proceeding from the eyepiece. This image P″P1″ is then the exit pupil of the combined system, and consequently the image of the entrance pupil of the combined system. As the exit pupil P′P1′ for the objective lies before the front focus of the eyepiece, generally at some distance and near the objective, the eyepiece projects a real image from it behind its image-side focus, so that if this point is accessible it is the exit pupil P″P1″. If, e.g. in the object-space the objective has telecentric transmission, the exit pupil must coincide with the back focal plane of the combined system, and it always lies behind the image-side focus of the eyepiece. The exit pupil, often called Ramsden’s circle, is thus accessible to the observer, who by head- and eye-movements may survey the whole field.
We can now understand the ray transmission in the compound microscope, shown in fig. 13. Points of a small object (compared with the focus of the objective) send to the objective wide pencils. The diaphragm limiting them, i.e. the entrance pupil, is placed so that the principal rays are either parallel or slightly inclined. The pencils producing the real image are very much more acute, and their inclination is the smaller the stronger the magnification. The eyepiece, which by means of narrow pencils represents the relatively large real image at infinity, transmits from all points of this real image parallel pencils, whereby the inclination of the principal rays becomes further increased. The point of intersection, i.e. the centre of the exit pupil, is accessible to the eye of the observer. In the case of the negative eyepiece, on the other hand, the divergence of the principal rays through the eyepiece is also further augmented, but their point of intersection is not accessible to the eye. This property shows the superiority of the collective eyepiece over the dispersive.
The increase of the inclination of the principal rays, which arises with the microscope, influences the perception of the relief of the object. In entocentric transmission this phenomenon appears in general as in the case of the contemplation of perspective representations at a too short distance, the objects appearing flattened. Although in the case of the spatial comprehension of a perspective representation experience plays a large part, in observing through a microscope it does not count, or only a little, for the object is presumably quite unknown. In telecentric and hypercentric transmission we obtain a false conception of the spatial arrangement of the objects or their details; in these cases one focusses by turns on the different details, and so obtains an approximate idea of their spatial arrangement.
While the limiting of the pencil is almost always effected by the objective, the limiting of the field of view is effected by the eyepiece, and indeed it is carried out by a real diaphragm DD arranged in the plane of the real image O′O1′ (fig. 13) projected from the objective. The entrance window is then the real image of this diaphragm projected by the objective in the surface conjugate to the plane focused for, and the exit window is the image projected by the eyepiece; this happens with the image of the object lying at infinity. The result must be that the field of view exhibits a sharp border. In the case of the dispersive eyepiece, on the contrary, no sharply limited field can arise, but vignetting must occur.
Illumination.—The dependence of the clearness of the image on the aperture of the system, i.e. on the angular aperture of the image-producing pencil, holds for all instruments.
The brightnesses of image points in a median section of the pencil are proportional to the aperture of the lens, supposing that the rays are completely reunited. This is valid so long as the pencil is in air; but if, on the other hand, the pencil passes from air through a plane surface into an optically denser medium, e.g. water or glass, the pencil becomes more acute and the aperture smaller. But since no rays are lost in this transmission (apart from the slight loss due to reflection) the brightness of the image point in the water is as large as that in air, although the apertures have become less. Fig. 17 shows a pencil in air, A, dispersing in water, W, from the semi-aperture u1, or a pencil in Water dispersing in air from the semi-aperture u2. If the value of the clearness in air be taken as sin u1, then by the law of refraction N＝sin u1/sin u2, the value for the clearness in water is N sin u2. This rule is general. The value of the clearness of an image-point in a median section is the sine of the semi-aperture of the pencil multiplied with the refractive index of the medium.
Fig. 17.Fig. 18.Fig. 19.
An illustration of this principle is the immersion experiment. A view taken under water from the point O (fig. 18) sees not only the whole horizon, but also a part of the bed of the sea. The whole field of view in air of 180° is compressed to one of 97·5° in water. The rays from O which have a greater inclination to the vertical than 48·75° cannot come out into the air, but are totally reflected. If pencils proceed from media of high optical density to media of low density, and have a semi-aperture greater than the critical angle, total reflection occurs; in such cases no plane surface can be employed, hence front lenses have small radii of curvature in order to permit the wide pencils to reach the air (see fig. 19, in which P is the preparation, O the object-point in it, D the cover slip, I the immersing fluid, and L the front lens).
The function n sin u＝A, for the microscope, has been called by Abbe the numerical aperture. In dry-systems only the sine of the semi-aperture is concerned; in immersion-systems it is the product of the refractive index of the immersion-liquid and the sine of the object-side semi-aperture. In the case of the brightness of large objects obviously the whole pencil is involved, and hence the clearness is the squares of these values, i.e. sin2u or n2sin2u. As the semi-aperture of a pencil proceeding from an object point cannot exceed 90°, the numerical aperture of a dry-system cannot be greater than 1. On the other hand, in immersion-systems the numerical aperture can almost amount to the refractive index, for A＝n sin u≦n. Dry systems of 0·98 numerical aperture, water immersion (n＝1·33) from A＝1·25, oil immersion (n＝1·51) from A＝1·40, and even α-bromnaphthalene immersions (n＝1·65) from A＝1·60, are available. In immersion-systems of such considerable aperture no medium of smaller refractive index than the immersion liquid may be placed between the surface of the front lens and the object, as otherwise total reflection would occur. This is especially inconvenient in the case of the α-bromnaphthalene immersion. As the embedding and immersing liquids must have equal refractive indexes, one must use α-bromnaphthalene for embedding; but this substance destroys organic preparations, so that one can employ this immersion-system only for examining inorganic materials, e.g. fine diatoms.
In immersion-systems a very much greater aggregate of rays is used in the representation than is possible in dry-systems. In addition to a considerable increase in brightness the losses due to reflection are avoided; losses which arise in passing to the back surface of the cover-slip and to the front surface of the front lens.
The Physical Theory
In order to fully understand the representation in the microscope, 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 δ＝λ/n sin u one can immediately deduce the limit to the diffraction constant δ, so that the banding by an objective of fixed numerical aperture can be perceived. The value n sin u equals the numerical aperture A, where n is the refractive index of the immersion-liquid, and u is the semi-aperture on the object-side. For microscopy the Fraunhofer formula is best written δ＝λ/A. This expresses δ as the resolving power in the case of direct lighting. All details of the object so resolved are perceived, if two diffraction maxima can be passed through the objective, so that the character of the object is seen in the image, even if an exact resemblance has not yet been attained.
The Fraunhofer diffraction phenomena, which take place in the
