IV -31 



For intermediate values of the correlation distance, i.e., 'f'^ond *"** ^ '^'^ ^> '■^^ 

 main contribution of the integral will come from the part of the spectrum in the 



2 TT ,, . 2rn 

 > 

 cond 



range — — <k<o so that; 



2 TT/>t _, 2nr/'t _, " 



r cond , /- cond . ^ r . ^-s /a 



B^(r)-2J dk(l-^^)k-^/^ = r^/^j d? (1 -^f'^-r^/M d? (1 -^)( 



2n/L 2TTr/L o 



(where ? = kr) (IV-70) 



For the Kolmogoroff law, the structure function behaves as the 2/3 power of the 

 distance of separation. Figure IV- 10 shows a number of measured structure 

 functions for the thermal fluctuations, which appear to substantiate the 2/3 power 

 law in the intermediate range. 



For purposes of calculation, and in the absence of better experimental 

 information, we shall make use of a fairly bold approximation to the spectrum of 

 the index of refraction as shown in Figure IV- 9, an approximation of the form: 



for k < e 



E^(k)-JE fore<k<kQ ^^^_^^^ 



k 5/3 



In other words, above some wave number kg we shall use the Kolmogoroff spectrum. 

 Usually there will be no need to introduce the conduction cutoff at very high fre- 

 quencies. For wave numbers below kg, we shall assume the spectrum to be flat, 

 down to some very small wave number e corresponding to the largest patch size to 

 be expected. The spectrum is assumed to vanish below e. We may usually take 

 the order of magnitude of e to be a factor of 10 smaller than kg. Based on the 

 spectrum (IV-71), we would, therefore, expect a mean square fluctuation of the 

 refractive index, see (IV- 63): 



/ 



<\f> = \ E(k)dk-'EQkQ + | E^k^ = 2.5 E^k^ (IV-72) 



We can, therefore, eliminate E^ from (IV-71) and write the spectrum strictly 

 in terms of the mean square fluctuation and the critical wave number k . 



artl)ur Jl.littlcJnt. 



5-7001-0307 



