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Rough interfaces

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.

Real-space model

Intensity image

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#!/usr/bin/env python3
"""
MultiLayer with correlated roughness
"""
import bornagain as ba
from bornagain import ba_plot as bp, deg, nm


def get_sample():
    """
    Returns a sample with two layers on a substrate, with correlated roughnesses.
    """
    # defining materials
    m_vacuum = ba.RefractiveMaterial("ambience", 0, 0)
    m_part_a = ba.RefractiveMaterial("PartA", 5e-6, 0)
    m_part_b = ba.RefractiveMaterial("PartB", 10e-6, 0)
    m_substrate = ba.RefractiveMaterial("substrate", 15e-6, 0)

    # defining layers
    l_ambience = ba.Layer(m_vacuum)
    l_part_a = ba.Layer(m_part_a, 2.5*nm)
    l_part_b = ba.Layer(m_part_b, 5*nm)
    l_substrate = ba.Layer(m_substrate)

    sigma, hurst, corrLength = 1*nm, 0.3, 5*nm
    roughness = ba.LayerRoughness(sigma, hurst, corrLength)

    my_sample = ba.MultiLayer()

    # adding layers
    my_sample.addLayer(l_ambience)

    n_repetitions = 5
    for _ 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)

    return my_sample


def get_simulation(sample):
    beam = ba.Beam(5e11, 0.1*nm, 0.2*deg)
    detector = ba.SphericalDetector(bp.simargs['n'], 1*deg, 0, 0.5*deg)
    simulation = ba.ScatteringSimulation(beam, sample, detector)
    return simulation


if __name__ == '__main__':
    bp.parse_args(sim_n=200)
    sample = get_sample()
    simulation = get_simulation(sample)
    result = simulation.simulate()
    bp.plot_simulation_result(result)
Examples/scatter2d/CorrelatedRoughness.py