From c93a0c66515080cfcb8215c7a1beb90211358c4e Mon Sep 17 00:00:00 2001
From: "Joachim Wuttke (h)" <j.wuttke@fz-juelich.de>
Date: Mon, 29 May 2023 21:22:07 +0200
Subject: [PATCH] multilayer, roughness, tanh

---
 Doc/DevRef/DevRef.pdf            | Bin 2236870 -> 2237528 bytes
 Doc/DevRef/Multilayers.tex       |  10 +-
 Doc/DevRef/Roughness.tex         |  37 +-
 Doc/DevRef/fig/funcplot/tanhc.ps | 674 ++++++++++++++++++++-----------
 4 files changed, 448 insertions(+), 273 deletions(-)

diff --git a/Doc/DevRef/DevRef.pdf b/Doc/DevRef/DevRef.pdf
index 9e9d911b180c3db1f4eb78357d25ae402b6ce15b..fc66d330f16feb85225709aef1277c0aa005350e 100644
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diff --git a/Doc/DevRef/Multilayers.tex b/Doc/DevRef/Multilayers.tex
index d9347c67ec8..27af97d8a50 100644
--- a/Doc/DevRef/Multilayers.tex
+++ b/Doc/DevRef/Multilayers.tex
@@ -540,7 +540,7 @@ the continuity conditions~\cref{Econtcond} take the form
   +A^-_l &+A^+_l
   &=&
   +A^-_{l-1}\delta_{l-1} &+A^+_{l-1}\delta_{l-1}^{-1},
-  \\
+  \\[1.3ex]
   -A^-_l \kappa_l  &+A^+_l \kappa_l
   &=&
   -A^-_{l-1}\delta_{l-1} \kappa_{l-1} &+A^+_{l-1}\delta_{l-1}^{-1} \kappa_{l-1}.
@@ -561,9 +561,9 @@ Parratt \cite{Par54}, unaware of Abelès, expressed the same relation as a recur
 \index{Parratt recursion}%
 }
 \begin{equation}\label{EMil}
-  M_l \coloneqq \Delta_{l-1} S_l
+  M_l \coloneqq \Delta_{l-1} S_l,
 \end{equation}
-where
+which we write using the phase rotation matrix
 \begin{equation}\label{DmatD}
   \Delta_l
    \coloneqq
@@ -572,14 +572,14 @@ where
        0 & \delta_{l}
    \end{array}\right)
 \end{equation}
-and
+and the refraction matrix
 \begin{equation}\label{DmatS}
   S_l
    \coloneqq
    \frac{1}{2}
    \left(\begin{array}{cc}
        s^+_l&s^-_l\\
-       s^-_l&s^+_l)
+       s^-_l&s^+_l
    \end{array}\right)
 \end{equation}
 with coefficients
diff --git a/Doc/DevRef/Roughness.tex b/Doc/DevRef/Roughness.tex
index 3e211a147f7..9190031845c 100644
--- a/Doc/DevRef/Roughness.tex
+++ b/Doc/DevRef/Roughness.tex
@@ -49,7 +49,7 @@ Graded interfaces have a smooth SLD profile,
 i.e.\ the function $\overline{v}(z)$ or $\kappa^2(z)$ evolves continuously
 from one bulk value to the other.
 Among the SLD profiles that can be solved analytically,
-the tanh profile is particularly important.
+the tanh (\cref{Ftanhc}a) profile is particularly important.
 A good summary of the solution can be found in Ch.~2.5 of Lekner \cite{Lek16}.\footnote
 {He credits Eckart (1930) and Epstein (1930) for the solution.
 For a short summary, see also \cite[\S~25, exercise~3]{LL3}.}
@@ -83,10 +83,6 @@ The transmission coefficient $t_{ab}$ is communicated in \cite{AnMR88}.
 Using various properties of the Gamma and sinh functions,
 one can verify flux conservation~\cref{EConservation}.
 
