# Dynamic light scattering and coherence of light

1. Apr 20, 2010

### gujax

Hi,
1. I am studying dynamic light scattering in which one experimentally measures intensity correlation i.e., <I(t)I(t+tau)>. Therefore, if the detector looks at a scattering signal from a solution of particles undergoing Brownian motion, under right experimental conditions one can detect the above correlation signal which looks like a reversed S shaped curve when plotted as correlation on Y axis and tau on X axis. Therefore one can obtain a sizable correlation all the way up to microseconds to milliseconds and then it decays.

2. I am told that more fundamental than intensity correlation is field correlation <E(t)E(t+tau)> and intensity correlation can be derived from field correlation. Great books such as "dynamic light scattering" by Carole and Pecora do begin with field correlation and then discuss intensity correlation.

3. This is my question: All the books assume a plane wave E0 exp(ik.r-wt). But in reality one should actually consider E0 exp(ikr-wt+theta(t)), where theta is the phase of the wave and will be a function of time, because each wave train emitted by a laser will have a different phase and there will be many such wave trains between t and t+tau. Therefore, shouldn't the term <E(t)E(t+tau)> average out to 0? why should there be any coherence between a wave at time t=0 and time t= tau which is as long as microseconds to milliseconds. Even the best lasers have a coherence length of few meters i.e., few nanoseconds. Milliseconds? - no way.
Appreciate any clue to my confusion.

2. Apr 21, 2010

### Cthugha

Generally speaking, intensity correlations contain more information than field correlations. Field correlations give you the coherence time. Intensity correlations also give you the kind of light (thermal, coherent, superradiant, nonclassical,...). However, if one already knows the kind of light present, one can derive intensity correlation from field correlation in some cases. For thermal/Gaussian processes you could use the Siegert relation for example.

The term <E(t)E(t+tau)> will indeed vanish for some field E with constant amplitude, if tau is large compared to the coherence time of the field. However, this is not the quantity investigated in dynamic light scattering experiments. The amplitude of the laser source is indeed constant and the field autocorrelation of this field will vanish soon. But what is really of interest is the scattered light. The spatial shape of this pattern will depend on the position of the particles inside your solution (or whatever you examine). So the amplitude of the scattered light field at some chosen angle will also depend on the position of the particles. As this position will slowly change with time due to Brownian motion, so will the amplitude of the scattered light at your chosen angle. You will see some decay in the autocorrelation and this is the timescale one usually measures in dynamic photon scattering experiments.

This also points out the difference between coherence time measurements and measurements of Brownian motion. For coherence time measurements you test the timescale after which the phase becomes unpredictable for a light field of constant amplitude, while for measurements of Brownian motion you measure the timescale after which the amplitude of the scattered light becomes unpredictable.

3. Apr 21, 2010

### gujax

Thanks Cthugha,
That was really really helpful. A big thank you.
I have to think more about your emphasis on amplitude rather than on the phase for dynamic light scattering (DLS) measurements. My next question would have been: If phase is not important for DLS, then why have formulations in terms of E and not Intensity I.
I did look up the book by Pecora et al and looks like what you are suggesting is correct. Phase information is not really needed (other than the phase which appears in k.r terms for scattering particles). Is that what is being done?

4. Apr 21, 2010

### Cthugha

First of all, I should add that my experience with dynamic light scattering is only rudimentary. The kind of intensity correlation experiments I am familiar with, happen on a somewhat faster timescale.

Having rethought the scenario, I think that phase is not really unimportant. There are two possibilities to create some spatially varying scattered light distribution:

1) The amplitude of the scattered light in a certain direction depends on the position of the particles. This is the case mentioned before.
2) Even if the amplitude of the scattered light did not depend on the position of the particles, the phase certainly would. So if you put a detector at some position and consider the case of completely isotropic scattering, the amplitude of the light field will not depend on the position of the scatterers, but the phase will. Accordingly also the intensity at the detector will depend on the phase difference of the fields scattered by two (or more) scatterers. This phase difference will of course also change, if the scatterers move around. In this case the coherence length of the laser should be longer than the the difference in the distances of the detector to the two (or more) scatterers. This should usually be well fulfilled. For long times the phase difference of the scattered fields at the detector $$\phi_2 - \phi_1$$ (for the example of two scatterers) will change as the scatterers move around and the intensity at the detector will vary correspondingly. The laser phase $$\phi_l$$ will of course vary strongly on this long timescale, but as it adds to the phase of both scatterers, the phase difference of the scattered fields at the detector will be something like $$\phi_1+\phi_l-(\phi_2+\phi_l)$$, so that this variation does not matter.

5. Apr 21, 2010

### gujax

I do know that if the particles diffuse in a very viscous medium, the intensity autocorrelation signal does extend out to several milliseconds...

Thank you for thinking this over. I appreciate it.

Also this time I must say sorry I am getting confused, but why is $$\phi_l$$(at time t=0) , same as $$\phi_l$$(at time t=Tau) . Aren't the two phi's from the equation $$\phi_2 - \phi_1$$ recorded at t=0 and t=tau? Therefore, the phase difference $$\phi_1+\phi_l-(\phi_2+\phi_l)$$ will get affected by the random phases of the laser emission at t=0 and tau.
Am I missing something obvious?

Last edited: Apr 21, 2010
6. Apr 22, 2010

### Cthugha

Yes, this is clear. I just wanted to point out that DLS is not my area of expertise.

$$\phi_l$$(at time t=0) is not the same as $$\phi_l$$(at time t=Tau). But that does not matter.

The $$\phi_1$$ and $$\phi_2$$ are just the phases of the light fields arriving at the detector at the same time t, but coming from two different scatterers. These will differ due to geometry (position of the scatterers). Whether interference at the detector will be constructive or destructive depends now just on the difference of these phases, so you get an intensity $$I(\phi_2-\phi_1)$$. As the scatterers move around, this phase difference will vary because the distance from the scatterers to the detector will vary, too. So you compare this phase difference at time t to this phase difference at time t=Tau.
This phase difference is of course uneffected by fluctuations in the laser phase as these fluctuations add to $$\phi_1$$ and $$\phi_2$$ alike and therefore cancel out in the difference.

7. Apr 22, 2010

### gujax

Thank you again Cthuga,
Got it this time.

To summarize my understanding:
1. At time t1, the field scatters off a particle and say also off a stationary medium such as wall of a tube carrying the particle.
2. The two will interfere because the two fields interfere in times less than coherence time of the laser.
3.As a result there will be beat pattern due to Doppler shift
4. This is at time t and the beat pattern has a phase proportional to the phase difference
$$\Delta\phi$$(t) which depends only on the particle motions and not field phase because that should cancel out.

5. After a time Tau (say milliseconds) the above phenomena repeats, only that this time the phase difference $$\Delta$$ $$\phi$$(t+tau) has changed but that is accounted by diffusional motion which the correlation signal should exhibit.

6. Therefore, the individual E's in the expression <|E(t)E(t+tau)|> result from interference (such as leading to Doppler shift) but the product has nothing to do with interference (i.e., vector nature of E at two different times are not important).

Last edited: Apr 22, 2010
8. Apr 6, 2011

### DavidSosa

My question is how are you planning to use dynamic light scattering in an experiment? Just interesing.

9. Apr 15, 2011

### gujax

Hi DavidSosa,