Revised Perturbation Theory for Shifting Geometric Interfaces in High-Contrast Nanophotonics

Abstract

Perturbation theory is a powerful technique for solving partial differential equations, efficiently bridging the gap between analytical solutions and numerical methods. While standard perturbation theory (SPT) is commonly applied in electromagnetism to account for the effects of small changes to material properties like nonlinear susceptibilities, losses, and thermo-optic effects, i.e., so-called “bulk” perturbations, it is well-known to produce poor results when trying to account for small changes to the location of the boundaries between materials. This arises from the vectorial nature of boundary conditions in Maxwell’s equations that are enforced at the location of the original boundary and can not be “moved” via the process of linearly combining the original modes. We present an alternative formulation called pacified perturbation theory (PPT), which is able to account for this and thus make an accurate prediction of both the effective indices and the mode profiles. We demonstrate this by benchmarking our proposed PPT using the case of a high-confinement multimode nanophotonic waveguide and verify that the error associated with a first-order PPT approximation scales quadratically, unlike SPT (and its recent improvements), which fails to predict the changes in mode profiles to first order. We anticipate that our pacification procedure can be extended to higher-order corrections in perturbation theory, as well as broadened to fields beyond electromagnetism for computationally efficient predictions, where interfaces between dissimilar materials or the noncompleteness of the original mode basis (e.g., for non-Hermitian systems) thwarts standard perturbation theory.