IV-36 



To obtain (IV-74), we multiply pi , as given in (IV-45), by its complex conjugate 

 and average over the ensemble of different micro- structures, making use of the 

 correlation function of the refractive index (IV-53). We may simplify (IV-74) by 

 introducing the difference coordinates ? = ? ^^^'| . since the integrand depends 

 only on these coordinates. Let us retain ^T") as the other coordinate of integration. 

 The only problem is then to determine the volume of integration for the £ coordinate. 

 We realize, however, that because the volume V is much larger than the 

 "correlation volume" of the correlation function R^J, the integration over £ may 

 just as well be taken over all of space. Making these substitutions in (IV-74), we 

 obtain: 



P^' JiW^ 



(|a.i(0) 



d.? e"'^-*-<u^>R^(i) (IV-75) 



The first integral in (IV-75) is seen to be the volume of the scattering region V . 

 The second integral is the spectrum of the refractive index refined in (IV- 57b). 

 We therefore obtain for the scattered power the following very simple relation 

 involving the spatial spectrum of the fluctuations of the refractive index: 



<lpil'>= ^ \f ^ S^(kd) (IV-76) 



The scattered power is seen to be proportional to the volume of the scattering 

 region; thus, the power scattered by two adjacent volumes must be added to give 

 the power scattered by the two volumes combined. For the far field, therefore, 

 the power scattered from two different volumes, each large compared to an individ- 

 ual inhomogeneity, is uncorrelated, as asserted in Chapter II. 



When the scattering volume is isotropic, we may use the isotropic spec- 

 trum E^, as defined in (IV- 60). Since the magnitude of the vector d is 2 sin 9/2, 

 the angular distribution of the scattered power is given by: 



4 E (2k sin 9/2) 



'IP'l=^ =y^ (L Sin e/2)- "^-"> 



For practical purposes, we can select the spectrum corresponding to the exponential 

 Gaussian, or Kolmogoroff correlation function from Table IV- 2 and substitute in 

 (IV-77). We observe that the directivity of the scattered power in the far field 

 scattered by a volume with a statistically distributed index of refraction is exactly 

 the same as the directivity resulting from scattering by a single inhomogeneity with 

 a determinate distribution of the refractive index equal to the correlation function. 



Arthur H.ILittleJnr. 



S-7001-0307 



