Simulating multimode nonlinearities at very high mode count via parallel factor decomposition

RESUMO

A variety of problems at the frontier of nonlinear optics involve strongly multimode interactions. Simulating these dynamics is often computationally challenging due to the complexity of evaluating a large number of nonlinear overlaps. This is especially acute in ultrafast Kerr effect physics, where simulations in a modal representation have a generic complexity scaling quartically with the number of modes. This has limited the largest number of modes retainable in simulations in practice to a few tens of modes (often fewer), even when run on modern GPUs. Here, I show how by taking advantage of parallel factor decompositions of the nonlinear overlap tensor, a far larger number of modes can be accommodated. Realistic values for generic index profiles without special symmetries or separability can be in the thousands, and with special symmetries, far larger. I show that simulations with $>$1,000 modes can be done on modern hardware, illustrating this with the example of a multimode soliton propagating over a meter of GRIN fiber. I further show that the improved scaling immediately makes it practical to consider statistical effects, illustrating with the example of noise amplification in soliton fission. These results, implemented in public code with examples, should enable access to simulating the most demanding regimes of multimode nonlinear optics. Compression of the nonlinear interaction tensor may provide a route for more efficient simulation of nonlinear wave dynamics across a variety of other systems including acoustics, water waves, phononics, magnonics, and Bose condensates.