Evolution equation of the field correlation.

For $q\ll k$, Eq. (3.5) can be approximated by:

$$E_z\left(\vec{q}\right) = E_0\left(\vec{q}\right) e^{\displaystyle i k z} e^{\displaystyle - i \frac{q^2}{2k} z}$$ (A.1)

In this approximation, neglecting the phase term $\exp\left(i k z\right)$, the field follows a Schröedinger equation:

$$i\frac{\partial}{\partial z} E\left(\vec{x},z\right) = -\frac{1}{2k}\nabla^2 E\left(\vec{x},z\right)$$ (A.2)

The three dimensional field correlation is defined as follows:

$$C_E\left(\Delta \vec{x},\Delta z\right) = \frac{1}{S}\int_S{E\lef... ...) E\left(\vec{x}+\Delta \vec{x},z+\Delta z\right) \mathrm{d}\vec{x}\mathrm{d}z}$$ (A.3)

In order to obtain an evolution equation for $C\left(\Delta \vec{x},\Delta z\right)$, as $\Delta z$ increases, we evaluate the first derivative of the correlation function:

$$\frac{\partial}{\partial \Delta z} C_E\left(\Delta \vec{x},\Delta... ... z} E\left(\vec{x}+\Delta\vec{x},z+\Delta z\right)\mathrm{d}\vec{x}\mathrm{d}z}$$ (A.4)

Using eq. (A.2):

$$\frac{\partial}{\partial \Delta z} C_E\left(\Delta \vec{x},\Delta... ...a^2 E\left(\vec{x}+\Delta\vec{x},z+\Delta z\right)\mathrm{d}\vec{x}\mathrm{d}z}$$ (A.5)

The operator $\nabla$ acts on the first argument of $E\left(\vec{x},z\right)$, thus it can be considered as acting on $\Delta \vec{x}$:

$$\frac{\partial}{\partial \Delta z} C_E\left(\Delta \vec{x},\Delta... ...ht) E\left(\vec{x}+\Delta\vec{x},z+\Delta z\right)\mathrm{d}\vec{x}\mathrm{d}z}$$ (A.6)

This proves that the evolution equation for $\left(\Delta \vec{x},\Delta z\right)$, as $\Delta z$ increases, is a Schröedinger equation:

$$i\frac{\partial}{\partial\Delta z}C_E\left(\Delta\vec{x},\Delta z\right)= -\frac{1}{2k} \nabla^2 C_E\left(\Delta \vec{x},\Delta z\right)$$ (A.7)

This equation can easily be solved in Fourier space:

$$C_E\left(\vec{q},z\right)=C_E\left(\vec{q},z=0\right) e^{\displaystyle -i\frac{q^2 z}{2k}}$$ (A.8)

We can now extend eq. (3.65) to the three dimensional case:

$$C_{I}\left(\Delta \vec{x},\Delta z\right) = \left< I\left(\vec{x}... ... I \right>^2 + \left\vert C_E\left(\Delta \vec{x},\Delta z\right) \right\vert^2$$ (A.9)

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