Specular scattering from a multilayered sample with surface roughness.

The sample is composed of a substrate on which is sitting a stack of layers (similar to Correlated Roughness). These layers consist in a repetition of 10 times two different superimposed layers (from bottom to top):

  •  layer A: 5 nm thick with a real refractive index n = 1-5e-6.
  •  layer B: 10 nm thick with a real refractive index n = 1-1e-5.

There is no added particle.

All layers present the same type of roughness on the top surface, which is characterized by (see Correlated Roughness for a definition of the "roughness" parameters) :

  • a rms roughness of the interfaces σ =1 nm,
  • a Hurst parameter H equal to 0.3,
  • a lateral correlation length ξ of 500 nm,
  • a cross correlation length ξ equal to 0 nm.

The incident beam is characterized by a wavelength of 1.54 Å.

The incident angle αi varies between 0 and 2°.

The numerical results are plots of the Fresnel coefficients |R| and |T| for different layers as functions of the incident angle αi. The layers are numbered starting from the top layer (air). Therefore

  • Layer 0 is the air layer, for which |T|=1,
  • Layer 1 is the first material layer (layer A),
  • Layer 20 is the last material layer (layer B),
  • Layre 21 is the substrate, for which |R|=0.
Real-space model: 
Intensity Image: 
Python Script: 
R and T coefficients in multilayer, ba.Specular simulation.
import numpy
import bornagain as ba
from bornagain import deg, angstrom, nm

alpha_i_min, alpha_i_max = 0.0, 2.0  # incoming beam

def get_sample():
    Returns a sample with two layers on a substrate, with correlated roughnesses.
    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)

    l_ambience = ba.Layer(m_ambience)
    l_part_a = ba.Layer(m_part_a, 5.0*nm)
    l_part_b = ba.Layer(m_part_b, 10.0*nm)
    l_substrate = ba.Layer(m_substrate)

    roughness = ba.LayerRoughness()

    my_sample = ba.MultiLayer()

    # adding layers

    n_repetitions = 10
    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)

    return my_sample

def get_simulation():
    Returns a specular simulation with beam and detector defined.
    simulation = ba.SpecularSimulation()
        1.54*angstrom, 500, alpha_i_min*deg, alpha_i_max*deg)
    return simulation

def run_simulation():
    Runs simulation and returns it.
    sample = get_sample()
    simulation = get_simulation()
    return simulation

def coefficientsRT(simulation):
    Returns coefficients
    R = [ [ abs(c) for c in simulation.getScalarR(i) ] for i in range(22) ]
    T = [ [ abs(c) for c in simulation.getScalarT(i) ] for i in range(22) ]
    return R, T

def plot(simulation):
    Plots results for several selected layers
    from matplotlib import pyplot as plt
    fig = plt.figure(figsize=(12.80, 10.24))

    alpha_angles = simulation.getAlphaAxis().getBinCenters()
    R, T = coefficientsRT(simulation)

    selected_layers = [0, 1, 20, 21]
    nplot = 1
    for layer_index in selected_layers:
        plt.subplot(2, 2, nplot)
        plt.ylim(ymax=50.0, ymin=1e-06)
        plt.xlabel(r'$\alpha_f$ (rad)', fontsize=16)
        plt.semilogy(alpha_angles, [ numpy.abs(coeff) for coeff in R[layer_index] ])
        plt.semilogy(alpha_angles, [ numpy.abs(coeff) for coeff in T[layer_index] ])
        plt.legend(['|R| layer #'+str(layer_index),
                    '|T| layer #'+str(layer_index)],
                   loc='upper right')
        nplot += 1


if __name__ == '__main__':
    results = run_simulation()