Focal length

“ In refraction, I added "The focal length f is considered negative for concave lenses." (Though this wouldn't be true for concave lenses with lower index of refraction than the surrounding medium.) I suggest also adding, "Incoming parallel rays are focused by a convex lens into an inverted real image one focal length from the lens, on the far side of the lens. Rays from an object at finite distance are focused further from the lens than the focal distance; the closer the object is to the lens, the further the image is from the lens. ”

“ As we recommend the use of long-focus lenses only, as will be seen in the chapter on lenses, and as no definite law can be laid down for this length, it is advisable to order a camera four or five inches longer than the focal length of the lens which is advertised to cover the next larger-sized plate to that which your dark slide holds. ”

“ If we suppose the focal length to be 66 ft., a single lens is practically perfect up to an aperture of 1·7 in.
Another obvious inference from the necessary imperfection of optical images is the uselessness of attempting anything like an absolute destruction of spherical aberration. An admissible error of phase of 1/4λ will correspond to an error of 1/8λ in a reflecting and 1/2λ in a (glass) refracting surface, the incidence in both cases being perpendicular. ”

“ The lines A B C represent parallel rays which pass through the lens, D D, and meet at F; this point being the principal focus, its distance from the lens is called the focal length. Those rays of light which are traversing a parallel course, when they enter the lens are brought to a focus nearer the lens than others. Hence the difficulty the operator sometimes experiences by not being able to “obtain a focus,” when he wishes to secure a picture of some very distant objects; he does not get his ground glass near enough to the lenses. ”

“ With a lens of such a focal length we shall require a camera with very long bellows to obtain much magnification of our object but, if our camera is one with only short bellows, we can still overcome our difficulties. If the extra expense is no object we can obtain one of the excellent Aldis lenses of only two inches focal length, especially designed for this work; if we are ingenious we can construct a device which will answer our purpose admirably and cost but a few pence. ”

“ A dry objective of the maximum air angle of 180° is only able to utilise a diameter of back lens equal to twice the focal length, while an immersion lens of even only 100° utilises a larger diameter, i.e., it is able to transmit more rays from the object to the image than any dry objective is capable of transmitting. Whenever the angle of an immersion lens exceeds twice the critical angle for the immersion fluid, i.e., 96° for water or 82° for oil, its aperture is in excess of that of a dry objective of 180°. ”

“ And were we to construct an object-glass of 100 feet focal length, we should require to apply an eye-glass, not less than 6 inches focus, which would produce a power of about 200 times; so that there is no possibility of producing a great power by single lenses, without extending the telescope to an immoderate length. ”

“ Probably the most useful lens to employ is a doublet of which the focal length is about a quarter more than the width of the plate, since it includes a fair angle and the margins of the photographs are not likely to be markedly different in general density to the centre, as is the case when wide-angle lenses are employed. In England a lens which will cover with a large stop, say f/8, is a desideratum, but in the Alps it is very rarely that such a large ratio of aperture to focal length is required. ”

“ If a lens be placed between the slit and the prism, at exactly its focal length from the former, the light entering the slit will, after passage through the lens, emerge as parallel rays, that is, they will emerge as they would do if the slit were placed at an infinite distance from the observer. ”

“ If a lens is now held an inch, or more, according to its focal length, over the part of the line A B, and the slightest amount out of its centre, that part of the line A B which passes under the lens will be seen in the direction of the figures 1.2; but if the lens be now moved a little out of its central position in the opposite direction, the line B C will be observed at 3.4, or below B C. A lens is a magnifying glass because it dilates or spreads out from its centre the objects observed through it Therefore whatever is magnified by it is seen a little out of its axis or centre. ”

“ Doubtless, defining power can be more satisfactorily determined by the examination of a suitable object, and the perfection of the image obtained; to assist in securing which, a solid axial cone of light equal to about three-fourths of the aperture of the objective must be employed. 263
To sum up, then, “the focal power of all objectives depends in their perfect definition, a property on which their converging power depends, and in turn their magnifying action is dependent; again, focal power is the curvature imprinted by the lens on a plane wave, and is reciprocal of the true focal length. ”

“ In the improved combination, the diameter is only sufficient to admit the proper pencil; the convex lenses are wrought to an edge, and the concave have only sufficient thickness to support their figure; consequently the combination is the thinnest possible, and it follows that there will be the greatest distance between the object and the object-glass. The focal length is one-eighth of an inch, having an angular aperture of 60°, with a distance of 1-25th of an inch, and a magnifying power of 970 times linear, with perfect definition on the most difficult Podura scales. ”

“ The curvature of the image is the thing which sets a limit to shortening the relative focus, as already noted, for the astigmatic image surfaces as we have seen, fall rapidly apart away from the axis, and both curvatures are considerable. The tangential is the greater, corresponding roughly to a radius notably less than 1⁄3 the focal length, while the radial fits a radius of less than 2⁄3 this length with all ordinary glasses, given forms correcting the ordinary aberrations. ”

“ If we know the refractive index of the glass n, the radii r1, r2 of the spherical surfaces, and the distances of the two lens-vertices (or the thickness of the lens d) we can determine all the properties of the lens. A biconvex lens is shown in fig. 8. Let F1 be the first principal focus of the first system of radius r1, and F1′ the second principal focus; and let S1 be its vertex. Denote the distance F1 S1 (the first principal focal length) by f1, and the corresponding distance F′1 S1 by f ′1. ”

“ To obtain a flatter field, and particularly one free from distortion the construction devised by Ramsden is commonly used. This consists, Fig. 103, of two plano convex lenses of equal focal length, placed with their plane faces outward, at a distance equal to, or somewhat less than, their common focal length. ”

“ An objective of 4.56′ inches aperture has a resolving constant of 1′′ and to develop this should take a magnification of say 300, about 65 to the inch of aperture, requiring a focal length of ocular about 0.20 to 0.25 inch for telescopes of normal relative aperture, and pushing the emergent pencil down to little more than 0.02 inch,—rather further than is physiologically desirable. Except for these extreme stunts of separation, half to two thirds this power is preferable and conditions under which one can advantageously go above this limit are very rare indeed. ”

“ For low powers the usable field is helped out by the accommodation of the eye, but in oculars of short focus the curvature of field is the limiting factor. The radius of curvature of the image is, in a single lens approximately 3/2 F, and in the common two lens forms about 3⁄4 F.
In considering this matter Conrady has shown (M. N. 78 445) that for a total field of 40° the sharpness of field fails at a [Pg 147] focal length of about 1 inch for normal power of accommodation. The best achromatic combinations reduce this limit to about 1⁄2 inch. ”

“ The photographic lens, on the other hand, is called upon to produce a clear image with light of a wide range of spectral composition, sharply defined over 47a flat surface of large area, and it must do this with an aperture that is large in comparison with the focal length, whereby the amount of light falling on the image surface shall be a maximum. ”