  We present a simulation capability for micro-scale light-emitting diodes
(uLEDs) that achieves comparable accuracy to CPU-based finite-difference
time-domain simulation but is more than 10^7 times faster. Our approach is
based on the Fourier modal method (FMM) -- which, as we demonstrate, is well
suited to modeling thousands of incoherent sources -- with extensions that
allow rapid convergence for uLED structures that are challenging to model with
standard approaches. The speed of our method makes the inverse design of uLEDs
tractable, which we demonstrate by designing a metasurface-enhanced uLED that
doubles the light extraction efficiency of an unoptimized device.
