@@ -100,25 +100,25 @@ If $\beta$ doesn't change much along the traveling wave structure (ultra relativ
\end{multline*}
Here $\beta c =2.9886774\cdot10^8\;\text{m s}^{-2}$, $\omega=2\pi\cdot1.4989534\cdot10^9$~Hz and, the cell length, $s =0.06\bar{6}$~m. To maximize this energy we have to take the derivative with respect to $\varphi$ and set the result to $0$. We split the field up into the core field, $E_z^{(1)}$ and the fringe fields (entry fringe field plus first half cell concatenated with the exit fringe field plus last half cell), $E_z^{(2)}$. The core fringe field is periodic with a period of $3\,s$. We thus find
In the last equal sign we used the fact that both functions, $n(\frac{\omega}{\beta c}z)$ and $E_z^{(1)}$ have a periodicity of $3\cdot s$ to shift the boundaries of the integral.
In the last equal sign we used the fact that both functions, $\sin(\frac{\omega}{\beta c}z)$ and $E_z^{(1)}$ have a periodicity of $3\cdot s$ to shift the boundaries of the integral.
Using the convolution theorem we find
\begin{equation*}
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@@ -129,7 +129,7 @@ where
\begin{align*}
g(z) & =
\begin{cases}
-n\left(\omega\left(\frac{z}{\beta c} + t_{0}\right)\right)\qquad& 0 \le z \le 3\cdot s\\
-\sin\left(\omega\left(\frac{z}{\beta c} + t_{0}\right)\right)\qquad& 0 \le z \le 3\cdot s\\
The phase lag [{rad}] (default: 0). In \textit{OPAL-t} this phase is in general relative to the phase at which the reference particle gains the most energy. This phase is determined using an auto-phasing algorithm (see~Appendix~\ref{autophasing}). This auto-phasing algorithm can be switched off, see \texttt{APVETO}.
\end{kdescription}
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@@ -1547,7 +1547,7 @@ placed using the \texttt{RingDefinition} element.
The phase lag [{rad}] (default: 0). In \textit{OPAL-t} this phase is in general relative to the phase at which the reference particle gains the most energy. This phase is determined using an auto-phasing algorithm (see~Appendix~\ref{autophasing}). This auto-phasing algorithm can be switched off, see \texttt{APVETO}.
\node at ($(1.4*\sinAlpha,-1+1.4*\cosAlpha)+(-0.05,0.2)$) {\Large{$E_2$}};
% Label reference trajector entry and exit points.
\node[fill=white] at (0.3, 0.15) {\Large{$O_{entrance}$}};
\node[fill=white, below=12pt, right=3pt] at ($(0,-1)+1.1*(nAlpha,\cosAlpha)$) {\Large{\textbf{$O_{exit}$}}};
\node[fill=white, below=12pt, right=3pt] at ($(0,-1)+1.1*(\sinAlpha,\cosAlpha)$) {\Large{\textbf{$O_{exit}$}}};
\end{tikzpicture}
\end{center}
\caption{Illustration of a rectangular bend (\texttt{RBEND}, see Section~\ref{RBend}) showing the entrance and exit fringe field regions. $\Delta_{1}$ is the perpendicular distance in front of the entrance edge of the magnet where the magnet fringe fields are non-negligible. $\Delta_{2}$ is the perpendicular distance behind the entrance edge of the magnet where the entrance Enge function stops being used to calculate the magnet field. The reference trajectory entrance point is indicated by $O_{entrance}$. $\Delta_{3}$ is the perpendicular distance in front of the exit edge of the magnet where the exit Enge function starts being used to calculate the magnet field. (In the region between $\Delta_{2}$ and $\Delta_{3}$ the field of the magnet is a constant value.) $\Delta_{4}$ is the perpendicular distance after the exit edge of the magnet where the magnet fringe fields are non-negligible. The reference trajectory exit point is indicated by $O_{exit}$}