-%==================================================================================================%
-\subsection{Fresnel coefficients with roughness factor}
-%==================================================================================================%
-
 In the limit $\tau\to0$, the phase factor $\varphi$ in \cref{ErTanh} goes to zero.
 For simplicity, we let $\varphi=0$ throughout.
 This approximation is equivalent to an adjustment of the interface position~$z_{ab}$
@@ -97,7 +93,7 @@ by an amount that can be expected to be small compared to the interface thicknes
 \includegraphics[width=0.41\textwidth]{fig/funcplot/tanhc.ps}
 \hfill
 \includegraphics[width=0.41\textwidth]{fig/funcplot/FresnelReductionTanh.ps}
-\caption{(a) Function tanhc. (b) Reflectivity reduction factor,
+\caption{(a) Functions tanh and tanhc. (b) Reflectivity reduction factor,
 obtained by dividing \cref{ErTanh2} through the Fresnel reflectivity~\cref{EtFresnel},
 as function of $\kappa_a \tau$ for ratios
 $\kappa_b / \kappa_a$ of 0.1, 0.2, 0.4, 0.9, 1.1, 2, and~5.}
@@ -140,15 +136,9 @@ Tests confirm that flux conservation is fulfilled at machine precision.\footnote
 Eq.~\cref{ESab2},
 which relates the refraction matrix to transmission and reflection coefficients,
 remains valid for our modified coefficients \cref{ErTanh2} and~\cref{EtTanh2}.
-Accordingly, \cref{ESab} must be replaced by
-\begin{equation}\label{ESabR}
-  S_{ab}
-   \coloneqq
-   \frac{1}{2\kappa_a}
-   \left(\begin{array}{ll}
-     (R_{ab}^{-1}\kappa_a+R_{ab}\kappa_b)&(R_{ab}^{-1}\kappa_a-R_{ab}\kappa_b)\\
-     (R_{ab}^{-1}\kappa_a-R_{ab}\kappa_b)&(R_{ab}^{-1}\kappa_a+R_{ab}\kappa_b)
-   \end{array}\right).
+Accordingly, the coefficients~\cref{Dslpm} must be replaced by
+\begin{equation}\label{EslpmTanh}
+  s^\pm_l = R_{l,l-1}^{-1} \pm R_{l,l-1}\kappa_{l-1}/\kappa_l.
 \end{equation}
 
 %==================================================================================================%
@@ -175,28 +165,23 @@ This interpretation is mentioned by N\'evot et al. \cite{NePC88}.
 More questionable is the simultaneous modification of the transmission coefficient.
 Currently BornAgain uses
 \begin{align}\label{EtNC}
-\tilde t_{ab} &= t_{ab}\, \e^{+\left( k_a - k_b \right)^2 \sigma_{ab}^2/2},
+\tilde t_{ab} &= t_{ab}\, \e^{+\left( k_a - k_b \right)^2 \sigma^2/2},
 \end{align}
 where $t_{ab}$ is the Fresnel coefficient \cref{EtFresnel}.
 This is the result obtained by Tolan \cite[Eq.~2.35]{Tol99},
 and is also given by de Boer \cite{BoLe96} as a result from formal perturbation theory
 in the limit of very small lateral correlation length.
-With \cref{ESab2}, we obtain the refraction matrix
-\begin{align}\label{ESabGibaud}
-\tilde S_{ab} &= \frac{1}{2\kappa_a} \begin{pmatrix}
-  \left( \kappa_a + \kappa_b \right) \NCm &
-  \left( \kappa_a - \kappa_b \right) \NCp\\[.2cm]
-  \left( \kappa_a - \kappa_b \right) \NCp &
-  \left( \kappa_a + \kappa_b \right) \NCm
-\end{pmatrix},
-\end{align}
+With \cref{ESab2}, we obtain the coefficients to replace~\cref{Dslpm},
+\begin{equation}\label{EslpmNC}
+  s^\pm_l = (1 \pm \kappa_{l-1}/\kappa_l) \exp(-(\kappa_{l-1}\mp\kappa_l)^2\sigma^2/2),
+\end{equation}
 which shows that the above is also consistent with \cite[Eq.~3.114]{GiVi09}.
 
