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The optical designs of resonant GaN light-emitting diodes (LEDs) have been determined for maximum extraction efficiency into typical plastic optical fiber of numerical aperture 0.5. An optimum extraction efficiency of 3.9% can be achieved for a practical resonant cavity LED (RCLED), taking account of current growth and processing considerations. The optimized device is a metal-active layer distributed Bragg reflector construction. Constructive interference effects from the top metal mirror are found to play the dominant role in efficiency enhancement. The extraction efficiency of an optimized resonant single-mirror LED is found to be 3.3%, indicating a small compromise in performance compared with the more complex RCLED structure.