ENFS data processing.

Once the experimental apparatus has been built, as described in Chapter 4, in the absence of the sample, the CCD should be illuminated in a quite uniform way. As a matter of fact, the illumination is never completely uniform, primarily because of the interference of the main beam with stray light. A typical image is shown in Fig. 6.1. We can easily see some sets of concentric circles, each due to reflections inside a lens, along with speckle patterns properly due to stray light.

Figure 6.1: Background image.

When the the sample is placed in the right position, we acquire about one hundred images for each measurement. 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 studied, an interlacement delay of $1/25\mathrm{s}$ is sufficient. Moreover, different images must be completely uncorrelated. For a $10\mathrm{\mu m}$ colloid, images must be grabbed at intervals longer that one minute, if only brownian movements are the source of decorrelation, while for the non equilibrium fluctuations we studied the images can be taken at intervals of $1\mathrm{s}$. In figure 6.2 and 6.3 we show two typical ENFS images, generated by the interference of the main beam with the light scattered by colloids of $5.2\mathrm{\mu m}$ and $10.0\mathrm{\mu m}$. The images show a mean intensity, modulated by the interference with the speckle pattern. We can notice the different typical size of the speckles. The set of concentric circles can be seen yet: the stray light will be removed with the following step.

Figure: ENFS image of a $10.0\mathrm{\mu m}$ colloid.

Figure: ENFS image of a $5.2\mathrm{\mu m}$ colloid.

Once the images $I_n\left(\vec{x}\right)$ have been acquired, they are averaged, in order to evaluate $\bar{I}\left(\vec{x}\right)$ and $\left<\bar{I}\right>$. By using Eq. (6.7), we evaluate $i\left(\vec{x}\right)$, the heterodyne signal. Figures 6.4 and 6.5 show the heterodyne signal: since $i\left(\vec{x}\right)$ is negative, for some points, a constant intensity has been added. The images thus simply represent the ENFS images, cleaned from stray light and optical imprefections.

Figure: ENFS signal of a $10.0\mathrm{\mu m}$ colloid.

Figure: ENFS signal of a $5.2\mathrm{\mu m}$ colloid.

The heterodyne signal of each image is then elaborated in order to obtain its power spectrum. Simple Fourier transforming of the signal would be uncorrect, due to border effects. First of all, 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 doing an FFT, evaluating the square modulus, and doing 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, well known tricks can be used, in order to correct the boundary effects [19]. Figure 6.6 and 6.7 show the correlation functions thus evaluated.

Figure: ENFS measurement of the field correlation function, for a $5.2\mathrm{\mu m}$ colloid.

Figure: ENFS measurement of the field correlation function, for a $10.0\mathrm{\mu m}$ colloid.

The correlation function evaluated following the above described algorithm suffers from shot and read noise, that is, for the noise due to the CCD light measurement and acquisition systems. Since such a noise is not correlated to the speckle field due to scattered light, the noise correlation function sums to the speckle correlation function. In order to evaluate the noise correlation function, we acquire a set of about one hundred images, before putting the sample in the system. Then, we apply the above described algorithm to the images, and obtain the correlation function of the noise signal. Figure 6.8 shows the correlation function of the noise signal. We can notice a marked peak in 0, quite narrow, representing the correlation inside a row, and a correlation between lines spaced by two pixels, due to interlacing.

Figure 6.8: Correlation function of the shot and read noise.

The correlation function of the noise signal is then subtracted by the overall correlation function.

Once the correlation function has been evaluated, through an FFT we obtain 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. 6.9 and 6.10 we show the measured $I\left(q\right)$.

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

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

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