Electrostatic quadrupole lens for focusing a cryogenic $^{205}$TlF molecular beam

Precision measurements with molecules require molecular beams with high flux and narrow divergence. Here we present the design, implementation, and characterization of an electrostatic quadrupole lens (EQL) for focusing a cryogenic beam of $^{205}$TlF molecules. The EQL consists of four electrodes operated at spatially alternating potentials up to $\pm 30$ kV. This configuration generates a transverse restoring force that focuses molecules prepared in selected manifold of hyperfine states. At the optimal applied voltage, the EQL increases the detected molecular signal by a factor of $14.4(4)$ at the current detection position. The measured lens gain, transverse Doppler spectra, and transverse spatial profiles are all reproduced by trajectory simulations using independently measured beam parameters as input. Based on this agreement, we project a gain of $16.5(8)$ for the full $7.9~\mathrm{m}$ CeNTREX beamline. These results establish the EQL as a key component for increasing sensitivity in precision experiments using cryogenic beams of polar molecules.