Analytical Markov Chain for Spatiotemporal Flux Evolution of the Inner Filter Effect in Fluorescent Media

Abstract

Characterizing emission and decay time spectra in multi-component fluorescent media is essential for identifying intrinsic material properties and optimizing detectors. However, wavelength evolution from the secondary inner filter effect (IFE) distorts these observable spectra. While Monte Carlo (MC) ray-tracing can simulate this distortion, accumulating adequate tracking statistics requires long computation times, which hinders parameter optimization within high-dimensional spaces. This paper presents an analytical Markovian transport model based on spatiotemporal decoupling. A Laplace transform converts the multi-nested convolution integrals over continuous domains into a discrete Markov transition matrix, reducing the computational complexity from an exponential scale with respect to wavelength bins $N_λ$ and cascade order $n$, $\mathcal{O}(N_λ^n)$, to a linear scale, $\mathcal{O}(N_λ + n)$. The resulting algebraic solutions evaluate transient decay time spectra as a continuum superposition of Gamma wave packets and predict steady-state wavelength spectrum distortions driven by the IFE within a sub-second timescale. Validations across orthogonal and front-face spectrometer configurations show that the calculated spectra match MC simulations in lineshape. This model can serve as a fast forward engine to accelerate parameter space screening, provide early-stage detector design references, and act as a physics-constrained input for event vertex reconstruction algorithms.