Spatial chirp in the focusing of few-optical-cycle pulses by a mirror

We calculate the electric field of few-optical-cycle pulses at the focus of a perfectly conducting mirror by adding coherently the Airy diffraction patterns for each pulse frequency. We will show that the pulse suffers temporal spreading generated by a change in the spectrum of the pulse as a function of position in the focal plane, which introduces spatial chirp to the pulse. A double pulse appears near to the diffraction minima of the carrier frequency due to the variations in the spectra.