Determination of the sign of the field correlation function.

The power spectrum, that is $C_E\left(\vec{q},z=0\right)$, is real. If the sample is isotropic, it is symetric with respect to the origin, and then the correlation function $C_E\left(\vec{x},z=0\right)$ is real. The knowledge of the intensity correlation function with $\Delta z=0$ gives the absolute value of the field correlation function. The sign of the field correlation function does not affect the intensity correlation function with $\Delta z=0$, but it can affect its value for $\Delta z\ne0$.

In figure A.1 and A.2 we see an example of this effect. The figures show the graphs of the square correlation functions. The first is such that $C_E\left(x,\Delta z=0\right)=\sin\left(x\right)/x$; in the second, the correlation function has the same absolute value, but alwais positive sign, for $\Delta z=0$. For $\Delta z=0$ the square correlation functions are equal; their evolution for other values of $\Delta z$ are different. We can explain this fact considering the evolution of the positive and negative parts of the correlation function. The two parts evolve, and overlap, as $\Delta z$ increases. The interference of the two parts depends on the initial phase.

Figure A.1: Comparison between two square correlation functions. $C_E\left(x,\Delta z=0\right)=\sin\left(x\right)/x$

Figure A.2: Comparison between two square correlation functions. $C_E\left(x,\Delta z=0\right)=\left\vert\sin\left(x\right)/x\right\vert$

The sign of the correlation function is alwais possible, in principle. The presence of errors could limit this possibility.

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