[m.2] Upd hugo crosscorr page ()

Merging branch 'm.2'  into 'main'.

See merge request !2956

hugo/content/ref/sample/roughness/interface.md

@@ -24,9 +24,9 @@ ba.SelfAffineFractalModel(sigma, hurst, lateral_corr_length, max_spatial_freq=0.
 ```
 
 where
-- `sigma`, $\sigma$, is the root-mean-square amptitude of out-of-plane (transversal) fluctuations,
-- `hurst`, $H$, is a fractal exponent `H` with `0<H<1`,
-- `lateral_corr_length`, $\xi$, is the lateral (in-plane) correlation length,
+- `sigma`, $\sigma$, is the root-mean-square amptitude of out-of-plane (transversal) fluctuation.
+- `hurst`, $H$, is a fractal exponent `H` with `0<H<1`. The smaller $H$ is, the more serrate the surface profile looks.
+- `lateral_corr_length`, $\xi$, is the lateral (in-plane) correlation length.
 - `max_spatial_freq`, $\nu_{max}$,  is a cut-off spatial frequency.
 
 This is the K-correlation model of
@@ -37,6 +37,14 @@ $$
 S(\nu)=\dfrac{4 \pi H \sigma^2\xi^2}{(1+(2\pi\nu)^2)^{1+H}} ,\quad  for \quad  \nu<\nu_{max}
 $$
 
+In case there is no cut-off, the real-space roughness correlation function at the interface is expressed as: 
+$$
+< U(x, y) U(x', y')> = \sigma^2 \frac{2^{1-H}}{\Gamma(H)} \left( \frac{\tau}{ξ} \right)^H K_H \left( \frac{\tau}{ξ} \right),
+\quad \tau=[(x-x')^2+(y-y')^2]^{\frac{1}{2}}
+$$
+
+where $U(x, y)$ is the height deviation at position $(x, y)$.
+
 The main property is that it remains nearly constant at low $\nu$
 and transitions into a straight line on a log-log scale at high $\nu$,
 exhibiting an inflection point between these two regions.
@@ -70,7 +78,7 @@ S_{above}(\nu)=S_{below}(\nu)e^{-b(\nu)t} + \Omega\dfrac{1-e^{-b(\nu)t}}{b(\nu)}
 $$
 where
 $t$ - film thickness,
-$b(\nu)$ - relaxation function
+$b(\nu) = \sum{a_i\nu^i}$ - relaxation function
 
 An essential property of the model is that it describes not only autocorrelation
 but also cross-correlation properties.

hugo/content/ref/sample/roughness/transient.md

@@ -21,6 +21,10 @@ value $\sigma$ is taken from [interface autocorrelation model](../interface)
 
 ### Erf transient
 
+```python
+ba.ErfTransient()
+```
+
 Corresponds to roughness with Gaussian height statistics.
 
 The profile function:
@@ -29,9 +33,12 @@ $$
 \dfrac{1}{2}\Big[1+erf\Big( \dfrac{x}{\sqrt{2}\sigma}\Big)\Big]
 $$
 
-
 ### Tanh transient
 
+```python
+ba.TanhTransient()
+```
+
 Corresponds to roughness with non-Gaussian height statistics, with longer tails.
 
 The profile function:

hugo/content/ref/sample/roughness/xcorr.md

-+++
-title = "Cross-correlation"
-weight = 13
-+++
-
-# Inter-interface correlation
-
-The correlation between different interfaces
-is an optional argument of the `Roughness` constructor.
-
-It can be specified through one of
-
-```python
-CommonDepthCrosscorrelation(cross_corr_depth)
-SpatialFrequencyCrosscorrelation(base_crosscorr_depth, base_frequency, power)
-```
\ No newline at end of file

hugo/content/ref/sample/roughness/xcorr/_index.md

++++
+title = "Cross-correlation"
+weight = 13
++++
+
+# Inter-interface correlation
+
+The correlation between different interfaces
+is an optional argument of the `Roughness` constructor.
+
+It can be specified through one of
+
+```python
+ba.SpatialFrequencyCrosscorrelation(base_crosscorr_depth, base_frequency, power)
+ba.CommonDepthCrosscorrelation(cross_corr_depth)
+```
+
+### "Spatial frequency" model 
+
+```python
+ba.SpatialFrequencyCrosscorrelation(base_crosscorr_depth, base_frequency, power)
+```
+
+- `base_crosscorr_depth`, $\xi\_{\perp base}$, is the vertical distance over
+which the correlation of rougness spectrum at base spatial frequency
+$\nu_{base}$ between interfaces is damped by a factor $1/e$,
+- `base_frequency`, $\nu_{base}$, is a spatial frequency, for which the
+correlation of rougness spectrum between interfaces is damped by a factor
+$1/e$ at vertical distance $\xi\_{\perp base}$,
+- `power`, $p$, is a degree that determines the dependence of cross-correlation on spatial frequency.
+
+Cross-correlation spectrum between interfaces $i$ and $j$ is
+
+$$
+S_{ij}(\nu)=\sqrt{S_i(\nu)S_j(\nu)} exp[-\dfrac{t}{\xi\_{\perp base}} \Big(\dfrac{\nu}{\nu_{base}}\Big)^p]
+$$
+
+where
+$t$ - distance between interfaces,
+$S_i(\nu),S_j(\nu)$ - autocorrelation spectra
+
+### "Common depth" model
+
+```python
+ba.CommonDepthCrosscorrelation(cross_corr_depth)
+```
+
+- `cross_corr_depth`, $\xi\_{\perp}$, is the vertical distance over
+which the correlation is damped by a factor $1/e$
+
+Subcase of "Spatial frequency" model without dependence on spatial frequency.
+
+$$
+S_{ij}(\nu)=\sqrt{S_i(\nu)S_j(\nu)} exp[-\dfrac{t}{\xi\_{\perp}}]
+$$
+
+[Example](../xcorr/scatter) of script using this model.
+

hugo/content/ref/sample/roughness/scatter.md → hugo/content/ref/sample/roughness/xcorr/scatter.md

 +++
-title = "Scattering"
+title = "Example: common depth crosscorr"
 weight = 40
 +++
 
@@ -9,7 +9,8 @@ weight = 40
 
 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):
+* 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.
@@ -17,18 +18,13 @@ Scattering from a multilayered sample with correlated roughness.
   * 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.
+  * a common cross correlation length $\xi\_{\perp}$ equal to $10$ nm,
+  * a height distribution is normal.
 * 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.
+So the model with [SelfAffineFractalModel](../interface), [ErfTransient](../transient),
+[CommonDepthCrosscorrelation](../xcorr) is used here.
 
 {{<figscg src="/img/auto/scatter2d/CorrelatedRoughness.png" width="350px">}}