We present a study of undoped AlGaN/GaN separate confinement heterostructures designed to operate as electron beam pumped ultraviolet lasers. We discuss the effect of spontaneous and piezoelectric polarization on carrier diffusion, comparing the results of cathodoluminescence with electronic simulations of the band structure and Monte Carlo calculations of the electron trajectories. Carrier collection is significantly improved using an asymmetric graded-index separate confinement heterostructure (GRINSCH). The graded layers avoid potential barriers induced by polarization differences in the heterostructure and serve as strain transition buffers which reduce the mosaicity of the active region and the linewidth of spontaneous emission.