Data processing algorithm.

Once the experimental apparatus has been built, as described in Chapter 4, and the sample is placed in it, we acquire one hundred images. The electronic shutter of the CCD and its interlacement time must be so short that no evident evolution of the system happens during the exposure: for the samples we measured, that is colloids some microns large, with brownian movements, and non equilibrium fluctuations in the free diffusion of simple liquids, an interlacement delay of $1/25\mathrm{s}$ is sufficient. Moreover, different images must be completely uncorrelated. For a $10.0\mathrm{\mu m}$ colloid, images must be grabbed at intervals longer than one minute, if only brownian movements are the source of decorrelation. For the non equilibrium fluctuations we studied, the interval was about one second.

In figure 5.1 and 5.2 we show two typical images of the near field intensity of the light scattered by colloids of $5.2\mathrm{\mu m}$ and $10.0\mathrm{\mu m}$. We can notice the different typical size of the speckles.

Figure: Near field intensity of the light scattered by a colloid of $5.2\mathrm{\mu m}$.

Figure: Near field intensity of the light scattered by a colloid of $10.0\mathrm{\mu m}$.

For each image, we evaluate the correlation function. This operation is quite fast, since we can use a Fast Fourier Transform (FFT) algorithm. An FFT algorithm allows to evaluate the Fourier tranform of an $M\times N$ matrix, with a number of arithmetic operations proportional to $MN\log\left(MN\right)$. By using Perceval relation, we can obtain the correlation function by making an FFT, evaluating the square modulus, and making an Inverse FFT (IFFT). This only requires a number of operation of the order of $MN\log\left(MN\right)$. By scanning every value of $\Delta x$, and averaging over every $N\times M$ pixels, the number of operations would be of the order of $\left(MN\right)^2$. Using FFT, care must be taken in order to correcly evaluate the correlations near the boundarys: FFT assumes periodic boundarys, so the image must be embedded in a bigger matrix, filled with zeroes. Since the FFT is faster if $N$ and $M$ are powers of 2 [19], we used a matrix of $1024\times 1024$ points. After the correlation function has been evaluated, we normalize it, by dividing by the number of independent couples used to evaluate the correlation function.

The correlation functions of every image are averaged, thus obtaining $C_I\left(\Delta \vec{x}\right)$. Fig. 5.3 and 5.4 show typical graphs of the intensity correlation function $C_I\left(\Delta \vec{x}\right)$, for a colloid made of polystyrene spheres with diameters of $5.2\mathrm{\mu m}$ and $10.0\mathrm{\mu m}$. We can notice that the correlation function has a maximum at $\Delta \vec{x}=0$, then decreases, until it reaches the plateau value, about one half the peak value. This behaviour is typical of every speckle field.

Figure: Intensity correlation function $C_I\left(\Delta \vec{x}\right)$, for a colloid made of polystyrene spheres with diameter of $\mathrm{5.2\mu m}$

Figure: Intensity correlation function $C_I\left(\Delta \vec{x}\right)$, for a colloid made of polystyrene spheres with diameter of $10.0\mathrm{\mu m}$

Neglecting the stray ligth, we could evaluate the field correlation function by using the Siegert relation, Eq. (3.65):

$$C_E\left(\Delta \vec{x}\right) = \sqrt{C_I\left(\Delta \vec{x}\right) - \left<I\right>^2}$$ (5.16)

where the mean intensity $\left\{\left<I\right>\right\}$, is obtained by averaging the measured intensity over every pixel of the image and over every image. In Fig. 5.5 and 5.6 are shown typical graphs of the field correlation function, calculated from the intensity correlation function, without any correction for the stray light. The correlation should vanish for $\Delta x \to \infty$, in absence of stray light.

Figure: Field correlation function $C_E\left(\Delta \vec{x}\right)$, not corrected for the stray light, for the colloid made of polystyrene spheres with diameter of $5.2\mathrm{\mu m}$

Figure: Field correlation function $C_E\left(\Delta \vec{x}\right)$, not corrected for the stray light, for the colloid made of polystyrene spheres with diameter of $10.0\mathrm{\mu m}$

In order to subtract the contribution of the stray light, we evaluate the correlation function of the average of all the images, thus obtaining $C_{\bar{I}}\left(\Delta \vec{x}\right)$. The evaluation of the correlation function is obtained with the above described algorithm. In Fig. 5.7 and 5.8 are shown typical graphs of the correlation function of the mean intensity, for the two colloids. The graphs are not flat, due to the stray light.

Figure: Correlation function of the mean intensity $C_{\bar{I}}\left(\Delta \vec{x}\right)$, for the colloid made of polystyrene spheres with diameter of $5.2\mathrm{\mu m}$

Figure: Correlation function of the mean intensity $C_{\bar{I}}\left(\Delta \vec{x}\right)$, for the colloid made of polystyrene spheres with diameter of $10.0\mathrm{\mu m}$

Through Eq. (5.23) we evaluate $C_E\left(\Delta \vec{x}\right)$, under the hypothesis that both the stray light field and the scattered light field have a real and positive correlation function. Typical field correlation function, corrected for the stray light using Eq. (5.23), are shown in figure 5.9 and 5.10: we can notice a signitificative increase in the smoothness of the graphs, with respect to Fig. 5.5 and 5.6.

Figure: Field correlation function $C_E\left(\Delta \vec{x}\right)$ for the colloid made of polystyrene spheres with diameter of $5.2\mathrm{\mu m}$

Figure: Field correlation function $C_E\left(\Delta \vec{x}\right)$ for the colloid made of polystyrene spheres with diameter of $10.0\mathrm{\mu m}$

We apply a Fourier tranform to the two dimensional correlation function $C_E\left(\Delta \vec{x}\right)$, thus obtaining the field power spectrum $S_E\left(q\right)$. Since our samples are isotropic, we make an angular average of the power spectra, and represent our data as a function of the modulus $q$ of $\vec{q}$. The scattered intensity $I\left(q\right)$ is then obtained by using Eq. (3.14), that is, simply relating each value of the power spectra, with wavelength $q$ to a value of $I\left(Q\right)$, where the relation $Q\left(q\right)$ is given by Eq. (3.13). In Fig. 5.11 and 5.12 we show the measured $I\left(q\right)$.

Figure: ONFS measurement of the scattered intensity of a $5.2\mathrm{\mu m}$ colloid.

Figure: ONFS measurement of the scattered intensity of a $10.0\mathrm{\mu m}$ colloid.

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