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Encyclopedia Britannica - Main :: TAV-THE |
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THE PHYSICAL THEORY In order to fully understand the representation in the micro-scope, the process must be investigated according to the wave-theory, especially in considering the representation of objects or object details having nearly the size of a wave-length. The rectilinear rays, which we have considered above, but which have no real existence, are nothing but the paths in which the light waves are transmitted. According to Huygens's principle (see DIFFRACTION) each aether particle, set vibrating by an incident wave, can itself act as a new centre of excitement, emitting a spherical wave; and similarly each particle on this wave itself produces wave systems. All systems which are emitted from a single source can by a suitable optical device be directed that they simultaneously influence one and the same aether particle. According to the phase of the vibrations at this common point, the waves mutually strengthen or weaken their action, and there arises greater clearness or obscurity. This phenomenon is called interference (q.v.). E. Abbe applied the Fraunhofer diffraction phenomena to the explanation of the representation in the microscope of uniformly illuminated objects. If a grating is placed as object before the microscope objective, Abbe showed that in the image there is intermittent clear and dark banding only, if at least two consecutive diffraction spectra enter into the objective and contribute towards the image. If the illuminating pencil is parallel to the axis of the microscope objective, the illumination is said to be direct. If in this case the aperture of the objective be so small, or the diffraction spectra lie so far from each other, that only the pencil parallel to the axis, i.e. the spectrum of zero order, can be admitted, no trace is generally found of the image of the grating. If, in addition to the principal maximum, the maximum of 1st order is admitted, the banding is distinctly seen, although the image does not yet accurately resemble the object. The resemblance is greater the more diffraction spectra enter the objective. From the Fraunhofer formula
formula
The Fraunhofer diffraction phenomena, which take place in the 0 back focal plane of the objective, can be conveniently seen with the naked eye by removing the eyepiece and looking into the tube, or better by focusing a weak auxiliary
auxiliary
It is possible to almost double the resolving power, as in the case (From Abbe, Theorie der Bilderzeugung im Mikroshop.) 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 6 =A/2A. Reverting to fig. 13, we suppose that a diffracting particle of such fineness is placed at 0 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 F'1; and the two diffraction maxima of the 1st order in P' and P'1. The waves proceeding from this point are united in the point 0'. Suppose that a well corrected objective is employed. The image 0' of the point 0 is then the interference effect of all waves proceeding from the exit pupil of the objective P1P1'. 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
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. I'. 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: V2=tan w'/d=2A tan 2'/X; N2=2A1 tan 2'/X; where l equals the sight range of lo 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: V4=2A tan 4'/X; N4=2A1 tan 4'/X. If o55 o is assumed for daylight observation, then according to Abbe (Journ. Roy. Soc., 1882, p. 463) we have the following table for the limits of the magnification numbers, for various microscope objectives, p=ooo1 mm.: A=n sin u. d in l4. N2. N4. 0I0 2.75 53 106 0.30 0.92 159 317 0.60 0.46 317 635 0.90 0.31 476 952 1.20 0.23 635 1270 1.40 0.19 741 1481 P60 0.17 847 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 ,u can be resolved, while the theoretical maximum of the resolving power is 0.167 , so that the theoretical maximum has almost been reached in practice. Still smaller particles cannot be portrayed by using ordinary day-light. In, order to increase the resolving power, A. Kohler (Zeit. f. Mikros., 1904, 21, pp. 129, 273) suggested employing ultra-violet light, of a wave-length 275 /.I g; he thus increased the resolving power to about double that which is reached with day-light, of which the mean wave-length is 550 p. Light of such short wave-length is, however, not visible, and therefore a photo-graphic 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 unnecessary to dye the preparations. Glycerin is chiefly used as immersion fluid. M. v. Rohr's monochromats are constructed with apertures up to P25. The smallest resolving detail with oblique lighting is I=X/2A, where a=275 . As the microscopist usually estimates the resolving power according to the aperture with ordinary day-light, Kohler 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 u. 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 I.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 ofthe exit pupil of the microscope is about 0.04 in. with the magnification N2, and about 0.02 in. with the magnification N4. Moreover, 'with such exceptionally narrow pencils shadows are formed on the retina of the observer's eye, from the irregularities in the eye itself. These disturbances are called " entoptical phenomena." From the section Regulation of the Rays (above) it is seen that the resolving power is opposed to the depth of definition, which is measured by the reciprocal of the numerical aperture, i/A. 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
