1. Introduction

Since Airy optical beams were first observed [1, 2] intense research has been carried out on this finite-energy solution of the paraxial Helmholtz equation. For example, based on the characteristics of the intensity pattern of Airy light beams, optical micromanipulation has recently been demonstrated [3, 4]. So due to the possibility of removing microparticles and cells from a section of a sample chamber, in these papers new perspectives were opened in the field of redistribution of particles and cells between different media. Furthermore, using femtosecond Airy beams, curved plasma channels have been experimentally shown to be generated in air, and these are parabolic for the dominant intensity of the beam [5]. Moreover, a new way of generating Airy beams using three-wave mixing processes in asymmetric nonlinear photonic crystals opens up new possibilities for all-optical switching and manipulation of Airy beams [6]. In this sense, this methodology allows the creation of Airy beams at new wavelengths and high intensities, which is impossible with the conventional (linear) methods.

Other interesting properties of these beams have been described such as the ballistic dynamics of the beams akin to projectiles moving under the action of gravity [7], the self restoration properties (self-healing) after passing small obstacles or propagating in adverse environments [8], the transformation by paraxial optical systems (that can be described by four complex parameters) [9], or the evolution of the Poynting vector and angular momentum when the beam propagates through space [10].

In all previous studies, analysis of the propagation of these beams is carried out by means of the paraxial Helmholtz equation with boundary conditions at the focal plane (see Fig. 1). Hence, the spatial evolution of Airy beams takes place only in +Z direction.

 

Fig. 1. Schematic diagram of the generation of Airy beams.

Download Full Size | PDF

Thus, in this paper in contrast to other studies, we use the nonparaxial vectorial diffraction integrals, which are an exact solution of the Maxwell equations and the complete Helmholtz wave equation given by:

The exact vectorial Helmholtz equation is solved using the boundary conditions on a diffraction aperture situated at z=0 plane (see Fig. 1). Therefore, using this methodology several effects (that in the paraxial Helmholtz equation solution are not obtained) can be analyzed such as the propagation in the whole space. Moreover, in this work we propose the existence of a new family of Airy beams (SAiry beams) with finite energy that can be generated at the focal plane of the lens due to diffraction by a circular aperture of a spherical wave modified by a cubic phase. For the SAiry beams, the finite energy is ensured by principle of conservation energy of a diffracted beam.

2. Nonparaxial vectorial diffraction integrals

The solution to the Maxwell equations (and as consequence the Helmholtz equation [Eq. (1)]), valid through the half-space z>0 (see Fig. 1), based on the knowledge of boundary conditions E⃗_o on the plane z=0:

is given by [11]:

where k is the wavenumber and R=[(x-xo)²+(y-yo)²+z ²]^1/2. The propagation properties of vectorial beams can be studied using the Eqs. (2) and (3). Assuming that after the lens (for analysis of experimental set-up see for example references [1, 3, 4]) we obtain an on-axis polarized, monochromatic spherical wave modified by an aperture function given by: $t(x_o,y_o)=\{\begin{array}{cc}\exp \left(i\beta \left(x_o^3+y_o^3\right)\right)\exp (-\frac{x_o^2+y_o^2}{{\mathrm{wo}}^2})& \mathrm{if}(x_o,y_o)\in D\\ 0& \mathrm{else}\end{array}$ where a is the aperture radius and D={(x_o,y_o)\(x ² _o+y ² _o)≤ a ²}.

The field at the plane after the aperture may be represented as:

where $\mathrm{zo}=\sqrt{f^2-a^2,}$, f is the focal length of the lens and α is the linearly polarized angle. In this work, we are going to analyze two types of diffraction patterns, the first one corresponds to those previously reported (see for example [1]) in which β≠0 and wo=finite (Airy beam) and the second one which is characterized by β≠0 and wo→∞ (SAiry beam). In both cases we assumed diffraction by a finite aperture as can be seen in the transmittance function.

The intensity distribution at any point (x,y) of a plane z>0 is given by:

The principle of conservation of energy in the solution to Maxwell’s equations ensures that total intensity must be equal and finite at all z-planes.

