Scattering measurements by microscopy techniques

By using Eq. (3.12), we can, in principle, evaluate the scattered intensities by measuring the field on a given plane. The above described microscopy techniques allow the determination of some functions of the electric field. Now, we assume that the scattered electric field is generated by a thin sample. From Eq. (3.19), we derive a property of the electric field:

$$\delta E\left(-\vec{q}\right)= -\delta E^*\left(\vec{q}\right) e^{\displaystyle 2i\left(\sqrt{k^2-q^2}-k\right)z}$$ (3.43)

For shadowgraph, we use Eq. (3.22), in order to evaluate the power spectrum $S_i$ of $i\left(\vec{x}\right)$:

$$S_i\left(\vec{q}\right) =\frac{4}{I_0} \sin^2\left[\left(k-\sqrt{k^2-q^2}\right)z\right] S_E\left(\vec{q}\right)$$ (3.44)

Using Eq. (3.12), we obtain:

$$S_i\left(q\right) =\frac{4}{I_0} \sin^2\left[\left(k-\sqrt{k^2-q^2}\right)z\right] I\left[Q\left(q\right)\right]$$ (3.45)

In order to use shadowgraph to evaluate the scattered intensity, the sample must be out of the focal plane. This technique has some disadvantages: some wavevectors cannot be seen, since the transfer function vanishes; moreover, if the sample is thick we cannot define a $z$: the oscillations of the transfer function are smeared, but it could be hard to know quantitatively how much.

For phase contrast, we use Eq. (3.25), in order to evaluate the power spectrum $S_i$ of $i\left(\vec{x}\right)$:

$$S_i\left(\vec{q}\right) =\frac{4}{I_0} \cos^2\left[\left(k-\sqrt{k^2-q^2}\right)z\right] S_E\left(\vec{q}\right)$$ (3.46)

Using Eq. (3.12), we obtain:

$$S_i\left(q\right) =\frac{4}{I_0} \cos^2\left[\left(k-\sqrt{k^2-q^2}\right)z\right] I\left(Q\left(q\right)\right)$$ (3.47)

The disadvantages of shadowgraph, as a scattering measurement technique, are also found in phase contrast. Phase contrast has a flat transfer function only if $z=0$: it's an ideal technique for thin samples.

Dark field doesn't allow to recover any information about the phase of the field. It is not suited to make scattered intensity measurements; a statistical approach, described in the following sections, will give interesting results.

We will describe in Section 3.11 the application of Schlieren technique to the measurement of the scattered intensity.

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