 However, the total reflected and transmitted flux
 $\kappa_a|\tilde r_{ab}|^2+\kappa_b|\tilde t_{ab}|^2$,
 computed as in \cref{EConservation},
 is \emph{greater} than the incoming flux~$\kappa_a$.
-This takes all credibility from \cref{EtNC} and \cref{ESabGibaud}.
+This takes all credibility from \cref{EtNC} and \cref{EslpmNC}.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Scattering by a rough interface}\label{Sroughscatter}
diff --git a/Doc/DevRef/fig/funcplot/tanhc.ps b/Doc/DevRef/fig/funcplot/tanhc.ps
index 70c14e68ffc..8effb78624d 100644
--- a/Doc/DevRef/fig/funcplot/tanhc.ps
+++ b/Doc/DevRef/fig/funcplot/tanhc.ps
@@ -1,5 +1,5 @@
 %!PS-Adobe-1.0 EPSF-1.0
-%%BoundingBox:    14   604   258   788
+%%BoundingBox:     7   604   259   788
 %%Comment: Bounding box extracted by bboxx
 %%+:       A program by Dov Grobgeld 2003
 %%Comment: Bounding box extracted by bboxx
@@ -1630,11 +1630,10 @@ WuGdict18a begin
    { 13 1 0 1. 1. pset  6 ipCol }
    ] def
 
-{ 3 aCol1 iColA } /icCol x bind def % number of colours and colour style
+{ 7 aCol3 iColA } /icCol x bind def % number of colours and colour style
 /cStyles [
-   { 1. [] lset 0 icCol }
-   { 1. [] lset 1 icCol }
-   { 1. [] lset 2 icCol }
+   { 2. [] lset 0 icCol }
+   { 2. [] lset 1 icCol }
    ] def
 
 Resets
@@ -1646,8 +1645,8 @@ BoxBackground
 
 %% output created by Frida version post-2.4.4a
 
-1 0.03 30 xSetCoord
-1 0.02 2 ySetCoord
+1 0.02 50 xSetCoord
+1 0.008 2 ySetCoord
 
 % x axis:
 [
@@ -1655,258 +1654,449 @@ BoxBackground
   1 wx {(1)}
   10 wx {(10)}
   ] SetTacVec
-0.01 wx 100 wx 5 9 SetTicVecLog
+0.01 wx 100 wx 5 3 SetTicVecLog
  0 10   0  0     0  90 OneAxx Axx Tic Tac xNumL %% low x axis
  0 10   0 10     0 270 OneAxx Axx Tic Tac       %% top x axis
 {(x)} xCL
 
 % y axis:
 [
+  0.01 wy {(0.01)}
   0.1 wy {(0.1)}
   1 wy {(1)}
   ] SetTacVec
-0.01 wy 10 wy 4 9 SetTicVecLog
+0.001 wy 10 wy 5 9 SetTicVecLog
  0 10   0  0    90   0 OneAxx Axx Tic Tac yNumL %% left y axis
  0 10  10  0    90 180 OneAxx Axx Tic Tac       %% right y axis
-{(tanhc x)} yCL
+{(f(x))} yCL
 
+24 setown
+3 5 50 {(tanh x)} rtextCM
+7 5 -50 {(tanhc x)} rtextCM
 
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+  1 [ ] zValues
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-  {(  cc tanh(t)/t # z from grid1x7)} TxLine
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+  {(  cca tanh(t))} TxLine
+  {(  tanh(t))} TxLine
+  {(  weighing: with reciprocal variance (data and curve))} TxLine
+  {(  j  oc chi^2                 1-R^2)} TxLine
+  1 {(  0  0                     1                     )} CvTxLine
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-  {(  data file: 3, weighing: constant)} TxLine
-  {(  j   z0           oc chi^2                 1-R^2)} TxLine
-  1 {(  0.1           0  0                     1                     )} CvTxLine
+  {(  weighing: with reciprocal variance (data and curve))} TxLine
+  {(  j  oc chi^2                 1-R^2)} TxLine
+  2 {(  0  0                     1                     )} CvTxLine
   {(plot -> /home/jwu/gnew/L1.ps)} TxLine
 } oooinfo 1 eq { exec } { pop } ifelse
 
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