Subtraction of the stray light

As in every heterodyne technique, we measure the overall intensity $I\left(\vec{x}\right)$, generated by the interference of an object field $\delta E\left(\vec{x}\right)$ with a more intense reference beam with amplitude $E_0$. In our case, the object field is generated by the scattered beams, and the reference beam is the transmitted one. We measure the heterodyne signal $i\left(\vec{x}\right)$:

$$i\left(\vec{x}\right) = \frac{I\left(\vec{x}\right) - I_0}{I_0},$$ (6.1)

where $I_0$ is the mean intensity. At first order in $\delta E/E_0$, Eq. (3.22) holds:

$$i\left(\vec{x}\right) = 2 \frac{\Re\left[\delta E\left(\vec{x}\right)\right]} {E_0},$$ (6.2)

where we have assumed that $E_0$ is real. Equation (3.72) states that, under the conditions in which NFS works, the power spectrum of $i\left(\vec{x}\right)$ is the power spectrum of the electric field, the quantity we must measure in order to evaluate the scattered intensities.

We developed an algorithm to subtract the contribution of the stray light, directly on each image, point by point. This is a noteworthy feature of the heterodyne techniques, since in dynamic light scattering and in ONFS the subtraction is possible only on the scattered intensity or on the correlation function, averages of square values. The scattered field can be decomposed into $E_{SL}\left(\vec{x}\right)$, the stray light field, and $\delta E\left(\vec{x}\right)$, the field of the light scattered by the sample. Both $E_{SL}\left(\vec{x}\right)$ and $\delta E\left(\vec{x}\right)$ are much less intense than $E_0$, the reference field. At the first order, the resulting intensity is:

$$I\left(\vec{x}\right) = E_0^2 + 2 E_0 \Re \left[E_{SL}\left(\vec{x}\right)\right] + 2 E_0 \Re \left[\delta E\left(\vec{x}\right)\right]$$ (6.3)

In many cases, $\delta E\left(\vec{x}\right)$ fluctuates in time and is correlated only on finite delays. On the contrary, stray light comes mainly from hard surfaces, and does not change as times go on. This is the case of the samples we studied. The spatial average of a scattered field is alwais wanishing; this property, along with the absence of correlation on different images, says that the average over many images of $\delta E\left(\vec{x}\right)$ vanishes.

In order to separate the contribution of the stray light, we average $I\left(\vec{x}\right)$ over many different images. Since the phase of $\delta E\left(\vec{x}\right)$ is random, its mean vanishes:

$$\left\{I\left(\vec{x}\right)\right\} = E_0^2 + 2 E_0 \Re \left[E_{SL}\left(\vec{x}\right)\right].$$ (6.4)

We use the symbol $\left\{\cdot\right\}$ for the mean over many samples, and the symbol $\left<\cdot\right>$ for the mean over $\vec{x}$. The fluctuation $I\left(\vec{x}\right) - \left\{I\left(\vec{x}\right) \right\}$ does not depend on $E_{SL}\left(\vec{x}\right)$:

$$I\left(\vec{x}\right) - \left\{I\left(\vec{x}\right)\right\} = 2 E_0 \Re \left[\delta E\left(\vec{x}\right)\right].$$ (6.5)

Because of the conservation of the total intensity during the scattering process, by averaging $\left\{I\left(\vec{x}\right)\right\}$ over the whole plane, we obtain $I_0$:

$$\left<\left\{I\left(\vec{x}\right)\right\}\right> = E_0^2 = I_0.$$ (6.6)

We can now evaluate the heterodyne signal $i\left(\vec{x}\right)$, subtracting the the stray light contribution:

$$i\left(\vec{x}\right) = \frac{I\left(\vec{x}\right) - \left\{I\left(\vec{x}\right)\right\}} {\left<\left\{I\left(\vec{x}\right)\right\}\right>}.$$ (6.7)

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