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.
Parallel rays incident upon the lens a (Fig. 39) used alone converge towards the focus f at a definite distance from the lens a ; similarly, parallel rays incident upon b diverge as from the virtual focus f', f' being the same distance from the lens b as both have the same focal length.
It is evident that when f1 is made to coincide with f2, incident parallel rays will emerge parallel.
In the particular case we have chosen as a preliminary illustration, on account of its simplicity, as the focal lengths of the two lenses are identical, they must, as is here seen, be placed in contact; but in general the focal length of the compound lens f is infinite, when positive and negative lenses are separated by a distance equal to the difference of their focal lengths (f1 - f2).
Let us now separate these lenses gradually and observe what happens:
Parallel incident rays upon the positive lens a (Fig. 40) are rendered convergent and proceed to form a focus at f1 ; if they are intercepted by the negative lens b separated by an interval a from a they are rendered less convergent, and proceed to form a focus at f, as though they came from a' and not from a. The distance a' f is now the focal length of the compound lens, when the two lenses composing it are separated by the interval d.
Note. - In general a (see Fig. 40) represents the measurement of the entire separation between positive and negative lenses.
f\ - f2 is the distance the component lenses must be separated in order that the focal length of the compound lens shall be infinite. (In the present case as f1-f2=o, the lenses are in contact.) d represents the measurement of any increase in the separation greater than the difference between the focal lengths of the component lenses f1-f2. (In the present case, d=a because f1=f2 .) (See Notes.)
If we know f1f2, the focal lengths of the component lenses, a the entire separation, and d the interval as defined, we can find the focal length of the compound lens f, and its back focal length b f (see figure) from the following simple formulae :
F = f1 x f2/d ; ................. (3)
BF = f2 (f1 - a)/d;........ (4) which tell us that:
The focal length of the compound lens is found by multiplying the focal lengths of its two components together, and dividing by the interval (d) of separation greater than the difference of their focal lengths.
The back focal length, or distance from the negative lens to the screen, is formed by multiplying the focal length of the negative lens by the difference between the focal length of the positive lens and the entire separation, and dividing by the interval (d) of separation greater than the difference of their focal lengths : o
Suppose the lenses a and b, Fig. 40, are both of 6 inches focal length, we see that:
d = 0,
F = ∞
BF = 8
= 1/10 ",
= 12 ;
One of the most valuable practical features of this lens arises from the fact that the back focal length is shorter than the true focal length of the compound lens, as illustrated above. This is much more pronounced where the negative lens has a shorter focal length than the positive lens, as will be shown later.
(1) The lens has an infinite focal length when the component lenses are separated by a distance equal to the difference of their focal lengths.
(2) Its focal length decreases from infinity to diminishing finite focal lengths as we increase this separation, until, equal to this (see Fig. 40) giving its shortest focal length. In other words, until the interval d is equal to the focal length of the negative lens, or the negative lens has been separated from the position for emergent parallel rays (as in 1) by an interval equal to its focal length.
(3) When the lenses are separated by a distance equal to the focal length of the positive lens, the focal length of the compound lens is
"Photograph of the Eclipse of January 22, 1898. Taken at Sahrdol, India, by the Astronomer Royal. Instrument used, Thompson photographic refractor, 9 in. in diameter, full aperture, focal length 8 ft. 6 in. Image of the Sun in primary focus 1 in. in diameter. Enlarged in telescope by a Dallmeyer telephoto lens to 4 in. Exposure § second." (Notes from the Royal Observatory, Greenwich.)
(4) The back focal length is always shorter than the true focal length, except in the positions of its longest and shortest focal lengths; but neither of these two positions is of practical value. We cannot have a camera of infinite length, nor should we place the plate in contact with the negative lens.
Let us now take a general case, applying the information we have gathered from our simple particular case, in order to examine conditions that are not included in it.
Let the focal length of the negative lens f2 be shorter than that of the positive lens f1 If the lenses are placed in contact, it is easy to see that no real image can be formed, for the combination would then be equivalent to a lens of negative focal length. For lenses in contact, the following simple relation holds good:
1/F = 1/f1 - 1/f2 so that if f2 is less than f1, f must be a negative quantity. (See Notes.) The focal length of the compound lens continues to be negative on separating the lenses, Fig. 41 (1), until the separation a is equal to the difference of their focal lengths, or until we arrive at a separation when d=o; here the rays emerge parallel or the focal length is infinite (2), as we have seen.