932 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



^1 = -r [Kl + k)S + VKl + ky& - (1 + k)@ coth e + l], 



(209) 

 K2 = -^ [-UI + A-)© + Vi(l + AO^e^ - (1 + k)@ coth + l], 



where as usual 



= a,h = (1 + t)/i/5i ;^ Kih . (210) 



If /c = 0, equations (208) and (209) evidently reduce to (158) and (159) 

 of the preceding section. For a stack of infinitesimally thin layers, the 

 constants T( and K are given by equations (93) and (94) of Section 

 III, namely 





(w lie — co>oeo) 



(-2ik)"d/di, (211) 



K = Tr/g = {-2ik)^/g,h. (212) 



Up to this point we have set no restrictions on the magnitude of A", 

 and we have not even assumed that k is necessarily real. Throughout 

 the rest of this section, however, we shall assume that A; is a positive or 

 negative real number, as it must be if there is no dielectric or magnetic 

 dissipation. 



In practice both the lamina thickness and the amount of dielectric 

 mismatch will be as small as it is feasible to make them. It will be useful, 

 therefore, to obtain approximate expressions for r, Ki , and Ko under 

 the assumptions 



I I « 1, I A; 1 « 1. (213) 



Then equation (208) yields 



sh' ir = i(ch - 1) - id + A;)0 sh 



^ _fc 2 _ (1 + 2k) 4 _ (214) 



4 48 



If I A: I « 1 we can neglect 2A; compared to unity in the coefficient of 

 , but since we have made no assumptions as to the relative magni- 

 tudes of I I and I A- I, we cannot drop either the term in kQ" or the 

 term in 0*. If we replace sh |r by §r in (214), we get 



