Pair production at the focus of two equal and oppositely directed laser beams: The effect of the pulse shape

We calculate the probability that an $e^{-}-e^{+}$ pair is created at rest from vacuum at the focus of two equal and oppositely directed pulsed laser beams. The effects of the finite duration and of the temporal behavior of the resulting laser pulse are taken into account perturbatively and the production probability is compared with the corresponding quantity calculated in the presence of an infinite laser beam with a constant pulse shape. By inserting theoretically achievable numerical values of the pulse parameters it is shown that the induced correction to the production probability cannot be neglected.

Figure 1

Semilog plot of the exact production probabilities (dots) calculated by evaluating Eq. (7) after integrating numerically the Schroedinger equation in the form (5) with the pulse shape (21) and of the approximated asymptotic production probabilities (continuous curve) calculated by means of Eq. (23). The probabilities are plotted as functions of the peak laser electric field $E$ expressed in unit of $E_1$ and the numerical values of all the parameters are those given below Eq. (24).