On the asymptotic evolution of finite energy Airy wavefunctions

In general, there is an inverse relation between the degree of localization of a wavefunction of a certain class and its transform representation dictated by the scaling property of the Fourier transform. We report that in the case of finite energy Airy wavepackets a simultaneous increase in their localization in the direct and transform domains can be obtained as the apodization parameter is varied. One consequence of this is that the far field diffraction rate of a finite energy Airy beam decreases as the beam localization at the launch plane increases. We analyse the asymptotic properties of finite energy Airy wavefunctions using the stationary phase method. We obtain one dominant contribution to the long term evolution that admits a Gaussian-like approximation, which displays the expected reduction of its broadening rate as the input localization is increased.