This section is from the book "Telephotography: An Elementary Treatise On The Construction And Application Of The Telephotographic Lens", by Thomas Rudolphus Dallmeyer. Also available from Amazon: Telephotography and Telephotographic Lens.
In general, if we take i as the limit of indistinctness permissible in the image, and we reproduce the object in the proportion of n : i, the circle of indistinctness for any point in the field of the object projected upon the object plane for which we have focused would be n times z, or n i. Referring to Fig. 52, the circles a' and a" projected on to the chief plane of the object from the points p p' are, of course, n times as great as the circles of indistinctness to which they correspond in the image. Using the same notation as before, calling d the distance between the chief object plane and the " Entrance Pupil," b the distance of the point p in front of this plane, and c the distance of the point p' behind it, a the diameter of the " Entrance Pupil," and i the limit of indistinctness in the image, b =dni/a + ni, c = dni/a - ni b is here the front depth of field, and c is the back depth of field, the whole depth being b + c = 2dnia/a2 - n2i2.
The size of the diverging circles a' and a" is not only dependent upon the diameter of the " Entrance Pupil," but also on its separation from the chief plane for which we have focused.
Fig. 54 clearly shows that these circles decrease when the separation d increases. To put this more clearly, we see from the former figure that a1 = b a/(d - b); now if we increase d by h the formula becomes a1 = ba/(d + h - b)
If d is very great compared with d - b, or, in other words, if the point p and the principal object plane are both widely separated from the "Entrance Pupil," the effect of further removing the "Entrance Pupil" becomes indifferent. Thus we see the advantage gained by the Telephotographic lens in producing sharper or better defined images of near objects, giving the effect of greater depth of focus, and that the advantage ceases for very distant objects. The reason for this greater depth of focus in the Telephotographic lens lies in the smaller angles subtended by the "Entrance Pupil" from points in the object, as its distance is greater than for a lens of ordinary construction. Supposing the angle subtended is 2 u, then tan u = a/ 2(d + h)
"A photograph taken with Dallmeyer's telephoto lens of highest amplification during war at a distance 01 upwards of two miles. The negative by an official photographer of the Japanese War Department. The vessel is the Tei-yen, one of the two largest in the Chinese Navy, 7300 tons displacement, coated with steel 14 in. thick. Vessel half sunk, having been damaged by torpedoes. The effect of the cannon shot is noticeable in the white irregular lines just above the water mark." (Y, Isawa, Editor of the " Shaskin Soma," Japan,)
In conclusion we find that a Telephotographic lens of the same intensity as an ordinary positive lens gives greater depth of focus than the latter when used to photograph near objects, although this advantage gradually diminishes when the distance of the object increases, and is finally lost when the distance of the object becomes very great. It must be remembered, however, that the greater separation of the "Entrance Pupil" of the Telephotographic lens from the object, in circumstances where this advantage occurs, will necessitate an increase of exposure depending upon the "law of inverse squares" as already pointed out.
Depth of Focus (usual interpretation). - Our method of treating the Telephotographic lens, under the heading B, enables us again to simplify the question of depth of focus applied to the construction. If the positive element of the system be considered alone, we may apply the well-known formulae for front and back "depth of field" in the usual manner, and if the result is to be regarded as a pictorial representation, we may neglect the effect produced by the magnification of the admissible circles of indistinctness which have been formed by the positive lens alone, because the more we magnify the primary image, the greater will be the distance from which the picture may be expected to be viewed. (This of course actually occurs in an ordinary " enlargement " from a photographic negative.) In selecting the circle of confusion of one-hundredth of an inch as sufficiently small when viewed at the normal distance of vision to appear practically sharp, we are guided by the fact that at a distance of about 12 inches this measurement subtends an angle of only three seconds of a degree. It is obvious that if we view the result obtained by the Telephotographic lens when a certain magnification has taken place, at a correspondingly greater distance than that at which we should view the result made by the positive lens alone, the increased circles of indistinctness will still only subtend the same angle at the eye, and therefore will appear equally well defined. If, on the other hand, we require the final image produced by the Telephotographic lens to have a definite degree of sharpness which shall not exceed one-hundredth of an inch, it will only be necessary to substitute the numeral m 100 (representing m times 100) for 100 in the formulae, wherever 100 occurs, in order to attain the same limit in the final image, whatever m may be ; M as before being the magnification.
For the sake of uniformity and greater simplicity we here give formulae for depth of focus which have reference to the relation between the size of object and image (or their distances from the focal planes of a positive lens), 1/n as before being " the magnification " of the image.
Calling f the focal length of the positive element, a its effective aperture, I = a/f its intensity, (n + 1) f the distance of the object from the lens, the distance beyond which all objects will be sufficiently well defined