DIELECTRIC-COATED WAVEGUIDE 1257 



values of 5 = 1 — p. This is done in Appendix 1. The perturbation of 

 the propagation constants for the various normal modes is: 



TM„. ^ = '-^ 8, (7) 



'Ynm € 



^■y ^ f 1 



-L-t!jnm ' ^ ;^ 5 ~-7Z TyT 0, (o) 



n^O Inm VnvC — W e{l — VunT) 



TEo,n ^ = ^ -, ^ 5'. (9) 



At ^ e - I Pom .3 



'Yom r Vom O 



In these equations pnm is the mth root of Jn{x) = for TMnm waves 

 and the mth root of J„'(x) = for TE„;„ waves. Furthermore, Vnm = 

 ^/Knm where Xc„m = 27ra/p„m . 



We note that the change in propagation constant is of first order in 

 5 for the TM„„i and TE„„j waves with n ?^ 0, but of third order in 5 for 

 TEo,„ waves. 



With e = e' — je" the perturbation of the propagation constant splits 

 into phase perturbation A/3 and dielectric attenuation ao . For a low 

 loss dielectric we may assume t" « e' and get : 



Pnm t 



^^^^ A^ ^ n"" t' - 1 ^ 



TE, 



'om 



Aj8 _ Pom e' - 1 



Pom O 1 Vom 



2 " ' 



TM ^ = — 5 



2 // 



TF ^ - ^ ^ ^ 



„j^0 Pnm Vnm — H" € "(1 — Vum') 



2 // 



T'T? Oio Pom C .a 



""^ /^ ^ F^ 2 • 



Pnm 'J -1- ''y/K 



Unfortunately the range in which (7), • • • (10) are good approxima- 

 tions is rather limited. Actually 



1 — Vnm 2Tra Al3 



Vnm, A Pnm 



has to be small compared to unity, at least not larger than 0.1. In the 



