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.
180° Oil Angle. (Numerical Aperture 1·52.)
180° Water Angle. (Numerical Aperture 1·33.)
180° Air Angle. 96° Water Angle. 82° Oil Angle. (Numerical Aperture 1·00.)
97° Air Angle. (Numerical Aperture ·75.)
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 (¼-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 ¼-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°.
Fig. 36.
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.
Fig. 37.
Fig. 37a.
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.
Fig. 38.
Fig. 38a.
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 u2), 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.