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Broad-area lasers (BALs) with external Fourier-optical cavities with spatial filter for transverse mode selection have been studied experimentally and theoretically. The transverse mode structure of BALs is modeled using a three mirror cavity approach. The model accounts for gain saturation, carrier diffusion, and anti-guiding effects. Near- and far-field distributions are calculated and compared to experimental results. Transverse mode selection is achieved for BALs with a residual front facet reflectivity of 10% at low pump currents. For BALs having a very low front facet reflectivity of 0.001%, transverse modes can be selectively excited up to pump currents more than 200% above laser threshold. Calculations show that BALs having a 0.001%-antireflection coating with an external Fourier-optical cavity high above threshold can operate in a self-Q-switched-like mode with pulse durations of 2-4 ns and repetition rates of 100-200 MHz. Experimental results obtained with a hybrid integrated-optical external cavity for transverse mode selection are also presented.