Determine whether smoothness generically causes low-rank four-wave-mixing tensors

Determine whether the low-rank nature of the four-wave-mixing tensor in multimode nonlinear waveguides generally follows from smoothness of the underlying index profile rather than from the special structure of graded-index fibers.

Background

The paper analytically explains the approximately linear CP-rank scaling of the four-wave-mixing tensor for graded-index fibers using Gauss–Hermite quadrature. For step-index and asymmetric fibers, numerical CP decompositions exhibit similar approximately linear rank scaling, but the paper does not provide an analogous exact analytical construction.

The proposed explanation is that these other geometries may possess an effective quadrature representation because their mode profiles and associated overlap integrals have sufficient smoothness. Establishing whether smoothness alone is the relevant general mechanism would clarify why low-rank compression persists beyond analytically tractable and highly symmetric geometries.

References

Going back to the finding of Figure 1a for step-index and asymmetric fiber, we can think that the tensor has some ideal quadrature representation albeit one that isn't analytically obvious unlike the GRIN case. In other words, the hypothesis is that the low-rank nature really just depends on some kind of smoothness not specific to GRIN.

— Simulating multimode nonlinearities at very high mode count via parallel factor decomposition  (2609.16502 - Rivera, 15 Sep 2026) in Appendix, subsection “Why is the four-wave-mixing tensor low rank?”