It is important to remark, that, from the analysis of Eq. (3), all the diffracted patterns on the half-space z>0 can be analyzed, including pre-focus (PRF) and post-focus (PF) regions (see Fig. 1). In this way, our method provides more complete information on the propagation characteristics of Airy beams which have not been previously analyzed due to the limitation of the paraxial Helmholtz equation (which can only be used at the PF region).

3. Numerical results and discussion

In order to analyze the propagation characteristics of the beams studied, we focus our attention on an electric field E⃗_o(x_o,y_o,0) which is x̂ polarized, so α=0, and Eq. (3) can be reduced to:

The parameters that we used in our numerical simulations are shown in Table 1 and were selected according to the experimental conditions mentioned in Ref. [1].

Table 1. Values of parameters used in the numerical simulations

View Table

3.1. Airy beams

These numerical simulations of Airy beams can be compared with the experimental and theoretical results reported in [1]. Thus, Figs. 2, 3 show the numerical simulations of diffraction patterns at different z planes in the PRF and PF zones for the Airy beam with parameters described in Table 1. The intensities have been normalized in such a way that the maximum value of intensity of the Airy beam at the focal plane is equal to 1.

As can be seen in Figs. 2(f)–2(h) an interesting result of the numerical simulations is that the Airy beam starts before the focal plane at the PRF region. This implies that the accelerating diffraction-free region that could be used in applications is 14 cm longer than predicted by the paraxial solution. These theoretical results justify the experimental measurements described in Ref. [5], where plasma channels start before the Fourier plane. Furthermore, it is important to remark that the behavior of the Airy beam is not symmetrical to the focal plane, indeed at the PRF region the diffraction free distance of the Airy beam is nearly 60 % of the PF region. In order to compare the numerical results with those obtained analytically using the paraxial Helmholtz solutions, Fig. 4 shows the normalized intensities obtained at different Z=(z-f) planes of the PF region. As can be observed when Z<19 cm the intensity patterns are identical in both methods. However (since the condition of paraxial approximation $\frac{{\partial }^{2}E}{\partial z^2}<<k^{2}E$ is not fulfilled), at Z>19 cm differences may be seen between the two method, such as the intensity distribution and the maximum intensity reached. However, these differences may greater when the a/f ratio tends to one, as was recently noted for spherical waves [1].

 

Fig. 2. Normalized intensity distribution for Airy beams at different Z=(z-f) planes of PRF region.

Download Full Size | PDF

In Fig. 5 the maximum values of intensity obtained at the PRF and PF regions are shown. As can be seen, the behavior of the maximum intensities is not symmetrical as regards the focal plane, and the slope of the growth zone in the PRF region is different to the decreasing slope in the PF region. Finally in Fig. 6 we show the parabolic trajectory of the Airy beam, where $r=\sqrt{x_{max}^2+y_{max}^2}$ is represented as a function of the propagation distance Z, and (x_max,y_max) are the Cartesian coordinates of the maximum intensity at each plane z. As can be seen the parabolic trajectory is observed in the -0.2 m 0.2 m interval, and there is no acceleration outside this region. It is important to remark that outside the -0.15 m, 0.15 m z interval the maximum intensity is lower than 80% of the maximum obtained at the focal plane, so the effects of the peak intensity for several applications could may be negligible although the trajectory is still parabolic in the -0.2 m 0.2 m interval as mentioned above. For values of |Z|>0.2 m the parabolic trajectory disappears.

3.2. SAiry beams

In the second part of this study we present the result of the numerical simulations for the SAiry beams. The proposal of this new type of beams is based on a simple idea. Since the Gaussian amplitude modulation used is selected to ensure the finite energy at different propagation planes, this can also be fulfilled by using an aperture that limits the incident wave and solving Maxwell’s equations with these boundary conditions. Figures 7 and 8 show the numerical simulations of diffraction patterns at different Z planes in the PRF and PF zones for the SAiry beam parameters described in Table 1. Since we did not use a Gaussian beam the amplitude of transmittance is equal to 1. Moreover, the intensities were normalized using the same conditions as for Airy beam. As result of this normalization, the maximum intensities are higher than 1 at the focal plane. This is a result of energy conservation because the total intensity at Z=0 plane is higher if |t(x_o,y_o)|=1onD than if $\mid t(x_o,y_o)\mid =\exp (-\frac{x_o^2+y_o^2}{{\mathrm{wo}}^2})\mathrm{onD}.$.

