An optical pump system for LED-pumped solid-state lasers is introduced to effectively couple the energy of pump sources to the active medium. The optical system is based on the coupling of optical rays from light-emitting diode sources to laser rod using high reflecting surfaces. Three different configurations with four, five, and six segments have been designed and their geometry is optimized with the 3D Monte-Carlo ray-tracing method to obtain the best performance. Moreover, a geometrical model is developed to describe the general behavior of the pump system. The pump systems have been fabricated and successfully applied to pump 3 mm diameter Ce:Nd:YAG laser rods with available white spectrum LEDs. The performance of the pump system is experimentally compared with the closed-coupled pumping configuration and white spectrum LEDs under equal pump energy and an identical optical resonator. The improved five and six segments pump systems when Epump>350mJ having more than 170 percent higher conversion efficiency than the closed-coupled configuration. Using six segments configuration and 500 mJ optical pump energy, the LED-pumped Ce:Nd:YAG laser oscillator produced laser spikes with a multi-mode beam structure and more than 700 μJ laser energy at 10 Hz repetition rate. A compact, high repetition rate >10 Hz, multi-milli joules LED-pumped solid-state lasers are possible by using the improved optical pump system and infrared LEDs.