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 back-ground, and can be easily seen. The cutting off of the chief
cone
cone
lLnuh l..wwufl,, nm.uunur ar n...,u AIEL X1111 1Io.mIIf , \ X11 W)/11,~ m "I (By permission of C. Zeiss.) The large loss of light, which is caused in dark-field illumination by the cutting off of the direct cone of rays, must be compensated by employing exceptionally strong sources. By dark-field illumination it is even possible to make such small details of objects perceptible as are below the limits of the resolving power. It is a similar phenomenon to that which arises when a ray of sunlight falls into a darkened room. The extremely small particles of dust (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 back-ground 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
Siedentopf employed two illuminating arrangements. With the orthogonal arrangement for illuminating and observing the beamof light traverses an extremely fine slit through a well-corrected system, whose optic axis is perpendicular to the axis of the micro-scope; the system reduces the dimensions of the beam to about 2 to 4 s in the focal plane of the objective. For the microscopic observation it is the same as if a thin section of a thickness of 2 to 4 had been shown. In this optical way it is possible to show thin sections even in liquid preparations. The inconvenience of orthogonal illumination, which certainly gives better results, is avoided in the coaxial apparatus. Care must here be taken, by using suitable dark-field screens, that no direct rays enter the observing system. The only sources of light are sunlight or the electric arc. The limit at which sub-microscopic particles are made visible is dependent upon the specific intensity of the source of light. With sunlight particles can be made visible to a size of about 0.004 . Production of the Image.As shown in LENS and ABERRATION, for reproduction through a single lens with spherical surfaces, a combination of the rays is only possible for an extremely small angular aperture. The aberrations, both spherical and chromatic, increase very rapidly with the aperture. If it were not possible to recombine in one image-point the rays leaving the objective and derived from one object-point, i.e. to eliminate the spherical and chromatic aberrations, the large angular aperture of the objective, which is necessary for its resolving power, would be valueless. Owing to these aberrations, the fine structure, which in consequence of the large aperture could be resolved, could not be perceived. In other words, a sufficiently good and distinct image as the resolving power permits cannot be arrived at, until the elimination, or a sufficient diminution, of the spherical and chromatic aberrations has been brought about. The objective and eyepiece have such different functions that as a rule it is not possible to correct the aberrations of one system by those of the other. Such a compensation is only possible for one single defect, as we shall see later. The demands made upon the eyepiece, which has to represent a relatively large field by narrow cones of rays, are not very considerable. It is therefore not very difficult to produce a usable eyepiece. On the other hand, the correction of the objective presents many difficulties. 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.The aberration of rays in which the outer rays intersect the axis at a shorter distance than the central rays is known as " under. correction." The reverse is known as " over-correction." By 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, under-corrected 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 aberration. The representation, free from aberration, of a small surface-element, is only possible, as Abbe has shown, if the objective simultaneously fulfils the " sine-condition," i.e. if the ratio of the sine of the aperture u on the object-side to the sine of the corresponding aperture u' on the image-side is constant, i.e. if n sin u/sin u'=C, in which C is a constant. The sine-condition is in contrast to the tangent-condition, which must be regarded as the point-by-point representation of the whole object-space in the image-space (see LENS), and according therefore the equation n tan u/tan u' =C must exist. These two conditions are only compatible when the representation is made with quite narrow pencils, and where the apertures are so small that the sines and tangents are of about the same value. Very large apertures occur in strong microscope objectives, and hence the two conditions are not compatible. The sine-condition is, however, the most important as far as the microscopic representation is concerned, because it must be possible to represent a surface element through the objective by wide cones of rays. The removal of the spherical aberration and the sine-condition can be accomplished only for two conjugate points. A well-corrected microscope objective with a wide aperture therefore can only represent, free from aberrations, one