Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures

Starting from some general and plausible assumptions based on geometrical optics and on a common feature of the truncated Bessel beams, a heuristic derivation is presented of very simple analytical expressions capable of describing the longitudinal (on-axis) evolution of axially symmetric nondiffracting pulses truncated by finite apertures. The analytical formulation is applied to several situations involving subluminal, luminal, or superluminal localized pulses, and the results are compared with those obtained by numerical simulations of the Rayleigh-Sommerfeld diffraction integrals. The results are in excellent agreement. The present approach can be rather useful, because it yields, in general, closed-form expressions, avoiding the need for time-consuming numerical simulations, and also because such expressions provide a powerful tool for exploring several important properties of the truncated localized pulses, such as their depth of fields, the longitudinal pulse behavior, and the decaying rates.