252 



BELL SYSTEM TECHNICAL JOURNAL 



phase is concerned, that is if the radiated energy has the same phase for all 

 points which are the same distance from a given point, then the reflector 

 should be parabolic. This can be proved by simple geometrical means. 

 In Fig. 18 let the point source .V coincide with the point .v = /", y = 

 of a coordinate system and let the uniphase wave front coincide with the 

 line X — f. Let us assume that one point of the reflector is at the origin. 

 Then it can be shown that any other point of the reflector must lie on the 

 curve 



A'2 = Afx 



20 40 60 80 



4>= MAXIMUM PHASE VARIATION IN DEGREES 

 Fig. 17 — Loss due to IMiase Variation in Antenna Wave Front. 



This is a parabola with focus at/, o and focal length/. 



The derivation based on Fig. 18 is two dimensional and therefore in 

 principle applies as it stands only to line source antennas employing para- 

 bolic cylinders bounded by parallel conducting planes (Fig. 24 and 25). If 

 Fig. 18 is rotated about the X axis the parabola generates a paraboloid of 

 revolution (Fig. 3). This paraboloid focusses energy spreading spherically 

 from the point source at .5 in such a way that a uniphase wave front over a 

 plane area is produced. Alternatively Fig. 18 can be translated in the Z 

 direction perjiendicular to the XY plane. The parabola then generates a 



