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BELL SYSTEM TECHNICAL JOURNAL 



3.5 Pattern in Terms of Antenna Wave Front 



If the relative phase and amplitude in a wave front are given by 

 E{x, y) = a(x, y)e"''^''' 



(31) 



the relative phase and amplitude at a distant point Q not necessarily on the 

 Z axis (Fig. 10) in the important case where the angle QOZ between the 

 direction of propagation and the direction to the point is small, is given from 

 (17) by adding the contributions at Q due to all parts of the wave front. 

 This gives 



Xa Js 



dS. 



(32) 



Fig. 10 — Geometry of Pattern Analysis. 



The quantity r in (32) is the distance from any point P with coordinates .r, 

 y, 0, in the XY, plane to the point Q (Fig. 10). Simple trigonometry shows 

 that when OQ is very large 



r = d — X sin a — y sin ^ (33) 



where d is the distance OQ, a is the angle ZOQ' between OZ and OQ' the 

 projection of OQ on the XZ plane and /3 is similarly the angle ZOQ". The 

 substitution of 33 into 32 gives 



Eo = 



• -i(2WX)d ^] 

 *^ i ^ t(2ir/X)(T8ino+i/sin/3) + i 



\d 



f 



*(!,!/) 



a{x, y) dS. 



(34) 



