture.

The fact is therefore made clear that the aperture of a dry objective of 180° does not represent, as was supposed, a maximum, but that aperture increases with the increase in the refractive index of the immersion fluid; and it should be borne in mind that this result has been arrived at in strict accordance with the ordinary propositions of geometrical optics, and without any reference to or deductions from the diffraction theory of Professor Abbe.

There still remains one other point for determination, namely, the proper function of aperture in respect to immersion objectives of large aperture. The explanation of the increased power of vision obtained by increase of aperture was, that by the greater obliquity of the rays to the object “shadow effects” were produced, a view which overlooked the fact, first, that the utilisation of increased aperture depends not only on the obliquity of the rays sent to the _object_, but also to the _axis of the microscope_; and exactly as there is no acoustic shadow produced by an obstacle, which is only a few multiples of the length of the sound waves, so there can be no shadow produced by minute objects, only a few multiples from the light waves, the latter then passing completely _round_ the object. The Abbe diffraction theory, however, supplies the true explanation of this, and shows that the increased performance of immersion objectives of large aperture is directly connected (as might have been anticipated) with the larger “openings” in the proper sense of the term, which, as we have already explained, such objectives really possess. Furthermore, in order that the image exactly corresponds with the object, all diffracted rays must be gathered up by the objective. Should any be lost we shall have not an actual image of the object, but a spurious one. Now, if we have a coarse object, the diffracted rays are all comprised within a narrow cone round the direct beam, and an objective of small aperture will transmit them all. With a minute object, however, the diffracted rays are widely spread out, so that a small aperture can admit only a fractional part--to admit the whole or a very large part, and consequently to see the minute structure of the object, or to see it truly, a large aperture is necessary, and in this lies the value of _aperture_ and of a _wide-angled immersion objective_ for the observation of minute structures.

Numerical Aperture.

=Measure of Apertures of Objectives. N.A.=--Numerical aperture, as it is termed, is measured by the scale of measurement calculated by the late Professor Abbe, and which has since been generally recognised and adopted. He showed that even in lenses made for the same medium (as air) their comparative aperture as compared with their focus was not correctly measured by the angle of the rays grasped, but by the actual diameters of the pencil of rays transmitted, which depend, as already seen, more upon the back of the lens than the front. To get a geometric measure for comparison, he took the radii, or half diameters (whose relative proportions would be the same), and which geometrically are the sines of the semi-angle of the outermost rays grasped. Abbe further showed that if this sine of half the outside angle were multiplied by the refractive index of the medium used we should have a number which would give the comparative _aperture_ of any lens, whatever the medium. This number, then, determines both the numerical aperture and the resolving power of the objective.

The following table of numerical apertures shows the respective angular pencils which they express in air, water and cedar oil, or glass.[16] The first column gives the numerical apertures from 0·20 to 1·33; the second, third, and fourth, the air, water and oil (or balsam) angles of aperture from 23° 4′ air angle to 180° balsam angle. The theoretical resolving power in lines to the inch is shown in the sixth column; the line E of the spectrum being taken from about the middle of the green, the column giving “illuminating power” being of less importance; while in using that of penetrating power, it must be remembered that several data beside that of 1/_a_ go to make up the total depth of vision with the microscope.

ABRIDGED NUMERICAL APERTURE TABLE.

