Analytical eigenmode framework for triplet-mediated fluorescence dynamics under pulsed excitation

An exact analytical solution to a three-state photophysical model is presented to explicitly map the time-resolved fluorescence dynamics of a fluorophore under pulsed excitation. As distinct from earlier analytical treatments that primarily focused on transient absorption and total ground-state depletion, the proposed eigenvalue-eigenvector framework captures both transient and steady-state emission responses. The photophysical model accounts for electronic transitions between the singlet ground state, the excited singlet state and a long-lived triplet state, enabling a treatment of excitation, radiative decay, intersystem crossing, together with triplet-state relaxation. By diagonalizing the rate matrix consisting of phenomenological transition rates, closed-form expressions for the time-dependent and steady-state populations are obtained, revealing a clear separation of time scales governing fluorescence dynamics. The analytical solution is then applied to the rhodamine-based CF640R dye with an aim to investigate population redistribution as a function of arbitrary pulse duration and excitation irradiance in the nanosecond-to-millisecond time window. The computational results demonstrate that pulsed excitation enables the resolution of triplet-state dynamics, which strongly influence the excitation-dependent fluorescence behavior arising from the singlet excited state. Beyond the specific case of CF640R, the presented analytical treatment establishes a general and physically adaptive framework for analyzing excitation-duration and irradiance-dependent fluorescence behavior of diverse fluorescent chromophores with long-lived, nonemissive electronic states in condensed phases.