The half power [BeamWidth] is the angular separation between the half power points on the antenna radiation pattern, where the gain is one half the maximum value. For a reflector antenna it may be expressed



HPBW = a = k l / D



where k is a factor that depends on the shape of the reflector and the method of illumination. For a typical antenna, k = 70° (1.22 if a is in radians). Thus the half power beamwidth decreases with decreasing wavelength and increasing diameter.



For example, in the case of the 2 meter antenna, the half power beamwidth at 6 GHz is approximately 1.75° . At 14 GHz, the half power beamwidth is approximately 0.75° . As an extreme example, the half power beamwidth of the Deep Space Network 64 meter antenna in Goldstone, California is only 0.04 ° at X-band (8.4 GHz).



The gain may be expressed directly in terms of the half power beamwidth by eliminating the factor D/l . Thus,



G = h (p k / a )2



Inserting the typical values h = 0.55 and k = 70° , one obtains



G = 27,000/ (a° )2



where a° is expressed in degrees. This is a well known engineering approximation for the gain (expressed as a numeric). It shows directly how the size of the beam automatically determines the gain. Although this relation was derived specifically for a reflector antenna with a circular beam, similar relations can be obtained for other antenna types and beam shapes. The value of the numerator will be somewhat different in each case.



For example, for a satellite antenna with a circular spot beam of diameter 1° , the gain is 27,000 or 44.3 dB. For a Ku-band downlink at 12 GHz, the required antenna diameter determined from either the gain or the half power beamwidth is 1.75 m.



A horn antenna would be used to provide full earth coverage from geostationary orbit, where the angular diameter of the earth is 17.4° . Thus, the required gain is 89.2 or 19.5 dB. Assuming an efficiency of 0.70, the horn diameter for a C-band downlink frequency of 4 GHz would be 27 cm. 



Taken From (http://www.aticourses.com/antennas_tutorial.htm ATI Space and Communications)