metrical optics expression for the phase. The analogous 

 expression for the amplitude in geometrical optics is 

 obtained by keeping the second-order transverse deriva- 

 tives in V as well. ' 



Returning to the general expression [Eq. (27)], let 

 us first evaluate < lA'{x) l^>. For convenience, we shall 

 choose x to lie along the x axis, so that x = (x, 0, 0). We 

 evidently have 



<|X(x)| = > 



<hl\"^ 



'^^Ya- 



jiilx-yilyjlx-yjl 

 x4?V(yi-y2)exp[i9(>'i + |x-yi| - x)] 

 xexp[- 29(3)2+ Ix-ysl -x)l , (29) 



where we have introduced the correlation function 



Wp(yi-yz)^(v(f,)v(y^)) 



(30) 



We assume p to be independent of ^(yj +y2) for the case 

 of a homogeneous background. 



It is convenient in Eq. (29) to shift to relative and 

 center- of- mass coordinates. We define 



y = yi-y2. Y=i(yi+y2). (3i) 



Then, if we assume that p(y) cuts off for values of y 

 t L, where L«x, we may expand in y/Y. Thus Eq. 

 (29) becomes 



xp(y) exp\(iqy). [Y-(x- Y)] , (32) 



where rand x- r stand for unit vectors in the direction 

 of Y and x-Y, respectively, and we have written 

 |Y±iy I =y and lx-Y±iy| = Ix-YI in the geometrical 

 factors multiplying the exponentials. This approxima- 

 tion introduces a negligible error. 



The integral over rf^Y may now be evaluated by sta- 

 tionary phase. The stationary phase path is the straight 

 line joining to x, and the result is 



<|x(x)|2> = 9^jrfj,„p(j,„,0), 



(33) 



where y„ refers to the component y in the direction 

 parallel to x. 



Introducing the Fourier transform of the correlation 

 function 



p(k)=\ rf^yexp(-/k. y)p(y) , 



(34) 



permits us to rewrite Eq. (33) in the sometimes more 

 convenient form 



<|x(x)|^)=0|rf^K,p(o,ir, 



(35) 



where "i " refers to the directions perpendicular to x. 



Next let us turn to (x' ). We now have, instead of 

 Eq. (29), the expression 



<^x)'> = ^J 'i'yr] '^\jJfZf^l^JfZf^ P'-yi-y^'>^Mi<liyi+\^-fi\ -^)lexp[/9(_V2+|x-y2| - x)] 



(36) 



We again shift to the variables Y and y, and appeal to the vanishing of p(y) for yi. Lto justify expanding in y/Y and 

 y/\K-Yl. We obtain 



<^^''> = 4$f'^'^FTlW^^^f2.-,(y.|x-Y|-.)lJrf\p(v)exp[f(^^^flj:^.^^^ . (37) 



jefore, we may evaluate the integral over d^Y by stationary phase. This yields 



(38) 



where, again, "11" and "i" refer to directions parallel 

 and perpendicular to x. 



At this point it is convenient to express p(y„, f^) in 

 terms of its Fourier transform, as given by Eq. (34). 

 The integral over dy,, d^y^ can then be carried out, and 

 we finally obtain the relatively simple expression 



(X(xf) = -^{ d%p(0,K)Vds 

 4t } Jo 



yexp[i{kl/q)(s-x)s/x] . 



(39) 



Equations (35) and (39) constitute our central results. 

 They express the quantities of interest as integrals 

 along unperturbed ray paths (in this case straight lines) 

 of the Fourier transform of the correlation function 

 p(!c) times rather simple geometrical factors. As we 

 shall see later, entirely parallel expressions obtain in 



the more difficult case of an inhomogeneous background 

 medium. 



The expression for < I A'(x) I ^), Eq. (35), is precisely 

 the same result for this quantity obtained by using geo- 

 metrical optics to compute X{x) itself, and then calcu- 

 lating ( I X{x)\^) from this. [This is easily seen by re- 

 ferring back to Eq. (28). 1 In contrast, Eq. (39) is «o; 

 what one obtains for (X{x)^) from geometrical optics. 

 Geometrical optics for this quantity is recovered if one 

 expands the exponential in Eq. (39), a procedure that 

 evidently is valid only if 



Q ' 



x)s 



«1 



Since k^-l/L and s, x-s~x, this condition can be more 

 familiarly written 



216 



