Stable <i>O</i> ( <i>N</i> ) recursive boundary mapping for robust parametric analysis of dispersion in 2D <i>N</i> -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.