Not yet reviewed by Pith; the record is open.

Laser-Wakefield Accelerators (LWFA) running at kHz repetition rates hold great potential for applications. They typically operate with low-energy, highly compressed laser pulses focused in high-pressure gas targets. Experiments have shown that the best-quality electron beams are achieved using Hydrogen gas targets. However, continuous operation with Hydrogen requires a dedicated pumping system. In this work, we present a method for designing a differential pumping system, which we successfully implemented in our experiments. This enabled the first demonstration of continuous operation of a kHz LWFA using a high-pressure Hydrogen gas jet. The system effectively maintained a pressure below 3e-4 mbar, even with a free-flowing gas jet operating at 140 bar backing pressure. Numerical fluid dynamics and optical simulations were used to guide and validate the system's design.