## Abstract

We investigate the finite power Airy beams generated by finite extent input beams such as a Gaussian beam, a uniform beam of finite extent, and an inverse Gaussian beam. Each has different propagation behavior: A finite Airy beam generated by a uniform input beam keeps its Airy profile much longer than the conventional finite Airy beam. Also, an inverse Gaussian beam generates a finite Airy beam with a good bent focusing in free space. In this paper, the analysis and experimental results of finite Airy beams are presented.

© 2013 Optical Society of America

## 1. Introduction

Since Berry and Balazs predicted the existence of non-spreading wave packets [1] and Siviloglou and Christodoulides studied their optical version [2], there have been very active researches on Airy beams [3–18]. To sum up their outcomes until now, we can present three unique characteristics of Airy beams. First, Airy beams are non-diffracting ones that satisfy the Helmholtz equation, like Bessel [19] and parabolic beams [20]. Second, Airy beams take up a bending trajectory in homogeneous media without any external forces. Lastly, Airy beams can recover their original shapes when a part of them is blocked by arbitrary obstacles.

To resolve the problem that ideal Airy beams are not square integrable (i.e., they carry infinite power), exponentially decaying terms are introduced to implement finite power Airy beams [2], which have similar features to those of the ideal one. They were generated by the optical Fourier transform of a Gaussian input beam on which cubic phase is imposed [3]. Although they can carry finite power, these *apodized* finite power Airy beams cannot maintain their shapes for a long time and are gradually spreading out during propagation [3]. Recently, based on these, various methods which tune a main lobe or side lobes of finite power Airy beams by using a flat-topped input Gaussian beam [21], a nonsymmetric apodization [22] and a sharp cutoff [23] were presented.

In this paper, we present some alternative methods for the apodization of Airy beams. Instead of Gaussian input beams (that involve exponentially decaying terms), we consider uniformly distributed beams of finite extent or beams having an inverse Gaussian distribution. In the case of a uniformly distributed finite-extent beam, we show that the resultant finite power Airy beam can preserve its Airy profile much farther than that generated by the Gaussian input beam. An inverse Gaussian apodization results in a unique propagation dynamics – a focused-bending beam. In each case, the solution of the finite power Airy beam is derived and verified by experiments.

## 2. Theoretical analysis of Airy beams

The (1 + 1)D potential-free paraxial Helmholtz equation can be written as

*s*and

*ξ*denote the transverse coordinate

*x*scaled by an arbitrary scaling factor

*x*

_{0}and the longitudinal coordinate

*z*scaled by

*k*

_{n}x_{0}

^{2}, respectively, where

*k*( = 2

_{n}*πn*/

*λ*) is a wavenumber in a medium with a refractive index

*n*, and

*λ*is the wavelength of light in free space. By solving Eq. (1), the Airy beam solution ${\varphi}_{0}$ can be obtained as follows [2]:

_{0}of Eq. (2) at

*ξ*= 0 is given byAs was mentioned in the introduction, Eq. (2) is not square integrable and thus ${\varphi}_{0}$ is not physically realizable. Usually, an exponentially decaying function exp(

*as*) is multiplied to ${\varphi}_{0}$ to obtain a finite power beam:which is square integrable and its Fourier transform Φ

_{1}becomes

*a*, $\mathrm{exp}\left(\frac{i}{3}(-3{a}^{2}k-i{a}^{3})\right)$ in Eq. (5), the only difference between Eqs. (3) and (5) is the Gaussian function [exp(-

*ak*

^{2})] which is originated from the apodization, i.e., exponentially decaying term exp(

*as*). Therefore, Eq. (3) can be understood as a plane wave (of infinite extent) with a cubic phase modulation, suggesting that the finite-extent feature of the input beam results in the finite power Airy beam.