 

Fig. 3. Normalized intensity distribution for Airy beams at different Z=(z-f) planes of PF region.

Download Full Size | PDF

From these figures it may be deduced that accelerating beams can be obtained introducing only a cubic phase in a spherical wave. The differences in the diffraction patterns shown in Figs. 7–8 and 2–3 are exclusively due to the transmittance values (see Fig. 9) which modify the weight of the electric field E_o(x_o,y_o,o) in the integral 7. As can be seen the most significant difference between the transmittance that generates the Airy and that which generates the SAiry beams is that the high and low spatial frequencies have the same weight in SAiry beams, whereas the high spatial frequencies are nearly null for the Airy beams, and the contrast is lower for SAiry beams in the other of spatial frequencies.

As can be seen in Figs. 7–8 the region where the diffraction pattern of SAiry beams can be observed varies from -23 to 38 cm and in all cases the intensity of the main peak is higher than that the obtained in the same plane with an Airy beam as can be clearly observed in Fig. 5. Moreover it is important to point out that the behavior of the intensity is quite different between the two kinds of beams, for example, the maximum intensity is not obtained at the focal plane, but at a plane situated at -14 cm from the focus. Also the growth and decreasing slopes are very different in PRF and PF regions. Figure 6 shows the parabolic trajectory of the SAiry beam, which as can be seen is greater than that the observed for the Airy beam and approximately coincides in the -20, 20 cm Z region. Moreover, an observed drawback with SAiry beams is that the spread of the diffraction pattern is always wider than for Airy beams. In any case, from Figs. 5 and 6 it may be deduced that for some applications where the spread of the diffraction pattern is not important and the peak maximum and parabolic trajectory are the main parameters, SAiry beams are better than Airy beams.

 

Fig. 4. Normalized intensity distribution for Airy beams at different Z=(z-f) planes of the PF region numerically calculated (a),(c),(e),(g) and analytically obtained by means of the paraxial Helmholtz solutions [1],(b),(d),(f), (h).

Download Full Size | PDF

 

Fig. 5. Normalized maximum intensities of Airy beams (blue) and SAiry Beams (red) at PRF and PF regions

Download Full Size | PDF

 

Fig. 6. Transverse acceleration of Airy beams (blue) and SAiry Beams (red) at PRF and PF regions.

Download Full Size | PDF

 

Fig. 7. Normalized intensity distribution for SAiry beams at different planes Z=(z-f) of PRF region.

Download Full Size | PDF

4. Conclusions

We have analyzed Airy beams by solving the exact vectorial Helmholtz equation using the boundary conditions at a diffraction aperture and as a consequence, the diffracted beams were obtained in the whole space. The proposed method may be very useful if a/f parameters become close to unity or for the study of different types of linear polarized beams. We demonstrate theoretically that the parabolic trajectories are larger that those previously reported using the paraxial Helmholtz equation, showing that the Airy beams start to form before the Fourier plane. We also propose the possibility of using a new type of Airy beams (SAiry beams) with finite energy that can be generated at the focal plane of the lens due to diffraction by a circular aperture of a spherical wave modified by a cubic phase, and the finite energy is ensured by principle of conservation of energy of a diffracted beam. These SAiry beams may be a good choice for some applications where the spread of the diffraction pattern is not important but the peak maximum and the parabolic trajectory are the main parameters to be considered.

 

Fig. 8. Normalized intensity distribution for SAiry beams at different planes Z=(z-f) of PF region.

Download Full Size | PDF

 

Fig. 9. Real (a)–(b) and Imaginary (c)–(d) part of t(x_o,y_o) for SAiry and Airy beams respectively

Download Full Size | PDF