THE OPTICS OF PHOTOGRAPHIC LENSES 25 



Hence the whole depth of acceptably sharp field would be from 8 ft. 8 in. to 11 ft. 

 10 in. 



By the approximate formula (12) we have 



Ri = Ri = 469 mm. = 18.5 in. = 1 ft. 6 in. 



giving a range from 8 ft. 6 in. to 11 ft. 6 in. 



Depth of Focus, Including Enlargement of the Print. — It can easily be shown that if 

 we photograph the same object with two lenses of different focal lengths and diameters, 

 if we subsequently enlarge the smaller picture to make it the same size as the larger 

 picture, and if we insist on equally sharp definition in the two final equal-sized pictures, 

 then the depths of focus of the two cameras will be proportional solely to the diameters 

 of the two lenses. Thus an //2 lens of 2-in. focus and an //4 lens of 4-in. focus both 

 have a diameter of 1 in. The 2-in. lens forms a picture half as large as the 4-in. lens, 

 but after enlargement to make them equal in size, the depth of focus of each will turn 

 out to be the same. This property constitutes the real advantage of the miniature 

 camera, in that it permits the use of a fast lens without loss of depth of focus. 



The Hyperfocal Distance. — In a fixed-focus hand camera, it is desirable to choose 

 the focused plane so that the extreme end of the beyond-focus depth just reaches 

 infinity. In this case, we write i^i = oo, whence md = c from Eq. (10). Now since 

 m = f/x [Eq. (8), page 16], our focused distance in this case will be given by a; = fd/c. 

 This is called the "hyperfocal distance." It should be noted that in a camera cor- 

 rectly focused for this distance, the within-focus depth just reaches x/2. As an 

 example, consider a camera lens of 100-mm. focal length and aperture //8. The 

 diameter of the pupil is d = '^^% = 12.5 mm., and the hyperfocal distance is given by 

 X = fd/c = (100 X 12.5)/0.25 = 5.0 m. (16 ft.), assuming the permissible circle of 

 confusion on the plate corresponds to c = 0.25 mm. The range of object distances 

 sensibly in focus then runs from » up to 2.5 m. (8 ft.). 



The Resolving Power of a Lens. — ^If we follow through all the implications of the 

 Huygens wave theory of light, we find that the image of a point source formed by a 

 perfect lens is not a true point, but a small disk of light surrounded by a series of very 

 faint rings of light, called an "Airy disk." The practical diameter of the central 

 circular patch is found to be 2\f/d, where X is the wavelength of the light used (approx- 

 imately 0.0005 mm.), /is the image distance from the lens, and d is the clear diameter 

 of the lens. Hence two close point sources will be just "resolved" if their separation 

 is equal to X//d or \A, where A is the aperture number of the lens. This quantity 

 ■ is so small that it scarcely ever enters into photographic problems, for even at //16, 

 as might be used for copying work, the least resolvable separation of two adjacent star 

 images is 16X = 0.008 mm., while the grain of even a process plate is at least twice as 

 large as that and with ordinary plates it may reach ten or twenty times as large. 



The Pinhole Camera. — A type of camera which should not be despised is the 

 common pinhole camera, which is simply an ordinary camera having a pinhole in 

 place of a lens. The size of the pinhole is of considerable importance, for if it is too 

 large the picture will be blurred owing to the spreading of the cones of light from the 

 various object points as they pass through the hole, but on the other hand if the pin- 

 hole is too small the light waves will spread out owing to diffraction effects, again 

 causing a blurred picture. There is thus an optimum size of hole to be used with 

 any given length of camera. It can be shown that the image of a single object point 

 will be as small as possible, as a result of interference effects between light waves from 

 the different parts of the hole, if the diameter of the hole A is given by the following 

 formula : 



A^ = 0.00007/ (13) 



