Z in kn 



-16* 14' -12' -to* -B" -6' -4' -2' 0" 2* «• 6* 8* '0* '2' 



lower loop Q upper loop 



FIG. 4. Plots of fjiy), proportional to the loop contributions 

 towards (X^). See Fig. 3. 



F2 is less peaked at the apex than F^ and decreases 

 monotonically with apex depth (Fig. 4). 



Let us next turn to the quantity (A'^). From Eqs. (75) 

 and (91) we find 



(107) 



G,(i?) = 4 -^ I / rf.vsec'e«V,(A)^(ft;- ) . (108) 



(109) 



{X') = -((5C/C)l)(j-')q'BRG,{R) , , 

 where 



r 



Here we have defined 



and we recall from Eq. (76) that 



= {n/Bf (n/uaf l/qA„ . 

 Note that as ^-0, g{0,jj~l and thus Gj-Fi and hence 



An approximate analytic expression for g(0,jj is 



-. , ^ [ exp(- i0j%)Ei(i0{ji + i)) - a(ig)] 

 ^'^'^*'" log(4/,.l) • 



Thus for small 0, we have 

 ,2 



(110) 



and hence for small /3, 



7 1 i^ 11/"" 



G,(R) = F,(R)-^ ^ _2^_L^^ / rf^ 



xsec^e«'/i(A)|3| ••• . 



For very long ranges where many loops are involved, 

 we may simplify the integral in Eq. (108) as follows. 

 First replace by the many loop long range value 0^^: 



0„^ = {u/Bf (n/n^f [x(R - x)/qR] tan^e/S = a^^yix) , 



(111) 



(112) 



where 



r{x) = x(R-x)/^Rg 



Next note that y varies rather little over one loop. 

 Then the integral from to i? may be broken up into a 

 sum of integrals over each of the loops. In any given 

 loop, say the /^th one, y has very nearly the value y(xj) 

 = y^ , where x,, is the position of the midpoint of the 

 loop. Thus we may write, in place of Eq. (108), 





where A"* is the number of ™^'^ loops, and where 



Ftir,) = 5 



1 





1 rv"'/^ 

 :5 b dx 



sec^e„yi(^)g{0,,ij 



(113) 



L with 



0,-TrHn/nof{t!in^e/5)y„ . 



Here x^±R*/2 are the positions of the two ends of the 

 loop. We note that as >j- 0, /^'(y^)— F* as defined 

 earlier. 



The variation of Ffiy) with y is shown in Fig. 3. 

 Large deviations from Ff begin to become apparent 

 when y is of order one. The maximum y that occurs 

 over a range R is R/4I^q; thus Gj is not very different 

 from Fi until ranges of order 4]fq. 



All of the foregoing results and definitions may be 

 summarized in the following relation: 



■ ix^y 



-(|A1^> 



b(iAr>j 



M> <'■■"- 



Gi 

 jOi^noFz, 



(114) 



The quantities of actual interest to us are not quite 

 (X^) and ( IXI^), but rather the mean-square phase and 

 intensity fluctuations. These, we recall from Eqs. (4), 

 (6), and (9), are related to (X^> and < lA'l^) through 



<*') = i((|x|^)-Re(.Y^)) 

 and 



{L')-(L)'=2((\x\')^Re{x')). 

 Thus we find 



^*'^ \ = {{6C/C)l){f')<fBRn 



(115) 



(116) 



(Fi±ReGi) 



(117) 



226 



