Stable O(N) recursive boundary mapping for robust parametric analysis of dispersion in 2D N-layered WGM resonators

A self-normalizing analytical framework is established to evaluate the exact dispersion relations of whispering
gallery modes (WGMs) in 2D N-layered cylindrical micro-resonators. By mapping a continuous radial boundary
function through a recursive propagator, the catastrophic numerical cancellations inherent to traditional transfer
matrix methods (TMMs) are avoided, particularly at high azimuthal orders in the strict evanescent regime. The
global dispersion equation is analytically recast as a Gauss-type continued fraction, establishing a strict discretization
scheme that asymptotically recovers the canonical Riccati equation for continuous gradient-index (GRIN)
media. While maintaining an O(N) temporal complexity, the proposed method reduces the spatial complexity
to a strict O(1) memory footprint during the iterative root-finding stage, while acknowledging that the subsequent
spatial field reconstruction inherently scales as O(N) This dichotomy completely eliminates dense matrix
allocations during massive dispersion sweeps. This framework is applied to a parametric study of silicon nitride
resonators, enabling rapid and stable forward analysis of anomalous dispersion regimes for soliton micro-comb
generation.