We study the grating-assisted light-emitting diode, an LED design for high brightness based on a resonant cavity containing one- or two-dimensionally periodically corrugated layers (grating). We give in detail a generally applicable electromagnetic analysis based on the rigorous coupled-wave theory to calculate the extraction efficiency of spontaneous emission in a periodically corrugated layer structure. This general model is then specified on the grating-assisted resonant-cavity LED, showing simulated efficiencies of more than 40%.