THE OPTICS OF PHOTOGRAPHIC LENSES 19 



It should be noted especially that this expression for E does not depend in any 

 way on the distance of the object or on the slope d of the entering ray but only on the 

 intrinsic brightness of the object and the slope of the emerging ray. This is because 

 there is a compensation here between the light-gathering power of the lens and the 

 magnification. Suppose for example the distance of the object shown in Fig. 15 

 from the first focal point of the lens were reduced to half. The angle 6 would be 

 doubled, and the amount of light entering the lens from each little element of the 

 object would become four times as great. But the image would now be twice as 

 large as it was, and hence it would have four times its former area. Thus four times 

 as much light would be spread over an image four times as large, and the resulting 

 illumination on the plate would be unchanged. 



Relation between Exposure and Aperture Ratio. — The relation between the 

 aperture ratio of a lens and the illumination in the image can be deduced from the 

 consideration that in a perfect lens the equivalent refracting surface ("principal 



plane") is a sphere centered about the focal point / ^ \ 



(Fig. 16). Hence, if h is the height of the incident X—l- ^ ^^^\ 



raj'^ above the lens axis, sin 0' is approximately equal h 7 /^ "-7- Ik 



to (h/f). Now the diameter of the entering beam i — j '^"~ --^fc^^F2 



(the entrance-pupil diameter) is equal to 2h, hence I ~Y% Tj J 



the "aperture number" or ratio of the focal length \ \ / 



to the diameter of the entrance pupil is equal to \ ^ / 



f/2h = 1/(2 sin 6'). If this number is represented \ i^—/ 



by A, e.g., A = 4.5 for an //4.5 lens, we see that Fio- 16. — The aperture ratio of a 

 sin d' = 1/2A, and hence the image illumination is perfect lens, 



given hy E — kirB/^A^ accurately for all apertures up to the very largest. Thus we 

 reach the familiar result that the required exposure is proportional to the brightness 

 of the object and inversely proportional to the square of the /-number and is inde- 

 pendent of the distance of the object. 



It is also interesting to see that the greatest possible aperture ratio ^ is f/0.5, for 

 at this value d' = 90° and the extreme ray would just graze the plate. Even this 

 ratio is, strictly speaking, unattainable, for there must be some space between the 

 back of the lens and the image plane. 



If the bellows of a camera is extended to focus a near object, then the value of 



/>/u J---1.J- ^- .X, _i- /original-image distanceX , , 



d becomes dimmished in proportion to the ratio I — - — ■■ ,. ^ ■ ), and the 



\ new-image distance / 



exposure required must be divided by the square of this ratio. Thus in changing 



from a distant object to equal conjugate distances (unit magnification), the aperture 



numbers must all be doubled, and the exposure made four times as great. If the 



magnification actually used is m, all marked /-numbers should be multiplied by 



(1 -|- m), and exposures by (1 -1- 7n)^. 



The effect of a change of bellows length on exposure is verj^ small except when the 

 object is quite close to the lens, as may be seen from Table III. 



The "Uniform Scale" (U. S.) sj^stem of designating the stops in a photographic 

 lens is based on the area of the iris opening rather than its diameter. When it was 

 introduced, //4 was felt to be the limit of large apertures and was called "U. S. 1." 

 Then the other apertures fell as shown in Table IV. This system is now practically 

 obsolete. 



1 Bracey has designed a lens for astronomical purposes consisting of a reversed oil-immersion micro- 

 scope objective, the photographic plate being attached by a layer of oil to the back (plane) surface of 

 the lens. The aperture of this is given as //0.36, such a speed being possible since for an immersion 

 lens of this type, aperture number is defined by f/27ih where /i is the index of the oil, say 1.52. Bracey, 

 Astrophys. J., 83, 179 (1936). 



