By solving the time-dependent Dirac equation numerically, we calculate the rate of electron-positron pairs that are created from the vacuum by an oscillating field. The data permit us to examine the validity of approximate, but analytical, expressions derived from the WKB-based description. If two characteristic parameters (E/c(3) and omega/c(2)) are less than 1 or the parameter gamma(=omega c/E) is small (tunneling region), this description predicts the true rate well, where E and omega are the amplitude and frequency of the external field. In the multi-photon region (where E/c(3) is small and gamma is large), perturbative methods become valid. (C) 2015 Optical Society of America
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