Scattering from a thin sample.

Let us consider a thin sample, with a non homogeneous refraction index, and a light plane wave, moving in the direction of the $z$ axis. For $z=0$, at the surface of the sample, the field will be:

$$E\left(\vec{x},z=0\right) = E_0 e^{\displaystyle i \delta l \left(\vec{x}\right) k}$$ (3.16)

where $\delta l \left(\vec{x}\right)$ is the difference between the light path, the integral of the refraction index along $z$, for a given point $\vec{x}$, and its mean value over the whole sample. If $\delta l$ is small compared to the light wavelength, we can consider a first order developement:

$$E\left(\vec{x},z=0\right) = E_0 \left[ 1 + i \delta l \left(\vec{x}\right) k\right]$$ (3.17)

Neglecting the higher order terms means that we are neglecting higher order diffracted beams than the first.

Using Eq. (3.5) we can find the field for every value of $z$:

$$E_z\left(\vec{x}\right) = E_0\left(z\right) + \delta E_z\left(\vec{x}\right)$$ (3.18)

where

$$\delta E_z\left(\vec{q}\right) = ikE_0\left(z\right) \delta l\left(\vec{q}\right) e^ {\displaystyle i\left(\sqrt{k^2-q^2}-k\right)z}$$ (3.19)

is the scattered field, and

$$E_0\left(z\right) = E_0 e^{\displaystyle ikz}$$ (3.20)

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