Trajectory analysis of phase effects associated with truncated Airy beams

. Airy wave packets constitute a very peculiar type of structured light: during their propagation, their transverse profile undergoes a self-accelerating displacement while it remains shape invariant. They are thus the only non-dispersive beam-type solution to the Helmholtz paraxial equation in free space. Such properties are possible by virtue of their infinite power content. However, experimentally, Airy beams can only be reproduced in an approximate manner, with a limited extension and hence a finite power content. To this end, di ff erent cuto ff procedures have been reported in the literature, based on a convenient tuning of the transmission properties of aperture functions. In this Communication, we present and discuss our latest advances in the analysis of the e ff ects that convolving an Airy beam with di ff erent aperture functions have on their propagation properties. More specifically, we make use of a trajectory-based methodology, which allows us to analyze and explain the beam propagation in terms of trajectories directly connected with the beam local phase variations.


Introduction
The last decade has seen the rise of a very active and promising field of research centered around the concept of structured light, motivated by the interesting fundamental and technological applications brought in by conveniently designed light beams [1,2].Hence, not only the methods aimed at designing and producing new types of structured light beams are important in the field, but also any analytical tool devised to investigate their properties.
Airy beams constitute a peculiar type of structured light, as they are the only non-diffracting solutions in free space of the paraxial Helmholtz equation.Specifically, during their propagation, their transverse intensity distribution undergoes a self-acceleration, while it remains shape invariant.This is possible due to their infinite power content.To make them practical, different types of finiteenergy Airy-type beams have been proposed in the literature [3][4][5] that still keep the essence of the original ones, although within a certain spatial range.
In this Communication, we discuss the behavior exhibited by some feasible finite-energy solutions with the aid of flux trajectories [6], which provide us with a neat trajectory-based picture of the beam degradation process as it propagates.Specifically, it is seen that such a degradation is related to a continuous back-flow due to the loss of the infinite tail present in non-truncated beams, which is the cause behind the accelerated push forwards of the whole of the wave packet.This is just the tip of the iceberg: the trajectories reveal us how small local variations in the phase lead to important consequences in the beam behavior.

Local phase variations and flow trajectories
Let us consider the one-dimensional Helmholtz equation in terms of the reduced coordinates x = x/x 0 (x 0 denotes an arbitrary typical value for the transverse coordinate) and z = z/kx 2 0 = λ 0 z/2πnx 2 0 [7], (to simplify notiation, tildes are removed).As Berry and Balazs showed [8], for wave packets with an initial amplitude described by an Airy function, propagating according to Eq. ( 1), their amplitude remains shape invariant and undergoes a transverse displacement that increases rapidly with z 2 : Alternatively, the propagation of these beams can be described at a local level in terms of trajectories associated with the transverse energy flux [9], Specifically, the trajectories arise from the integration along z of the equation of "motion" where v(x, z) is the local field describing the energy flow along the transverse direction (x) at a given value of the longitudinal coordinate (z), and Because the field ( 4) is directly related to the beam local phase variations along the transverse direction, it becomes a suitable tool to explore any property associated with the phase and, more specifically, with effects due to possible variations induced on it [6,10].Note that for Airy beams, v(x, z) depends linearly on the z-coordinate and the associated trajectories are fully analytical: These flux trajectories are parabolas, in agreement with the renowned parabolic acceleration undergone by the beam.
If an exponentially decaying cutoff is introduced [3], then the solution of Eq. (1) becomes with y = x − z 2 /4 + iγz, which leads to the appearance of an extra phase factor in v(x, z): This extra factor is directly connected with the new local phase configuration induced by the exponential cutoff.Apart from making finite the beam power content, the above extra phase factor is also responsible for the gradual dispersion of the beam, which will progressively lose both the shape invariance and the transverse self-acceleration.In this regard, the trajectory-based approach is going to constitute a suitable tool to detect and explore any change directly related to the phase at a local level in the case of finite-energy Airy-type beams with an initial functional form These beams represent the convolution of the usual Airy beam with a given window function w(x), satisfying two properties: (i) it is L 2 in order to warrant the finiteness of the power content of ψ(x), and (ii) under some control parameter it approaches a Dirac δ-function, which allows us to recover the original infinite energy Airy beam.After propagation, the amplitude (8) becomes which leads to an expression for v(x, z) with the same functional form as (7), with the extra phase factor containing information about the window function and its effects.
Accordingly, we have investigated the dynamics generated by the following window functions Given the non-analiticity of the problem in all these cases, we have carried out a series of numerical simulations with the values considered in [3,6].

Concluding remarks
We have investigated the phase effects induced by the truncation of Airy beams by means of different window functions.Unlike other more conventional approaches, here we have focused on the energy flux at a local level, i.e., on what happens at precise spatial positions with the flow of energy, which is possible because of the trajectory-based formulation considered.Thus, by using these transverse flux trajectories, we have been able to elucidate how fast the well-known properties associated with Airy beams are lost, which gives us a measure, to some extent, of the efficiency of the designed truncated Airy-type beams.Moreover, the trajectories also allow us to investigate the form in which the flux is dissipated along the back direction (opposite to the beam's forward self-accelerated displacement), which depends on the particular shape of the window function.