Light pulse propagation through a dielectric slab. II. Two roads to the same field: numerical cross-validation of local (FDTD-ADE) and non-local (retarded integral-equation) solutions
Abstract
Paper I of this series (arXiv:2610.29336) solves, via the retarded Green's functions, the local differential equations governing light propagation in a plane-parallel dielectric slab, obtaining a pair of retarded integral equations that represent, at the continuum level, the microscopic multiple-scattering picture underlying the local formulation. The two formulations are thus proven to be analytically equivalent. The local system of differential equations is then solved numerically with a finite-difference time-domain scheme (FDTD-ADE). This report closes the loop by solving the retarded integral equations directly, without passing through the local formulation, and comparing the result with the FDTD-ADE solution under the same physical and material conditions. The two independently implemented, structurally different numerical schemes agree within 0.02-1.3% for the reported reflected and transmitted energy fractions; the absorbed fraction, computed independently in both schemes from the Lorentz damping term rather than merely inferred from 1-R-T, agrees closely between the two methods, to the precision reported. Both solutions are also compared with an exact reference, the Airy formula weighted over the pulse spectrum: the direct solution reproduces it to within 0.3%, and its Richardson extrapolation to all digits shown. The agreement persists for a few-cycle pulse and for a sub-wavelength slab, where naive fixed-point iteration of the integral equations is not a practical convergent strategy. A complementary set of movie-style, nine-frame visualizations is also presented, illustrating the same equivalence directly in the field rather than only in integrated observables. To the author's knowledge, a direct numerical cross-validation of these two formulations in this genuinely time-dependent regime has not previously been reported.
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BibTeX
@misc{reali2026light,
title = {{Light pulse propagation through a dielectric slab. II. Two roads to the same field: numerical cross-validation of local (FDTD-ADE) and non-local (retarded integral-equation) solutions}},
author = {Giancarlo Reali},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30833},
eprint = {2610.30833},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30833},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Light pulse propagation through a dielectric slab. II. Two roads to the same field: numerical cross-validation of local (FDTD-ADE) and non-local (retarded integral-equation) solutions
AU - Giancarlo Reali
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30833
UR - https://arxiv.org/abs/2610.30833
AB - Paper I of this series (arXiv:2610.29336) solves, via the retarded Green's functions, the local differential equations governing light propagation in a plane-parallel dielectric slab, obtaining a pair of retarded integral equations that represent, at the continuum level, the microscopic multiple-scattering picture underlying the local formulation. The two formulations are thus proven to be analytically equivalent. The local system of differential equations is then solved numerically with a finite-difference time-domain scheme (FDTD-ADE). This report closes the loop by solving the retarded integral equations directly, without passing through the local formulation, and comparing the result with the FDTD-ADE solution under the same physical and material conditions. The two independently implemented, structurally different numerical schemes agree within 0.02-1.3% for the reported reflected and transmitted energy fractions; the absorbed fraction, computed independently in both schemes from the Lorentz damping term rather than merely inferred from 1-R-T, agrees closely between the two methods, to the precision reported. Both solutions are also compared with an exact reference, the Airy formula weighted over the pulse spectrum: the direct solution reproduces it to within 0.3%, and its Richardson extrapolation to all digits shown. The agreement persists for a few-cycle pulse and for a sub-wavelength slab, where naive fixed-point iteration of the integral equations is not a practical convergent strategy. A complementary set of movie-style, nine-frame visualizations is also presented, illustrating the same equivalence directly in the field rather than only in integrated observables. To the author's knowledge, a direct numerical cross-validation of these two formulations in this genuinely time-dependent regime has not previously been reported.
ER -