RADAR ANTENNAS 239 



In most practical cases this equation can be simplified by the assumptions 



cf>(x,y) = <t>'{x) + ct>"iy) 



a{x,y) = a'ix)a''{y) 

 from which it follows that 



I £q I = Fid)Fia)F(fi) (35) 



where F{d) is an amplitude factor which does not depend on angle, 



F{a) = j e*'^-'^''^"'""+'*'^^^a'(x)^x- (36) 



is a directional factor which depends only on the angle a and not on the angle 

 (8 or d, and F(/3) similarly depends on /3 but not on a or d. The pattern of 

 an antenna can be calculated with the help of the simple integrals as in 36, 

 and illustrations of such calculations will be given in the following sections. 



3.6 Pattern of an Ideal Rectangular Antenna 



Let the wave front be that of an ideal rectangular antenna of dimensions 

 a, b ; with linear polarization and uniform phase and amplitude. The dimen- 

 sions a and b can be placed parallel to the .Y and F axes respectively as 

 sketched in Fig. 9. Equation 36 then gives 



F{a) = r'\'''-'^''^'''" dx = a'-^ (37) 



J-al2 W 



, , X a sin a 

 where ^ = . 



Similarly 



F^0)=b'^ (38) 



where i/' = 



, _ TT 6 sin /3 



The pattern of the ideal rectangular aperture, in other words the distribution 

 of electrical field in angle is thus given approximately by 



F(a)F(ff) = ai'^'^ . (39) 



The function is plotted in Fig. 11. It is perhaps the most useful 



function of antenna theory, not because ideal antennas as defined above are 

 particularly desirable in practice but because they provide a simple stand- 



