The baseline design of the SwissFEL injector foresees the "PSI Gun 1", a 2.6-cell RF photo-cathode gun operating at S-band frequency, as the electron source. In this paper a new design is presented where a 5.6-cell C-band gun could replace the PSI Gun 1 with no impact on the rest of the injector setup. A conservative maximum gradient of 135 MV/m at the cathode is assumed which drives the electron beam faster into the relativistic regime and therefore allows to tolerate larger charge densities. The presented solution also foresees a coaxial RF coupling from the cathode side in order to place the gun solenoid as near to the photo-cathode as possible, improving the emittance compensation. Astra simulations showed that the transverse beam brightness can be doubled before the first bunch compressor preserving the low transverse emittance value as compared to the current design for the S-band injector configuration of SwissFEL.
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