Siegert relation for the near field speckles.

Each scatterer of the sample generates a diffraction pattern which, at least in its far field, becomes larger and larger linearly, as the distance $z$ from the screen and the sample is made longer. So, for $z$ longer than a given distance, many diffraction patterns overlap: the Wick formulas should become valid. Unfortunately, the considerations of the previous section cannot be applied directly, since the area the diffraction pattern has not been already defined in a quantitative way.

Now we will prove the Wick formula for the case of Siegert relation, by using the formalism developed in the previous section.

From the results of the previous section, we can evaluate the intensity correlation function of the sum of the patterns, at a given $z$:

$$\left<\left\vert E_z\left(\vec{x}\right)\right\vert^2 \left\vert E_z\left(\vec{x}+\Delta \vec{x}\right)\right\vert^2\right> =$$ (3.50)

$$\mathcal{N} \int{ \left\vert\tilde{E}_z\left(\vec{x}\right)\right... ...eft\vert\tilde{E}_z\left(\vec{x}+\Delta \vec{x}\right)\right\vert^2 d\vec{x}} +$$

$$\mathcal{N}^2 \int{ \left\vert\tilde{E}_z\left(\vec{x}\right)\rig... ...tilde{E}_z\left(\vec{y}+\Delta \vec{x}\right)\right\vert^2 d\vec{x} d\vec{y}} +$$

$$\mathcal{N}^2 \int{ \tilde{E}_z\left(\vec{x}\right) \tilde{E}^*_z... ...c{y}\right) \tilde{E}_z\left(\vec{y}+\Delta \vec{x}\right) d\vec{x} d\vec{y}} +$$

$$\mathcal{N}^2 \int{ \tilde{E}_z\left(\vec{x}\right) \tilde{E}_z\l... ...c{y}\right) \tilde{E}^*_z\left(\vec{y}+\Delta \vec{x}\right) d\vec{x} d\vec{y}}$$

where $\tilde{E}_z\left(\vec{x}\right)$ is the field from a single scatterer, at a distance $z$. Assuming that the phases are random, and considering that the correlation function of any field does not change with $z$:

$$\left<I_z\left(\vec{x}\right) I_z\left(\vec{x}+\Delta \vec{x}\rig... ...\left(\vec{x}\right) \tilde{I}_z\left(\vec{x}+\Delta \vec{x}\right) d\vec{x}} +$$ (3.51)

$$\mathcal{N}^2 \left[\int{ \tilde{I}_0\left(\vec{x}\right) d\vec{x... ...\right) \tilde{E}^*_0\left(\vec{x}+\Delta \vec{x}\right) d\vec{x}}\right\vert^2$$

The first term on the right hand side depends on $z$: it is the intensity correlation function of the diffraction pattern. Since the diffraction pattern becomes larger as $z$ increases, while the total intensity keeps its value, the intensity correlation function of the diffraction pattern decreases, and vanishes as $z\to \infty$.

In order that Siegert relation holds, for a finite value of $z$, we must impose that the term with the four point correlation function is negligible compared to the two point ones:

$$\mathcal{N} \int{ \tilde{I}_z\left(\vec{x}\right) \tilde{I}_z\lef... ...ll \mathcal{N}^2 \left[\int{ \tilde{I}_0\left(\vec{x}\right) d\vec{x}}\right]^2$$ (3.52)

The first term can be substituted by its higher value, the one with $\Delta \vec{x}=0$:

$$\mathcal{N} \frac{ \displaystyle \left[\int{ \tilde{I}_z\left(\ve... ...t]^2 }{ \displaystyle \int{ \tilde{I}^2_z\left(\vec{x}\right) d\vec{x}} } \gg 1$$ (3.53)

The fraction represents the area $A$ covered by the diffraction pattern:

$$\mathcal{N}A \gg 1$$ (3.54)

In order that Siegert relation holds, we need that many particles scatter light inside a single diffraction pattern, that is, any point of the screen must be hit by light coming from many particles. This can be obtained without changing $\mathcal{N}$, but simply increasing $z$, thus increasing $A$.

It should be noted that the validity of Vick formulas for a given $z$ does not mean that the field is completely gaussian. For example, we have shown that $\tilde{C}^z_I\left(\Delta \vec{x}\right) \to 0$ for $z\to \infty$, where $\tilde{C}^z_I\left(\Delta \vec{x}\right)$ is the intensity correlation function of the diffraction pattern, defined as $\int{\tilde{I}_z\left(\vec{x}\right) \tilde{I}_z\left(\vec{x}+\Delta \vec{x}\right) d\vec{x}}$. But this does not imply any uniform convergence. Its integral, $\int {\tilde{C}^z_I\left(\Delta \vec{x}\right) d\Delta \vec{x}}$, for example, is a constant, and does not vanishes as $z\to \infty$. This means that we can build suitable linear operators, acting on the field, yelding quantities which do not have a gaussian distribution. A dramatic example can be obtained considering the scattering from a two dimensional screen, with many holes of a given shape. As $z\to \infty$, the field meets the Vick formulas ever better. But it is alwais possible to analyze an area, bigger than the diffraction pattern of each hole, and to recover the shape of the holes. This can be done by deconvolving the field by a suitable function: it's the operation made by a lens, which creates an image of the holes. The deconvolution gives any information about the sample, including the fourth order correlations: the deconvolved field is not gaussian. The gaussianity is only local: once we defined an area, corresponding to the aperture of a lens, there's a distance beyond which we are not able to recover the shape of each hole, and so informations on higher order correlation functions than second order ones are lost.

We can conclude that Eq. (3.55) implies only a local gaussianity; gaussianity is valid only when considering points inside an area small compered with the diffraction pattern of each scatterer. On the other hand, the knowledge of the field on a whole plane allows to recover any information on the correlation function of any order.

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