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We develop a two-stage end-pumped Nd∶YVO4 amplifier seeded by a passively Q-switched microchip laser.An average output power of 13.5W with repetition rate up to 7kHz and pulse duration of ～1.24ns is obtained,corresponding to a pump extraction efficiency of 16.1％ (19.5％ for the second stage) and peak power of ～1.5 MW.The beam quality factors at maximum output power are measured to be Mx2 =1.56 and My2 =1.48.We introduce an analytical model to estimate gain and beam quality after amplification.This model focuses on the influence of ratio of seed spot radius to pump spot radius when designing an amplifier.Moreover,our experiments reveal that the re-imaging system in the double-pass configuration can be used to enhance the beam quality.