=====+=========================+==========================+======+======
| Corresponding Angle |Limit of Resolving Power, | |
| (2 _u_) for | in Lines to an Inch. | |
+--------+-------+--------+--------+-------+---------+ |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
-----+--------+-------+--------+--------+-------+---------+------+------
1·33 | ... |180° 0′|122° 6′| 128,225| 138,989|168,907 | 1·769| ·752
1·32 | ... |165° 56′|120° 33′| 127,261| 137,944|167,637 | 1·742| ·758
1·30 | ... |155° 38′|117° 35′| 125,333| 135,854|165,097 | 1·690| ·769
1·28 | ... |148° 42′|114° 44′| 123,405| 133,764|162,557 | 1·638| ·781
1·26 | ... |142° 39′|111° 59′| 121,477| 131,674|160,017 | 1·588| ·794
1·24 | ... |137° 36′|109° 20′| 119,548| 129,584|157,477 | 1·538| ·806
1·22 | ... |133° 4′|106° 45′| 117,620| 127,494|154,937 | 1·488| ·820
1·20 | ... |128° 55′|104° 15′| 115,692| 125,404|152,397 | 1·440| ·833
1·18 | ... |125° 3′|101° 50′| 113,764| 123,314|149,857 | 1·392| ·847
1·16 | ... |121° 26′| 99° 29′| 111,835| 121,224|147,317 | 1·346| ·862
1·14 | ... |118° 0′| 97° 11′| 109,907| 119,134|144,777 | 1·300| ·877
1·12 | ... |114° 44′| 94° 55′| 107,979| 117,044|142,237 | 1·254| ·893
1·10 | ... |111° 36′| 92° 43′| 106,051| 114,954|139,698 | 1·210| ·909
1·08 | ... |108° 36′| 90° 34′| 104,123| 112,864|137,158 | 1·166| ·926
1·06 | ... |105° 42′| 88° 27′| 102,195| 110,774|134,618 | 1·124| ·943
1·04 | ... |102° 53′| 86° 21′| 100,266| 108,684|132,078 | 1·082| ·962
1·02 | ... |100° 10′| 84° 18′| 98,338| 106,593|129,538 | 1·040| ·980
1·00 | 180° 0′| 97° 31′| 82° 17′| 96,410| 104,503|126,998 | 1·000| 1·000
0·98 | 157° 2′| 94° 56′| 80° 17′| 94,482| 102,413|124,458 | ·960| 1·020
0·96 | 147° 29′| 92° 24′| 78° 20′| 92,554| 100,323|121,918 | ·922| 1·042
0·94 | 140° 6′| 89° 56′| 76° 24′| 90,625| 98,223|119,378 | ·884| 1·064
0·92 | 133° 51′| 87° 32′| 74° 30′| 88,697| 96,143|116,838 | ·846| 1·087
0·90 | 128° 19′| 85° 10′| 72° 36′| 86,769| 94,053|114,298 | ·810| 1·111
0·88 | 123° 17′| 82° 51′| 70° 44′| 84,841| 91,963|111,758 | ·774| 1·136
0·86 | 118° 38′| 80° 34′| 68° 54′| 82,913| 89,873|109,218 | ·740| 1·163
0·84 | 114° 17′| 78° 20′| 67° 6′| 80,984| 87,783|106,678 | ·706| 1·190
0·82 | 110° 10′| 76° 8′| 65° 18′| 79,056| 85,693|104,138 | ·672| 1·220
0·80 | 106° 16′| 73° 58′| 63° 31′| 77,128| 83,603|101,598 | ·640| 1·250
0·78 | 102° 31′| 71° 49′| 61° 45′| 75,200| 81,513| 99,058 | ·608| 1·282
0·76 | 98° 56′| 69° 42′| 60° 0′| 73,272| 79,423| 96,518 | ·578| 1·316
0·74 | 95° 28′| 67° 37′| 58° 16′| 71,343| 77,333| 93,979 | ·548| 1·351
0·72 | 92° 6′| 65° 32′| 56° 32′| 69,415| 75,242| 91,439 | ·518| 1·389
0·70 | 88° 51′| 63° 31′| 54° 50′| 67,487| 73,152| 88,899 | ·490| 1·429
0·68 | 85° 41′| 61° 30′| 53° 9′| 65,559| 71,062| 86,359 | ·462| 1·471
0·66 | 82° 36′| 59° 30′| 51° 28′| 63,631| 68,972| 83,819 | ·436| 1·515
0·64 | 79° 36′| 57° 31′| 49° 48′| 61,702| 66,882| 81,279 | ·410| 1·562
0·62 | 76° 38′| 55° 34′| 48° 9′| 59,774| 64,792| 78,739 | ·384| 1·613
0·60 | 73° 44′| 53° 38′| 46° 30′| 57,846| 62,702| 76,199 | ·360| 1·667
0·58 | 70° 54′| 51° 42′| 44° 51′| 55,918| 60,612| 73,659 | ·336| 1·724
0·56 | 68° 6′| 49° 48′| 43° 14′| 53,990| 58,522| 71,119 | ·314| 1·786
0·54 | 65° 22′| 47° 54′| 41° 37′| 52,061| 56,432| 68,579 | ·292| 1·852
0·52 | 62° 40′| 46° 2′| 40° 0′| 50,133| 54,342| 66,039 | ·270| 1·923
0·50 | 60° 0′| 44° 10′| 38° 24′| 48,205| 52,252| 63,499 | ·250| 2·000
0·45 | 53° 30′| 39° 33′| 34° 27′| 43,385| 47,026| 57,149 | ·203| 2·222
0·40 | 47° 9′| 35° 0′| 30° 31′| 38,564| 41,801| 50,799 | ·160| 2·500
0·35 | 40° 58′| 30° 30′| 26° 38′| 33,744| 36,576| 44,449 | ·123| 2·857
0·30 | 34° 56′| 26° 4′| 22° 46′| 28,923| 31,351| 38,099 | ·090| 3·333
0·25 | 28° 58′| 21° 40′| 18° 56′| 24,103| 26,126| 31,749 | ·063| 4·000
0·20 | 23° 4′| 17° 18′| 15° 7′| 19,282| 20,901| 25,400 | ·040| 5·000
=====+========+========+=======+========+========+========+======+======

