22 BELL SYSTEM TECHNICAL JOURNAL 



The differential equations 17, 18 • • • for the aberration coefficients 

 are linear and subject to solution in the usual manner when A and v 

 are known functions of z. The solutions for the higher coefficients 

 would of course be quite involved. The higher terms are, however, 

 small compared to the second term, which causes most of the aberra- 

 tion, and the approximate distortion is given by the second term alone. 

 This term is called the second order aberration term. 



The Reduction of A herration 



The coefficient of any aberration term vanishes when conditions 

 are arranged so that the last member of its differential equation is zero, 

 for the coefficient may be arbitrarily set equal to zero at the first side 

 of the lens, and the solution of its linear equation is then zero through- 

 out the lens. 



The important second order aberration term can thus be made to 

 vanish by arranging conditions so that 



V"" {A'Y 



16 



= 0. (75) 



In a lens that is not too thick compared to the focal distances involved, 

 we have seen that the term A"^ may be neglected in the differential 

 equation for A , and 



The substitution of this value for A' in the above equation gives 



(v"r 



V — 



= (77) 



as the differential equation for electric fields that are approximately 

 free from second order aberration, when the focal distances are reason- 

 ably large compared to the length of the field along the axis. 



The general solution of this equation is a series solution, but several 

 particular solutions have been obtained in terms of known functions. 

 The potentials corresponding to these particular solutions are given 

 by the following equations : 



$ = ae±"Vo(aj/'), (78) 



$ = (a sin co2 -f 6 cos o)z)Jo{io:r), (79) 



$ = (a sinh u)Z -{- b cosh cos)7o(wr), (80) 



^ = 3az'''\ 4-4(7^V + i^ ' ^^ 



[3 4 



2z ' 64\2z/ 



(81) 



