NORMAL MODE BENDS FOR CIRCULAR ELECTRIC WAVES 1295 



7i,2 = j^i.2 + ai,2 = propagation constants of modes 1 and 2, respec- 

 tively, (the small perturbations of 71 and 72 

 caused by the coupling may be neglected here); 

 and 

 c{z) = coupling coefficient between modes 1 and 2. 

 In the curved waveguide the coupling coefficient is proportional to the 

 curvature k: 



ciz) = c'm ^ c~, (2) 



dz 



in which d is the direction of the guide axis. The coupled line equations 

 (1) with varying coupling coefficient have been solved by W. H. Louisell 

 and we shall borrow freely from his treatment. 

 We define local normal modes 1^1(5;) and Wi{z) : 



(3) 



in which 



El = [wi cos ^ ^ — If 2 sin f ^] e '^\ 

 Ei = [wi sin I ^ -\- w-i cos | ^] e^'^% 



7i + 72 

 7 = 7^ 



tan^ = j2—^- =i2-f . 

 72 - 7i A7 



Substituting (3) into (1), we find that ^1(2) and W2{z) must satisfy 



dwi ^, . 1 d^ 



dz 2 dz 



(4) 



where T{z) = f \/A7- — 4c-. In (4) the local normal modes are coupled 

 only through the terms poportional to d^/dz. When ^ is constant they are 

 uncoupled and true normal modes. For small values of d^/dz or more 

 specifically when 



2rdz 



« 1 (5) 



approximate solutions of (4) can be written do^^^l, which proceed essen- 

 tialh^ in powers of d^/dz. These solutions are: 



