back focal plane of the objective, can be conveniently seen with FIG. 27. FIG. 24. FIG. 25. FIG. 26. } (From Abbe, Theorie der Bilderzeugung im Mikroskop.) of direct lighting, so that a banding of double the fineness can be perceived, by inclining the illuminating pencil to the axis; this is controlled by moving the diaphragm laterally. If the obliquity of illumination be so great that the principal maximum passes through the outermost edge of the objective, while a spectrum of 1st order passes the opposite edge, so that in the back focal plane the diffraction phenomenon shown in fig. 22 arises, banding is still to be seen. The resolution in the case of oblique illumination is given by the formula & =\/2A. Reverting to fig. 13, we suppose that a diffracting particle of such fineness is placed at O that the diffracted pencils of the 1st order make an angle w with the axis; the principal maximum of the Fraunhofer diffraction phenomena lies in Fi; and the two diffraction maxima of the 1st order in P' and P'. The waves proceeding from this point are united in the point O'. Suppose that a well corrected objective is employed. The image O' of the point O is then the interference effect of all waves proceeding from the exit pupil of the objective PP. Abbe showed that for the production of an image the diffraction maxima must lie within the exit pupil of the objective. In the silvering of a glass plate lines are ruled as shown in fig. 23, one set traversing the field while the intermediate set extends only half-way across. If this object be viewed by the objective, so that at least the diffraction spectra of 1st order pass the finer divisions, then the corresponding diffraction phenomenon in the back focal plane of the objective has the appearance shown in fig. 21, while the diffraction figure corresponding to the coarser ruling appears as given in fig. 20. If one cuts out by a diaphragm in the back focal plane of the objective all diffraction spectra except the principal maximum, one sees in the image a field divided into two halves, which show with different clearness, but no banding. By choosing a somewhat broader diaphragm, so that the spectra of 1st order can pass the larger division, there arises in the one half of the field of view the image of the larger division, the other half being clear without any such structure. By using a yet wider diaphragm which admits the spectra of 2nd order of the larger division and also the spectra of 1st order of the fine division, an image is obtained which is similar to the object, i.e. it shows bands one half a division double as fine as on the other. If now the spectrum of 1st order of the larger division be cut out from the diffraction figure, as is shown in fig. 24, an image is obtained which over the whole field shows a similar division (fig. 25), although in the one half of the object the represented banding does not occur. Still more strikingly is this phenomenon shown by Abbe's diffraction plate (fig. 26). This is a so-called cross grating formed by two perpendicular gratings. Through a suitable diaphragm in the back focal plane, banding can easily be produced in the image, which contains neither the vertical nor the horizontal lines of the two gratings, but there exist streaks, whose direction halves the angle under which the two gratings intersect (fig. 27). There can thus be shown structures which are not present in the object. Colonel Dr Woodward of the United States army showed that interference effects appear to produce details in the image which do not exist in the object. For example, two to five rows of globules were produced, and photographed, between the bristles of mosquito wings by using oblique illumination. In observing with strong systems it is therefore necessary cautiously to distinguish between spectral and real marks. To determine the utility of an objective for resolving fine details, one experiments with definite objects, which are usually employed simultaneously for examining its other properties. Most important are the fine structures of diatoms such as Surirella gemma and Amphipleura pellucida or artificial fine divisions as in a Nobert's grating. The examination of the objectives can only be attempted when the different faults of the objective are known. If microscopic preparations are observed by diffused daylight or by the more or less white light of the usual artificial sources, then an objective of fixed numerical aperture will only represent details of a definite fineness. All smaller details are not portrayed. The Fraunhofer formula permits the determination of the most useful magnification of such an objective in order to utilize its full resolving power. As we saw above, the apparent size of a detail of an object must be greater than the angular range of vision, i.e. 1'. Therefore we can assume that a detail which appears under an angle of 2' can be surely perceived. Supposing, however, there is oblique illumination, then formula (5) can always be applied to determine the magnifying power attainable with at least one objective. By substituting y, the size of the object, for d, the smallest value which a single object can have in order to be analysed, and the angle w' by 2', we obtain the magnifying power and the magnification number: V1=tan w'/d=2A tan 2'/λ; N=2A/ tan 2'/^; where I equals the sight range of 10 in. Even if the details can be recognized with an apparent magnification of 2', the observation may still be inconvenient. This may be improved when the magnification is so increased that the angle under which the object, when still just recognizable, is raised to 4'. The magnification and magnifying number which are most necessary for a microscope with an objective of a given aperture can then be calculated from the formulae: V1=2A tan 4'/^; N.