Dynamics of microstructure and Diffusing Wave Spectroscopy (DWS)

Diffusing wave spectroscopy (DWS) is a light scattering technology adapted to globally study the dynamic of a complex material inside its volume. Is is just like a "radar" dedicated to microscopic objects ! It has been introduced first by George Maret [Maret 87] and the DWS term was coined by D.J. Pine and D. Weitz [Pine 1988]. Maret modelized it for Brownian motion and Weitz generalized for any motion. It is adapted to opaque material since its takes into account the multiple scattering of the photons inside the material. The sensitivity is very high: typically the range of movements that can be detected is in the range 0.1nm to 100nm, depending on the product and the set-up. It gives an averaged measurement of the motion of many millions of microscopic objects. The principle is simple: we send a coherent light inside the material. The light will diffuse inside the volume up to a depth depending on the optical properties of this material (diffusion and absorption). The photons experience many scattering whenever they encounter optical index inhomogeneities or microscopic objects like particles, fibers, etc...

multiple scattering

Figure: multiple scattering of light in opaque media gives diffusion of light.

  At each scattering, if the object is moving, the photon wavelength change slightly due to the well known Doppler effect. The wavelength can change in plus or minus but it will always change if the object is moving in a direction not perpendicular to photon direction. Let's remind again that in such opaque media the photon is scattering multiply, and so its direction is random. Well, the wavelength is changing by a random value at each scattering event and this for many scattering events. It's time to do some statistics: you add many times positive or negative values to the wavelength (with equal probability both side) the resulting wavelength will on average not be modified but the photon that have diffused in our medium will get their wavelength spread around the laser wavelength. If we plot the probability density function (PDF) of the photons wavelength after they have traveled inside the medium, we obtain a typical bell curve.

wavelength dispersion

Figure: after traveling inside the medium, the moving microscopic scattering objects imply a dispersion of light wavelength by Doppler effect.

  The thickness of this bell curve is typically few nm. Now let's observe the light that can be collected: I mean: if we put a white screen at a place reached by the diffused light, what do we see ? To better understand let's mentally "freeze the product" for a moment ! It means that all the microscopic object inside our product becomes static. At each point of the screen, the light intensity is the result of interferences between all the photons coming from our diffusive product. Each photon has traveled with a variety of path length. And so the resulting light intensity is the result of the sum of many waves having the same frequency (here no Doppler effect because we have frozen the product) but not the same phase shift (due to differences of path length). It is know that such interferences produce "fringes" called speckle [wiki speckle]. These fringes look like small grains of light. The contrast is one: it means that some points receive no light at all ! These points are so lucky that all the light waves coming to them interfere destructively... These points are quite interesting for other aspects, we'll speak about that maybe in another post. Thus our white screen looks like this:

montage DWS retro rendu

Figure: simple setup of DWS in backscattering. A laser send photons inside the medium and the backscattered rays create the speckle pattern on the screen (the speckle is taken from real experimental data). Black zone are due to destructive interferences and bright spots to constructive interferences.

 

Video: animation showing the case of 2 photon paths that interfere constructively and then create a bright speckle grain.

  Now it's time to detect our light after it has gone through the product. The simplest experiment just used a single sensor. Imagine a point-like sensor for the moment. And also,let's "unfreeze" our product: the microscopic objects inside start to move again. The rays scattered by theses objects will have their paths modified due to these movements ans so the interference state that comes from the phase shift of each ray coming to the sensor will change with time. Our sensor will detect variations of light intensity. And we now understand that these variations are directly connected to the movement of the microscopic objects inside the product! In fact we have demodulated the Doppler frequency shift (small) from the main laser frequency (large). And this thanks to the interference of rays coming from the same laser but having traveled through different paths inside the product. We can here refer to a Moiré effect: if we sum 2 waves having a little difference in frequencies and the result is a wave of main frequency modulated by a low frequency which is the difference of the 2 initial frequencies. Here we have much more than 2 waves but the idea is the same.

Video: simplified view of the variation of interferences as a consequence of the movement of the diffusing objects.

