Lastly, it should also be noted that it is numerical and not angular aperture which measures the quantity of light admitted to the objective by different pencils.
Fig. 38.
Fig. 38a.
First take the case of the medium being the same. The popular notion of a pencil of light may be illustrated by [Fig. 38], which assumes that there is equal intensity of emission in all directions, so that the quantity of light contained in any given pencils may be compared by simply comparing the contents of the solid cones. The Bouguer-Lambert law, however, shows that the quantity of light emitted by any bright point varies with the obliquity of the direction of emission, being greater in a perpendicular than in an oblique direction. The rays are less intense in proportion as they are more inclined to the surface which emits them, so that a pencil is not correctly represented by [Fig. 38], but by [Fig. 38]a, the density of the rays decreasing continuously from the vertical to the horizontal, and the squares of the sines of the semi-angles (i.e., of the numerical aperture) constituting the true measure of the quantity of light contained in any solid pencil.
If, again, the media are of different refractive indices, as air (1·0), water (1·33), and oil (1·52), the total amount of light emitted over the whole 180° from radiant points in these media under a given illumination is not the same, but is greater in the case of the media of greater refractive indices in the ratio of the squares of those indices (i.e., as 1·0, 1·77 and 2·25). The quantity of light in pencils of different angle and in different media must therefore be compared by squaring the product of the sines and the refractive indices, i.e. (n Sin u2), for the square of the numerical aperture.
The fact is therefore made clear that the aperture of a dry objective of 180° does not represent, as was supposed, a maximum, but that aperture increases with the increase in the refractive index of the immersion fluid; and it should be borne in mind that this result has been arrived at in strict accordance with the ordinary propositions of geometrical optics, and without any reference to or deductions from the diffraction theory of Professor Abbe.
There still remains one other point for determination, namely, the proper function of aperture in respect to immersion objectives of large aperture. The explanation of the increased power of vision obtained by increase of aperture was, that by the greater obliquity of the rays to the object “shadow effects” were produced, a view which overlooked the fact, first, that the utilisation of increased aperture depends not only on the obliquity of the rays sent to the object, but also to the axis of the microscope; and exactly as there is no acoustic shadow produced by an obstacle, which is only a few multiples of the length of the sound waves, so there can be no shadow produced by minute objects, only a few multiples from the light waves, the latter then passing completely round the object. The Abbe diffraction theory, however, supplies the true explanation of this, and shows that the increased performance of immersion objectives of large aperture is directly connected (as might have been anticipated) with the larger “openings” in the proper sense of the term, which, as we have already explained, such objectives really possess. Furthermore, in order that the image exactly corresponds with the object, all diffracted rays must be gathered up by the objective. Should any be lost we shall have not an actual image of the object, but a spurious one. Now, if we have a coarse object, the diffracted rays are all comprised within a narrow cone round the direct beam, and an objective of small aperture will transmit them all. With a minute object, however, the diffracted rays are widely spread out, so that a small aperture can admit only a fractional part—to admit the whole or a very large part, and consequently to see the minute structure of the object, or to see it truly, a large aperture is necessary, and in this lies the value of aperture and of a wide-angled immersion objective for the observation of minute structures.