=2A/ tan 4'/λ. Abbe Journ. Roy. Soc., 1882, p. 463) we have the following table for N2. N.. 0.10 2.75 53 106 1.60 0.17 741 1693 From this it can be seen that, as a rule, quite slight magnifications suffice to bring all representable details into observation. If the magnification is below the given numbers, the details can either not be seen at all, or only very indistinctly; if, on the contrary, the given magnification is increased, there will still be no more details visible. The table shows at the same time the great superiority of the immersion-system over the dry-system with reference to the resolving power. With the best immersion-system, having a numerical aperture of 1.6, details of the size 0-17 μ can be resolved, while the theoretical maximum of the resolving power is 0.167 μ, so that the theoretical maximum has almost been reached in practice. Still smaller particles cannot be portrayed by using ordinary daylight. In order to increase the resolving power, A. Köhler (Zeit. f. Mikros., 1904, 21, pp. 129, 273) suggested employing ultra-violet light, of a wave-length 275 ; he thus increased the resolving power to about double that which is reached with day-light. of which the mean wave-length is 550 . Light of such short wave-length is, however, not visible, and therefore a photographic plate must be employed. Since glass does not transmit the ultra-violet light, quartz is used, but such lenses can only be spherically corrected and not chromatically. For this reason the objectives have been called monochromats, as they have only been corrected for light of one wave-length. Further, the different transparencies of the cells for the ultra-violet rays render it unneces sary to dye the preparations. Glycerin is chiefly used as immersion fluid. M. v. Rohr's monochromats are constructed with apertures up to 1.25. The smallest resolving detail with oblique lighting is 8-X/2A, where λ=275 μ. As the microscopist usually estimates the resolving power according to the aperture with ordinary day-light, Köhler introduced the "relative resolving power for ultra-violet light. The power of the microscope is thus represented by presupposing day-light with a wave-length of 550. Then the denominator of the fraction, the numerical aperture, must be correspondingly increased, in order to ascertain the real resolving power. In this way a monochromat for glycerin of a numerical aperture 1.25 gives a relative numerical aperture of 2.50. If the magnification be greater than the resolving power demands. the observation is not only needlessly made more difficult, but the entrance pupil is diminished, and with it a very considerable decrease of clearness, for with an objective of a certain aperture the size of the exit pupil depends upon the magnification. The diameter of MICROSCOPE the exit pupil of the microscope is about 0.04 in. with the magnifica- Dark-field Illumination. It is sometimes desirable to make minutest objects in a preparation specially visible. This can be done by cutting off the chief maximum and using only the diffracted spectra for producing the image. At least two successive diffraction maxima must be admitted through the objective for there to be any image of the objects. With this device these particles appear bright against a dark background, and can be easily seen. The cutting off of the chief maximum can be effected by a suitable diaphragm in the back focal plane of the objective. But, owing to the various partial reflections which the illuminating cone of rays undergoes when traversing the surfaces of the lenses, a portion of the light comes again into the preparation, and into the eye of the observer, thus veiling the image. This defect can be avoided (after Abbe) if a small central portion of the back surface of the front lens be ground away and blackened; this portion should exactly catch the direct cone of rays, whilst the edges of the lens let the deflected cone of rays pass through (fig. 28). (By permission of C. Zeiss.) D FIG. 28. It is a The large loss of light, which is caused in dark-field illumination By dark-field illuminaby the cutting off of the direct cone of rays, must be compensated by employing exceptionally strong sources. tion it is even possible to make such small details of objects perceptible as are below the limits of the resolving power. similar phenomenon to that which arises when a ray of sunlight falls The extremely small particles of dust into a darkened room. (motes in a sunbeam) in the rays are made perceptible by the diffracted light, whilst by ordinary illumination they are invisible. The same observation can be made with the cone of rays of a reflector, and in the same way the fine rain-drops upon a dark background and the fixed stars in the sky become visible. It is not possible to recognize the exact form of the minute objects because their apparent size is much too small; only their presence is observable. In addition, the particles can only be recognized as separate objects if their apparent distance from one another is greater than the angular definition of sight. Ultramicroscopy. This method of illumination has been used by H. Siedentopf in his ultramicroscope. The image consists of a diffraction disk from whose form and size certain conclusions may be drawn as to the size and form of the object. It is impossible to get a representation as from an object. Very finely divided sub-microscopic particles in liquids or in transparent solids can be examined; and the method has proved exceptionally valuable in the investigation of colloidal solutions. Siedentopf employed two illuminating arrangements. With the orthogonal arrangement for illuminating and observing the beam | of light traverses an extremely fine slit through a well-corrected We will now examine