\section{Electron Beam Injection}

The electron beams with a charge of 4 nC and a normalized emittance of less than 10 mm mrad are expected to be generated in the photocathode RF gun for injector linac of SuperKEKB HER injection. To high-current, low-emittance beams generation, the laser source with mJ pulse energy, center wavelength of 260 nm and a pulse width of 20 ps are required at the A-1 unit (Fig.~\ref{electron1}). Further its pulse shape should be reshaped to rectangle structure to obtain the required energy spread of 0.1 \%.

The thermionic electron gun is rearranged to the upstairs for the 10 nC primary electron beam for the positron generation as shown in Fig.~\ref{electron2}.

 \begin{figure}[hbt]
 \begin{center}
  \includegraphics[width=0.8\textwidth]{fig/electron1.jpeg}
  \caption{Layout of RF-gun and thermionic electron gun.}
  \label{electron1}
 \end{center}
\end{figure}

 \begin{figure}[hbt]
 \begin{center}
  \includegraphics[width=0.8\textwidth]{fig/electron2.pdf}
  \caption{Layout of injector part.}
  \label{electron2}
 \end{center}
\end{figure}

\subsection{High charge and low emittance electron beam}

 The acceptable normalized emittance for the SuperKEKB injection is 20 mm mrad. To achieve this emittance with the initial emittance of 10 mm mrad from RF-gun, the emittance dilution of 10 mm mrad is acceptable for the entire linac.

The advanced RF-Gun, which adopts the quasi-traveling wave annular coupled cavity, provides the high charge and low emittance beam for SuperKEKB. The optimum bunch length at RF-Gun for the low emittance electron beam is around 20 ps against the space charge effect. The longitudinal bunch shape was controlled by the laser system to obtain the required energy spread. After the RF-Gun, it is compressed to 10 ps using a chicane section.

The transverse wakefield becomes the main source of the emittance dilution for the high bunch charge of 4 nC after the beam orbit is well corrected in the LINAC beam optics. And the energy spread is determined by the longitudinal wakefield strength. The smaller alignment tolerance and the shorter bunch length can reduce the transverse wakefield strength. However, the shorter bunch length causes the stronger longitudinal wakefield.

Figure~\ref{electron3} shows the analytical calculation of the projected emittance in several cases of the bunch length and the alignment error. The bunch lengthes of 4 ps and 10 ps are required for the alignment error of 0.3 mm and 0.1 mm respectively from Fig.~\ref{electron3}. Since our normal bunch length is 10 ps, we are working to reach the alignment error of 0.1 mm for each local section which consists of 8 units. However, it seems very hard to keep this small alignment error in our old support structure and the frequent earthquake. Thus we have to consider the 4 ps option for the alignment error of 0.3 mm.

 \begin{figure}[hbt]
 \begin{center}
  \includegraphics[width=0.75\textwidth]{fig/electron3.pdf}
  \caption{Analytical calculation of the projected emittance in several cases of the bunch length and the alignment error.}
  \label{electron3}
 \end{center}
\end{figure}

Further, the initial offset can compensate the transverse wakefield to recover the emittance dilution. The initial offset is taken as the two steering magnets with 90 degree phase difference of the beta function for each axis. Figure~\ref{electron4} shows the emittance optimization using the initial offset scan. It is realized using the RF deflector to observe the longitudinally sliced bunch shape.

 \begin{figure}[hbt]
 \begin{center}
  \includegraphics[width=0.4\textwidth]{fig/electron4.pdf}
  \caption{Initial offset to optimise the projected emittance.}
  \label{electron4}
 \end{center}
\end{figure}

The shorter bunch can suppress the emittance dilution due to the transverse wakefield. However, it causes a stronger longitudinal wakefield. The RF phase becomes deeper and deeper corresponding to the longitudinal wakefield strength to suppress the energy spread. Thus the bunch length was chosen by those balances.

