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When a laser wake field passes a sharp downward plasma density transition, a significant amount of plasma electrons are trapped and accelerated to relativistic high energies over a distance of the plasma wavelength. Phase space area of the plasma electrons remains constant during acceleration in the ion channel, but it was found that the phase space area, which is related with emittance, increases rapidly by the nonlinear space-charge field when the electron beam moves out of the plasma. As a result, the emittance of the accelerated electron beam grows severely during propagation in free space, and ultimately this limits the beam intensity. The emittance growth problem is investigated with two-dimensional particle-in-cell simulations, and several methods are suggested to suppress such a rapid emittance growth.