592 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



Et and Ht may now be eliminated yielding two simultaneous second 

 order ecjuations for Ez and H, . These, in turn, may be combined to 

 produce two independent second order equations each of which is satisfied 

 by an appropriate linear combination of Ez and H^ . These equations 

 may be solved and Ez and Hz expressed as linear combinations of the 

 solutions. The transverse fields are then written in terms of E^ and Hz 

 and, finally, the boundary conditions are applied leaving a transcendental 

 equation in /3 . 



Operating on (10a) and (10c) withV- and taking account of (10b), 

 (lOd), one finds that 



j^V-Ht* = veUV -Et + jpeHz) = -iS^. , and 



jl3^-Et* = -VH{j'7-Ht+ JphEz) = m^ 



Operating on (10a) and (10c) withV*-, usingV*- V* = V^ and so on, 

 one obtains, using (lOb) and (lOd), 



V'^. + j^y-Ht = ve{-H, + PEV-Et), and 



(12) 

 V'Ez+Jl3V-Et = -ph(Ez + pnV-Ht). 



Now, elimination oi V -Et and V -Ht between (11) and (12) yields 

 V'Hz -\- pe (l - Pe^ - ^)Hz = jKpe + Ph)Ez , and 



; "r; (13) 



V'-E/. + VH il - Ph" - ^]E, = -Mpe + Ph)Hz, 



equations which demonstrate that pure TE or TM fields no longer exist, 

 as the result of the presence of p's. Hz or Ez might now be eliminated 

 between these equations giving a single equation in V" and ( V')", but it 

 is more convenient to find those linear combinations of Ez and Hz which 

 satisfy a first order equation in V . Writing such a linear combination as 



xP = Ez-h jKHz , (14) 



and adding jA times the first of equations (13) to the second, it is found 

 that this is an ecjuation in xp alone of the form 



VV + xV = 0, (15) 



provided that A is a root of the quadratic 



A - 1 = 0. (16) 



