CIRCULAR ELECTRIC WAVE IN SERPENTINE BENDS 1281 



mode. For low attenuation in the coupled mode the power transfer may- 

 even be essentially complete. 



II. SOLUTION OF THE COUPLED LINE EQUATIONS 



In a curved round waveguide the TEoi mode couples to the TMn 

 mode and the infinite set of TEi„ waves. Consequently, the wave propa- 

 gation is described by an infinite system of simultaneous first order 

 linear differential equations. An adequate procedure is to consider only 

 couphng between TEni and one of the spurious modes at a time. Thus, 

 the infinite system of equations reduces to the well known coupled line 

 equations, 



"^^^ + 71^1 - kE^ = 0, 



dz 



dE2 



dz 



(1) 



+ 72^2 - kEi = 0; 



in which 



Ei,i{z) = wave amplitudes in mode 1 (here always TEoi) and mode 

 2 (TMii or one of the TEi„), respectively; 

 7i,2 = propagation constant of modes 1 and 2, respectively, 

 (The small perturbation of y-i , 72 caused by the coupling 

 may be neglected here) ; and 

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



In the curved waveguide the coupling is proportional to the curvature; 



Co dd . . 



in which R = radius of curvature, and Q = direction of guide axis. (The 

 various coupling coefficients are listed in the appendix.) Without loss of 

 generality we start with 9(0) = 0. We will use the average propagation 



constant 7 , 



7 = 1(71 + 72), 



(3) 

 A7 = hill - 72). 



Several coordinate transformations will change (1) to a form which can 

 be solved approximately. A similar procedure has been used to solve the 



