Approximate Solutions for the Coupled Line Equations

Consider the coupled line equations: h'{z) //(«) = = - Yoh(z) +jc(z)I0(z) + jc{z)h{z), AAO). (1) These equations are of interest in many applications. Our particular interest in them in a companion paper 1 is that they describe the effects 1011 1012 T H E BELL SYSTEM TECHNICAL J O U R N A L , MAY 19(52 of coupling between the TE 0 i signal mode, represented by the complex wave amplitude / 0 , and a single spurious mode, represented by the complex wave amplitude 11, caused by geometric imperfections in circular waveguide. We have, of course, assumed that only a single spurious mode has significant magnitude, so that all other spurious modes may be neglected. For example, we may consider copper waveguide with a rather general straightness deviation; the most important spurious mode under many conditions will be the forward TE^ 1 (both polarizations must, of course, be considered unless the straightness deviation is confined to a single plane). However, these equations apply to a variety of other problems which may be described by only two modes with varying degrees of accuracy. 1 In copper waveguide if the wall losses may be neglected the propagation constants T 0 and Ti are pure imaginary and the coupling coefficient c(z) is pure real. In helix guide, where loss is added to the spurious mode, the propagation constant Ti has a significant negative real part; further, as shown by H. G. Unger, 2 the coupling coefficient c(z) also becomes complex. The case where the geometric imperfection (e.g., straightness deviation) and hence the coupling coefficient is a stationary random process, perhaps Gaussian, is of great interest; here it is desired to compute the statistics of the TE 0 i transmission / 0 in terms of the statistics of the coupling coefficient c(z).