An optimization of light extraction efficiency (LEE) from light-emitting diodes (LEDs) is provided. The optimization is achieved when both the cavity and waveguide modes are in their best conditions and their emitted waves interfere constructively in the grating-assisted structures. In other words, high LEE appears when the structural parameters results in good cavity modes in both planar and grating structures, and also when the propagation direction of the fundamental waveguide mode is significantly converted to the direction normal to the grating surface by the gratings. In this paper, we use two simple equations, one for cavity modes and the other for waveguide modes, to predict the loci of the optimal structural parameters of gratings. The prediction is verified with the rigorous coupled wave analysis (RCWA). We systematically present an investigation of the various structural effects of gratings on the LEE of the thin film vertical gallium-nitride (GaN) based LEDs. The results show that ultra-high LEE can be achieved when the equivalent cavity thickness is good for fundamental cavity mode, the source is positioned around antinodes for both cavity and waveguide modes, there is significant diffraction, and the fundamental waveguide mode is largely converted to the normal direction to the air, and the emitted waves from fundamental cavity and waveguide modes interfere constructively.
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