This section is from the book "Photography", by E. O. Hoppe, et al.. Also available from Amazon: Photography.

Any one of the lenses 1, 2, 3, Fig. 8, is able to form a reproduction ba or image of an object AB, Fig. 9, on the opposite side of the lens, provided that the object AB is more than a certain distance from the lens. Three typical rays from the point A are shown converging to the point a. Such lenses are known as Positive or Converging, and can be recognised easily, because they are always thicker at the centre than at the periphery. The lenses 4, 5, 6, Fig. 8, appear to form an image, ab, Fig. 10, on the same side of the lens as the object AB. Such an image of course does not exist, and is called Virtual. The lenses are known as Negative or Diverging. They are always thinner at the centre than at the periphery. Negative lenses would appear in themselves to be of little use to the photographer, since no image can be received on a screen. Such lenses are, however, absolutely indispensable to the construction of a good lens, and also form an essential part of a lens which is becoming extremely popular - viz. the Telephoto lens.

Fig. 9.

Before considering the formula connecting the distances of the object and image from a lens, we must have a clear idea how distances are to be measured. Unfortunately the convention adopted in Photographic Optics is not the same as that adopted in English treatises on general optics. The usual convention in Photographic Optics is adopted here, but the difference must be borne in mind if more advanced information is required from the larger works. The convention is that distances measured from the lens in the same direction as the incident light are positive, and in the opposite direction negative. Thus in Fig. 9 the distance of the image is positive, and the distance of the object negative. In Fig. 10 both distances are negative.

"Gossips".

J. Craig Annan.

In Fig. ii a biconvex lens, LL', is shown in cross-section passing through the centres of curvature CC' of the surfaces LPL', LP'L' respectively. The straight line CC' is called the Principal Axis of the lens. Let the image of an object, AB, at right angles to this axis be formed at ba. Then if the distance of AB from the lens, which is supposed to be infinitely thin, is u, and the distance of the image ba is v, it can be shown that 1/v + 1/u = a/f constant, denoted 1.

Fig. 10.

Fig. 11.

In this formula the signs of u, v, f have been allowed for, 3 so that it is only necessary to substitute the numerical values of the various distances.

The constant f is called the Focal Length of the lens, and is the distance of the image when the object is infinitely remote. Rays from such an object are. shown in Fig 12 The point F, where the image cuts the Principal Axis is called the Principal Focus. The focal length of a thin lens can easily be determined by sharply focussing some distant object such as a church spire, and measuring the distance of the focussing screen from the lens.

Now an infinite number of values of u and v can be found to satisfy the above formula, and the value of u corresponding to any value of v is said to be conjugate to it. The corresponding points where the object and image cut the Principal Axis are called the Conjugate Foci. Again considering the formula 1/v + 1/u = 1/f we see that, as u increases, v decreases, so that as the object recedes from the lens the image approaches it. When u is less than f a virtual image is formed, but with this we are not concerned.

Fig. 12.

It can also be shown that the relative sizes of image and object are in the ratio v: u, so that the magnification or reduction is v u .

The solution of some examples will make clear the use of these formulae.

A 6 in. lens is sharply focussed on an object distant 10 ft. What is the distance of the lens from the plate?

Here f = 1/2 (6 in. is 1/2 ft.) u = 10 and v is required.

Substituting these values in 1/v + 1/u = 1/f we have 1/v + 1/10 = 2, so that 1/v = 19/10 and v = 10/19 ft.

It is desired to enlarge from quarter-plate to whole-plate with a 6 in. lens. What must be the distance of the paper and negative respectively from the lens?

In this case the magnification is 2, so that v/u = 2.

Substituting in 1/v + 1/u = 1/f we have 1/2u + 1/u = 2. so that 3/ 2u = 2 and u = 3/4 ft. v = 1 1/2 ft.

Example 2 is another form of the frequently occurring question in which it is desired to know whether a lens of given focus can be used in a studio of certain length to obtain portraits of a particular size.

The angle of view included by a lens of focal length f when used with a plate whose longer side is n can easily be found by drawing a right-angled triangle, in which the sides subtending the right angle are f and n/2. The angle opposite the side of length - is half the angle of view, and its value can easily be ascertained by measurement with a protractor. When the focal length of a lens is the same or less than the shorter side of the plate it is used with, it is a wide-angle lens for that plate. Obviously it could be used on a smaller plate, and would not then include such a large field of view. A normal angle of view would be included by a lens of focal length about equal to the diagonal of the plate. Lenses of longer relative focal length are known as long-focus or narrow-angle lenses. It must be remembered that the resulting picture will be the same if it is taken with a wide-angle, medium-angle, or narrow-angle lens of the same focal length on the same plate. The wide-angle and medium-angle lenses can, however, be used with larger plates, and so have a more extended field of usefulness. As an example it may be mentioned that a 6 in. lens can be obtained which will sharply cover a whole plate at full aperture. Such a lens is, however, of suitable focal length for use on a 5 x 4 in. plate as a medium-angle lens. The medium-angle 6 in. lens recommended for a 5 x 4 in. plate will, however, not be suitable for use on a whole plate, because it will not illuminate it all over. The angle of the cone of rays transmitted by different lenses of the same focal length is not the same, owing to the different constructions and mountings employed. That transmitted by the 6 in. wide-angle lens is much greater than the angle of the cone of rays transmitted by the medium-angle lens. (It may here be mentioned that the perspective of a picture becomes more pleasing as the focal length of the lens increases.

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