Study of Gaussian and Bessel beam propagation using a new analytic approach

The main feature of Bessel beams realized in practice is their ability to resist diffractive effects over distances exceeding the usual diffraction length. The theory and experimental demonstration of such waves can be traced back to the seminal work of Durnin and co-workers already in 1987.Despite that fact, to the best of our knowledge, the study of propagation of apertured Bessel beams found no solution in closed analytic form and it often leads to the numerical evaluation of diffraction integrals, which can be very awkward. In the context of paraxial optics, wave propagation in lossless media is described by an equation similar to the non-relativistic Schrödinger equation of quantum mechanics, but replacing the time t in quantum mechanics by the longitudinal coordinate z. Thus, the same mathematical methods can be employed in both cases. Using Bessel functions of the first kind as basis functions in a Hilbert space, here we present a new approach where it is possible to expand the optical wave field in a series, allowing to obtain analytic expressions for the propagation of any given initial field distribution. To demonstrate the robustness of the method two cases were taken into account: Gaussian and zeroth-order Bessel beam propagation.