Increasing the luminosity of relativistic hadron beams is critical for the advancement of nuclear physics. Coherent electron cooling (CEC) promises to cool such beams significantly faster than alternative methods. We present simulations of 40 GeV/nucleon Au+79 ions through the first (modulator) section of a coherent electron cooler. In the modulator, the electron beam copropagates with the ion beam, which perturbs the electron beam density and velocity via anisotropic Debye shielding. In contrast to previous simulations, where the electron density was constant in time and space, here the electron beam has a finite transverse extent, and undergoes focusing by quadrupoles as it passes through the modulator. The peak density in the modulator increases by a factor of 3, as specified by the beam Twiss parameters. The inherently 3D particle and field dynamics is modeled with the parallel VSim framework using a $delta$f PIC algorithm. Physical parameters are taken from the CEC proof-of-principle experiment under development at Brookhaven National Lab.
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