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We demonstrate a high power,high brightness,slab amplifier based on face-pumped Nd∶ YAG slab gain modules,having a high efficient hybrid cooling system of the conduction cooling and forced convection cooling.Using a single gain module,a laser output power up to 4.5 kW with a remarkable optical-optical conversion efficiency of 51% is realized,indicating an excellent lasing performance of the Nd∶YAG slab module.The amplifier operates at a repetition rate of 700Hz and delivers a maximum average output power exceeding 10.5 kW with pulse duration of 150μs.A good beam quality factor is measured to be β =1.9.To the best of our knowledge,this is the highest brightness for a 10 kW level Nd∶YAG slab amplifier.