The initial bunch length of 20 ps generated by the photocathode RF-Gun is compressed to 10 ps using the chicane at the low energy section. And the J-ARC section is placed at the energy of 1.5 GeV. This J-ARC optics was changed to the achromatic condition from the original isochronous design to obtain certain R56 to compress the bunch as shown in Fig.~\ref{electron5}.

 \begin{figure}[hbt]
 \begin{center}
  \includegraphics[width=0.9\textwidth]{fig/electron5.pdf}
  \caption{Achromatic design of J-ARC section.}
  \label{electron5}
 \end{center}
\end{figure}

 We chose the positive R56 of 0.3, since negative R56 makes low energy tail and it consumes more RF slope to recover the energy spread against the longitudinal wakefield. Figure~\ref{electron7} shows the measured R56 of J-ARC using the RF phase variation of C-1 unit after J-ARC corresponding to the energy knob of B-5 and B-6 unit. This inclination leads the R56 of 0.23. It is slightly different from the designed value due to the fudge factor of the quadrupole magnets. However, the measured value is enough to determine the RF voltage for the designed bunch compression factor. One RF unit (B-7) with zero crossing RF phase is used to make the energy slope for the bunch compression. The bunch compression is performed only for the SuperKEKB HER injection since the timing of this RF unit for the energy slope is thrown away to the standby timing for the positron injection and PF injection. To compress the bunch length from 10 ps to 4 ps, the RF voltage becomes 60 MV from eq. 1. Thus the RF power of B-7 unit reduced to 8 MW.
 
$$
\sigma_z - \sigma_z^\prime = R_{56} \delta, 
$$
$$
\delta = \cfrac{\Delta p}{p} = \cfrac{V_{1unit} \times 2\pi \times \cfrac{\sigma_z}{\lambda}}{p},
$$
$$
\cfrac{V_{1unit}}{p} = \cfrac{{\lambda} \left( 1 - \cfrac{\sigma_z^\prime}{\sigma_z} \right) }{2{\pi}R_{56}}.
$$

 \begin{figure}[hbt]
 \begin{center}
  \includegraphics[width=0.7\textwidth]{fig/electron7.pdf}
  \caption{Measured R56 of J-ARC using the RF phase variation.}
  \label{electron7}
 \end{center}
\end{figure}

Figure~\ref{electron8} shows the beam tracking simulation of the transverse emittance growth with the alignment error of 0.3 mm with and without the bunch compression in the J-ARC~\cite{electron2}. The compressed bunch length of 4 ps leads to suppress the emittance growth.

 \begin{figure}[hbt]
 \begin{center}
  \includegraphics[width=0.6\textwidth]{fig/electron8.pdf}
  \caption{Beam tracking simulation with and without bunch compression in the J-ARC.}
  \label{electron8}
 \end{center}
\end{figure}


\subsection{Longitudinal wakefield}

The required energy spread of 0.1 \% was achieved for the bunch charge of 1 nC at the previous KEKB operation. However, the required bunch charge becomes 4 nC for SuperKEKB and it induces the stronger wakefield. Further, the stronger longitudinal wakefield induced by the shorter bunch causes the larger energy spread~\cite{electron1}. Fortunately, the energy spread at the optimum RF phase dramatically decreases using the uniform charge density bunch as shown in Fig.~\ref{electron9}.

As described before, the shorter bunch length suppresses the emittance dilution due to the transverse wakefield. However, it causes a larger energy spread due to the longitudinal wakefield. Also, the shorter bunch length requires the deeper RF phase to compensate the longitudinal wakefield and it causes a lower accelerating gradient.

Thus the uniform charge distribution must be required for any bunch length to reach the energy spread of 0.1 \% at 4 nC operation. The longitudinal bunch shape is controlled by the laser system using the pulse stacking method or the chirped pulse amplification method.

 \begin{figure}[hbt]
 \begin{center}
  \includegraphics[width=0.8\textwidth]{fig/electron9.pdf}
  \caption{Minimum energy spread and optimum phase.}
  \label{electron9}
 \end{center}
\end{figure}


