Schlieren-like near field speckles.

For Schlieren technique, we can use Eq. (3.34), in order to evaluate the power spectrum $S_i$ of the signal $i\left(\vec{x}\right)$:

$$S_i\left(\vec{q}\right) = \left\{ \begin{array}{ll} \frac{2}{I_0}... ...ac{2}{I_0}S_E\left(\vec{q}\right) &\vec{q}\cdot\vec{n}\ge 0 \end{array} \right.$$ (3.72)

We assume the sample is isotropic, so that $I\left(\vec{q}\right)$ depends only on $\left\vert\vec{q}\right\vert$. Using Eq. (3.14):

$$I\left[Q\left(\vec{q}\right)\right] = \frac{1}{2} I_0 S_i\left(\vec{q}\right)$$ (3.73)

Schlieren technique can alwais be applied to measure the scatterered intensity, no matter how long the misfocusing is. The sample can be thick or thin, in the focal plane or away from it: the result is never affected.

Once the dimension of the image has been selected, in order to observe an interesting range of wavevectors, the diameter of the sample must fulfill the condition expressed by Eq. (3.64), as in ONFS and ENFS, but in this case there's no limitation on $z$. The diameter will be, in general, sufficient to give a good statistical sample of the particles we are measuring; $z$ will be as small as we can.

In general, for a thick sample, some of the objects will be too small, or too far from the focal plane, to be completely resolved. But their presence will prodece a speckle field, analogous to that of NFS. We will call this technique Schlieren-like Near Fiels Speckles, since it behaves like a true Schlieren technique only for big objects in the focal plane, while for the other cases it allows to measure the statistical properties of a speckle field.

Equation (3.74) must be compared with Eq. (3.45), that holds for values of $z$ much less than those imposed by Eq. (3.62), and without the blade. 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. In SNFS, the phase relation is destroied, because one of the beams scattered at symmetric angles is stopped.

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