## Principal Planes

It was stated in the last chapter that for purposes of calculation a thick lens can be replaced by a single thin lens of negligible thickness. Now, no single point can be found at which such a lens could be situated, but it is possible to find two points at each of which the lens must be imagined to be situated. These points are shown at P, P', Fig. 13, and the planes through them at right angles to the Principal Axis are called Principal Planes. The distances u of the objects are measured from the point P, and the distances v of the images from the point P'. We must therefore imagine the thin lens as shifting instantaneously from P to P' as the light passes through it. The focal length is measured from the point P'. A simple and yet accurate method of determining the position of p' is given in the next paragraph.

Fig. 13.

## Determination Of The Focal Length Of A Thick Lens

It is always desirable to know the exact focal length of a lens. An idea finds currency that it would be sufficiently accurate to measure the distance of the focal plane from the diaphragm, DD'; but whilst this may do for the rectilinear type of lens, it is not sufficiently accurate for some anastigmats, and of course quite wrong for lenses like the Bis-Telar, in which p' is outside the lens. The Back Focus, which is the distance of the focal plane from the back surface of the lens, is equally inaccurate. The following method is one of the simplest yet suggested.

Focus the lens on a distant object, and note the position of any suitable point on the moving part of the baseboard with respect to any suitable point on the fixed part. Now sharply focus on a scale placed sufficiently near the camera to give a large image. The selected point on the moving part of the baseboard will have changed its position with respect to the point on the fixed part, and its change of position must be measured accurately. Suppose it has moved e inches. It is also necessary to measure the size of the image of the scale, and from this to determine the magnification or reduction as the case may be. Suppose the reduction is r. Then the Focal length = e/r.

The lens should now be focussed on a distant object, and the position of the point P', or rather the plane passing through it, which is e/r inches from the focussing screen, indicated by a mark on the lens mount.

## Measurement Of The Effective Aperture Of A Lens

All lenses are fitted with means for varying at will the amount of light passing through them. In a good lens this should take the form of the Iris Diaphragm, which will be familiar to all photographers. The effective aperture of such a diaphragm for a doublet cannot be determined by unscrewing the front combination, and accurately measuring the diameter of the opening in the diaphragm. The reason for this is shown in Fig. 13. In that figure the extreme rays parallel to the Principal Axis are shown as just clearing the diaphragm after refraction. Clearly the diameter of this cylinder of rays before refraction is the effective aperture of the diaphragm. Another ray has also been shown which just clears the diaphragm, and the intersection E of the original direction of this ray with the original direction of the extreme parallel ray is also shown. This point E can be regarded as the Virtual Image of the point e on the diaphragm formed by the front component of the lens. If the complete virtual image EE' is obtained, we get what is called the Entrance Pupil of the system. Similarly the virtual image of the diaphragm formed by the back combination is known as the Exit Pupil. When we measure the effective aperture of a lens we are really measuring the diameter of these pupils. To find the effective aperture the lens must first be sharply focussed on a distant object. The focussing screen is then removed, and replaced by a piece of card which prevents any light entering the camera except through a small perforated hole at its centre. The camera is now taken into a room in which the only source of illumination is a candle flame, and the flame is placed close to the perforation. A finely divided scale is placed touching the lens mount, and the diameter of the cylindrical beam of light passing through the lens measured. This value is then usually expressed in terms of the focal length f, e.g. if it is 2 inches, and the focal length is 8 inches, it would be f/8/2, i.e. f/4. The value so obtained should correspond with that indicated by the pointer on the diaphragm. A recent test on a cheap lens showed that the aperture marked f/8 was in reality f/16. This result is exceptionally bad.

## Depth Of Field

In addition to limiting the amount of light passing through the lens, the diaphragm performs the very important duty of controlling the definition. From the last paragraph it is evident that it is not the size of the actual aperture, but the size of the entrance pupil, etc., that we have to consider, so in Fig. 14 the pupils only are shown. The lens is supposed to be sharply focussed on the plane AB, and the image to be formed at ba. Now consider any other point, P, on any plane, e.g. on a plane more distant from the lens, as in Fig. 14. The cone of rays from this point will cut the plane AB in a circle XY, and this circle will be reproduced as a circle yx on the image plane ba. The circle on the image plane is called the Circle of Confusion due to the point P. Now it is agreed that circles of less diameter than 1/100 inch may be considered as points, so that if our circle on the image plane is less than this value, we shall have the distant plane containing P in focus at the same time as the plane AB. Obviously, however, there is a limit to the position of this plane on each side of the plane AB. The distance apart of these limiting planes, CD, EF, is called the Depth of Field for that aperture. By reducing the size of the aperture we increase the distance apart of these planes. The effect is shown in dotted lines in Fig. 14. The depth of field for a lens of known focal length can be very easily calculated for any aperture; but, before giving the formula, we must consider what is known as Hyperfocal Distance. From consideration of Fig. 8 it will be obvious that when sharp focus is secured on a very distant object, other objects much nearer to the lens are also in focus. The distance of the nearest plane to the lens which is in focus is called the Hyperfocal Distance, and is usually denoted by H. It can be shown that F being the focal length in inches, f the aperture (8, 11, 16, and so on).