the conditions which must be fulfilled by an objective, and then how far these conditions have been realized. Consider the aberrations which may arise from the representation by a system of wide aperture with monochromatic light, i.e. the spherical aberrations. The rays emitted from an axial object-point are not combined into one image-point by an ordinary biconvex lens of fixed aperture, but the central rays come to a more distant focus than the outer rays. The so-called "caustic" occupies a definite position in the image-space. The spherical aberrations, however, can be overcome, or at least so diminished that they are quite harmless, by forming appropriate combinations of lenses. "under. The aberration of rays in which the outer rays intersect the axis By at a shorter distance than the central rays is known as over-correction." The reverse is known as correction." selecting the radii of the surfaces and the kind of glass the under- or over-correction can be regulated. Thus it is possible to correct a system by combining a convex and a concave lens, if both have aberrations of the same amount but of opposite signs. In this case the power of the crown lens must preponderate so that the resulting lens is of the same sign, but of a little less power. Correction of the spherical aberration in strong systems with very large aperture can not be brought about by means of a single combination of two lenses, but several partial systems are necessary. Further, undercorrected systems must be combined with over-corrected ones. Another way of correcting this system is to alter the distances. If, by these methods, a point in the optic axis has been freed from aberration, it does not follow that a point situated only a very small distance from the optic axis can also be represented without spherical surface-element, is only possible, as Abbe has shown, if the aberration. The representation, free from aberration, of a small ratio of the sine of the aperture u on the object-side to the sine of objective simultaneously fulfils the "sine-condition," ie. if the n sin u/sin u'-Č, in which C is a constant. the corresponding aperture u' on the image-side is constant, i.e. if as the point-by-point representation of the whole object-space in the image-space (see LENS), and according therefore the equation is in contrast to the tangent-condition, which must be regarded n tan u/tan u'=C must exist. These two conditions are only comand where the apertures are so small that the sines and tangents are of about the same value. patible when the representation is made with quite narrow pencils, The sine-condition Very large apertures occur in strong microscope objectives, and is, however, the most important as far as the microscopic representahence the two conditions are not compatible. The sine-condition element through the objective by wide cones of rays. The removal of tion is concerned, because it must be possible to represent a surfacethe spherical aberration and the sine-condition can be accomplished only for two conjugate points. A well-corrected microscope objective Hence the importance As soon as the object is moved a short distance away from this with a wide aperture therefore can only represent, free from aberrations, one object-element situated on a definite spot on the axis. spot the representation is quite useless. of observing the length of the tube in strong systems. If the sine-condition is not fulfilled but the spherical aberrations in the axis have been removed, then the image shown in fig. 19 results. | and with a high refraction a low dispersion. By using these glasses The cones of rays issuing from a point situated only a little to the FIG. 29. The lens is spherically corrected for OO', but the sine- side, which traverse different zones of the objective, have a and employing minerals with special optical properties, it is possible to correct objectives so that three colours can be combined, leaving only a quite slight tertiary spectrum, and removing the spherical aberration for two colours. Abbe called such systems "apochromats." Good apochromats often have as many as twelve lenses, whilst systems of simpler construction are only achromatic, and are therefore called "achromats." Even in apochromats it is not possible to entirely remove the chromatic difference of magnification, i.e. the images produced by the red rays are somewhat smaller than the images produced by the blue. A white object is represented with blue streaks and a black one with red streaks. This aberration can, however, be successfully controlled by a suitable eyepiece (see below). A further aberration which can only be overcome with difficulty, and even then only partially, is the "curvature of the field," i.e. the points situated in the middle and at the edge of the plane object can not be seen clearly at the same focusing." Historical Development.-The first real improvement in the microscope objective dates from 1830 when V. and C. Chevalier, at first after the designs of Selligue, produced objectives, consisting of several achromatic systems arranged one above the other. The systems could be used separately or in any combination. A second method for diminishing the spherical aberration was to alter the distances of the single systems, a method still used. Selligue had no particular comprehension of the problem, for his achromatic single systems were simply telescope objectives corrected for an infinitely distant point, and were placed so that the same surface was turned towards the object in the microscope objective as in the telescope objective; in the microscope objective is small in proportion to the distance although contrary to the telescope, the distance of the object of the image. It would have been more correct to have employed these objectives in a reverse position. must be so arranged that the aplanatic (virtual) image-point O' A second method of correcting the spherical aberration depends FIG. 30.