### Correlated roughness

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 $1$ $\unicode{x212B}$.
• The incident angles are $\alpha_i = 0.2 ^{\circ}$ and $\phi_i = 0^{\circ}$.

Note:

The roughness profile is described by a normally-distributed random function. The roughness correlation function at the jth interface is expressed as: $$< U_j (x, y) U_j (x’, y’)> = \sigma^2 e^{-\frac{\tau}{ξ}2H}, \tau=[(x-x’)^2+(y-y’)^2]^{\frac{1}{2}}$$

• $U_j(x, y)$ 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.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72  """ MultiLayer with correlated roughness """ import bornagain as ba from bornagain import deg, angstrom, nm def get_sample(): """ Returns a sample with two layers on a substrate, with correlated roughnesses. """ # defining materials m_ambience = ba.HomogeneousMaterial("ambience", 0.0, 0.0) m_part_a = ba.HomogeneousMaterial("PartA", 5e-6, 0.0) m_part_b = ba.HomogeneousMaterial("PartB", 10e-6, 0.0) m_substrate = ba.HomogeneousMaterial("substrate", 15e-6, 0.0) # defining layers l_ambience = ba.Layer(m_ambience) l_part_a = ba.Layer(m_part_a, 2.5*nm) l_part_b = ba.Layer(m_part_b, 5.0*nm) l_substrate = ba.Layer(m_substrate) roughness = ba.LayerRoughness() roughness.setSigma(1.0*nm) roughness.setHurstParameter(0.3) roughness.setLatteralCorrLength(5.0*nm) my_sample = ba.MultiLayer() # adding layers my_sample.addLayer(l_ambience) n_repetitions = 5 for i in range(n_repetitions): my_sample.addLayerWithTopRoughness(l_part_a, roughness) my_sample.addLayerWithTopRoughness(l_part_b, roughness) my_sample.addLayerWithTopRoughness(l_substrate, roughness) my_sample.setCrossCorrLength(10*nm) print(my_sample.treeToString()) return my_sample def get_simulation(): """ Characterizing the input beam and output detector """ simulation = ba.GISASSimulation() simulation.setDetectorParameters(200, -0.5*deg, 0.5*deg, 200, 0.0*deg, 1.0*deg) simulation.setBeamParameters(1.0*angstrom, 0.2*deg, 0.0*deg) simulation.setBeamIntensity(5e11) return simulation def run_simulation(): """ Runs simulation and returns intensity map. """ simulation = get_simulation() simulation.setSample(get_sample()) simulation.runSimulation() return simulation.result() if __name__ == '__main__': result = run_simulation() ba.plot_simulation_result(result) 
CorrelatedRoughness.py