246 BELL S YSTEM TECH NIC A L JOURNA L 



aperture can always be expressed as a Fourier sum of this form this solution 

 can in princij)le always be found. 



Alternatively in section 3.10 the integral was evaluated as a sum of inte- 

 grals of the general type / x"g''"^</.v. Since d'(.v)e'* ^'^' for finite amplitudes 



in a finite aperture can always be expressed in terms of a power series, 

 this solution can also in principle always be found. 



3.12 xirrays 



When the aperture consists of an array of component or unit apertures 

 the evaluation of (36) must be made in part through a summation. When all 

 of the elementary apertures are ulike this summation can be reduced to the 

 determination of an 'Array Factor'. The pattern of the array is given by 

 multiplying the array factor by the pattern of a single unit. 



The pattern of an array of identical units spaced equally at distances some- 

 what less than a wavelength can be proved to be usually almost equivalent 

 to the pattern of a continuous wave front with the same average energy 

 density and phase in each region. 



3.13 Limitations to Antenna Wave Front Analysis 



Through the analysis of antenna characteristics by means of wave front 

 theory as based on equation (17) we have been able to demonstrate some of 

 the fundamental theoretical principles of antenna design. The use of this 

 simple approach is justified fully by its relative simplicity and by its applica- 

 bility to the majority of radar antennas. Nevertheless it cannot always be 

 used. It will certainly be inaccurate or inapplicable in the following cases: 



(1) When any dimension of the aperture is of the order of a wavelength 

 or smaller (as in many primary feeds). 



(2) Where large variations in the amplitude or phase in the aperture occur 

 in distances which are of the order of a wavelength or smaller (as in 

 dipole arrays). 



(3) Where the antenna to be considered does not act essentially through 

 the generation of a plane wave front (as in an end lire antenna or a 

 cosecant antenna). 



When the wave front analysis breaks down alternative satisfactory ap- 

 proaches based on Maxwell's equation are sometimes but not always fruit- 

 ful. Literature on more classical antenna theory is available in a variety of 

 sources. For much fundamental and relevant theoretical work the reader 

 is referred to Schelkunoff.'' 



" S. A. Schelkunoff, Loc. Cit. 



