ingest subsection from Stratified.tex

Doc/DevRef/Multilayers.tex

@@ -781,6 +781,45 @@ The following cases are treated seperately:
     of non-vacuum layer always contains an absorptive component.
 \end{itemize}
 
+%==================================================================================================%
+\subsection{Opaque layers and evanescent waves}\label{Sevawa}
+%==================================================================================================%
+
+TODO: Rework this fragment (ingested 29may23 from ba-intern/theory/Stratified.tex).
+
+For incident angles below the critical angle $\alpha_i < \alpha_c$
+(also at interfaces inside the sample), the $k_z$-component of the wave vector turns imaginary.
+This situation then corresponds to an evanescent wave.
+As a consequence, the phase factor $\delta$ \eqref{Ddell}
+turns real and describes the exponential decrease
+of the amplitudes $t_j$, $r_j$ in the corresponding layer.
+For large layer thicknesses, this means that $\delta$ rapidly approaches zero, and its inverse becomes very large.
+This leads to an increasingly ill-conditioned transfer matrix \eqref{Etransfer_matrix_BA}, until,
+at some point, both quantities underflow or overflow, leading to invalid numerical results.
+
+As the simulation approaches this singular situation, the amplitudes $t_j$ and $r_j$ will
+rapidly increase towards the top of a sample and potentially overflow at some point
+before the computation reaches the top layer.
+Intuitively, this can easily be understood as follows.
+As we impose the boundary condition $t_J = 1$, $r_J = 0$ in the substrate for incident
+angles below the critical angle with $T = 0$, $R = 1$, this means,
+that we must find $t_0 \to \infty$.
+Obviously, this cannot be implemented numerically in a sane manner.
+
+In order to detect and handle this over/underflow, BornAgain checks for an
+overflow of $t_j$.
+%As soon as this situation is encountered, the computation is stopped and
+%restarted in order to find the deepest layer where a valid numerical result is obtained.
+%Practically, this is done by a bisection algorithm for minimizing the computational load.
+
+It must be mentioned though, that this algorithm still fails if a very thick layer is encountered.
+This will lead to the immediate underflow of $\delta$ and hence to an overflow of~$\delta^{-1}$.
+This means, that the transmission of a single layer is within the numerical precision zero,
+but the mathematical formulation applied cannot handle this corner case and the computation still crashes.
+Therefore, the only way to circumvent this problem is by restarting the computation from the
+current layer by reapplying the boundary condition $t_j = 1, r_j = 0$ and
+setting all amplitudes in the layers below to zero.
+
 %===============================================================================
 \subsection{Flux, evanescent waves}\label{SSpecial}
 %===============================================================================