Recurrence and Photon Statistics in Fluorescence Fluctuation Spectroscopy

We report on fluorescence fluctuations of nanoparticles diffusing through a laser focus. Subject to an intensity threshold the fluorescence signal is transformed into time traces of on and off periods. The distribution functions of the experimental on and off times follow power laws $t^{-\alpha }$ over several orders of magnitude with exponents $\alpha \simeq 1.5-2$. At long times the distribution functions cross over to exponential decays. For the interpretation of the experimental data a diffusion-reaction equation is proposed which covers both, the diffusion controlled recurrence and the photon statistics as the relevant processes.

Figure 1

Experimental intensity time traces and on and off periods of 200 nm beads. Intensities are given in units of photon counts. On the intermediate panels the on periods are given by black bars for $\theta =3$ on consecutively zoomed-in scales with magnification factors of approximately 20. The top two and the lowest two panels refer to the top and bottom time scale, respectively.

Figure 2

Experimental and calculated on and off PDFs. Pairs of curves give on PDFs (lower line) and off PDFs (upper line) for thresholds $\theta$, as indicated, and are shifted vertically for clarity. The wiggly lines give the experimental data of Fig. 1 and the smooth lines are the calculations. Dotted and dashed lines are power laws with exponents $\alpha =3/2$ and $2$, respectively.

Figure 3

First-passage PDFs for two limiting cases. (a) $⟨{\psi }_{\pm }(t)⟩$ of Eqs. (6, 7) based exclusively on diffusion. The ratio of the initial displacement and the radius of the absorbing sphere is $\Delta :r_a=1:10$ and the particle density is given by $4\pi c_0{r}_a^3/3=0.02$. Time $t$ is given in units of $t_0=r_a^2/D$. The dash-dotted line is a power law with exponent $\alpha =3/2$. The inset illustrates a diffusion path and first passages in 2D with a permeable circle. The shaded area indicates the homogeneous initial density $C(r,0)$ of bulk particles. (b) $⟨{\psi }_{\pm }(t)⟩$ based exclusively on photon statistics. Calculations of Eq. (12) are given by dashed lines for ${\Lambda }_0=100$ and for $\theta$ values, as indicated. The thin lines indicate power laws $t^{-\alpha }$ from bottom to top with exponents: 2.25, 2.15, 2.0, 1.9.

Figure 4

Change of state probabilities $p_{-}{p}_{+}$ in the focal zone for $\Lambda (\rho ,z)$ of Eq. (1) and for ${\Lambda }_0=100$ and $z_0/w_0=3/2$. The outer and the inner shaded area result from thresholds $\theta$ as indicated. On the left and top the full and dashed lines denote the probabilities $p_{-}{p}_{+}$ in the focal plane and along the $z$ axis, respectively. The transverse and longitudinal IRFs $\Lambda (\rho ,0)$ and $\Lambda (0,z)$ are given by dash-dotted lines.