# Autocorrelation of intensity fluctuations

Hi Guys,

I am working on some fluorescence correlation spectroscopy and I have some data generated by the machine I use. The data has the correlation plot of G(tau) vs the delay time (tau) which is an exponential decay.

The data also has the count rate given for the duration for the experiment (240 readings over a 120 second experiment).

Ive read about the auto correlation function and would like to generate it by hand (using matlab or excel or whatever) to get the same graph. The problem I have is all of the formulas I have been using dont seem to give a nice graph that starts nicely correlated, then decays down to 0 (like this: http://www.fcsxpert.com/classroom/theory/whatisfcs_pics/image038.gif)
(The formulas are like the one on this site http://www.fcsxpert.com/classroom/theory/what-is-autocorrelation.html)

Could anyone please explain how i would set up a program to generate that decay curve from the intensity fluctuations using the auto correlation function?

Thanks

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Cthugha
I assume that you are intending to characterize the auto correlation of a CW beam and not a pulsed one.

If you want to generate it by hand, you need to model the fluctuations around the mean as given in the equations shown in your like. Note that the intensities given there are momentary intensities which change fro moment to moment and only the averages <I> are really fixed numbers. So which approach did you take to simulate the fluctuations?

Hey, thanks for the replay.

Yes it is the autocorrelation I want. Just the correlation of the value and itself at a later time.

For the ACF i want to generate, I am modeling some point charges moving in random 2D ways (in matlab) and the resultant electric field at each step. Say I make them move 100 times I get an electric field for 100 locations (each step = 1).

So i have a columb of values that when plotted vs step I get fluctuations. From here I have tried to make a for loop in matlab but I just cant get a decaying function.

Thanks

Cthugha
Hmm, it is not quite clear to me what you are calculating exactly, so let me ask a few questions.

So you have some fixed number of point charges moving around on some 2D grid, right? From this info you get the electric field. Are you calculating the total field at each spatial position and then calculating the autocorrelation at each position individually? Are you then calculating the intensity corresponding to the field at each position and calculating that intensity correlation or do you intend to model something different? And most important: How exactly do your point charges move around? Gaussian random walk? Correlated Gaussian random walk?

Sorry for asking many questions, but there are many experimental situations for which correlation spectroscopy can be used and I am not quite sure I get the situation you try to model.

So you have some fixed number of point charges moving around on some 2D grid, right? From this info you get the electric field.
Yes, right now I can tell the program how many I want but just for the meantime say I have 2 charges. Each step they move around and the electric field (at a certain point of my specification) is calculated. So I get a table of values like
1 - 343
2 - 434
3 - 263
4 - 645
... up to a N steps of my choosing. Therefore I can plot steps vs E Field and get a fluctuating signal.

(just making those up)

Are you then calculating the intensity corresponding to the field at each position and calculating that intensity correlation or do you intend to model something different?
Yes, as above, the E field at a position of my choosing is based on the positions of the 2 particles at each step. (or how ever many I want).

And most important: How exactly do your point charges move around? Gaussian random walk? Correlated Gaussian random walk?
The charges move around based on brownian motion, with a diffusion D. I set up the partcile size and temperature which gives a calculated diffusion. This is then incorporated into their movement. (Think particles in a solution). I can then calculate the simulated Diffusion value, with a more accurate value closer to the theoretical one being obtained the more steps I used.

But right now I calculate it just using a formula. I want calculate a correlation decay curve and then fit the theoretical (exp(-D*q^2*tau)) to work out the diffusion.

So right now Im stuck at getting the decay curve.

Cthugha
Ok, did you try varying your observation volume or your time steps?

If your charges are moving too rapidly in the simulation or you sum over too few of your positions on the grid (or have too few grid points) the decay will also wash out. I assume that your field will have some r^-2 dependence on the charge position. What is your range (in terms of grid points) over which the field decays in space?

Ok, did you try varying your observation volume or your time steps?
Right now the electric field is being calculated for all the charges in the grid. I do not have a boundry (or illuminated section).

If your charges are moving too rapidly in the simulation or you sum over too few of your positions on the grid (or have too few grid points) the decay will also wash out. I assume that your field will have some r^-2 dependence on the charge position. What is your range (in terms of grid points) over which the field decays in space?
The charges move under a simulated brownain speed of a particles which would have a diameter of 1 um. Yes the field depends on where my observation point is.

Im not too sure what you mean by the range over which the field decays, do you mean where i am calculated the electric field in relation to where the charges are?

I think the other problem may be the actual maths. I have 2 columbs of just steps (1: N) and the intensity at each one. Just from this I have some for loops trying to do the equations but cant seem to get anything that resembles a decay function.

Thanks