object-element situated on a definite spot on the axis. As soon as the object is moved a short distance away from this spot the representation is quite useless. Hence the importance of observing the length of the tube in strong systems. If the sine-condition is not fulfilled but the spherical aberrations in the side, which traverse different zones of the objective, have a different magnification. The sine-condition can therefore also be understood as follows: that all objective zones must have the same magnification for the plane-element. According to Abbe, a system can only be regarded as aplanatic if it is spherically corrected for not only one axial point, but when it also fulfils the sine-condition and thus magnifies equally in all zones a surface-element situated vertically on the axis at this point. A second method of correcting the spherical aberration depends of 0 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 from one pointthe image-point (see fig. 30). With this, the object-point 0, and consequently the image-point 0' 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 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, and with a high refraction a low dispersion. By using these glasses and employing minerals with special
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 183o 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; although contrary to the telescope, the distance of the object in the microscope objective is small in proportion to the distance of the image. It would have been more correct to have employed these objectives in a reverse position. These circumstances were considered by Chevalier and Lister. Lister showed that a combination of lenses can be achromatic for only two points on the axis, and therefore that the single systems must be so arranged that the aplanatic (virtual) image-point 0' (fig. 32) of the first system coincides with the object-point of the next system. This system will always be aplanatic. These objectives permitted a much larger aperture than a simple achromatic system. Although such systems have been made recently for special
axis have been removed, then the image shown in fig. 19 results. The cones of rays issuing from a point situated only a little to the O IF noticeable. For this reason Amici constructed objectives of a similar aperture and focus for different thicknesses of glass covers. This expensive method was simplified in 1837 by Andrew Ross by making the upper and lower portion of the objective variable by means of a so-called correction-collar, and so giving the objective a corresponding under-correction according to the thickness of the glass cover. The alteration of the focus and the aperture are little influenced. The correction-collar was improved by Wenham and Zeiss, by working the upper system upon the lower, and not the reverse; for in this way the preparation remains almost exactly focused during the operation (see fig. 34).The injurious influence of the glass cover is substantially lessened if no air is admitted to the space between the glass cover and the tive fitted with cor- matic objective for r e c t i o n collar homogeneous immer- (Zeiss). sion. front lens (as in the dry-system) but if the intervening space is filled with an immersion-liquid. Amici was likewise the first to produce practical and good immersion-systems. The slight difference of the refractive indexes of the glass cover and the immersion-liquid involves a diminution of the aberrations, by which the objective will become less sensitive to the differences in thickness of the glass covers and admits of a more perfect adjustment. Water-immersion was introduced by Amici in 184o, and was improved by E. Hartnack in 1855. The advantages of the immersion over the dry-systems are greatest when the embedding-liquid, the glass cover, the immersion-liquid and the front lens have the same refractive index. Such systems with a so-called homogeneous immersion were first constructed after the plan of E. Abbe in 1878 inthe Zeiss workshops at the instigation of J. W. Stephenson. Cedarwood oil (Canada balsam), which has a refractive index of 1.515, is the immersion-liquid. The structure of a modern system of this type, with a numerical aperture of 1.30, is shown in fig. 35. The most perfect microscope objective was invented by E. Abbe in 1886 in the so-called apochromatic objective. In this, the secondary spectrum is so much lessened that for all practical purposes it is unnoticeable. In the apochromats the chromatic difference of the spherical aberrations is eliminated, for the spherical aberration is completely avoided for three colours. Since in these systems the sine-condition can be fulfilled for several colours, the quality of the images of points beyond the axis is better. There still remains a slight chromatic difference in magnification, for although the magnification consequent upon the fulfilment of the sine-condition is the same for all zones for one colour, it is impossible to avoid a change of the magnification with the colour. Abbe overcame this defect by using the so-called compensation ocular, made with Jena glasses. Fig. 36 shows ,an apochromat of a numerical aperture of P40. End of Article: THE PHYSICAL If you wish, you can link directly to this article.
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