Conversion Formulas between the WLP Spectrum and the Frequency Spectrum for WLP-FDTD Analysis

The direct output of the weighted Laguerre polynomial (WLP) finite-difference time-domain method consists of the expansion coefficients of the electromagnetic response in terms of WLPs, i.e., the discrete WLP spectrum. In this study, we derive closed-form conversion formulas between the discrete WLP spectrum and the frequency spectrum. These formulas enable direct transformation between the discrete WLP spectrum and the frequency spectrum without reconstructing the time-domain response. The transformation of the frequency spectrum into the discrete WLP spectrum is simpler than transforming the time-domain function because the former involves differentiation, while the latter requires integration. These formulas offer a significant advantage by simplifying spectral analysis, because the basis functions in the frequency domain retain a constant amplitude regardless of order. The validity of the derived formulas is verified through analytical and numerical calculations on example functions.