Lifetime-unified quadratic and cubic approximations of fluorescence peak time for a time-domain diffuse-light response
Abstract
We study the temporal maximum of a boundary response generated by a coupled diffusion system in a half-space with Robin boundary conditions and a point fluorophore. The fluorescence lifetime enters through a one-sided exponential time convolution. Previous peak-time approximations were derived under separate short- and long-lifetime scalings. We construct a single reduced peak-time formula for every positive ratio of lifetime to the intrinsic width of the transport peak, thereby describing the transition between these regimes without switching formulas. Starting from a known large-distance approximation of the zero-lifetime response, we normalize at its unique maximum and approximate its logarithm quadratically. The resulting scalar mode equation has a unique solution, expressed through the inverse scaled complementary error function. Its endpoint expansions recover the leading structures of the earlier approximations in the corresponding joint large-distance regimes. A cubic correction accounts for the leading local asymmetry. Under a stated bound on the normalized cubic coefficient, we prove uniqueness of its positive mode and agreement with the exact reduced-profile peak expansion through second order in the lifetime. A conditional peak-location stability estimate is combined with a decomposition into transport-profile and local truncation errors. Numerical comparisons with the full diffusion response support the accuracy of the construction. The lifetime-unified formula is distinguished from the fixed-geometry, infinite-lifetime limit, in which the transport tail yields a different peak-time growth law.
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BibTeX
@misc{eom2026lifetimeunified,
title = {{Lifetime-unified quadratic and cubic approximations of fluorescence peak time for a time-domain diffuse-light response}},
author = {Junyong Eom},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21443},
eprint = {2610.21443},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21443},
}
RIS
TY - UNPB
TI - Lifetime-unified quadratic and cubic approximations of fluorescence peak time for a time-domain diffuse-light response
AU - Junyong Eom
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21443
UR - https://arxiv.org/abs/2610.21443
AB - We study the temporal maximum of a boundary response generated by a coupled diffusion system in a half-space with Robin boundary conditions and a point fluorophore. The fluorescence lifetime enters through a one-sided exponential time convolution. Previous peak-time approximations were derived under separate short- and long-lifetime scalings. We construct a single reduced peak-time formula for every positive ratio of lifetime to the intrinsic width of the transport peak, thereby describing the transition between these regimes without switching formulas. Starting from a known large-distance approximation of the zero-lifetime response, we normalize at its unique maximum and approximate its logarithm quadratically. The resulting scalar mode equation has a unique solution, expressed through the inverse scaled complementary error function. Its endpoint expansions recover the leading structures of the earlier approximations in the corresponding joint large-distance regimes. A cubic correction accounts for the leading local asymmetry. Under a stated bound on the normalized cubic coefficient, we prove uniqueness of its positive mode and agreement with the exact reduced-profile peak expansion through second order in the lifetime. A conditional peak-location stability estimate is combined with a decomposition into transport-profile and local truncation errors. Numerical comparisons with the full diffusion response support the accuracy of the construction. The lifetime-unified formula is distinguished from the fixed-geometry, infinite-lifetime limit, in which the transport tail yields a different peak-time growth law.
ER -