Model for Relating Coupled Power Equations to Coupled Amplitude Equations

Consider the coupled line equations: hz) = - Y M z ) +jc(z)h{z) Ix'(z) = jc{z)h(z) - YMz). (1) (2) These equations are useful in describing effects of coupling between a signal mode, represented by a complex wave amplitude Io(z), and a single spurious mode, represented by Ii(z), caused by geometric imperfections in a multimode transmission line. These equations may be derived in two ways from basic principles: 1,2,3,4 direct conversion of Maxwell's equations to generalized telegraphist's equations, or allowing discrete converters to become continuous. Exact solutions are known in only a few special cases, one of which is the case of constant c(z). Making use of this solution and assuming c(z) is a stationary Gaussian random function with a white noise spectrum S 0 and zero mean value, it is 2761 27G2 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1963 possible to derive coupled differential equations with the expected values of the power in h(z) and Ij(z) as independent parameters. T h a t is P0'(z) Pi(z) where P0(z) = (I0(z)I0*(z)). = = (--2ao SoP0(z) So)Po(z) + SoPi(z) + (-2a, SQ)PI(Z)