This paper presents a fast analytical model that predicts how self‑absorption and re‑emission — the inner filter effect (IFE) — change the color and timing of light from mixed fluorescent materials. The authors replace slow Monte Carlo (MC) ray‑tracing with a Markovian transport description. Using a Laplace transform, they turn many nested integrals over time, distance and wavelength into a discrete transition matrix. That change cuts the computational scaling from exponential in the number of wavelength bins and cascade order to a linear cost, and yields sub‑second evaluations in the cases shown.

To build the model the authors split the problem into space, time and wavelength parts. They represent a simple but representative three‑component system: solvent (S), a primary fluorophore (F), and a secondary wavelength shifter (W). An interaction kernel I(λin→λout,τ) encodes the chance that light of wavelength λin is absorbed and, after a delay τ, is re‑emitted at λout. The paper shows how cascaded non‑radiative transfers and radiative decays are combined into that kernel. After the Laplace transform, the continuous convolutions become a low‑dimensional Markov transition matrix. The authors report algebraic solutions that describe transient decay spectra as a superposition of Gamma‑shaped wave packets and predict steady‑state spectrum changes driven by the IFE.

Why this matters: MC simulations can be precise but are slow to run when many wavelengths and cascade steps are needed. That slowness makes it hard to search a large parameter space or to fit experiments when the IFE hides intrinsic material properties. The analytical model is offered as a fast forward engine to screen parameters, to give early detector‑design guidance, and to provide a physics‑constrained input for event reconstruction algorithms. The authors validated their formulas against MC ray‑tracing for common measurement setups — 90° (orthogonal) and front‑face spectrometers — and found the calculated spectra match the MC lineshape.