# 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 = 1-5e-6.

- *layer B:* 5 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:

- a rms roughness of the interfaces σ =1 nm,

- a Hurst parameter H equal to 0.3,

- a lateral correlation length ξ of 5 nm,

- a cross correlation length ξ_{⊥} equal to 1e-4 nm.

- The incident beam is characterized by a wavelength of 1 Å.
- The incident angles are α
_{i}= 0.2° and Φ_{i}= 0°.

**Note:**

The roughness profile is described by a normally-distributed random function. The

roughness correlation function at j

^{th}interface is expressed as

<U

_{j}(x, y)U

_{j}(x', y')>= σ

^{2}exp(-(τ/ξ)

^{2H}), τ= [(x-x')

^{2}+(y-y')

^{2}]

^{1/2},

where U

_{j}(x, y) is the height deviation of the j

^{th}interface at position (x, y).

σ 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 ξ » τ the surface looks smooth.

The cross correlation length ξ

_{⊥ }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 ξ

_{⊥}= 0 there is no correlations between layers. If ξ

_{⊥}is much larger than the layer

thickness, the layers are perfectly correlated.

Python Script:

""" MultiLayer with correlated roughness """ import numpy import bornagain as ba from bornagain import deg, angstrom, nm phi_min, phi_max = -0.5, 0.5 alpha_min, alpha_max = 0.0, 1.0 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(1e-4) print(my_sample.treeToString()) return my_sample def get_simulation(): """ Characterizing the input beam and output detector """ simulation = ba.GISASSimulation() simulation.setDetectorParameters(200, phi_min*deg, phi_max*deg, 200, alpha_min*deg, alpha_max*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. """ sample = get_sample() simulation = get_simulation() simulation.setSample(sample) simulation.runSimulation() return simulation.getIntensityData() if __name__ == '__main__': result = run_simulation() ba.plot_intensity_data(result)