Analytic description of Airy-type beams when truncated by finite apertures

In this paper, we have developed an analytic method for describing Airy-Type beams truncated by finite apertures. This new approach is based on suitable superposition of exponentially decaying Airy beams. Regarding both theoretical and numerical aspects, the results here shown are interesting because they have been quickly evaluated through a simple analytic solution, whose characteristics of propagation has agreed with those already published in literature through the use of numerical methods. To demonstrate the method's potentiality, three different truncated beams have been analyzed: ideal Airy, Airy-Gauss and Airy-Exponential.


Introduction
In the last five years, a new nondiffracting wave has called attention: the Airy Beam [1][2][3][4]. Solution of the wave equation under paraxial approximation, the Airy Beam has ( †) E-mail addresses for contacts: mzamboni@dmo.fee.unicamp.br slightly different properties. Contrary to Bessel and Mathieu beams that keep their transverse shapes only, the Airy beam maintains that property but also it is observed that its main spot propagates according to a parabolic trajectory, which gives the idea of a bent propagation or a self-accelerated beam.
Like every ideal nondiffracting wave, the ideal Airy Beam can propagate over infinite distance resisting the diffraction effects, but it presents an infinite power flux through any plane orthogonal to the propagation direction.
To overcome this problem, Siviloglou and Christodoulides [1] obtained a finite energy Airy beam solution given by an ideal Airy beam modulated by an exponentially decaying function at z = 0.
Another possibility is to perform a spatial truncation on the ideal Airy beam (finite aperture generation). Actually, the spatial truncation is the most effective and realistic option, since every beam must be generated by finite apertures.
Until now, to the best of our knowledge, no one has ever studied propagation characteristics of truncated Airy-Type beams using any kind of analytic method once that all results reported in literature were done numerically [2] or experimentally [3]. In this paper we will present the first effort to this direction, describing the propagation of truncated Airy-Type beams in a homogeneous medium through an analytic approach.
The method here described is based on suitable superposition of exponentially decaying Airy beams. To support the method three different truncated beams have been analyzed: ideal Airy, Airy-Gauss and Airy-Exponential.
The results here presented are of interest in any possible application, theoretical or practical, that makes use of Airy-Type beams.

Analytic Description for Truncated Airy-Type Beams
In [1], Siviloglou et al. considered an initial field profile, at ζ = 0, given by as an initial condition to the electric field envelope equation in the paraxial regime and (1+1)D, i∂ ζ ψ + 1/2∂ 2 s ψ = 0, obtaining the following finite energy Airy beam where s = x/x 0 and ζ = z/kx 2 0 are the dimensionless transverse and longitudinal coordinates, with x 0 being the spatial spotlight and k = 2πn/λ 0 the wavenumber of the optical wave. We shall named the solution (2) as Airy-Exponential beam.
Before expose our method, it is important to notice that our goal is to describe, analytically, Airy-Type beams truncated by finite apertures at ζ = 0, i.e., we wish a solution describing (approximately) the evolution of fields of the type Ψ(s, ζ) = where m(s) is a function that modulates the Airy function within the finite aperture, −S ≤ s ≤ S, which is represented by the difference between the Heaviside functions θ(s + S) and θ(s − S). Here, S = X/x 0 is the dimensionless width of the aperture. Now we start to develop our method by considering the initial field profile, at ζ = 0, as the following superposition where B n and a n are complex constants.
Let us return to the initial field (3) and make the choice a n = a R + i2π n/L , where a R and L are positive constants. It is important to notice that a R is the same for all terms in the summatory. Therefore, the initial field is rewritten as: and we define a function I as the whole expression of summatory in eq. (6), that clearly is a Fourier's Series with L as the period. Now, we chose I(s), within −L/2 ≤ s ≤ L/2, as given by: that results on the following B n in eqs.(7,6): where, as we are going to see, m(s) is a function that will modulate the Airy function within the finite aperture.
The reason for choosing I(s), eq.(7), as given by (8) will become clear in the next step.
By using our choices (5) and (9) into eq.(6) we get the following expression to the initial field: where I(s) is the same of eq.(7) and so it repeats its values in periodic space intervals.
Since L/2 > S, for appropriated choices of L and a R we have that Ai(s) exp(a R s)I(s) ≈ 0 for |s| ≥ L/2 due to the behavior of the functions Ai(s) and exp(a R s) for positive and negative values of s, respectively.
In this way, we have shown that the initial field given by (3), with a n and B n given by (5) and (9), can represent at z = 0 an Airy-Type beam truncated by a finite aperture.
Finally, once the truncated Airy-Type beam at z = 0 is described by eq.(3), the resulting beam emanated from the finite aperture will be given by the solution (4).

Applying the method to three truncated Airy-Type beams
In this Section we shall apply our method to three situations involving Airy-Type beams truncated by finite apertures: the ideal Airy beam, the Airy-Exponential beam and the Airy-Gauss beam. Obviously, we will use a finite number 2N + 1 in (3) and (4), It is important to notice that the choice of the values of L and a R in (5) is not unique, actually there are many alternative sets of those values that yield excellent results.
In all cases we shall assume a wavelength of 500 nm.
At ζ = 0 this field is described by eq.(3), with a n and B n given by eqs. (5) and (9) respectively and with m(s) = 1. In this example, an excellent result can be obtained by the choice L = 50, a R = 0.1 and N = 60. Figure 1 shows the intensity of the field given by eq.(3) and we can see that it represents, at ζ = 0, the truncated Ideal Airy beam with high fidelity The field emanated by the aperture is given by solution (4), and its intensity is shown in Fig.2a. This result corresponds to an Ideal Airy beam truncated by a finite aperture.   At ζ = 0 the field is approximately described by eq.(3), with a n and B n given by eqs.(5) and (9) respectively and with m(s) = exp(qs). Here we obtain quite good result by choosing, again, L = 50, a R = 0.1 and N = 60. Figure 3 shows the intensity of the field given by eq.(3) and we can see that it represents very well, at z = 0, the truncated Airy-Exponential beam.
The resulting beam emanated by the aperture is given by solution (4), and its intensity is shown in Fig.4a. Figure 4b shows the orthogonal projection of this case.  At ζ = 0 this field is described by eq.(3), with a n and B n given by eqs. (5) and (9) respectively and with m(s) = exp(−qs 2 ). In this case, an excellent result can be obtained by the choice L = 50, a R = 0.1 and N = 60. Figure 5 shows the intensity of the field given by eq.(3) and we can see that it represents, at z = 0, the truncated Airy-Gauss beam with high fidelity  The beam emanated by the aperture is given by solution (4), and its intensity is shown in Fig.6a. This result corresponds to an Airy-Gauss beam truncated by a finite aperture. Figure 2b shows the orthogonal projection of this case.

Acknowledgments
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Conclusion
In this paper we developed an analytic method capable to describe Airy-Type beams truncated by finite apertures.
This new approach is based on suitable superposition of Airy-Exponential beams and presents a simple analytic solution. The results agree with those already published in literature through the use of numerical methods. We demonstrated the method's potentiality applying it to three different truncated Airy type beams: the Ideal Airy, the Airy-Gauss and the Airy-Exponential, but many others Airy-Type beams can easily be described with this analytic approach.