(1) Numerical Aperture. (_n_ sin _u_ = _a_.)
(2) _Air_ (_n_ = 1·00).
(3) _Water_ (_n_ = 1·33).
(4) _Homogeneous Immersion_ (_n_ = 1·52).
(5) White Light. (λ = 0·5269 μ, Line E.)
(6) Monochromatic (Blue) Light.(λ = 0·4861 μ, Line F.)
(7) Photography. (λ = 0·4000 μ, Near Line _h__k_.)
(8) Illuminating Power (a^2.)
(9) Penetrating Power (1/a.)

Abbe’s Apertometer.

The apertometer is an auxiliary piece of apparatus invented by Abbe, for testing the fundamental properties of objectives and determining their numerical and angular apertures. This accessory of the microscope involves the same principles as that of Tolles, which the late Mr. J. Mayall and myself brought to the notice of the Royal Microscopical Society of London in 1876. Abbe’s apertometer (Fig. 39) consists of a flat cylinder of glass, about three inches in diameter, and half an inch thick, with a large chord cut off, so that the portion left is somewhat more than a semicircle; the part where the segment is cut is bevelled from above downwards, to an angle of 45°, and it will be seen that there is a small disc with an aperture in it denoting the centre of the semicircle. To use this instrument the microscope is placed in a vertical position, and the apertometer is placed upon the stage with its circular part to the front and the chord to the back. Diffused light, either from the sun or lamp, is assumed to be in front and on both sides. Suppose the lens to be measured is a dry one-quarter inch; then with a one-inch eye-piece having a large field, the centre disc, with its aperture on the apertometer, is brought into focus. The eye-piece and the draw-tube are now removed, leaving the focal arrangement undisturbed, and a lens supplied with the apertometer is screwed into the end of the draw-tube. This lens, with the eye-piece in the draw-tube, forms a low-power compound microscope. This is now inserted into the body-tube, and the back lens of the objective whose aperture we desire to measure is brought into focus. In the image of the back lens will be seen stretched across, as it were, the image of the circular part of the apertometer. It will appear as a bright band, because the light which enters normally at the surface is reflected by the bevelled part of the chord in a vertical direction, so that in reality a fan of 180° in air is formed. There are two sliding screens seen on either side of the figure of the apertometer; they slide on the vertical circular portion of the instrument. The images of these screens can be seen in the image of the bright bands. _These screens should now be moved so that their edges just touch the periphery of the back lens._ They act, as it were, as a diaphragm to cut the fan and reduce it, so that its angle just equals the aperture of the objective and no more.

This angle is now determined by the arc of glass between the screens; thus we get an angle in _glass_ the exact equivalent of the aperture of the objective. As the numerical apertures of these arcs are engraved on the apertometer, they can be read off by inspection. A difficulty is not infrequently experienced from the fact that it is not easy to determine the exact point at which the edge of the screen touches the periphery of the back lens, or rather the limit of the aperture. Zeiss, to meet this difficulty, made a change in the form of the apparatus--furnished a glass disc mounted on a metal plate, with a slot for the purpose of its more accurate adjustment.[17]

Stereoscopic Binocular Vision.

Professor Wheatstone’s remarkable discovery of stereoscopic vision led, at no distant period, to the application of the principle to the microscope. It may therefore prove of interest to inquire how stereoscopic binocular vision is brought about. Indeed, the curious results obtained in the stereoscope cannot be well understood without a previous knowledge of the fundamental optical principles involved in this contrivance, whereby two slightly dissimilar pictures of any object become fused into one image, having the actual appearance of relief. The invention of the stereoscope by Sir Charles Wheatstone, F.R.S., 1838, and improved by Brewster, was characterised by Sir John Herschel as “one of the most curious discoveries, and beautiful for its simplicity, in the entire range of experimental optics,” led to a more general appreciation of the value of the conjoint use of both eyes in conveying to the mind impressions of the relative form and position of an object, such as the use of either eye singly does not convey with anything like the same precision. When a near object having three dimensions is looked at, a different perspective representation is seen with each eye. Certain parts are seen by the right eye, the left being closed, that are invisible to the left eye, the right being closed, and the relative positions of the portions visible to each eye in succession differ. These two visual impressions are simultaneously perceived by both eyes, and combined in the brain into one image, producing the effect of perspective and relief. If truthful right-and-left monocular pictures of an object be so presented to the two eyes that the optic axes when directed to them shall converge at the same angle as when directed to the object itself, a solid image will be at once perceived. The perception of relief referred to is closely connected with the doubleness of vision which takes place when the images on corresponding portions of the two retina are not similar. But, if in place of looking at the solid object itself we look with the right and left eyes respectively at pictures of the object corresponding to those which would be formed by it on the retina of the two eyes if it were placed at a moderate distance in front of them, and these visual pictures brought into coincidence, the same conception of a solid form is generated in the mind just as if the object itself were there.

