1256 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



Here we have used the following definitions: 



Un(x, px) = Jn{px)Nn{x) - Nn{px)Jn(x), 



Vnix, px) = pX%J,:{pX)Nn'{x) - Nr:{pX)Jn'{x)], 



Wn{x, px) = px[J,Xx)N,/(px) - Jr.'(pX)Nnix)], 



Zn(x, px) = x[Jn'ix)Nn{pX) - J n{px) N n' (x)] . 



I 



(2) 



Jn and A^„ are cylinder functions of the first and second kind, respec- 

 tively, and the prime marks differentiation with respect to the argument. 

 The other symbols are: 



p = b/a ratio of radii, 



xi = ^itt , (3) 



X2 = ^20. . 



The relative dielectric constant of region 2 is indicated by e which is 

 assumed to be complex to take dielectric losses into account. The radial 

 propagation constants ^i and ^2 are related to the axial propagation 

 constant 7 and the free-space propagation constant k = 27r/X by 



(4) 



^f = k' + Y, 



$2' = ek" + 7'. 



The circumferential order of the solution is indicated by n. 

 For w = 0, equation (1) splits into the two equations 



J_ Jo'(p.ri) _e_ Wq{x2 , PX2) ^ Q /^N 



xi Jo{pXi) p.r2^ Uo{x2 , PX2) 



and 



j_ Jo'ipxi) J\_ Fo(-r2 , p.r2) ^ Q /gN 



Xi JoipXi) pX2' Zo{X2 , PX2) 



representing the TMo,„ and TEo,„ waves, respectively. 



Except for n = the solutions of (1) and the modes in the waveguide 

 do not have transverse character as in the case of a uniformly filled 

 waveguide. They are hybrid modes. However, it is reasonable to label 

 them as TE„,„ or TM„;„ , according to the limit which they approach as 

 the dielectric layer becomes very thin. 



Modes in the dielectric-coated guide with a very thin coat may be 

 treated as perturbed TE„;„ or TM„,„ modes of the ideal circular wave- 

 guide. The perturbation terms are found by expanding (1) for small 



