Scattering from a multilayered sample with correlated roughness.
The sample is composed of a substrate on which is sitting a stack of layers. These layers consist in a repetition of 5 times two different superimposed layers (from bottom to top):
layer A: $2.5$ nm thick with a real refractive index $n = 5 \cdot 10^{-6}$.
layer B: $5$ nm thick with a real refractive index $n = 10 \cdot 10^{-6}$.
There is no added particle.
All layers present the same type of roughness on the top surface, which is characterized by:
a rms roughness of the interfaces $\sigma = 1$ nm,
a Hurst parameter $H$ equal to $0.3$,
a lateral correlation length $\xi$ of $5$ nm,
a cross correlation length $\xi_{\perp}$ equal to $10^{-4}$ nm.
The incident beam is characterized by a wavelength of 0.1 nm.
The incident angles are $\alpha_i = 0.2 ^{\circ}$ and $\varphi_i = 0^{\circ}$.
Note:
The roughness profile is described by a normally-distributed random function. The roughness correlation function at the jth interface is expressed as: $$ < U_j (x, y) U_j (x’, y’)> = \sigma^2 \frac{2^{1-H}}{\Gamma(H)} \left( \frac{\tau}{ξ} \right)^H K_H \left( \frac{\tau}{ξ} \right), \tau=[(x-x’)^2+(y-y’)^2]^{\frac{1}{2}}$$
$U_j(x, y)$ is the height deviation of the jth interface at position $(x, y)$.
$\sigma$ gives the rms roughness of the interface. The Hurst parameter $H$, comprised between $0$ and $1$ is connected to the fractal dimension $D=3-H$ of the interface. The smaller $H$ is, the more serrate the surface profile looks. If $H = 1$, the interface has a non fractal nature.
The lateral correlation length ξ acts as a cut-off for the lateral length scale on which an interface begins to look smooth. If $\xi \gg \tau$ the surface looks smooth.
The cross correlation length $\xi_{\perp}$ is the vertical distance over which the correlation between layers is damped by a factor $1/e$. It is assumed to be the same for all interfaces. If $\xi_{\perp} = 0$ there is no correlations between layers. If $\xi_{\perp}$ is much larger than the layer thickness, the layers are perfectly correlated.