Professor Abbe, however, contended that the method by which dissimilar images are formed in the binocular microscope differs materially from that of ordinary stereoscopic vision, and that the pictures are united solely by the activity of the brain, not by the prisms which ordinarily give rise to sensations of _solidity_. This can be only partially true, as binocularity in the microscope is due to difference of projection exhibited by the different parallax displacement of the images, and also to the perception of depth imparted by the instrument.

Wheatstone was firmly convinced that his stereoscopic principle could be applied to the microscope, and he therefore applied first to Ross and then to Powell to assist him in its adaptation. But whether either of these opticians made any attempt to give effect to his wishes and suggestions is not known. In the year 1851 Professor Riddell, of America, succeeded in constructing a binocular microscope by employing two rectangular prisms behind the objective. M. Nachet also constructed a binocular with two body-tubes and a series of prisms. But neither Riddell’s nor Nachet’s instrument was ever brought into use; they were either too complicated or too costly.

It will be understood, however, that the binocular stereoscope combines two dissimilar pictures, while the binocular microscope simply enables the observer to look with both eyes at images which are essentially identical. Stereoscopic vision, to be effective, requires that the delineating pencil shall be equally separated, so that one portion of the admitted cone of light is conducted to one eye, and the other portion to the other eye.

Select any object lying in an inclined position, and place it in the centre of the field of view of the microscope; then, with a card held close to the object-glass, stop off alternately the right or left hand portion of the front lens: it will then appear that during each alternate change certain parts of the object will change their relative positions.

To illustrate this, Fig. 40 _a_, _b_ are enlarged drawings of a portion of the egg of the common bed-bug (_Cimex lecticularis_), the operculum which should cover the opening having been forced off at the time the young was hatched. The figures exactly represent the two positions that the inclined orifice will occupy when the right- and left-hand portions of the object-glass are stopped off. This object is viewed as an opaque object, and drawn under a two-thirds object-glass of about 28° aperture. If this experiment is repeated, by holding the card over the eye-piece, and stopping off alternately the right and left half of the ultimate emergent pencil, exactly the same changes and appearances will be observed in the object under view. The two different images just produced are such as are required for obtaining stereoscopic vision. It is therefore evident that if instead of bringing them confusedly together into one eye we can separate them so as to bring together _a_, _b_ into the left and right eye, in the combined effect of the two projections we obtain at once all that is necessary to enable us to form a correct judgment of the solidity and distance of the several parts of the object.

Nearly all objectives from the one inch upwards of any considerable aperture give images of the object seen from a different point of view with the two opposite extremes of the margin of the cone of rays; the resulting effect is that there are a number of dissimilar perspectives of the object blended together at one and the same time on the retina. For this reason, if the object under view possesses bulk, a more accurate image will be obtained by reducing the aperture of the objective.

Diagram 3, Fig. 41, represents the method employed by Mr. Wenham for bringing the two eyes sufficiently close to each other to enable them both to see through the double eye-piece at the same moment. _a a a_ are rays converging from the field lens of the eye-piece; after passing the eye-lens _b_, if not intercepted, they would come to a focus at _c_; but they are arrested by the inclined surfaces, _d d_, of two solid glass prisms. From the refraction of the under incident surface of the prisms, the focus of the eye-piece becomes elongated, and falls within the substance of the glass at _e_. The rays then diverge, and after being reflected by the second inclined surface _f_, emerge from the upper side of the prism, when their course is rendered still more divergent, as shown by the figure. The reflecting angle given to the prisms is 47-1/2°, to accommodate which it is necessary to grind away the contact edges of the prisms, as represented, otherwise they prevent the extreme margins of the reflecting surfaces from coming into operation, which are seldom made quite perfect.