Gaussian speckles.

In this section we consider gaussian speckles, and we evaluate their three dimensional correlation function.

Far field speckles are often generated by scattering a gaussian beam, so that the far field speckles have a gaussian correlation function. We consider gaussian speckles in near field, since the case is analitically solvable, and involves some calculations used in quantum mechanics.

The field correlation function of the scattered light, in the plane orthogonal to $z$, is gaussian:

$$C_E\left(\Delta \vec{x},\Delta z=0\right)= C e^{\displaystyle -\frac{\Delta\vec{x}^2}{2\sigma^2}}.$$ (A.10)

In the Fourier space:

$$C_E\left(\vec{q},\Delta z=0\right)= 2\pi\sigma^2C e^{\displaystyle -\frac{1}{2}\sigma^2q^2}.$$ (A.11)

Using eq. (A.8):

$$C_E\left(\vec{q},z\right)= 2\pi\sigma^2C e^{\displaystyle -\frac{1}{2}\sigma^2q^2 -i\frac{q^2z}{2k}}.$$ (A.12)

Coming back to real space:

$$C_E\left(\vec{x},z\right)= C\frac{\sigma^2} {\sigma^2+iz/k} e^{\displaystyle -\frac{x^2}{2\left(\sigma^2+iz/k\right)}}.$$ (A.13)

Now we evaluate the modulus of the field correlation function, the quantity needed in eq. (A.9) to determine the intensity correlation function:

$$\left\vert C_E\left(\vec{x},z\right)\right\vert^2= C^2 \frac{\sigma^4} {\sigma^4+z^2/k^2} e^{\displaystyle -\frac{x^2\sigma^2}{\sigma^4+z^2/k^2}}.$$ (A.14)

We can now evaluate the intensity correlation function for $\vec{x}=0$:

$$C_I\left(\vec{x}=0,z\right)= C^2 \left(1+ \frac{\sigma^4} {\sigma^4+z^2/k^2}\right),$$ (A.15)

and for $z=0$:

$$C_I\left(\vec{x},z=0\right)= C^2 \left(1+ e^{\displaystyle \frac{x^2}{\sigma^2}}\right).$$ (A.16)

While the transverse correlation function follows a gaussian law, the longitudinal one is a Lorentzian, The diameter of the speckles is about $\sigma$, while their length is $\sigma^2k$.

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