Propagation of time-resolved fluorescence in a diffuse medium: complex analytical derivation.

The spatiotemporal evolution of fluorescence in an optically diffusive medium following ultrashort laser pulse excitation is evaluated using complex analytical methods. When expressed as a Fourier integral, the integrand of the time-resolved diffuse fluorescence with embedded fluorophores is shown to exhibit branch points and simple pole singularities in the lower-half complex-frequency plane. Applying Cauchy's integral theorem to solve the Fourier integral, we calculate the time-resolved signal for fluorescence lifetimes that are both shorter and longer compared to the intrinsic absorption timescale of the medium. These expressions are derived for sources and detectors that are in the form of localized points and wide-field harmonic spatial patterns. The accuracy of the expressions derived from complex analysis is validated against the numerically computed, full time-resolved fluorescence signal. The complex analysis shows that the branch points and simple poles contribute to two physically distinct terms in the net fluorescence signal. While the branch points result in a diffusive term that exhibits spatial broadening (corresponding to a narrowing with time in the spatial Fourier domain), the simple poles lead to fluorescence decay terms with spatial/spatial-frequency distributions that are independent of time. This distinct spatiotemporal behavior between the diffuse and fluorescence signals forms the basis for direct measurement of lifetimes shorter than the intrinsic optical diffusion timescales in a turbid medium.