THE DEVELOPMENT OF THE PHOTOGRAPHIC OBJECTIVE 



39 



well-known Rapid Rectilinear or Aplanat lens (Fig. 6), covering a field of 45° at //8, 

 and giving excellent definition at the center of the picture because of the good 

 spherical correction. 



The Petzval Theorem.— Actually, the astigmatism in the Rapid Rectilinear lens 

 was deliberately not reduced to zero because that would give rather a badly curved 

 field. There is a simple theorem named after Petzval, which states that the radius of 

 curvature of the central part of the astigmatism-free field of a lens is given by p in 



1 _ ^(n' — n\ 

 p -^V nn' r } 



(2) 



where r is the radius of curvature of a surface in the lens separating materials of 

 refractive index n and n', the summation to be made for all the refracting surfaces in 



PS T T SP 



PS T 



(a) (b) Cc) (d) (e) 



P+z.= + + o o + 



Ast. = - + - o O 



Fig. 7. — The Petzval surface and astigmatism curves. 



the lens system. This sum is independent of the object distance, the thicknesses and 

 airspaces in the system, and the stop position. It is therefore a very inflexible quan- 

 tity which it is hard to vary or control. The surface whose radius p is given by Eq. (2) 

 is called the "Petzval surface" and represents the shape of the field if astigmatism 

 is corrected. 



If astigmatism is present, however, it is found that the longitudinal distances from 

 this Petzval surface to the radial (sagittal) and tangential (meridional) focal lines, 

 respectively, are in a ratio of 1 : 3, as indicated in the various cases illustrated in Fig. 7. 

 In case a of this figure, it is clear that the introduction of a little negative (over- 

 corrected) astigmatism has flattened the effective field, as compared with case e 

 in which the astigmatism is zero. The ideal case is, of course, 

 zero Petzval sum and zero astigmatism; this condition is 

 realized approximately in the modern "anastigmat" lenses 

 (case d). 



Attempts to Reduce the Petzval Sum. — It soon became 

 apparent that a flat field free from astigmatism could only be 

 obtained if the Petzval sum were drastically reduced in 

 magnitude. This could be done in three different ways: (1) A 

 single lens could be made to have a low sum by giving it a meniscus form with equal 

 outside radii and considerable thickness. This shape appears commonly in manj^ 

 types of anastigmat, reaching its limit in the nonachromatic Hypergon (Fig. 8) which is 

 designed to cover a field of 140° at//22. (2) In an achromatic lens, if the crown and 

 flint components are separated by a fuiite distance, the flint must be strengthened to 

 compensate for its smaller effective diameter, and this will at once reduce the Petzval 

 sum. (3) To fulfill the Petzval sum and also the achromatic condition in a reason- 



FiG. 8. — Hypergon 

 wide-angle lens. 



