Homodyne near field speckles.

This technique has been presented very recently [1,2]. The device for the measurement of the Near Field Speckles is, basically, a misfocused dark field microscope. The transmitted beam is removed, and the image is due only to the light scattered from the sample.

Some of the parameters of the system must be selected: the distance $z$ from the sample to the focal plane of the objective, or from the CCD, if the objective is missing; the diameter $D$ of the sample and of the incident beam; the superficial particle density of the sample. The parameters must be selected on the basis of the required wavevector range $\left[q_{min},q_{max}\right]$. The ratio $q_{max}/q_{min}$ cannot exceed two decades, due to the finite size and discretization introduced by the CCD sensor. The range $\left[q_{min},q_{max}\right]$ is generally selected in order to cover interesting wavelenghts of the sample: for example, from one tenth to ten diameters of the particles, in the case of the scattering from a monodisperse colloid. Three conditions must be fulfilled.

1. The misfocusing $z$ must be selected in order to meet the condition expressed by Eq. (3.60), with $q_1=q_{min}$, in order that the correlations $\left <EE\right >$ vanish:

   $$z\gg \frac{k}{q_{min}^2}$$ (3.61)

   
   
   This condition is stronger than $z\gg k/\left(q_{typ}q_{min}\right)$, where $2\pi/q_{typ}$ is the typical diameter of the particles. Since $2\pi/q_{min}$ is the side $L$ of the images we take, Eq. (3.62) implies that $q_{typ}z/k\gg L$: the diffraction pattern of each particle covers a surface much bigger than the observed one, as required in order that the Fourier transform of the field can be considered gaussian.
2. In order that the field is gaussian, many diffraction patterns must overlap, in each point. Under the condition of Eq. (3.62), in order to fulfill Eq. (3.55), we must only provide that there are many particles in the surface $S$ covered by an image:

   $$\mathcal{N}S\gg 1$$ (3.62)

   
   
   . For particles suspended in a three dimensional volume, we can define the superficial particle density by multipling the volumetric particle density by the thickness of the cell. In order to fulfill condition (3.63), we can increase the volumetric particle density, or increase the thickness of the sample. Care must be teken in order to avoid multiple scattering.
3. The images we take must collect light scattered at any angle by the sample. The highest wavevector we want to measure is $q_{max}$; in order that the sensor collect light scattered by that wavevector, coming from any area of the sample, its diameter $D$ must satisfy:

   $$D\gg \frac{q_{max}}{k}z$$ (3.63)

   
   
   This condition ensures that the sensor cannot see the sample boundaries: the sample can be considered as infinite. If the diameter is much less than the one imposed by Eq. (3.64), the speckles are governed by the classical, Van Cittert and Zernike theorem.

Under Eq. (3.62), (3.63) and (3.64), Siegert relation holds:

$$C_I\left(\Delta \vec{x}\right) = \left< I\left(\vec{x}\right) I\l... ... = \left< I \right>^2 + \left\vert C_E\left(\Delta \vec{x}\right) \right\vert^2$$ (3.64)

where the $\left<\cdot\right>$ is the mean over $\vec{x}$ and $C_E\left(\Delta \vec{x}\right)$ is the field correlation function. The intensity we measure in a point is not directly connected with any physical part of the sample: each speckle is generate by the superposition of many diffraction patterns.

The measurement of the intensity allows to recover the modulus of the field correlation function through Eq. (3.65). Since $C_E\left(\Delta \vec{x}\right)$ is the Fourier transform of the power spectrum, which is symmetric and real, $C_E\left(\Delta \vec{x}\right)$ is also real. Moreover, if we think $C_E\left(\Delta \vec{x}\right)$ is alwais positive, we can calculate it by extracting a square root. Then, we Fourier transform $C_E\left(\Delta \vec{x}\right)$, thus obtaining $S_E\left(\vec{q}\right)$:

$$S_E\left(q\right) = \mathcal{F}\left[ \sqrt{C_I\left(\vec{x}\right)-\left<I\right>^2} \right]\left(q\right)$$ (3.65)

Using Eq. (3.19), we obtain, for a thin sample:

$$k^2S_{\delta l}\left(q\right) = \frac{1}{\left<I\right>} \mathcal{F}\left[ \sqrt{C_I\left(\vec{x}\right)-\left<I\right>^2} \right]\left(q\right)$$ (3.66)

Using Eq. (3.14), we obtain the scattered intensity:

$$I\left[Q\left(q\right)\right] = \mathcal{F}\left[ \sqrt{C_I\left(\vec{x}\right)-\left<I\right>^2} \right]\left(q\right)$$ (3.67)

The results do not depend on $z$. The misfocusing $z$ must be enough, in order that the field is gaussian, but its value does not affect the results.

The extraction of the square root of the difference between two experimental data is a dangerous operation, since the difference could be negative. In general, as any other inversion of experimental data, it involves an increase and a distorsion of noise. Chapter 7 provides a detailed description of this problem. This kind of problems are avoided by using ENFS or SNFS, described in Sects. 3.10 and 3.11.

We must notice that Eqs. (3.62) and (3.64) give $D\gg q_{max}/q_{min}^2$. We can think $q_{max}/q_{min}$ as the wavevector dynamic range we want to measure; hopefully it can be about one hundred for spatial measurements. On the other hand, $2\pi/q_{min}$ is of the order of some length of the particles, for example ten times. For example, if we consider $10\mathrm{\mu m}$ colloids, each image must cover about $100\mathrm{\mu m}$. In order to cover two decades in wavelength, we need a $D$ about one hundred times wider: about $1\mathrm{cm}$. This is not a huge length; moreover, from the industrial point of view, there's no problem in making many acquisitions, with different magnifications and $z$, for every wavevector range, since each acquisition needs no accurate positioning.

On the other hand, in some cases, for scientific purposes, $D$ should be too wide. This is the case of measurements of non-equilibrium fluctuations, described in Chapt. 9. In this case, we want to evaluate power spectra on two decades in spatial frequencies. The dimension of the largest fluctuations is about one tenth of millimeter: a good statistical sample is about a millimeter large, and the whole sample must be two decades bigger: about $10\mathrm{cm}$. The building of a Soret cell, or a free diffusion cell, with such an big diameter can be avoided by using SNFS, which will be described in Section 3.11

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