That expresses my problem exactly. Afaics, the other modes will just change the phase of the resultant throughput (or possibly, for a broadband input, produce extra peaks or notches.) As an RF man, I am often confused by the way Optical guys look at things. After all, it's exactly the same stuff!!Sorry, I thought you wanted to generalize the formalism in the paper you linked to on ring resonators for multiple transmission line modes. Ring resonators are passive linear devices which don’t alter the input frequency in any way whatsoever.
A simpler example is single versus multi mode fiber. If fed with a monochromatic source neither modifies the frequency of the light, ever.