We report results of studies of laser beam divergence, laser beam intensity profiles and spatial and temporal laser pulse characteristics for a rectangular bore strontium recombination laser operating on the 430.5 nm Sr transition at high pulse rates (3-10 kHz). Beam divergence is found to be very nearly diffraction limited for a flat-flat optical cavity, with a very uniform spatial distribution of laser intensity under optimum operating conditions. Laser pulse characteristics provide an indication of plasma conditions across the laser tube.