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 microscope on the back focal plane of the objective. If one has, e.g. in the case of a grating, telecentric transmission on the object-side, and in the front focal plane of the illuminating system a small circular aperture is arranged, then by the help of the auxiliary microscope one sees in the middle of the back focal plane the round white image O (fig. 20) and to the right and left the diffraction spectra, the images of different colours partially overlapping. If a resolvable grating is considered, the diffraction phenomenon has the appearance shown in fig. 21.
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| Fig. 20.Fig. 21.Fig. 22.Fig. 23. |
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| Fig. 24.Fig. 25.Fig. 26.Fig. 27. |
| (From Abbe, Theorie der Bilderzeugung im Mikroskop.) |
It is possible to almost double the resolving power, as in the case 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 δ＝/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 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 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 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 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:
V2＝tan w ′/d＝2A tan 2′/λ; N2＝2Al tan 2′/λ;
where l 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:
V4＝2A tan 4′/λ; N4＝2Al tan 4′/λ.
If 0·55 μ 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, μ＝0·001 mm.:—
| A＝n sin u. | d in μ. | N2. | N4. |
| 0·10 | 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 |
| 1·60 | 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 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 μμ; he thus increased the resolving power to about double that which is reached with day-light, of which the mean wave-length is 550 μμ. 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 unnecessary 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 δ＝λ/2A, where λ＝275 μμ. 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 μμ. 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
