Theory

Theory

A laser beam is focused through an objective lens to create a small excitation region at the focus through which particles will diffuse. We approximate the spatial distribution of the laser at this focal point as a two-dimensional Gaussian function with radial parameter ω0, commonly referred to as the beam waist. ω0 is then approximated as the radius of a diffraction limited spot,

(1)

Where λ is the excitation wavelength, and NA is the numerical aperture.

In order to determine the diffusion coefficient and size of diffusing particles, we define an autocorrelation function [1],

(2)

Where <F(t)> is the average fluorescence intensity over the entire measurement period, and dF(t) = F(t) - <F(t)>. An example autocorrelation curve is shown in Figure 2. Then, using the previous approximation of the spatial distribution of the laser as well as Fick's Second Law (a differential relationship between concentration (C) and the diffusion coefficient), we derive a theoretical model for the previously defined autocorrelation function [2],

(3)

where Veff is the effective volume of the excitation region, and a new constant has been introduced,

(4)

known as the characteristic decay time--the average time it takes for a particle to diffuse across the excitation region. A quick analysis of the theoretical function show, that for τ = 0,

(5)

and we should expect to see a linear relationship between the amplitude of the autocorrelation curve and concentration. By fitting the theoretical model (3) to our experimental data (2), we can determine τD, and therefore the diffusion coefficient. Finally, using the Stoke's Einstein relation [2],

(6)

(where k is boltzmann's constant, T is the temperature, η is the viscosity of the solution, and R is the radius), which allows us to determine the radius of the particle from the diffusion coefficient.

Figure 2: Autocorrelation curve obtained from fluorescent nanospheres diffusing through the excitation region. Original figure