To obtain the magnification of the complete microscope we must combine the objective magnification M with the action of the eyepiece. If we replace y ′ in equation (4) by the value given by (3), we obtain
| tan w ″/y＝Δ/f1′. 1/f2″＝V, | (5) |
the magnification of the complete microscope. The magnification therefore equals the power of the joint system.
The magnification is also expressed as the ratio of the apparent size of the object observed through the microscope to the apparent size of the object seen with the naked eye. As the conventional distance for clear vision with naked eye is 10 in., it results from fig. 1 that the apparent size is tan w=y/l. If this value of y be inserted in equation (5), we obtain the magnification number of the compound microscope:—
| N＝tan w″/ tan w＝Δl/f1f2′＝Vl. | (6) |
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Fig. 14.—Ray transmission in compound microscope with a negative eyepiece.
L1=weak achromatic objective. L2=negative eyepiece. F1, F1′＝object- and image-side foci of objective. F2, F2′＝object- and image-side foci of eyepiece. P′P1′＝exit pupil of objective. P″P1′＝virtual image of P1P1′＝exit pupil of complete microscope.
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The magnification number increases then with the optical tube-length and with the diminution of the focal lengths of objective and eyepiece.
As with the simple microscope, different observers see differently in the same compound microscope; and hence the magnification varies with the power of accommodation.
The image produced by a microscope formed of two positive systems (fig. 13) is inverted, the objective L1 tracing from the object OO1 a real inverted image O′O′1, and the eyepiece L2L3 maintaining this arrangement. For many purposes it is immaterial whether the image is inverted or upright; but in some cases an upright image lightens the work, or may be indispensable.
The simplest microscope which produces an upright image has a negative lens as eyepiece. As shown in fig. 14, the real image formed by the objective must fall on the object-side focal plane of the eyepiece F2, where a normal eye without accommodation can observe it. But as the object-side focus F2 lies behind the eyepiece, the real image is not produced, but the converging pencils from the objective are changed by the eyepiece into parallels; and the point O1 in the top of the object y appears at the top to the eye, i.e. the image is upright.
The erection of inverted images by prisms, which was applied to the simple telescope by Porro, and to the binocular (q.v.) by A. A. Boulanger was employed by K. Bratuscheck in the Greenough double microscope; these (inverting prisms permit a convenient adaptation of the instrument to the interpupillary distance of the observer. Double microscopes, which produce a correct impression of the solidity of the object, must project upright images. The terrestrial eyepiece (see Telescope), which likewise ensures an upright image, but which involves an inconvenient lengthening, has also been employed in the binocular microscope.
Regulation of the Rays.—Weak and medium microscope objectives work like photographic objectives in episcopic or diascopic projection; in the microscope, however, the projected image is not intercepted on a screen, but a real image in air is formed. This must lie in the front focal plane of the eyepiece if we retain the supposition that it is to be viewed by a normal eye with passive accommodation. The plane in the object conjugate to the focal plane of the eye-piece is the plane focused for; and all points in it are sharply portrayed (a perfect objective being assumed). Object points lying out of the focal plane, on the other hand, are projected as circles of confusion on the plane focused for, the centre of the entrance pupil being the centre of projection and the circles of confusion constituting, with the points of the focal plane, the object-side imago. As the pencils used in the representations are of wide aperture on the object-side, only such points as are proportionately very near the focal plane can produce such small dispersion circles on the plane focused for, that they, so far as the objective- and eyepiece-magnification permit, appear as points to the eye. It follows that the depth of definition of the microscope is in general very trifling. As it is entirely a function of the aperture and the magnification, it can be increased by diminishing the entrance pupil, the magnification remaining unchanged. A diminution of the aperture, however, would injure a very much more important property, viz. the resolving power (see below). With powerful systems, object-points lying quite near the plane focused for would be represented by such large dispersion circles that practically only the points lying in one plane appear simultaneously sharp; and it is only by varying the Focus that the object-points lying in other planes can be observed.
The position of the diaphragm limiting the pencils proceeding from the object-points is not constant in the compound microscope. In all microscopes the rays are limited, not in the eyepiece, but in the objective, or before the objective when using a condenser. If the pencils are limited in the objective, the restriction of the pencil proceeding from the object-point is effected by either the front lens itself, by the boundary of a lens lying behind, by a real diaphragm placed between or behind the objective, or by a diaphragm-image.
The centre of the entrance pupil is the point of intersection of the principal rays; and it is therefore determinative for the perspective representation on the plane focused for. In fig. 15 the centre of the entrance pupil lies behind the focal plane, and consequently nearer objects appear larger, and farther objects smaller (“entocentric transmission,” see below).
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| (After M. v. Rohr.)
Fig. 15.—Entocentric transmission through a microscope objective. |
| E＝plane focused for; O1*, O2*=projections of O1O2 on E; Z＝centre of projection; P P1＝a virtual image of real diaphragm P′P1′ with regard to the preceding part of the objective is the entrance pupil. |
If a diaphragm lying in the back focal plane of the objective forms the exit pupil for the objective, as in figs. 13 and 14, so that its image, the entrance pupil, lies at infinity, all the principal rays in the object-space are parallel to the axis, and we have on the object-side “telecentric” transmission. The size of the imago on the focal plane is always equal to its actual size, and is independent of the distance of the object from the plane focused for. This representation acquires a special importance if the object be micrometrically measured, for an inaccuracy in focusing does not involve an alteration of the size of the image. To ensure the telecentric transmission, the diaphragm in the back focus of the objective may be replaced by a diaphragm in the front focal plane of the condenser, supposing that uniformly illuminated objects are being dealt with; for in this case all the principal rays in the object-space are transmitted parallel to the axis.
With uniformly illuminated objects it may happen that the pencil in the object-space may be limited before passing the object, either through the size of the source of light employed or through a diaphragm connected with the illuminating system. In fig. 16 the intersection of the principal rays lies in front of the object, and consequently objects in front of the plane focused for will be projected on E magnified and the objects lying behind it diminished (“hypercentric” transmission). It produces a perspective representation entirely opposed to ordinary vision. As objects lying near us appear smaller in the case of hypercentric transmission than those lying farther from us, we receive a false impression of the spatial arrangement of the object.
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| (After M. v. Rohr.) |
Fig. 16.—Hypercentric transmission in a microscope
objective. |
Whether the entrance pupil be before or behind the object, in general its position is such that it lies not too near the object, so that the principal rays will have in the object space only trifling inclinations towards one another or are strictly parallel. This is specially important, for otherwise pencils from points placed somewhat laterally to the axis arrive with diminished aperture at the image.
We see from fig. 13 that the objective’s exit pupil P′P1′ is
portrayed by the positive eyepiece, the image P″P1″ limits the pencils
