760 cox [CHAP. 22 



If the radiation is isotropic (^93 = 77), then 



R exp(^■0) = jQ{kx). 

 Even in this extreme case (Fig. 27) the coherence is large for kx<^\.2. For 



ny 



(0,0) 



•(^-y) 



Fig. 26. Waves within the fan- 

 shaped beam of half angle 

 A(p incident upon observ- 

 ing positions at 0,0 and 



1.0 



0.5 

 



0.25 



CYCLES PER MINUTE 



Fig. 27. Coherences i?2i, Rz2 ^.nd 

 i?3l from Fig. 25 compared 

 with values calculated for 

 waves propagated in a fan- 

 shaped beam with half angle 

 A(p as indicated. When half 

 angle is tt, the radiation is 

 isotropic. 



kx>\.2 the coherence oscillates with decreasing amplitude. The opposite 

 extreme occurs for a narrow pencil beam of radiation. When A(p<^ 1 then 



i? exp(i^) = {Acpy^ eyi^Y — ixk{\ — \(p^)'\coQ{ykq))dcp-\-0{q)^) 

 Jo 



= {Arpy'^A eyi^{ — ixk), 



where 



A = eji^{\ixkcp'^) cos {ykcp) dcp 



jo 



can be evaluated in terms of Fresnel integrals. 



Observed values are shown in Fig. 27 compared to computed values which 

 take into account the reduction in coherence associated with finite angular 