-O' is the virtual image on the notion of aplanatic points. of O formed at a spherical sur- If there are two transparent face of centre C and radius CS. substances separated from one another by a spherical surface, then there are two points on the axis where they can be reproduced free from error by monochromatic light, and these are called "aplanatic points." The first is the centre of the sphere. All rays issuing from this point pass unrefracted through the dividing surface; its image-point coincides with it. Besides this there is a second point on the axis, from which all issuing rays are so refracted at the surface of the sphere that, after the refraction, they appear to originate Lister showed that a combination of lenses can be achromatic for These circumstances were considered by Chevalier and Lister. from one point-the image-point (see fig. 30). With this, the object-only two points on the axis, and therefore that the single systems point O, and consequently the image-point O' also, will be at a quite definite distance from the centre. If however the object-point does not lie in the medium with the index n, but before it, and the medium is, for example, like a front lens, still limited by a plane surface, just in front of which is the object-point, then in traversing the plane surface spherical aberrations of the under-corrected type again arise, and must be removed. By homogeneous immersion the object-point can readily be reduced to an aplanatic point. By experiment Abbe proved that old, good microscope objectives, which by mere testing had become so corrected that they produced usable images, were not only free from spherical aberrations, but also fulfilled the sine-condition, and were therefore really aplanatic systems. The second aberration which must be removed from microscope objectives are the chromatic. To diminish these a collective lens of crown-glass is combined with a dispersing lens of flint; in such a system the red and the blue rays intersect at a point (see ABERRATION). In systems employed for visual observation (to which class the microscope belongs) the red and blue rays, which include the physiologically most active part of the spectrum, are combined; but rays other than the two selected are not united in one point. The transverse sections of these cones of rays diverge more or less from the transverse section of the chosen blue and red cones, and produce a secondary spectrum in the image, and the images still appear to have a slightly coloured edge, mostly greenish-yellow or purple; in other words, a chromatic difference of the spherical aberrations arises (see fig. 31). This refers to systems with small apertures, but still more so to systems with large ones; chromatic aberrations are exceptionally increased by large apertures. The new glasses produced at Schott's glass works, Jena, possessed in part optical qualities which differed considerably from those of the older kinds of glass. In the old crown and flint glass a high 0.. FIG. 31.-Showing a system with chromatic difference of spherical aberration. O"- image of O for red light; O"" for blue. The system is under-corrected for red, and over-corrected for blue rays. refractive index was always connected with a strong dispersion and the reverse. Schott succeeded, however, in producing glasses which with a comparatively low refraction have a high dispersion, was FIG. 32. DD DD FIG. 33. more complex one adopted, which also made the production of better objectives possible; this is the principle of the compensation of the aberrations produced in the different parts of proceeded on quite different lines, the objective. Even Lister, who hinted at the possibility of such a compensation. This method makes it specially possible to orders and to fulfil the sinc-condition, and the chief merit of overcome the chromatic and spherical aberrations of higher this improvement belongs to Amici. He had recognized that the good operation of a microscope objective depended essentially upon the size of the aperture, and he therefore endeavoured to produce systems with wide aperture and good correction. He used chiefly a highly curved plano-convex front lens, which has since always been employed in strong systems. Even if the object-point on the axis cannot be reproduced quite free from aberration through have been produced by the first plane outer limiting surface, yet such a lens, because aberrations of the type of an under-correction the defects with the strong refraction are relatively small and can be well compensated by other systems. Amici chiefly employed cemented pairs of lenses consisting of a plano-convex flint lens and a biconvex crown lens (fig. 33),and constructed objectives with an aperture of 135°. He also showed the influence of the cover-slip on pencils of such wide aperture. The lower surface of the slip causes undercorrection on being traversed by the pencil, with over-correction when it leaves it; and since the aberration of the surface lying farthest from the object, i.e. those caused by the upper surface preponderate, an over-corrected cone of rays enters the objective. The over-correction increases when the glass is thickened. In order to counteract this aberration the whole objective must be correspondingly under-corrected. Objectives with definite undercorrection can however only produce really good images with glass covers of a specified thickness. With apertures of 0-90-0-95 differences of even 0-004-0-008 in. in the glass covers can be noticed by the deterioration of the image. In systems with smaller apertures variations of the thickness of the glass cover are not so |