the 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, 1/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 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).
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| (By permission of C. Zeiss.)
Fig. 28. |
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 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 system, whose optic axis is perpendicular to the axis of the microscope; the system reduces the dimensions of the beam to about 2 to 4 μ 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 μ 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 μ.
Production of the Image.—As shown in Lens and Abberation, 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
