484 BELL SYSTEM TECHNICAL JOURNAL 



the physical Hmitation that the illumination in any part of the beam 

 must be positive, that is, the illumination from one part of the beam 

 must always add to that from another part and cannot subtract from 

 it.^ This observation enables one to define the resolution of two 

 apertures of different shapes as being equal along a certain direction 

 when their transfer admittances in the useful frequency range show 

 the same filtering effect, if that direction is used as the direction of 

 scanning. This will occur when the radii of gyration (about a normal 

 axis in the plane of each aperture) are equal. According to this 

 definition all the apertures illustrated in Fig. 12 have the same resolu- 

 tion along a horizontal direction. 



When the picture is analyzed as a two-dimensional Fourier series 

 the equations which have been given above become 



Ei{x, y)=22 zZ Amn exp nr[ r -7- ) > 



Ei{x + k,y + ri) 



= E L ^^™exp«x — !--- - + -^^ ^ ' (18) 



Fiix, y)= f fr^a, v)E,{x -{- ^,y + v)d^drj, (19) 



•^ •^aperture 



Fi{x, y) = Z Z Y,(m, n)Amn exp tV f — -f ^ \ , (20) 

 where 



Yi(m, n)= f fr^i^, r?) exp ^tt ( ^ -f ^ ) d^dr,. (20') 



ty t^ aperture \ ^ '' / 



For an aperture symmetrical about both | and 77 axes 



Fi(m, n) = f fr.a, v) cos ^('^-\-^) d^drj. (20") 



*J ^'aperture \ ^ '^ ' 



^ The shape of the transfer admittance curve near n = depends upon the power 

 of n in the first variable term of the Taylor expansion for Y{n) about n = 0, and 

 upon the sign of this term. Assuming a symmetrical aperture, the expansion from 

 equation (17") is 



Y{n) = fTdi - ^ fe-Td^ + ^ f^Td^ . 



Since T is everywhere positive the first variable term is always in ti' and negative. 

 The shape of the curve near w = is, therefore, always a parabola (indicated in Fig. 

 12), which can be made the same parabola by suitably choosing the two disposable 

 constants in the aperture. Even after departing from this common parabola, the 

 curves maintain the same general shape over a substantial range; for the next variable 

 term is in n* and positive, and has the same order of magnitude for all usual types 

 of apertures. Consequently, the curves for these apertures have approximately 

 the same shape over a wide range extending uj) from n = 0. The results are the 

 same for an unsymmetrical aperture, but the reasoning is more involved. 



