IV-8 



From this equation, we would like to eliminate those terms which we can legiti- 

 mately regard as being of second-order importance even if ifi itself is not 

 necessarily small compared to unity. To this end, we shall make two approxi- 

 mations. 



1. Since the variations of the index of refraction are small, we may 

 certainly make the approximation: 



n^ - 1 = u (2 + u) ~ 2u (IV-27) 



If we make this approximation, we are dropping a term u^ k^ from (IV-26), and 

 we are, therefore, making explicitly the assumption: 



U<<2 (IV-28) 



It is clear that this approximation is always permissible if the inhomogeneities 

 are indeed weak. The approximation corresponds entirely to the one made in 

 (IV- 13) and does not restrict us to the small scattering approximation which we 

 wish to avoid. 



2. If we want to remove those terms from (IV-26) which would be of 

 second order if i|)i were small, we must also remove the square of the gradient 

 of i|ii , i.e. , (V \[(i )^ . This entails the assumption that the size of the gradient is 

 comparable in order of magnitude to the term just neglected above: 



(V 'i/^f ^ u^k^ (IV- 29) 



Because of (IV-28), this assumption is tantamount to the condition on the 

 gradient of >lfi: 



I V (l(il << 2k (IV-30) 



Let us examine the physical meaning of (IV-30). If the magnitude of the gradient 

 of the complex phase function is to be sufficiently small, both its real part and 

 its imaginary part must be small enough to satisfy (IV-30). In other words: 



A 2 TT 



i V tn — — I < < 2k, or since k = -r — , we require that 

 o 



X|v^n4— I <<4Tr (IV-31a) 



O 



1 V (S - S )1 << 2k (IV-31b) 



Arthur B.HittleJnt. 



S-7001-0307 



