Concave Lenses.
The refracting influence of a concave lens ([Fig. 14]) will be precisely the opposite of that of a convex. Rays which fall upon it in a parallel direction will be made to diverge as if from the principal focus, which is here called the negative focus. This will be, for a plano-concave lens, at the distance of the diameter of the sphere of curvature; and for a double-concave, in the centre of that sphere.
Fig. 14.—A Virtual Image formed by Concave Lens.
In [Fig. 14] A B is the object and a b the image. Rays incident from A and B parallel to the principal axis will emerge as if they came from the principal focus F; hence, the points a b are determined by the intersections of the dotted lines in the figure with the secondary axis, O A, O B. An eye on the other side of the lens sees the image a b, which is always virtual, erect and diminished.
In the construction of the microscope, either simple or compound, the curvature of the lenses employed is usually spherical. Convergent lenses, with spherical curvatures, have the defect of not bringing all the rays of light which pass through them to one and the same focus. Each circle of rays from the axis of the lens to its circumference has a different focus, as shown in [Fig. 15]. The rays a a, which pass through the lens near its circumference, are seen to be more refracted, or come to a focus at a shorter distance behind it than the rays b b, which pass through near its centre or axis, and are less refracted. The consequence of this defect of lenses with spherical curvatures, which is called spherical aberration, is that a well-defined image or picture is not formed by them, for when the object is focussed, for the circumferential rays, the picture projected to the eye is rendered indistinct by a halo or confusion produced by the central rays falling in a circle of dissipation, before they have come to a focus. On the other hand, when placed in the focus of the central rays, the picture formed by them is rendered indistinct by the halo produced by the circumferential rays, which have already come to a focus and crossed, and now fall in a state of divergence, forming a circle of dissipation. The grosser defects of spherical aberration are corrected by cutting off the passage of the rays a a, through the circumferences of the lens, by means of a stop diaphragm, so that the central rays, b b, only are concerned in the formation of the image. This defect is reduced to a minimum, by using the meniscus form of lens, which is the segment of an ellipsoid instead of a sphere.
Fig. 15.—Spherical Aberration of Lens.
The ellipse and the hyperbola are forms of lenses in which the curvature diminishes from the central ray, or axis, to the circumference b; and mathematicians have shown that spherical aberration may be practically got rid of by employing lenses whose sections are ellipses or hyperbolas. The remarkable discovery of these forms of lenses is attributed to Descartes, who mathematically demonstrated the fact.
If a l, a l′, for example ([Fig. 16]) be part of an ellipse whose greater axis is to the distance between its foci f f as the index of refraction is to unity, then parallel rays r l′, r′′ l incident upon the elliptical surface l′ a l, will be refracted by the single action of that surface into lines which would meet exactly in the farther focus f, if there were no second surface intervening between l a l′ and f. But as every useful lens must have two surfaces, we have only to describe a circle l a′ l′ round f as a centre, for the second surface of the lens l′ l.
Fig. 16.—Converging Meniscus.
As all the rays refracted at the surface l a l′ converge accurately to f, and as the circular surface l a′ l′ is perpendicular to every one of the refracted rays, all these rays will go on to f without suffering any refraction at the circular surface. Hence it should follow, that a meniscus whose convex surface is part of an ellipsoid, and whose concave surface is part of any spherical surface whose centre is in the farther focus, will have no appreciable spherical aberration, and will refract parallel rays incident on its convex surface to the farther focus.
Fig. 17.—Aplanatic Doublet.
The spherical form of lens is that most generally used in the construction of the microscope. If a true elliptical or hyperbolic curve could be ground, lenses would very nearly approach perfection, and spherical aberration would be considerably reduced. Even this defect can be further reduced in practice by observing a certain ratio between the radii of the anterior and posterior surfaces of lenses; thus the spherical aberration of a lens, the radius of one surface of which is six or seven times greater than that of the other, will be much reduced when its more convex surface is turned forward to receive parallel rays, than when its less convex surface is turned forwards. It should be borne in mind that in lenses having curvatures of the kind the object would only be correctly seen in focus at one point—the mathematical or geometrical axis of the lens.
Chromatic Aberration.—We have yet to deal with one of the most important of the phenomena of light, CHROMATIC ABERRATION, upon the correction of which, in convex lenses in particular, the perfection of the objective of the microscope so much depends. Chromatism arises from the unequal refrangibility and length of the different coloured rays of light that together go to make up white light; but which, when treated of in optics, is always associated with achromatism, so that a combination of prisms, or lenses, is said to be achromatic when the coloured rays arising from the dispersion of the pencil of light refracted through them are combined in due proportions as they are in perfectly white light.
A lens, however, of uniform material will not form a single white image, but a series of images of all colours of the spectrum, arranged at different distances, the violet being nearest, and the red the most remote, every other colour giving a blurred image; the superposition of these and the blending of the different elementary rays furnishing a complete explanation of the beautiful phenomenon of the rainbow. Sharpness of outline is rendered quite impossible in such a case, and this source of confusion is known as chromatic aberration.
In order to ascertain whether it is possible to remedy this evil by combining lenses of two different materials, Newton made some trials with a compound prism composed of glass and water (the latter containing a little sugar of lead), and he found it impossible by any arrangement of these two, or by other substances, to produce deviation of the transmitted light without separation into its component colours. If this ratio were the same for all substances, as Newton supposed, achromatism would be impossible; but, in fact, its value varies greatly, and is far greater for flint than for crown glass. If two prisms of these substances, of small refracting angles, be combined into one, with their edges turned in opposite directions, they will achromatise each other.
