Nonlinear propagation equations in fibers with multiple modes - Transitions between representation bases

The transverse pattern of the field that propagates in a fiber supporting multiple modes can always be described as a superposition of the patterns of the individual fiber modes. Yet, the use of other bases is often found to be more convenient, with the most famous example being that of linearly polarized modes in weakly guiding fibers. The nonlinear propagation equations contain coefficients that involve overlap integrals between the lateral profiles of multiple propagation modes. A fundamental question that has been raised in this context is whether it is legitimate to compute these coefficients from the overlap integrals between elements of alternative bases for the field representation. In this paper, we show that the answer to this question is positive in the most general sense. This result is significant in the context of space-division multiplexed transmission in multi-mode and multi-core fibers.