A constructive interference gives a bright speckle grain. A destructive interference gives a dark speckle grain.  To optimize the detection let's a sensor that has more or less the size of a speckle grain. If the sensor is bigger it will cover many grains having variations usually uncorrelated on to each other (this is a property of the speckle) and then we get a lower signal: an average of many speckle grain. If the sensor is smaller than a speckle grain, we collect less photons and our electrical signal will be noisy. Another thing to take care about is the bandwidth of the sensor: it must cover the highest frequency of the light intensity oscillations. It won't be be the same sensor for thick product (slow speckle few Hz bandwidth) than red blood cell circulation detection (up to 1kHz needed). The signal processing: What we have to do now is to extract the correct information from the light intensity signal. In fact what we want to know is the "rate of change" or "speed of change" of our speckle pattern. There is a correct tool to do that in statistic: the auto-correlation [wiki autocorrelation]. Basically, the autocorrelation tells how much the signal seems to itself after a period of time has passed. Let's call this time laps "dt". Mathematically, we just multiply the signal by itself after giving an offset of dt (horizontal translation of graph). The next figure show the operation with real speckle data:  

autocorrelation

Figure: the autocorrelation of the light intensity versus time is calculated for each time laps dt by multiplication of this signal by itself. (Real DWS data)

  We do that for short time laps and long time laps. For dt=0 of course the correlation is complete. And for large dt's, there will be always a dt when the signal has no correlation with itself. The result is a curve called g2(dt) that starts at 2 (fully correlated) and ends at 1 (no correlated at all). Usually we take the curve g2(dt)-1 to get a function ending at 0. For example, on can take the inverse of the dt when g2-1 equals 1/2 to give a value of the "speed" of the microscopic objects. Averaging: The calculation of g2(dt)-1 needed quite a lot of data to get a clean curve. For example, one calculate the autocorrelation for many samples of 1000 points and takes the average. Maybe 100 to 1000 or more samples are needed to get a clean autocorrelation curve. For example if you acquire the signal at 1kHz of sampling rate, it may take 1000s to get a correct curve, not counting the calculation time. It the product is a fast drying paint, or you to see a 1 second phenomena, it is just too long! ... but there is a solution for that: by using a camera: Multi-speckle Diffusing Wave Spectroscopy (MSDWS) MSDWS is an extension of DWS. It just put many different sensors, ideally one for each speckle grain and gather the information in one signal. The averaging in time is replaced by the averaging in space (sensors). The formula of the correlation is still the same, replacing time by pixel for the averaging simbolized by <...> : g2(t)=<I(t)I(t+dt>/ [<I(t)><I(t+dt>] Applications: DWS can be applied to: - Drying monitoring (paint, varnish, glue, etc): the "speed" of the speckle can be monitored - Microrheology: a more careful study of this curve at different frequency can allows detection of the visco-elastic state of the medium. For example, in a gel, there is a network of polymer molecules. If we have nanoparticules inside and we follow them with DWS we will see that the freely move for short dt's, they are blocked by the polymer at a certain dt, giving the sign that the material has an elastic component! [Mason 1995] [wiki] - Foam coarsening monitoring. For example, a shaving foam coarsen by the elimination of bubbles one by one. DWS can monitor this [Durian 1991]. - Blood circulation in skin. - etc..

References:

G. Maret, P. E. Wolf (1987). "Multiple light scattering from disordered media. The effect of brownian motion of scatterers". Zeitschrift für Physik B65

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer (1988). "Diffusing wave spectroscopy". Physical Review Letters60 (12): 1134

L. Brunel, A. Brun, P. Snabre, and L. Cipelletti (2007). "Adaptive Speckle Imaging Interferometry: a new technique for the analysis of microstructure dynamics, drying processes and coating formation". Optics Express15 (23): 15250–15259

Durian,D.J.,Weitz,D.A.andPine,D.J.,Phys.Rev.A44,R7092(1991)

Mason Weitz (1995).Physical_Review_Letters