t 



1282 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



related problem of the tapered mode coupler/ With the first transforma- 

 tion 



E^{z) = he"'' [u,(z)e'''' - U2{z)e~''''], 

 the new coordinates satisfy the equations: 



-— - + Aye " ii2 = 0, 

 dz 



— — + A7e ° Ui = 0. 

 dz 



With the second transformation 



ui(z) = i[v,{z) e-^'' + V2{z) e-^^^1, 

 U2(z) = i[v^(z) e-^'' - V2{z) e+^n, 



Vi{z) and ^2(2) satisfy 



-— ^ — 2A7 sin Codvi + jAy sin 2co6e^ '^'V2 = 0, 

 dz 



—- + 2A7 sin' Co^t'2 — 7A7 sin 2co0e Vi = 0. 

 dz 



With the third transformation 



Viiz) = Wi{z) exp f 2A7 / sin" CoS dz j , 

 Viiz) = iVo(z) exp f — 2A7 / sin" Cod dz' j . 



We finally get the equations 



dwi 

 dz 



dw2 

 dz 



in which 



+ ^1(2)^^2 = 0, 

 + ^2(2)^1 = 0, 



(4) 



(5) 



(6) 



(7) 



(8) 



(9) 



