Heterodyne near field speckles.

We developed this technique very recently; the device has been patented [3,4]. The setup for hEterodyne Near Field Speckles measurement is identical to ONFS one, but the beam stop is missing. The transmitted beam is not removed, and the image is due to the interference of the light scattered from the sample with the transmitted beam.

Two 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 and the diameter $D$ of the sample and of the incident beam. In ENFS, the superficial particle density of the sample plays no role. 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 a monodisperse colloid. Two conditions must be fulfilled.

1. The misfocusing $z$ must be selected to meet Eq. (3.62), in order that the correlations $\left <EE\right >$ vanish. 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. 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 Eq. (3.64). This condition ensures that the sensor cannot see the sample boundaries: the cample 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.

We can notice that the conditions expressed by Eq. (3.62) and (3.64) must hold for both ONFS and ENFS. On the contrary, in ENFS no condition is imposed on the particle density, in analogy with (3.63), since the field does not need to be gaussian.

In general, Eq. (3.63) is fulfilled by the sample; in that case, the field is gaussian, and the ENFS image represents the interference of a gaussian field with a plane wave. The particle density can be so small that the field is not gaussian; this does not mean that the speckles we see represent real objects in the sample. Each speckle is due to the interference between light scattered by many different particles.

Care must be taken in order to avoid multiple scattering. Since we want avoid multiple scattering, the scattered intensity is small compared to the transmitted beam intensity: the second order effects in $\delta E/E_0$ can be neglected, and Eq. 3.21 holds.

$$C_i\left(\Delta \vec{x}\right) = +\frac{2}{I_0} \Re C_{\delta E}\... ...left(\vec{x}\right) \delta E\left(\vec{x} + \Delta \vec{x}\right)\right>\right]$$ (3.68)

Under Eq. (3.62), the $\left <EE\right >$ correlations vanish:

$$C_i\left(\Delta \vec{x}\right) = \frac{2}{I_0}\Re C_{\delta E}\left(\Delta \vec{x}\right)$$ (3.69)

This means that $S_i\left(q\right)=\frac{2}{I_0} S_E\left(q\right)$. The measurement of the intensity allows to recover the field power spectrum.

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

$$2k^2S_{\delta l}\left(q\right) = S_i\left(q\right)$$ (3.70)

Under (3.64), Eq. (3.14) holds. We obtain the scattered intensity:

$$I\left[Q\left(q\right)\right] = \frac{1}{2}I_0 S_i\left(q\right)$$ (3.71)

The results do not depend on $z$. The misfocusing $z$ must be sufficent, in order that the correlations $\left <EE\right >$ vanish, but its value does not affect the results.

The considerations about the diameter $D$ of the sample hold also for ENFS: Eq. (3.62) and (3.64) give $D\gg q_{max}/q_{min}^2$. This problem has been discussed in Section 3.9. The result is that, in some cases, the sample and the laser beam have to be extremely large. In that cases, SNFS can be used instead of ENFS: that technique will be described in Section 3.11.

Equation (3.72) must be compared with Eq. (3.45), that holds for values of $z$ much less than those imposed by Eq. (3.62). The oscillations in the sensibility of shadowgraph technique come from the non vanishing of $\left <EE\right >$ correlations, essentially due to the phase relation of the beams scattered at symmetric angles by a thin sample. For example, the zeroes of the transfer function are due to the distructive interference of the symmetrically scattered beams. In ENFS, the phase relation is destroied, because the light that hit the sensor at symmetric angles comes from different regions.

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