The chromatism of lenses may, however, be somewhat further reduced by stopping out the marginal rays, but as the most perfect correction possible is required when lenses are combined for microscopic uses, other means of correction are resorted to, as will be seen hereafter. I shall first proceed to show the deviations which rays of white light undergo in traversing a lens.
If parallel rays of light pass through a double-convex lens the violet rays, the most refrangible of them, will come to a focus at a point much nearer to the lens than the focus of the red rays, which are the least refrangible; and the intermediate rays of the spectrum will be focussed at points between the red and the violet. A screen held at either of these foci will show an image with prismatic fringes. The white light, A A′′ ([Fig. 18]), falling on the marginal portion of the lens is so far decomposed that the violet rays are brought to a focus at C, and crossing there, diverge again and pass on to F F′, while the red rays, B B′′, do not come to a focus until they reach the point D, and cross the divergent violet rays, E E′. The foci of the intermediary rays of the spectrum (red, green, and blue) are intermediate between these extremes. The distance, C D, limiting the blue or violet, and the red is termed the longitudinal chromatic aberration of the lens. If the image be received upon a screen placed at C, violet will predominate and appear surrounded by a prismatic fringe, in which violet will predominate. If the screen be now shifted to D, the image will have a predominant red tint, surrounded by a series of coloured fringes in an inverted order to those seen in the former experiment. The line E E′ joins the points of intersection between the violet and red rays, and this marks the mean focus, the point where the coloured rays will be least apparent.
Fig. 18.—Chromatic Aberration of Lens.
In the early part of this century the optical correction of chromatic aberration was partially brought about by combining a convex lens of crown-glass with a concave lens of flint-glass, in the proportion of which these two kinds of glass respectively refract and disperse rays of light; so that the one medium may by equal and contrary dispersion neutralise the dispersion caused by the other, without at the same time wholly neutralising its refraction. It is a curious fact that the media found most available for the purpose should be a combination of crown and flint-glass, of crown-glass whose index of refraction is 1·519, and dispersive power 0·036, and of flint-glass whose index of refraction is 1·589, and dispersive power 0·0393. The focal length of the convex crown-glass lens must be 41⁄3 inches, and that of the concave flint-glass lens 72⁄3 inches, and the combined focal length 10 inches. The diagram ([Fig. 19]) shows how rays of light are brought to a focus, nearly free from colour. The small amount of residual colour in such a combination is termed the secondary spectrum; the violet ray F Y, crossing the axis of the lens at V, and going to the upper end P of the spectrum, the red ray F B going to the lower end T. But as the flint-glass lens l l, on the prism A a C, which receives the rays F V, F R, at the same points, is interposed, these rays will unite at f, and form a small circle of white light, the ray S F being now refracted without colour from its primitive direction S F Y into the direction F f. In like manner, the corresponding ray S F′ will be refracted to f, and a white colourless image be the result.
Fig. 19.—Correction of Chromatic Aberration.
The achromatic aplanatic objective constructed on the optical formula enunciated, did not meet all the difficulties experienced by the skilled microscopist, in obtaining resolution of the finest test objects, and whereby the intrinsic value of the objective (in his estimation) must stand or fall. There were other disturbing residuary elements besides those of the secondary spectrum, and which at a later period were met by the practical skill of the optician, who applied the screw-collar, and by means of which the back lens of the objective is made to approach the front lens, thus more accurately shortening the distance between the eye-piece, where the image is eventually formed, and the back lens of the objective.
In this diagram L L is a convex lens of crown-glass, and l l a concave one of flint-glass. A convex lens will refract a ray of light (S) falling at F on it exactly in the same manner as the prism A B C, whose faces touch the two surfaces of the lens at the points where the ray enters, and quits. The ray S F, thus refracted by the lens L L, or prism A B C, would have formed a spectrum (P T) on a screen or wall, had there been no other lens.
Fig. 20.—Virtual Image formed by Convex Lens.
Formation of Virtual Images.—The normal eye possesses a considerable power of adjusting itself to form a distinct image of objects placed at varying distances; the nearer, within a certain limit, the larger it appears, and the more distinctly the details are brought out. When brought within a distance of two or three inches, the images become blurred or quite indistinct, and when brought closer to the eye, cannot be seen at all, and it simply obstructs the light. Now the utility of a convex lens, when interposed between the object and the eye, consists in reducing the divergence of the rays forming the several pencils which issue from it, and send images to the retina in a state of moderate divergence, that is, as if they had issued from an object beyond the nearest point of distinct vision, and so that a more clearly defined image may reach the sensitive membrane of the eye. But, not only is the course of the several rays in each pencil altered as regards the rest, but the course of the pencils themselves is changed, so that they enter the eye under an angle corresponding with that under which they would have arrived from a larger object situated at a greater distance, and thus the picture formed by any object corresponds in all respects with one which would have been made by the same object increased in its dimensions and viewed at the smallest ordinary distance of distinct vision. For instance, let an object A B ([Fig. 20]) be placed between a convex lens and its principal focus. Then the foci conjugate to the points A B are virtual, and their positions can be found by construction from the consideration that rays through A, B, parallel to the principal axis, will be refracted to F, the principal focus on the other side. The refracted rays, if produced backwards, must meet the secondary axis O A, O B in the required points. An eye placed on the other side of the lens will accordingly see a virtual image erect, magnified, and at a greater distance from the lens than the object. This is the principle of the simple microscope.