This discussion, i.e., a finite power Airy beam can be generated by the Fourier transform of a cubic phase-modulated beam of finite extent, implies that there can be other methods for the apodization of ideal Airy beams. That is, we can use other types of finite-extent beam than the Gaussian one. Here, we consider finite power Airy beams generated by three different input beams: those having a conventional Gaussian distribution (*CASE* I), a uniform distribution of finite extent (*CASE* II), and an inverse Gaussian distribution (*CASE* III). Throughout this paper, we assume that the wavelength of an incident beam *λ* is 633 nm, *x*_{0} is 50 μm, the focal length of the lens *f* is 50 cm, and the SLM has 1080 × 1080 pixels with an 8 μm pixel pitch.

First, we start with the *CASE* I, which adopts a Gaussian beam as an input beam. Due to the finite SLM size, the incident Gaussian beam is truncated. Therefore, Eq. (5) becomes

*l*is the length of the SLM along the one dimension given by the product of the pixel pitch and the number of pixels. The propagation dynamics of the

*CASE*I Airy beam can now be written as follows using the Fresnel diffraction form of Eq. (4):

*a*is chosen to be 0.1 throughout this paper. Calculation results of Eq. (7) are plotted in Fig. 1(a). In this case, the number of side lobes in the initial plane (

*z*= 0 cm) is decreased because the Gaussian distribution of the input beam cannot fully retain the high spatial frequency components. As a result, unlike the ideal case shown in Fig. 1(d), the Airy beam is diffracted or spreads out due to insufficient power flows from the side lobes.

Second, under the same conditions, we adopt a beam with a uniform intensity distribution of finite extent or a truncated plane wave as an input beam (the *CASE* II), In this case, we have

From the property of the Fourier transform, the (inverse) Fourier transform of Eq. (8) is a convolution of the sinc function and the Airy function as follows:

*CASE*II Airy beam can be expressed as:

*CASE*I. Figure 1(b) also shows that the incidence of a truncated plane wave and its Fourier transform after the cubic phase modulation can generate a finite power Airy beam which takes up a bending trajectory with the acceleration toward +

*x*direction. What is interesting is that this finite Airy beam can preserve its Airy profile much farther than that generated by the Gaussian input beam. This is because more high-frequency components are retained in Eq. (8) than in Eq. (6).

Lastly, in the *CASE* III, we use a beam with an inverse Gaussian intensity distribution as an input beam. We have

*x*direction. That is, Eq. (11) also describes a finite power Airy beam. Actually, this case can be taken as an extreme example of the self-healing property [7]: side lobes without the main lobe can regenerate the Airy profile during the propagation.

To compare the propagation characteristics of these three finite power Airy beams, their cross-correlations with the ideal Airy beam are calculated along the propagation direction *z*. In Fig. 2, we plotted the variation of the maximum value of this cross-correlation, i.e., *C*_{1}(*z*) defined by

*CASE*I, II, and III, respectively. The cross-correlation values in Fig. 2 are normalized by those at the initial plane (

*z*= 0 cm). For the

*CASE*I and II, the cross-correlation becomes maximum at the initial plane and gradually decreases. This means that the finite power Airy beams spread out or are diffracted so that they lose their initial Airy shape during propagation. However, we would like to point out that the cross-correlation value of the

*CASE*II Airy beam is always higher than that of the

*CASE*I at every

*z*. This means that the

*CASE*II Airy beam can preserve its original shape much farther than the

*CASE*I Airy beam. In the

*CASE*III, however, the cross-correlation increases after some propagation distance. This indicates the recovery process of the Airy profile: the regeneration of the main lobe and its acceleration along the transverse coordinate.

Now, let us consider (2 + 1)D finite power Airy beams. Equations (7), (10), and (11) are modified as

*CASE*I, II, and III Airy beams, respectively, where

*l*and

_{x}*l*are the horizontal and vertical lengths of the SLM and

_{y}*y*

_{0}is an arbitrary scaling factor along the

*y*coordinate.

