GUIDED-WAVE PROPAGATION TllUOl (ill O YUO.MAGXKTIC MKDIA 021 



crosses all contours of G{T(\i), a), in particular G(T{\i), a) = g. All 

 the additional solution curves arising in this way start at c = 1, Xi = 1; 

 the ?i"' of them threads its way from one blank region to another, first 

 through the intersection of /„+i with (Ih')t , then through the intersec- 

 tion of 0„ with {Oi)t , and finally comes to an end at the intersection of 

 /„ with (Ia')t ■ At the end point (a- = 1, Xi = 1), X2 and, therefore, /S" are 

 infinite, (just as for the TEn solution curve). At the end point (/„ , 

 (7.4') 7), Xo , and, therefore, (3' are zero. The a and Xi values corresponding 

 to the latter are obtained from the equations 



l-J^rn' ^ J? . ^^^ = a^ + p. (40) 



1 — ffnXln ro" 



It is possible to derive the slope 3(3' /8a of the (f — a curves at these cut- 

 off points. Near cut-off, the infinity /„ of G{Xi , a) is matched by the 

 infinity /./ of G(Xo , a). The G'-equation therefore degenerates to 



Fin) = 



Xo , /l — Xi^ . Xl„ jn 



Writing a = an — .rXo , Xi = Xi„ — //X2 , expansion of the right hand 

 side of this equation to order I/X2 furnishes one relation between x and ij\ 

 the Polder equation furnishes another. The two can be solved for .r, and 

 so, since to first order 



2 x^ -.0" — an \ ^^ 



6/3 — —AlvA2 ~ Xin = Xi — 



jO jo 



d(3^/8a may be found. It is found that for convenience in computation, 

 the results of this calculation are best presented parametrically. Ecjua- 

 tions (46-8) represent equations (40) and 8l3'/8a in this way. Fig. 12 

 (a) and (b) show the result of some computations. Near a = 1, /3" = 

 CO , these added solution curves behave rather like the TEn curve. The 



2 1 . 



leading term in the expansion of jS in powers of _ ^ is now 



K-^-^1 



<T - 1 



for the solution curve ending at /„ — {Ia')t ■ Here Zn+i is the (?i -f 1)*' 

 root of F(z) = 1, not counting 0. 



It will turn out later that the infinity of solution curves just discussed 

 represents an incipient form of the whole mode spectrum; the reservoir 



