Longitudinal correlation.

We want to derive the field correlation along the $z$ axis. We consider its Fourier transform:

$$C_E\left(\Delta \vec{x}=0,q_z\right)= \frac{1}{\left(2 \pi\right)... ...C_E\left(\vec{q},z\right)e^{\displaystyle -iq_zz} \mathrm{d}\vec{q}\mathrm{d}z}$$ (A.17)

Using eq. (A.8):

$$C_E\left(\Delta \vec{x}=0,q_z\right)= \frac{1}{\left(2 \pi\right)... ...\right)e^{\displaystyle -i\frac{q^2z}{2k} -iq_zz} \mathrm{d}\vec{q}\mathrm{d}z}$$ (A.18)

The integration over $z$ gives a Dirac delta:

$$C_E\left(\Delta \vec{x}=0,q_z\right)= \int{C_E\left(\vec{q},z=0\right)\delta\left( \frac{q^2}{2k} + iq_z\right) \mathrm{d}\vec{q}}$$ (A.19)

In radial coordinates:

$$C_E\left(\Delta \vec{x}=0,q_z\right)= \int{C_E\left(q,\varphi,z=0\right)q\delta\left( \frac{q^2}{2k} + iq_z\right) \mathrm{d}q\mathrm{d}\varphi}$$ (A.20)

If the sample is isotropic:

$$C_E\left(\Delta \vec{x}=0,q_z\right)=2\pi \int{C_E\left(q,z=0\right)q\delta\left( \frac{q^2}{2k} + iq_z\right) \mathrm{d}q}$$ (A.21)

The integral can be evaluated:

$$C_E\left(\Delta \vec{x}=0,q_z\right)=2\pi C_E\left(\sqrt{2kq_z},z=0\right)\sqrt{2kq_z}$$ (A.22)

The dynamic of an instrument measuring the longitudinal correlation function is twice that obtained with the transversal one. This facts closely mirrors the quadratic relation between the diameter and the length of the speckles.

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