The stability of a plane wave propagating in single- and double-pass broad-area diode laser amplifiers is analyzed. It is found that laterally periodic perturbations exhibit a longitudinal variation that is a mixture of exponential growth and periodic oscillation. The latter oscillation has a spatial period related to the Talbot distance. Due to the interaction between these two effects, double-pass amplifiers become unstable against lateral perturbations, and the field spontaneously collapses into a periodic array of filaments. Experimental results are presented that show good agreement with the theoretical analysis.