Calculated 2D intensity distributions of (2 + 1)D finite power Airy beams are shown in Figs. 3(a) and 3(b) (${\varphi}_{1}$; *CASE* I), Figs. 3(c) and 3(d) (${\varphi}_{2}$; *CASE* II), and Figs. 3(e) and 3(f) (${\varphi}_{3}$; *CASE* III). Figures 3(a), 3(c), 3(e) and 3(b), 3(d), 3(f) are the results at *z* = 0 cm and *z* = 15 cm, respectively. Comparing Figs. 3(a), 3(c) and 3(e), we can find more side lobes in Fig. 3(c) than in Fig. 3(a) while the main lobe disappears in Fig. 3(e). After some propagation (*z* = 15 cm), the finite power Airy beam generated by a Gaussian beam (*CASE* I) does not preserve its initial Airy profile anymore. On the other hand, the finite power Airy beam generated by a truncated plane wave (or a uniform beam of finite extent; *CASE* II) maintains its initial profile although the beam is broadened due to the diffraction. Therefore, the uniform beam of the *CASE* II is more advantageous than the Gaussian beam of the *CASE* I. In the case of the inverse Gaussian beam (*CASE* III), although the main lobe is missing at the initial plane (Fig. 3(e)), side lobes recover the main lobe after some propagations as can be found in Fig. 3(f).

## 3. Experiments of the finite power Airy beams

Figures 4(a) and 4(b) show the schematic diagram and the experimental setup for the generation of finite power Airy beams. A laser beam with a 633 nm wavelength is expanded to form a collimated plane wave. To change this input wave into the Gaussian, inverse Gaussian, or uniform beam of finite extent, the SLM 1 (Mitsubishi Electric XL9U LCD projector with 1024 × 768 pixels of 11.9 μm pixel pitch) is placed between two orthogonal linear polarizers (P1 and P2) [24]. These polarizers allow the SLM 1 to perform intensity modulations. The beams from the SLM 1 have a Gaussian distribution (*CASE* I), a uniform distribution of finite extent (*CASE* II), or an inverse Gaussian distribution (*CASE* III). Passing these beams through the optical Fourier transform (2-*f*) system, we can obtain finite power Airy beams at the initial plane (*z* = 0 cm). The optical Fourier transform system consists of a lens (L1; *f =* 50 cm) and a phase-only SLM 2 (Holoeye Pluto with 1920 × 1080 pixels of 8 μm pixel pitch) which is placed in front of the lens and imposes cubic (*k*^{3}) phase. The same cubic phase mask is used to all *CASES* because the higher-order phase terms can be ignored due to the relatively small constant *a*. Another polarizer P3 is used to attenuate the output beam so that a twin image can be eliminated.

The images of the finite power Airy beams were captured by the charge-coupled device (CCD) at *z* = 0 cm and *z* = 15 cm and shown in Figs. 5(a) - 5(f). All images are obtained under the same conditions that no adjustments in intensities were made. From these results, we can conclude that the *CASE* II Airy beam [Figs. 5(c) and 5(d)] retains the Airy profile much longer than the *CASE* I Airy beam [Figs. 5(a) and 5(b)]. Meanwhile, the *CASE* III Airy beam recovers the main lobe at *z* = 15 cm as shown in Figs. 5(e) and 5(f). These experimental results coincide well with the calculation results shown in Figs. 3(a) - 3(f).

## 4. Conclusion

In this paper, finite power Airy beams generated by the Fourier transform of a cubic phase-modulated beam of finite extent such as a Gaussian beam (*CASE* I), a uniform beam of finite extent (*CASE* II), and an inverse Gaussian beam (*CASE* III) were discussed. The propagation dynamics of resultant finite power Airy beams were analyzed and compared. We showed both theoretically and experimentally that the finite Airy beam generated by the use of a uniform input beam (*CASE* II) retains the Airy profile much longer than the conventional finite Airy beam (*CASE* I). Also, the finite Airy beam via an inverse Gaussian beam (*CASE* III) builds up a focused-bending beam. We expect that our works in this paper can be utilized to particle tweezing and optical trapping [4–6].

## Acknowledgment

This work was supported by the National Research Foundation of Korea grant funded by the Korean government (MSIP) through the National Creative Research Initiatives Program (#2007-0054847).

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